*164*
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*English*
*Pages 232*
*Year 2007*

*Table of contents : 010 - 04287162......Page 1020 - 04287163......Page 3030 - [email protected] 4040 - [email protected] 12050 - [email protected] 20060 - [email protected] 30070 - [email protected] 43080 - [email protected] 52090 - [email protected] 61100 - [email protected] 68110 - [email protected] 78120 - [email protected] 85130 - [email protected] 92140 - [email protected] 104150 - [email protected] 108160 - [email protected] 116170 - [email protected] 123180 - [email protected] 128190 - [email protected] 136200 - [email protected] 142210 - [email protected] 149220 - [email protected] 158230 - [email protected] 166240 - [email protected] 175250 - [email protected] 183260 - [email protected] 190270 - [email protected] 199280 - [email protected] 205290 - [email protected] 213300 - [email protected] 220310 - [email protected] 230320 - 04287168......Page 231330 - 04287164......Page 232*

AUGUST 2007

VOLUME 55

NUMBER 8

IETMAB

(ISSN 0018-9480)

PAPERS

Linear and Nonlinear Device Modeling Exact Analysis of the Wire-Bonded Multiconductor Transmission Line . ........ ......... ......... ........ ......... ......... .. .. ........ ..... J. E. Page, E. Márquez-Segura, F. P. Casares-Miranda, J. Esteban, P. Otero, and C. Camacho-Peñalosa Active Circuits, Semiconductor Devices, and Integrated Circuits A 1-V Wideband Low-Power CMOS Active Differential Power Splitter for Wireless Communication ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .... S. Lee and C. C. Lai Analysis and Design of Bandpass Single-Pole–Double-Throw FET Filter-Integrated Switches . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . Z.-M. Tsai, Y.-S. Jiang, J. Lee, K.-Y. Lin, and H. Wang Printed and Integrated CMOS Positive/Negative Refractive-Index Phase Shifters Using Tunable Active Inductors ..... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... M. A. Y. Abdalla, K. Phang, and G. V. Eleftheriades A 20-Gb/s 1 : 2 Demultiplexer With Capacitive-Splitting Current-Mode-Logic Latches ........ J.-C. Chien and L.-H. Lu Signal Generation, Frequency Conversion, and Control Multitone Fast Frequency-Hopping Synthesizer for UWB Radio ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... .... K. Stadius, T. Rapinoja, J. Kaukovuori, J. Ryynänen, and K. A. I. Halonen A New -Band Low Phase-Noise Multiple-Device Oscillator Based on the Extended-Resonance Technique . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... J. Choi and A. Mortazawi Design and Analysis of a Millimeter-Wave Direct Injection-Locked Frequency Divider With Large Frequency Locking Range .. ......... ......... ........ ......... ......... ...... ... ......... ......... ........ ......... ......... ... C.-Y. Wu and C.-Y. Yu Millimeter-Wave and Terahertz Technologies -Band Waveguide Impedance Tuner Utilizing Dielectric-Based Backshorts .. ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... T. Kiuru, V. S. Möttönen, and A. V. Räisänen

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Wireless Communication Systems Variable Antenna Load for Transmitter Efficiency Improvement ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .. V. Kaajakari, A. Alastalo, K. Jaakkola, and H. Seppä

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Field Analysis and Guided Waves Rigorous Analysis of a Metallic Circular Post in a Rectangular Waveguide With Step Discontinuity of Sidewalls ...... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ... C. A. Valagiannopoulos and N. K. Uzunoglu Reactance of Posts in Circular Waveguide ........ ........ ......... ......... ... Q. C. Zhu, A. G. Williamson, and M. J. Neve Fourier Decomposition Analysis of Anisotropic Inhomogeneous Dielectric Waveguide Structures ..... ....... R. Pashaie Analysis of a 118-GHz Quasi-Optical Mode Converter .. .. H. O. Prinz, A. Arnold, G. Dammertz, and M. Thumm

1673 1685 1689 1697

CAD Algorithms and Numerical Techniques Mixed-Mode Chain Scattering Parameters: Theory and Verification ..... ........ ......... .. H. Erkens and H. Heuermann A General Multigrid-Subgridding Formulation for the Transmission Line Matrix Method ... . L. Pierantoni and T. Rozzi Resonance Absorption in Nonsymmetrical Lossy Dielectric Inserts in Rectangular Waveguides ........ . ........ L. A. Rud An Efficient Application of the Discrete Complex Image Method for Quasi-3-D Microwave Circuits in Layered Media .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... W.-H. Tang and S. D. Gedney New Series Expansions for the 3-D Green’s Function of Multilayered Media With 1-D Periodicity Based on Perfectly Matched Layers ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ........ H. Rogier Interpolated Coarse Models for Microwave Design Optimization With Space Mapping .... .. S. Koziel and J. W. Bandler Compact Planar Quasi-Elliptic Function Filter With Inline Stepped-Impedance Resonators ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ........ J.-T. Kuo, C.-L. Hsu, and E. Shih Filters and Multiplexers Balanced Coupled-Resonator Bandpass Filters Using Multisection Resonators for Common-Mode Suppression and Stopband Extension .... ........ ......... ......... ........ ......... ......... ........ C.-H. Wu, C.-H. Wang, and C. H. Chen Dual-Mode Microstrip Open-Loop Resonators and Filters ....... ......... ........ J.-S. Hong, H. Shaman, and Y.-H. Chun Design of Vertically Stacked Waveguide Filters in LTCC ........ .... T.-M. Shen, C.-F. Chen, T.-Y. Huang, and R.-B. Wu Wideband Bandstop Filter With Cross-Coupling . ........ ......... ......... ........ ......... ...... H. Shaman and J.-S. Hong Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements 60-GHz System-on-Package Transmitter Integrating Sub-Harmonic Frequency Amplitude Shift-Keying Modulator ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ....... D. Y. Jung, W. Chang, K. C. Eun, and C. S. Park Left-Handed Metamaterial Coplanar Waveguide Components and Circuits in GaAs MMIC Technology ....... ......... .. .. ........ ......... ......... ........ ......... .... W. Tong, Z. Hu, H. S. Chua, P. D. Curtis, A. A. P. Gibson, and M. Missous Instrumentation and Measurement Techniques Virtual Auxiliary Termination for Multiport Scattering Matrix Measurement Using Two-Port Network Analyzer ...... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... C.-J. Chen and T.-H. Chu

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LETTERS

Corrections on “Low-Loss Patterned Ground Shield Interconnect Transmission Lines in Advanced IC Processes” ..... .. .. ........ ......... ......... ........ ......... ......... ........ ... L. F. Tiemeijer, R. M. T. Pijper, R. J. Havens, and O. Hubert

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Information for Authors .. ........ ......... ......... ........ ......... .......... ........ ......... ......... ........ ......... ......... .

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Digital Object Identifier 10.1109/TMTT.2007.905568

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 8, AUGUST 2007

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Exact Analysis of the Wire-Bonded Multiconductor Transmission Line Juan E. Page, Enrique Márquez-Segura, Senior Member, IEEE, Francisco P. Casares-Miranda, Student Member, IEEE, Jaime Esteban, Pablo Otero, Member, IEEE, and Carlos Camacho-Peñalosa, Member, IEEE

Abstract—In this paper, the exact model of a multiconductor transmission line with bonding wires is presented. The model is based on the multiport admittance matrix, and is valid for any number of conductors in the structure. The model shows how to compute the two-port admittance matrix of the wire-bonded structure with direct application to the wire-bonded interdigital capacitor. The model has been validated by means of method-of-moments-based numerical simulation and by experimental work.

Fig. 1. (a) Interdigital capacitor. (b) Wire-bonded interdigital capacitor.

Index Terms—Interdigital capacitor, microwave integrated circuits, microwave passive circuits, multiconductor transmission lines, wire-bonded interdigital capacitor, wire bonding.

I. INTRODUCTION

ICROSTRIP interdigital capacitors have been improved by the connection of bonding wires across the open ends of the capacitor fingers [1], as sketched in Fig. 1. The resulting device is the so-called wire-bonded interdigital capacitor. This device has several advantages over the conventional interdigital capacitor: the undesired resonances at higher frequencies are removed, which results in a larger bandwidth of operation, and there is a slight increment in nominal capacitance. A recent application of the wire-bonded interdigital capacitor can be found in [2]. Of course, the process of connecting bonding wires across the open ends of the fingers of the interdigital capacitor is cumbersome. An analytical model of the wire-bonded interdigital capacitor has been recently published [3]. Nevertheless, this device can be considered as well, i.e., a rectilinear multiconductor guiding structure in a heterogeneous dielectric. In this paper, the exact analysis of a multiconductor guiding structure with short-circuited alternate conductors is presented. The analysis, which is based on the multiport admittance matrix, is derived in Section II. The model is valid for any number of

M

Manuscript received August 9, 2006; revised May 10, 2007. This work was supported by the Spanish Ministry of Education and Science and by the European Union under European Regional Development Funds Project TEC200604771. J. E. Page and J. Esteban are with the Departamento Electromagnetismo y Teoría de Circuitos, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: [email protected]). E. Márquez-Segura, F. P. Casares-Miranda, Pablo Otero, and C. CamachoPeñalosa are with the Departamento Ingeniería de Comunicaciones, Escuela Técnica Superior de Ingeniería de Telecomunicación, Universidad de Málaga, 29071 Málaga, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.902084

Fig. 2. Multiconductor guiding structure.

coupled transmission lines, even or odd. Particular equations for homogeneous and heterogeneous lossless dielectrics have been obtained and presented, respectively, in Sections III and IV. Section V deals with the analysis validation by means of both numerical simulations using a full-wave computer program based on the method of moments (MoM) and by experimental work. II. EXACT ANALYSIS OF THE WIRE-BONDED INTERDIGITAL CAPACITOR Consider a finite length guiding structure consisting of conductors, as shown in Fig. 2. It is not necessary that all the conductors have the same shape in the transversal plane. This structure can propagate TEM modes in the homogeneous dielectric case or quasi-TEM modes in the heterogeneous case. Without loss of generality, one of the conductors, i.e., the con, can be considered the voltage reference. ductor numbered Voltages and currents at both ends of the conductors are related through the admittance matrix equation (1) where and are vectors of length , their elements being the voltages and currents at both ends of the circuit of Fig. 2 as follows:

0018-9480/$25.00 © 2007 IEEE

(2) (3)

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The wire-bonded interdigital capacitor is a two-port device, therefore, (1) becomes (7) where

is the two-port admittance matrix, given by (8)

, are obtained by means of the and the matrices following equations, depending on the number of fingers of the wire-bonded interdigital capacitor. even 1) Fig. 3. Multiconductor transmission line structure of the wire-bonded interdigital capacitor.

odd

where superscript denotes transpose. The admittance matrix is square of dimension as follows:

.. .

.. .

..

.

.. .

even

odd

even

(4) odd

(9)

even

When the ends of the odd-numbered conductors are connected on one side of the structure, and the ends of the even-numbered conductors are connected on the opposite side, as shown in Fig. 1(a), an interdigital capacitor is obtained, and the following conditions apply:

odd

even (5a) even

odd

(10)

even (5b) when is odd and four equations similar to (5a) and (5b) when is even. If the fingers of the interdigital capacitor are now connected together on the open end using bonding wires, as shown in Fig. 1(b), the multiconductor structure shown in Fig. 2 becomes that of Fig. 3 (the so-called wire-bonded interdigital capacitor), and additional constraints on voltages and currents are imposed as follows:

even

odd

odd 2)

(11)

odd even

(6a) even

even (6b) when is odd and four equations similar to (6a) and (6b) for the case of are even.

odd

even

(12)

PAGE et al.: EXACT ANALYSIS OF WIRE-BONDED MULTICONDUCTOR TRANSMISSION LINE

odd

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even

odd

even (a)

odd

even

(13)

even

even

even

(14)

Equations (9)–(14) show that once the admittance matrix in (4) is known, the two-port admittance in (8) can be calcufor the lossless homogeneous lated. The admittance matrix and heterogeneous dielectric cases are obtained below, but the presence of moderate losses hardly changes either the algebra or the results [4].

(b) Fig. 4. (a) Pair of coupled lines in bandpass filter configuration. (b) Equivalent circuit.

and, for

odd, by

odd III. HOMOGENEOUS LOSSLESS DIELECTRIC Providing the dielectric around the conductors in Fig. 2 is homogeneous and lossless, TEM modes will be supported by the guiding structure and the admittance matrix can be written as [5]

odd

odd

(18)

(15) is the phase constant of all the TEM modes, where is the length of the fingers, and is the characteristic admittance matrix of dimension , where is the static capacitance matrix per unit length. Using (9)–(15), the two-port admittance matrix in (8) can be obtained as

A. Equivalent Circuit of the Structure obtained in (16) is the admittance matrix of The matrix two coupled lines in the bandpass filter configuration, as shown in Fig. 4(a) [6]. The static capacitance coefficients of that cell are those obtained with (17) or (18), and the equivalent circuit is shown in Fig. 4(b) [7]. The values of the characteristic impedances, and the voltage transform ratio, are given by

(19)

(16) where the

coefficients are given, for

(20)

even, by and the coefficients

odd

odd

even

even

and

are given by

(21) (17)

The behavior of the circuit shown in Fig. 4 can be qualitatively determined by means of a couple of examples. Consider

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Fig. 7. Low-frequency equivalent circuit of Fig. 4.

has to be used if the desired response is that of the dashed line in Fig. 5, but, if the desired response is that of the solid line in Fig. 5 (e.g., when a wideband dc coupling capacitor is to be designed), a capacitor narrower than the stripline could be necessary. Some results confirming that behavior in microstrip wire-bonded interdigital capacitors are included in Section V. B. Electrically Short Multiconductor Structure When the conductors are electrically short, i.e., when (22) where is the electrical length shown in Fig. 4, the equivalent circuit is simplified to that shown in Fig. 7. The values of the elements in Fig. 7 are Fig. 5. Scattering matrix elements of the coupled line bandpass filter section.

(23a) (23b) and the approximation obtained by means of this simplified circuit is accurate up to frequencies where the length is smaller than one-eighth of the guided wavelength. IV. HETEROGENEOUS LOSSLESS DIELECTRIC CASE

Fig. 6. Two coupled line bandpass filter sections.

two multiconductor circuits of equal length. Fig. 5 shows the magnitudes of the scattering elements when both circuits are loaded with the reference impedance . When , a single minimum is obtained, but not a reflection zero. On , then two reflection zeros the other hand, if are obtained. Which case corresponds to a particular structure? It should be noted that, in the case of two coupled striplines with a total width equal to that of the connecting stripline, as shown in Fig. 6(a) (even if the two fingers had unequal widths), it al, where is the characterways happens that istic impedance of the connecting stripline. Only when the connecting stripline is wider than the coupled pair, as in Fig. 6(b), can be fulfilled. the condition Obviously, the performance of a wire-bonded interdigital capacitor depends on its geometry. Nevertheless, in all the cases analyzed thus far, the behavior stated above has been found. A structure of the same width or wider than the stripline

When the multiconductor structure is built with a heterogeneous lossless dielectric, the quasi-TEM modes are described by means of the eigenvalues and eigenvectors of the matrix , where is the static capacitance matrix per unit length, and is the same matrix of the multiconductor line in free space [8]. The eigenvalues of that matrix are the effective permittivities corresponding to each single mode. The admittance matrix can be computed by means of the following relation: (24) with

(25) where is the matrix of eigenvectors arranged in columns, and is the length of the multiconductor structure. Any quasi-TEM mode has a different phase constant and no simple equivalent circuit seems to exist, but an approximation is possible if a single phase velocity is used for all modes.

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Fig. 9. (a) Layout of the microstrip wire-bonded interdigital capacitor. (b) Drawing of the ideal microstrip transmission line model of the wire-bonded interdigital capacitor. (c) Photograph of the five fingers wire-bonded interdigital capacitor prototype (Rogers Ultralam 2000 substrate, thickness: 1.52 mm, relative permittivity: 2.4).

TABLE I RELEVANT PARAMETERS OF THE MEASURED WIRE-BONDED INTERDIGITAL CAPACITOR [SEE (20) AND FIG. 9]

matrix is approximated by

(26) Fig. 8. Calculated scattering matrix elements of microstrip wire-bonded interdigital capacitors (FR4 substrate, thickness: 1.52 mm, relative permittivity: 4.2).

An effective phase velocity can be computed as some type of mean of the phase velocities of all the modes. Once the effective phase velocity has been computed, the performance of the heterogeneous dielectric structure can be determined in the same way as if the dielectric were homogeneous so that all the previous results apply. As shown in Section V, there is an excellent agreement between experimental results and the exact computed values obtained using an effective permittivity as the mean value of the effective permittivities of the different propagating modes. In that case, the capacitance

V. MODEL VALIDATION A number of multiconductor structures have been designed, simulated, built, and measured to validate the model, which is valid for any number of conductors, even or odd (the analytical model in [3] only supported an even number of conductors, and the reference conductor). The frequency responses of several microstrip wire-bonded interdigital capacitors, obtained with a MoM simulation, are presented in Fig. 8. The results confirm the presence of zero reflection frequencies only for structures narrower than the transmission line of reference impedance. For the experimental validation, a structure with five conductors,asshowninFig.9,has beenchosen.TableIshowstherelevant dimensions and values of the measured and simulated structures.

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Fig. 10. Calculated scattering matrix elements of the five conductors (plus common) structure described in Table I, computed using models of Figs. 4 and 7, the model in [2], and the MoM (Rogers Ultralam 2000 substrate, thickness: 1.52 mm, relative permittivity: 2.4).

To fully validate the model, two sets of graphics are presented. In Fig. 10, the comparison between different models is presented, while Fig. 11 compares calculated to measured results. Fig. 10 shows the scattering matrix of that structure. The four lines correspond, respectively, to the analytical model, the

Fig. 11. Measured and calculated (MoM) scattering matrix elements of the same capacitor of Fig. 10. IDC denotes the interdigital capacitor and WBIDC denotes the wire-bonded interdigital capacitor.

coupled transmission line model, the low-frequency circuital model, and the MoM analysis of the wire-bonded structure. LINPAR 2.0 was used to calculate the static capacitance matrices of the multiconductor structure [9]. Nonadjacent capacitance terms were included in the simulations. The MoM analysis has been conducted with Agilent’s ADS Momentum. It can be

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Fig. 12. Calculated and measured argument of the scattering matrix element S 21 of the five conductors (plus common) structure described in Table I. Magnitude values shown in Figs. 10 and 11.

observed that the model results agree with the full-wave electromagnetic simulation. It must be noted that the simulation results obtained with the aforementioned three methods only consider the ideal (lossless) multiconductor structure, as shown in Fig. 9(b). On the other hand, simulation results obtained with the MoM include the parasitic effects of the junction between the access transmission line. multiconductor structure and the Fig. 11 shows the measured results of the multiconductor structure, with and without bonding wires, of the prototype shown in Fig. 9(c). Fig. 12 shows the computed and measured -parameter. Measurements have been argument of the performed with the vector network analyzer Hewlett-Packard HP8510C with thru-reflect-line (TRL) calibration. There is a slight increase in the value of capacitance of the wire-bonded interdigital capacitor, as already explained in [1]. The inherent resonance of the multiconductor structure [10] is eliminated when the bonding wires are connected [1]. Comparing Figs. 10 and 11, it can be seen that the results obtained with the presented model show a very good agreement with the MoM and measured results. Similar agreement between the results obtained using the presented model and the measured results has been obtained with a number of conductors from three to ten (and the reference conductor). Fig. 13 shows the scattering matrix elements of the same structure, but having ten fingers instead of five. The same structure here means that the length of the fingers and the width of the wire-bonded interdigital capacitor are kept constant and equal to those in Table I, which also means that the fingers and slots widths and , respectively, decrease with the number of fingers. The values of and are indicated in the caption of Fig 13. An identical behavior is found in similar structures with 3–10 fingers. In Fig. 13, the models results are compared to the MoM results. The agreement is good with a slight deviation at higher frequencies, which is mainly due to parasitic effects not considered in the models, as the capacitances of the fingers ends, and the approximation of the value of the effective permittivity of (26). The resonance at high frequencies, evident

Fig. 13. Calculated scattering matrix elements of the ten conductors (plus common) structure, computed using models of Figs. 4 and 7, the model in [2], and the MoM (Rogers Ultralam 2000 substrate, thickness: 1.52 mm, relative permittivity: 2.4). Fingers and slots widths: w = s = 239 m.

in the MoM simulation of Fig. 13, is due to the inductance of the bonding wires [3]. VI. CONCLUSION A particularly useful capacitor device in microstrip technology isthewire-bondedinterdigitalcapacitor,whichpresentsanumber

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of advantages over the interdigital capacitor. The wire-bonded device can be viewed as a multiconductor guiding structure in a heterogeneous dielectric. In this paper, the exact model of such a structure has been presented. The model is based on the multiport general admittance matrix, and is valid for any number of conductors in the structure. The model allows a designer to compute the two-port admittance matrix of the guiding structure. An equivalent circuit of the structure is presented as well. The model has been validated by means of MoM-based numerical simulation and experimental study. Different microwave substrates and number of fingers have been considered. The model results are in good agreement with the measured results. REFERENCES [1] F. P. Casares-Miranda, P. Otero, E. Márquez-Segura, and C. CamachoPeñalosa, “Wire bonded interdigital capacitor,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 700–702, Oct. 2005. [2] F. P. Casares-Miranda, E. Márquez-Segura, P. Otero, and C. Camacho-Peñalosa, “Composite right/left-handed transmission line with wire bonded interdigital capacitor,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 11, pp. 624–626, Nov. 2006. [3] E. Márquez-Segura, F. P. Casares-Miranda, P. Otero, C. CamachoPeñalosa, and J. E. Page, “Analytical model of the wire bonded interdigital capacitor,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 748–754, Feb. 2006. [4] R. F. Harrington and C. Wei, “Losses on multiconductor transmission lines in multilayered dielectric media,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 7, pp. 705–710, Jul. 1984. [5] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: Wiley, 1994. [6] E. M. T. Jones and J. T. Bolljhan, “Coupled-strip transmission-line filters and directional couplers,” IRE Trans. Microw. Theory Tech., vol. MTT-4, no. 4, pp. 75–81, Apr. 1956. [7] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Microwave-Matching Networks, and Coupling Structures. New York: McGraw-Hill, 1964. [8] R. Mongia, I. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA: Artech House, 1999. [9] A. R. Djordjevic, M. B. Bazdar, T. K. Sarkar, and R. F. Harrington, LINPAR for Windows: Matrix Parameters for Multiconductor Transmission Lines, Software and User’s Manual, Version 2.0. Norwood, MA: Artech House, 1999. [10] N. Dib, Q. Zhang, and U. Rhode, “New CAD model of the microstrip interdigital capacitor,” Active and Passive Electron. Compon., vol. 27, pp. 237–245, Dec. 2004. Juan E. Page was born in Madrid, Spain, in 1946. He received the Ingeniero de Telecomunicación and Doctor Ingeniero degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1971 and 1974, respectively. Since 1983, he has been a Professor with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. His research activities include the teaching of electromagnetic and circuit theories and research in the field of computer-aided design (CAD) of microwave devices and systems Enrique Márquez-Segura (S’93–M’95–SM’06) was born in Málaga, Spain, in April 1970. He received the Ingeniero de Telecomunicación and Doctor Ingeniero degrees from the Universidad de Málaga, Málaga, Spain, in 1993 and 1998, respectively. In 1994, he joined the Departamento de Ingeniería de Comunicaciones, Escuela Técnica Superior de Ingeniería (ETSI) de Telecomunicación, Universidad de Málaga, where, in 2001, he became an Associate Professor. His current research interests include electromagnetic material characterization, measurement techniques, and RF and microwave circuits design for communication applications.

Francisco P. Casares-Miranda (S’05) was born in Granada, Spain, in 1978. He received the Ingeniero de Telecomunicación degree from the Universidad de Málaga, Málaga, Spain, in 2003, and is currently working toward the Ph.D. degree at the Universidad de Málaga. Since 2004, he has been a Graduate Student Researcher with the Departamento de Ingeniería de Comunicaciones, Universidad de Málaga. His current research is focused on the analysis and applications of composite right/left-handed metamaterials. Mr. Casares was the recipient of a Spanish Ministry of Education and Science Scholarship (2004–2008). Jaime Esteban was born in Madrid, Spain, in 1963. He received the Ingeniero de Telecomunicación and Dr.Eng. degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1987 and 1990, respectively. Since January 1988, he has been with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. In 1990, he became Profesor Interino, and in 1992, Profesor Titular de Universidad. His research topics include the analysis and characterization of waveguides, transmission lines, planar structures and periodic structures, the analysis and design of microwave and millimeter-wave passive devices, and numerical optimization techniques (genetic algorithms and evolution programs). His current research is focused on the analysis and applications of left-handed double-negative, metamaterials and on the biological effects of exposure to RF/microwave fields of radar and modulated signals. Dr. Esteban was the recipient of a Spanish Ministry of Education and Science Scholarship (1988–1990). Pablo Otero (S’84–M’93) was born in Seville, Spain, in 1958. He received the Ingeniero de telecomunicación degree from the Universidad Politécnica de Madrid, Madrid, Spain, in 1983, and the Ph.D. degree from the Swiss Federal Institute of Technology at Lausanne (EPFL), Zürich, Switzerland, in 1998. From 1983 to 1993, he was with the Spanish companies Standard Eléctrica, E.N. Bazán, and Telefónica, where he was involved with communications and radar systems. In 1993, he joined the Universidad de Sevilla, Seville, Spain, where he was a Lecturer for two years. In 1996, he joined the Laboratory of Electromagnetism and Acoustics, EPFL, where he was Research Associate, working under a grant of the Spanish Government. In 1998, he joined the Escuela Técnica Superior de Ingeniería (ETSI) de Telecomunicación, Universidad de Málaga, Málaga, Spain, where he is currently an Associate Professor. His research interests include electromagnetic theory and printed microwave circuits and antennas. Carlos Camacho-Peñalosa (S’80–M’82) received the Ingeniero de telecomunicación and Doctor Ingeniero degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1976 and 1982, respectively. From 1976 to 1989, he was with the Escuela Técnica Superior de Ingenieros (ETSI) de Telecomunicación, Universidad Politécnica de Madrid, as Research Assistant, Assistant Professor, and Associate Professor. From September 1984 to July 1985, he was a Visiting Researcher with the Department of Electronics, Chelsea College (now King’s College), University of London, London, U.K. In 1989, he became a Professor with the Universidad de Málaga, Málaga, Spain. He was the Director of the ETSI de Telecomunicación (1991–1993), and the Vice-Rector (1993–1994), and Deputy Rector (1994) of the Universidad de Málaga. From 1996 to 2004, he was the Director of the Departamento de Ingeniería de Comunicaciones, ETSI de Telecomunicación, Universidad de Málaga. From 2000 to 2003, he was Co-Head of the Nokia Mobile Communications Competence Centre, Málaga, Spain. His research interests include microwave and millimeter solid-state circuits, nonlinear systems, and applied electromagnetism. He has been responsible for several research projects on nonlinear microwave circuit analysis, microwave semiconductor device modeling, and applied electromagnetics.

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A 1-V Wideband Low-Power CMOS Active Differential Power Splitter for Wireless Communication Shuenn-Yuh Lee, Member, IEEE, and Chun-Cheng Lai

Abstract—A 1-V wideband CMOS phase and power splitter (PPS) with an RLC network load and frequency compensation capacitor is proposed. Adopting the RLC network load and the frequency compensation capacitor, the gain and phase imbalances of the output can be improved and the wideband response can be achieved, respectively. Moreover, this architecture can not only offer high transfer power gain, but also adopts the tuning current source to overcome the power imbalance caused by process variation. An example with a differential phase condition (180 ) has been designed and fabricated. Based on our measured results, in the frequency range from 3.5 to 6 GHz, the phase error is less than 7 and the power imbalance is less than 1.4 dB. For wireless local area network 802.11a applications in the frequency range from 5.15 to 5.35 GHz, the phase error is less than 0.6 , and the power imbalances are less than 0.27 dB, respectively. In addition, the transfer power gain is 9.66 dB under the power consumption of 15 mW and 1-V supply voltage. This architecture is different from the passive PPS circuit, and it has the advantages of no conversion loss and a small chip area with 0.8 mm 0.7 mm. Compared with the conventional active PPSs, such as using the GaAs and BiCMOS process, this architecture implemented by the TSMC 0.18- m CMOS process is competitive in cost and possesses the characteristics of low voltage and low power, and it is more easily integrated and suitable for system-on-a-chip applications. Index Terms—Balun, CMOS, low power, low voltage, phase splitter, power splitter.

I. INTRODUCTION

I

N RECENT years, the wireless communications market and the integrated circuit industry have grown vigorously. Accordingly, RF integrated circuits (RFICs) for wireless communications with CMOS technology become an important trend in RFIC design due to their advantages of low price and small size. However, there are some design challenges including high performance and wideband design in the CMOS RFICs. Although the performance of CMOS technology is usually less than that of a GaAs field-effect transistor (FET) or BiCMOS, especially in wideband system design, its characteristics of low cost and low power are contributive to being integrated in a system-on-a-chip (SoC) system with less discrete components.

Manuscript received September 7, 2006; revised February 8, 2007. This work was supported in part by the Chip Implementation Center, by National Nano Device Laboratories, by Wireless Communication Laboratories, by the Microwave Multi-Layer Circuit Laboratory, and by the National Science Council, Taiwan, R.O.C., under Grant NSC 95-2221-E-194-091. The authors are with the Department of Electrical Engineering, National Chung Cheng University, Ming-Hsiung, Chia-Yi, Taiwan 62102, R.O.C. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.901130

Fig. 1. Splitter in a direct conversion receiver with fully differential mixer.

Currently, wireless communication techniques are employed not only in portable devices, but also in many new applications such as biochips, identifications, and intelligent electric appliances. For those applications, low power is an increasingly important design issue. However, the performance, including the conversion gain and third intercept point (IP3), will be degraded when the power or supply voltage are reduced. On the other hand, due to using fewer RF components, the receivers adopting direct-conversion or low-IF architectures are suitable for low-power and high-integration CMOS implementations [1]. In reviewing direct-conversion architectures, the critical design issues are dc offset, in-phase/quadrature (I-Q) mismatch, and even-order distortion [2]. The low-power even harmonic mixer (EHM) adopting a double frequency circuit has been proposed to overcome the problems including dc offset and evenorder distortion [3], [4]. In order to improve I-Q mismatch, we need a accurate splitter (balun) circuit to achieve lower phase error and power imbalance, as shown in Fig. 1. Currently, active phase and power splitter (PPS) circuits in [5]–[8] can fulfill this requirement, although they suffer from large imbalance or high power consumption. In this paper, we propose a differential CMOS architecture, with an RLC network to implement the 180 phase shift and with a current tuner to overcome the gain imbalance suffered from the process variation in low-voltage operation. Moreover, theoretical analyses of the RLC network and transfer function are also presented to facilitate the design including 180 phase shift and transfer power gain. The remainder of this paper is organized as follows. Section II presents the analysis of the proposed architecture. The measured results of an active differential PPS circuit are described in Section III to demonstrate the theoretical analysis. Finally, Section IV presents a brief conclusion.

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Fig. 3. Operation of a PPS circuit as R is infinite in the resonant frequency.

Fig. 2. Simple PPS adopting differential amplifier topology.

II. SPLITTER ARCHITECTURE

B. Basic Circuit Analysis

A. Circuit Principle There is currently a splitter using passive components that is operated in Marchand-type or multiport couplers [10], but for lower frequency applications, its size increases rapidly and it is difficult to integrate precisely. Many active PPSs [5]–[9] have been proposed to overcome these problems. However, although the distributed gate line method [5] is suitable for wideband applications, its many extra components will increase the power consumption and chip area. For narrowband design, the single transistor, and common-source common-gate (CS-CG) architectures are commonly used, but they have the drawback that the parasitic components could degrade the performance in the high-frequency response, which increases the circuit imbalance [6], [7]. In addition, the reverse transfer function between two output ports will limit the phase difference to near 180 . Therefore, it is difficult for these methods to provide arbitrary phases. For programming phase difference, the differential amplifier architecture has been proposed, as shown in Fig. 2 [9]. Here, the impedance, sizes of phase difference can be adjusted by the and , and the load impedances and . transistors However, since this method still suffers from serious process by a current tuner to restore the variation, we replace effect of the process variation. , the virtual ground For an ideal current source, i.e., can separate the input signal as node

(1) can be implemented by a current The high-impedance sink with an additional bandstop resonator, as illustrated in Fig. 3. However, it is difficult in practice to design an infinite impedance in the required RF range even using the resonator. To compensate for this effect, the required transistor ratio (W/L) in the differential pair and the load impedances in the differential outputs should be imbalanced and carefully designed. In addition, in order to reduce the insertion loss, the is used to adjust the bias current and further contransistor trol the conversion gain . The detailed analysis and design issues will be presented in Section II-B.

To analyze the behavior of the phase difference and gain imbalance, as shown in Fig. 3, the inherent channel square-law current is used to derive the voltage gain transfer functions. Assume and are that the source impedances of transistors and , while their load impedances are and , respectively. The voltage gain transfer functions can then be derived as

(2)

(3) W/L . From (2) and (3), where it is clear that the power gain is dominated by the bias conditions of transistors and , load impedances, and tuning . Therefore, the transistor can act as a power voltage tuning controller. Moreover, for the symmetric structure, we can . obtain the condition It means this condition can also be achieved by the different output impedances and input transconductances, and can be regarded as a pseudosymmetric structure. According to this concept, the half circuit can be adopted on the rough analysis and further acquire the design guideline. Consequently, in the real implementation, the input transistor sizes of the proposed structure are designed in mismatch to compensate for the unbalance output impedance and to be as a pseudosymmetric structure, i.e., in the actual design and the asymmetric effect is dominated by the mismatch of the frequency response and . Based on the half-circuit analysis between in the differential pair, the arbitrary phase relationship between the output and input can be approximately derived as [11] (4) (5) The phase difference between two outputs can be illustrated as

(6)

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Fig. 4. High-impedance Z implemented by RLC network.

Assuming

and , thus, Fig. 5. Proposed differential wideband power splitter with feedback capacitor.

(7) which can be implemented as a constant phase controlled by and independent of . On the other hand, the high-impedance can be implemented in an RLC network [7], as shown load in Fig. 4, where

In Fig. 5, as compared with Fig. 3, an extra feedback capacitor is connected from the drain to gate to compensate conversion gain. According to (2), (3), and (7), the output the conversion gain and phase imbalance of the proposed differential power splitter with a feedback capacitor can be modified as

(10) (8) Assume the imaginary part is zero, the resonated frequency can be found where and . At the resonated frequency, the impedance value can be derived in (9). From (2) and (3), the voltage gain is proportional to the load impedances and , thus and can be used as a power imbalance tuning load under constant impedance , but being insensitive to the phase imbalance

(9) The arbitrary phase difference relationship is very important in an RF system such as a balun, power divider, or I-Q generator. According to (7), for the given and , the arbitrary phase can be developed by the PPS circuit, followed by the differential output loads . C. Gain and Bandwidth Improved Circuit Although the bandstop resonator, as shown in Fig. 3, can increase the impedance to balance the output power, the drawbacks of this structure include that it is not appropriate for wideband applications and the conversion gain will be limited as the output power is balanced. In addition, the RLC network will also limit the desired bandwidth. In order to improve the bandwidth, is implemented in the feedback path, as shown a capacitor can not only improve the in Fig. 5. The feedback capacitor bandwidth, but also enhance the conversion gain. The detailed description will be explained as follows.

(11) (12) is the equivalent capacwhere itor series with , and is and output impedance the functions of feedback capacitor . Clearly, the phase imbalance can also be implemented as and independent of . Aca constant phase controlled by cording to (10) and (11), and can be simultaneously improved by increasing the feedback capacitor. The power gain comparison between the with and without capacitor feedback is shown in Fig. 6. As mentioned above, the transfer power gain can be compensated from 5 to 9.66 dB in the frequency range from 5.15 to 5.35 GHz for 802.11a applications. Moreover, in order to compare the power imbalance and phase error between the two outputs, the power imbalance and phase error are defined in (13) and (14) as follows through all of the descriptions in this paper: Power imbalance Phase error

(dBm)

(dBm)

(13) (14)

Therefore, in addition to the power gain enhancement, according to the simulation result with the input frequency of 5.2 GHz, as shown in Fig. 7, the power imbalance and phase error can also be improved by the feedback capacitor without increasing extra power consumption and is more flat than that of without the feedback capacitor as the tuning voltage is varied.

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Fig. 6. Power gain comparison between with and without feedback capacitor in the proposed differential power splitter.

Fig. 8. Proposed differential PPS.

TABLE I SUMMARY OF THE COMPONENT VALUES

Fig. 7. Power imbalance and phase error comparisons between with and without feedback capacitor in the proposed differential power splitter.

III. SIMULATION AND MEASUREMENT The proposed full monolithic differential PPS implemented in TSMC 0.18- m 1P6 M process is shown in Fig. 8. All of the impedance transfer network circuits are integrated on a chip, and the on-chip inductor is employed in the symmetrical structure provided by TSMC to save the chip size. The total component values are illustrated in Table I. The bond-wire model is included in the simulation. According to Fig. 9, the tuning voltage is proportional to the power imbalance and the phase error based on the differential condition (180 ). Therefore, if the power and phase shift caused by the process variation are toward to positive (negative) simultaneously, the imbalance can be restored by the tuning voltage. versus power and phase imbalance The tuning voltage with and without compensation network , based on the input frequency of 5.2 GHz, is also shown in Fig. 10. The simuis insensilated result is agreement with our study in that tive to the phase difference, i.e., if the required phase difference can be used to improve the is firstly designed by , and

power imbalance, but without destroying the phase error. The chip is mounted on the printed circuit board (PCB) through the bonding wire for measurement. The bond-wire effect has been concerned in the initial design. The characteristic impedance should be matched in the design of off-chip transmission lines, connectors, and cables on the PCB board. The -parameter measured results are shown in Fig. 11. The conversion gain is larger than 6.75 dB and the input return loss is less than 5 dB in the wideband range from 3.5 to 6 GHz. The and are also illusisolations between two output ports trated in Fig. 12, which is larger than 9 dB in the frequency range from 3.5 to 6 GHz. The large differences between the measurement and simulation are mainly caused not only by inaccuracy on the device model and parasitic model on the layout, but also the process variation and impedance match for the wideband system. Actually, in Figs. 11 and 12, their bias conditions in the measurement and simulation are different. Therefore, a more accurate model and estimated methods should be concerned if the measured results would like to be consistent with the simulated results in the wideband RFICs. The measured phase errors of less than 7 and the power imbalances of less than 1.4 dB compared with the simulation results are, respectively, shown in Fig. 13, but in the frequency range between 5.15–5.35 GHz for 802.11a applications, the better phase errors of less than 0.6 and the power imbalances of less than 0.27 dB can be obtained. In addition, the transfer power gain of 9.66 dB can be obtained under the power consumption of 15 mW and 1-V supply

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Fig. 11. Measured and simulated results of gain and input return loss.

Fig. 9. Simulation of tuning voltage versus: (a) power imbalance and (b) phase error in 802.11a frequency range.

Fig. 12. Measured and simulated results of the isolation between two output ports V and V .

Fig. 13. Comparison of simulated and measured results of power imbalance and phase error. Fig. 10. Power imbalance and phase error comparisons between the with and without RLC network in the proposed differential power splitter.

voltage. The measured results reveal that this splitter is superior for narrowband WLAN 802.11a applications.

Some performances, including conversion gain, input return loss, power imbalance, and phase error with temperature variation from 0 to 85 are also concerned by the simulation. According to the simulation results from 3.5 to 6 GHz, the observed

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Fig. 16. Measured CMRR of the proposed PPS. Fig. 14. Measurement of noise figure.

Fig. 17. Measured result of the IIP performance. TABLE II SUMMARY OF THE PROPOSED PSS Fig. 15. Measured power imbalance and phase error for 10% supply voltage variation.

variations of conversion gain, input return loss, power imbalance, and phase error are less than 1.88, 0.69, and 1.68 dB and 2 , respectively. The measured noise figure less than 4.35 dB in the frequency range from 3.5 to 6 GHz is shown in Fig. 14. The noise figure is low enough [12] for ultra-wideband (UWB) applications. The power imbalance and phase error for 10% voltage variation with 5.2 GHz are also illustrated in Fig. 15. It reveals that this splitter is insensitive to the supply voltage variation. Moreover, as shown in Fig. 15, the little performance degradation caused by the supply voltage or process variation can also be further improved by reducing the tuning voltage according to Fig. 9. Fig. 16 illustrates the measured result of common-mode rejection ratio (CMRR). A peak CMRR of 65 dB is coming up at approximately 5 GHz, which is the resonant frequency of the bandstop resonator. Moreover, it still remains larger than 20 dB from 3.5 to 6 GHz. The design issues in the high CMRR and wideband system are a tradeoff of this proposed architecture.

Further, the input third intercept point is measured by two-tone testing using a center frequency of 5.25 GHz with 10-MHz space, as plotted in Fig. 17. Apparently, after getting

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IV. CONCLUSION In this paper, we have proposed a PPS circuit. The circuit imbalance can be overcome by adopting the current tuner, high-impedance RLC load, and feedback network. The basic mathematical expressions reveal how to implement the phase difference without destroying the output power balance. The measured results demonstrate the theoretical analysis. This chip not only achieves high conversion gain and low imbalance, but also provides acceptable linearity, noise figure, and CMRR. Finally, the comparison with other active PPSs shown in Table III reveals that the proposed splitter has lower power consumption and larger conversion gain. Even adopting the CMOS process under a 1-V supply voltage, this active splitter has proven competitive in power imbalance and phase error with others fabricated in a high-cost process such as GaAs and BiCMOS, which inherently have better high-frequency characteristics. In addition, another benefit of the proposed splitter is easily integrated with other CMOS wireless components such as a voltage-controlled oscillator (VCO) and mixer or applied in I-Q modulator/demodulator circuits without the extra discrete component, and then the integrated cost without a monolithic microwave integrated circuit (MMIC) can be reduced.

Fig. 18. Photomicrograph of the proposed PPS.

TABLE III COMPARISON OF THIS STUDY WITH METHODS REPORTED IN THE LITERATURE

ACKNOWLEDGMENT The authors thank the anonymous reviewers for the valuable comments in improving the quality of this paper. REFERENCES

17.19-dBm fundamental signal 71.93-dBm modu, can be obtained [see (15)]. Hence, lated signal of 4 dBm is measured at the 9.66-dB conversion gain.

(15) The measured specifications for the wideband and WLAN 802.11a band, respectively, are summarized in Table II. This reveals that the proposed PPS circuit can not only achieve high conversion gain and high isolation, but also provide low phase error and power imbalance. Furthermore, the power consumption is only approximately 15 mW. A photomicrograph of the proposed PPS is also illustrated in Fig. 18.

[1] B. Razavi, “Design considerations for direct-conversion receivers,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 44, no. 6, pp. 428–435, Jun. 1997. [2] B. Razavi, RF Microeletronics. Englewood Cliffs, NJ: Prentice-Hall, 1998. [3] S.-Y. Lee, M. F. Huan, and C. J. Kuo, “Analysis and implementation of a CMOS even harmonic mixer with current reuse for heterodyne/direct conversion receivers,” IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 52, no. 9, pp. 1741–1751, Sep. 2005. [4] M. F. Huan, C. J. Kuo, and S.-Y. Lee, “A 5.25 GHz CMOS folded-cascode even harmonic mixer for low voltage applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 660–669, Feb. 2006. [5] A. H. Baree and I. D. Robertson, “Monolithic MESFET distributed baluns based on the distributed amplifier gate-line termination technique,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 2, pp. 188–195, Feb. 1997. [6] H. Kamitsuna and H. Ogawa, “Ultra-wideband MMIC active power splitters with arbitrary phase relationships,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 9, pp. 1519–1523, Sep. 1993. [7] J. Lin, C. Zelley, O. B. Lubeeke, P. Gould, and R. Yan, “A silicon MMIC active balun/buffer amplifier with high linearity and low residual phase noise,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 1289–1292. [8] C. Viallon, D. Venturin, J. Graffeuil, and T. Parra, “Design of an original -band active balun with improved broadband balanced behavior,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 280–282, Apr. 2005. [9] H. Ma, S. J. Fang, F. Lin, and H. Nakamura, “Novel active differential phase splitters in RFIC for wireless applications,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2597–2603, Dec. 1998. [10] C. W. Tang, J. W. Sheen, and C. Y. Chang, “Chip-type LTCC-MLC baluns using the stepped impedance method,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2342–2349, Dec. 2001. [11] M. Kawashima, T. Nakagawa, and K. Araki, “A novel broadband active balun,” in Eur. Microw. Conf., 2003, pp. 495–498. [12] C.-W. Kim, M.-S. Kang, P. T. Anh, H.-T. Kim, and S.-G. Lee, “An ultra-wideband CMOS low noise amplifier for 3–5 GHz UWB system,” IEEE J. Solid-State Circuits, vol. 40, no. 2, pp. 544–547, Feb. 2005.

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Shuenn-Yuh Lee (S’98–M’00) was born in Taichung, Taiwan, R.O.C., in 1966. He received the B.S. degree from National Taiwan Ocean University, Chilung, Taiwan, R.O.C., in 1988, and the M.S. and Ph.D. degrees from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1994 and 1999, respectively. Since 2002 and 2006, he has been an Assistant Professor and Associate Professor with the Institute of Electrical Engineering, National Chung Cheng University, Chia-Yi, Taiwan, R.O.C. His current research activities involve the design of analog and mixed-signal integrated circuits including filters, high-speed ADCs/DACs, sigma–delta ADCs/DACs, biomedical circuits and systems, low-power and low-voltage analog circuits, and RF front-end integrated circuits for wireless communications. Dr. Lee is a member of the IEEE Circuits and Systems (CAS) Society, the IEEE Solid-State Circuits Society, and the IEEE Communication Society.

Chun-Cheng Lai was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electronic engineering from I-Shou University, Kaohsiung, Taiwan, R.O.C., in 2004, and the M.S. degree in electrical engineering from National Chung Cheng University, Chia-Yi, Taiwan, R.O.C., in 2006. He is currently with the Department of Electrical Engineering, National Chung Cheng University. His current research interests include RF modules, microwave multilayer circuits, and RF front-end integrated circuits for wireless communications.

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Analysis and Design of Bandpass Single-Pole–Double-Throw FET Filter-Integrated Switches Zuo-Min Tsai, Member, IEEE, Yu-Sian Jiang, Jeffrey Lee, Student Member, IEEE, Kun-You Lin, Member, IEEE, and Huei Wang, Fellow, IEEE

Abstract—This paper proposes a method to integrate a single-pole–double-throw (SPDT) switch and a quarter-wavelength bandpass filter. A 1-GHz SPDT hybrid switch and a 60-GHz pseudomorphic HEMT monolithic-microwave integrated-circuit SPDT switch with 30% fractional bandwidth are demonstrated. The 1-GHz SPDT switch achieves 1.5-dB insertion loss and 20-dB isolation at center frequency. For the 60-GHz SPDT switch, the measured insertion loss is lower than 2.5 dB and the isolation is higher than 27 dB. The low insertion loss and high isolation show that no performance is degraded when integrating the filter function. The analysis of the power performance is also described. Using the device dc–IV curves, the power compression point can be predicted. Index Terms—Bandpass filter, monolithic microwave integrated circuit (MMIC), power compression point, single-pole double-throw (SPDT).

I. INTRODUCTION

R

ADIO-FREQUENCY (RF) switches are important in time division duplex wireless communication systems. Since a filter is required in front of a switch when integrating a wireless system, the integration of filter and switch functions has the advantages of reduction of insertion loss, as well as complexity. To achieve good switching performance, the nonideal off-state capacitance and on-state resistance need to be taken into considerations. In [1], the nonideal effects of the device are first applied for the 2.69-GHz ring-filter integrated with a single-pole–single-throw (SPST) switch. In [2], the concept of the filter-integrated switch (FIS), which gives the systematic design approach to implement the switch using the filter synthesis of a 1-GHz quarter-wavelength bandpass filter, was proposed. To design a single-pole–double-throw (SPDT) switch, an impedance transformer is usually required to combine two SPST switches so that the impedance of the off-state SPST switch will not influence the insertion loss of the on-state SPST switch [3], [4], [7], [8] in cases when the input/output Manuscript received November 14, 2006; revised March 21, 2007. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC 93-2752-E-002-002-PAE, Grant NSC 93-2219-E-002-016, Grant NSC 93-2219-E-002-024, and Grant NSC 93-2213-E-002-033 and by National Taiwan University under Excellent Research Project 95R0062-AE00-01. The authors are with the Graduate Institute of Communication Engineering and the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.901132

impedances of off-state switches are low. In [3], the image filter synthesis was applied in the design of an SPST traveling-wave switch and quarter-wavelength transmission lines applied as an impedance transformer. However, the frequency response of the entire SPDT switch cannot be synthesized. In [12], a narrowband hybrid SPDT switch using diodes with integration of a coupled-resonator filter was demonstrated by the shared resonator technique. The hybrid switch exhibits 5% fractional bandwidth (FBW) because of the filter topology. However, the topology in [12] is not suitable for wide-bandwidth application. In this paper, we expand the concept of field-effect transistor (FET) FISs from SPST [2] to SPDT switches. The quarter-wavelength impedance transformers are integrated in the quarter-wavelength bandpass filter. A 1-GHz hybrid and a 60-GHz monolithic-microwave integrated-circuit (MMIC) SPDT switches are both implemented to verify the design concept. The SPDT switches exhibit 30% FBW and are suitable for the wide-bandwidth communication system. A simple and general analysis of the power compression is also described. The analysis in [2] is only derived for SPST switches. On the other hand, a more general analysis for -port passive nonlinear circuits is proposed here. This analysis can not only be applied to this SPDT switch, but also to other -port passive switches. II. CONCEPT OF SPDT FIS The concept of the SPST FIS has been proposed in [2]. To implement an SPDT FIS, the impedance of the off-state SPST switch and the impedance transformer that combines two SPST switches have to be designed as part of the filter. Fig. 1(a) illustrates the circuit block diagram of the conventional SPDT switch. It is composed of two SPST switches and one impedance transformer. For the shunt switch, quarter-wavelength transformers are used to transfer the low impedance of the off-state SPST switch to high impedance [4]. Although the filter analysis was used in the low-pass SPST switch design, the frequency responses of the entire SPDT switch were not synthesized due to the bandpass impedance transformer. However, in our proposed filter-integrated SPDT switch, the impedance transformer and the SPST switches are designed for bandpass frequency response, as shown in Fig. 1(b). Fig. 2(a) shows the circuit schematic of the SPDT FIS. It is composed of five shunted resonators and four series quarterwavelength transmission lines. One short stub shunted resonator directly connects to Port 1. Other four shunted resonators are constructed by parallel transistors and short stubs.

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Fig. 1. (a) Circuit block diagram of conventional SPDT switches. (b) Concept of the filter-integrated SPDT switch.

Fig. 2(b) illustrates the equivalent circuit of the SPDT switch is below the threshold voltage and is higher when is below the threshold than the threshold voltage. Since voltage, the transistors are turned off and can be modeled to and ). Similarly, the high turned capacitors ( on the transistors and the transistors are modeled as resistors and ). Comparing Fig. 2(b) and (c), we can treat ( the through path of the SPDT switch (Port 2 to Port 1) as the third-order quarter-wavelength short-circuited stubs bandpass filter shown in Fig. 2(c). To make the frequency response of the third-order quarter-wavelength short stubs bandpass filter equal to the through path of the SPDT switch, two series quarter-wavelength transmission lines between Port 1 and Port 2 in Fig. 2(b) correspond to the series transmission lines in Fig. 2(c). Furthermore, the susceptance and their frequency of three shunt resonators derivatives at the center frequency and ) of the switch have to be identical to those ( and ). Thus, the relation of the bandpass filter ( of the design parameters between the switch and bandpass filter can be obtained as

Fig. 2. (a) Proposed SPDT FIS. (b) Equivalent circuit of the SPDT switch. (c) Circuit schematic of a third-order quarter-wavelength short-circuited stubs bandpass filter.

Since the frequency derivatives of the susceptances should be equal, we can obtain

(1) (2)

(5)

(3)

(7)

is the admittance from the turned-on transistor to In (3), the isolation port and is expressed with the system characteristic admittance . Since the transistors are turned on, and are large conductances, and thus, it is reasonable to to as follows: approximate

(4)

(6)

Applying the filter synthesis, the design parameters of the bandpass filter ( and ) can be obtained. With and are degiven device size, termined. Consequently, the design parameters of the switch and ) are obtained from (1)–(7). ( For the insertion loss of the through path at , only one is required to be taken into consideralossy component tion. From (1) to (3), and are all zero at , the

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both the susceptance and frequency differentiation of the susceptance can be equal to those of the quarter-wavelength bandpass filter and, thus, the frequency response of the switch may be synthesized. III. POWER ANALYSIS

Fig. 3. Circuit schematic for the calculation of isolation.

insertion loss follows:

is expressed in terms of

and

as

(8)

There are two steps to analyze the power performance. The first step is to describe the nonlinear device, and the second is to define the relation between power compression and the nonlinear conductance. In [2], the relation between power compression and the nonlinear conductance was derived only for the SPST FIS, but is not suitable for SPDT switches. To analyze our SPDT switches, a more general analysis is performed for an arbitrary -port network. Thus, this method is not only suitable for this SPDT switches, but also for -port passive switches. It is noticed that the piecewise linear approximation is used for the power analysis, which can estimate the power handling of the circuit fairly accurately. However, the piecewise linear approximation cannot be used to estimate the intermodulation since the third-order nonlinear effect is needed in the calculation of the intermodulation. A. Piecewise Linear Approximation

Similarly, the isolation of the isolation path can be calculated. From Fig. 3, the input admittance from the off-state transistor to Port 2 is expressed as

Fig. 4(a) illustrates the measured IV curves of the off-state passive HEMT [13]. By approximating the IV curves with two segments, can be expressed as if

is (10), shown at the bottom of this From (9), the isolation page. From (8) and (10), it is observed that the insertion loss and and . It is isolation can be improved by increasing also noticed that both the insertion loss and isolation can be imand proved at the same time. One constraint is that have to be in a reasonable range to make (1)–(7) solvable. Since and are the on-state channel conductances and off-state capacitances of the transistor and are all proportional to the gatewidth of the device, the device sizes in the switch need to be selected to make (1)–(7) solvable. It is worth mentioning that in the conventional switch design, there are tradeoffs between insertion loss and the isolation for the device size. Due to the off-state capacitance, the insertion loss will be higher if the device size is increasing. However, in our new approach, the off-state capacitance is one of the design parameters and is to be resonated with the short stubs. Furthermore, by adjusting the impedance and length of the short stubs,

(11)

if

(9) Assuming

is a sinusoid voltage signal, (12)

The amplitude of the fundamental term of by the Fourier transformation

can be evaluated

(13) In (13),

is the conduction angle and is defined as (14)

From (13), the nonlinear conductance of the off-state trancan be expressed in terms of and as sistor follows: if (15) if

(10)

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Fig. 5. Schematic of a general source.

n-port nonlinear

circuit with driving power

Fig. 6. Simplified passive HEMT small-signal model of NE32584C.

Fig. 4. Measured and piecewise linear approximation of the FET (NE32584C [13]) in: (a) turned-off state and (b) turned-on state.

For the measured on-state passive HEMT shown in Fig. 4(b), the IV curves of the on-state transistor can be expressed as Fig. 7. Circuit schematic of the 1-GHz hybrid SPDT switch.

if

In (14),

if

and

are defined as if

if

if

if

if

if if

if (16)

if if

Therefore, the nonlinear conductance of the on-state tranis derived as sistor

if

(18)

With the piecewise linear parameters and the implementation of (15) and (17), the relation between voltage amplitude and and ) can be analytically the equivalent admittance ( evaluated. B. Analysis of the Power Compression

(17)

Fig. 5 illustrates the general schematic of an -port nondriving to Port 1. The linear circuit with power source

TSAI et al.: ANALYSIS AND DESIGN OF BANDPASS SPDT FET FISs

Fig. 8. Admittance of the resonator of filter (Y onator of switch (Y ; Y ; and Y ).

;Y

;

and

Y

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) and res-

Fig. 9. Comparison of the simulated frequency responses between the 1-GHz ideal third-order filter and the 1-GHz hybrid SPDT switch.

Fig. 11. Simulated and measured results of the 1-GHz hybrid SPDT switch. (a) Insertion loss and output return loss of the on-state switch. (b) Isolation and output return loss of the off-state switch. (c) Input return of the switch.

Fig. 10. 1-GHz hybrid SPDT switch.

schematic is composed of one linear network and several . The nonlinear conducnonlinear conductances tance are assumed to be constants when the voltage swings are small, and after the voltage swings become large, the components are driven into the nonlinear region since are weakly nonlinear at the input -dB compression point . The effect of can be discussed separately. Considering only input,

can be obtained by sweeping for the -port network parameters since is the function of , while is also a function of , at -dB compression dominated can be determined. After investigating the conductance by of the -port nonlinear circuit can one by one, the input be determined by the lowest -dB compression point and the dominant nonlinear conductance can be decided. It is worth noticing that no nonlinear simulation tool is used to find the -dB compression point in this method. For the passive HEMT, the nonlinear effect of the is analyt-

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TABLE I DEVICE PARAMETERS OF THE DEVICE OBTAINED IN PIECEWISE LINEAR APPROXIMATION (NE32584C)

Fig. 12. Schematic of the 1-GHz SPDT hybrid switch for power analysis.

ically expressed in (15) and (17). Thus, only the IV curves and linear network analysis tool are required to obtain the power performance. IV. EXPERIMENTAL RESULTS A. 1-GHz Hybrid SPDT Switch

Fig. 13. (a) Compression of Port 2 to Port 1 insertion loss versus nonlinear conductances of 1-GHz SPDT hybrid switch. (b) Calculated nonlinear conductance versus input power (P ) at Port 2 of 1-GHz SPDT hybrid switch.

The devices used for this experiment are a packaged heterojunction field-effect transistor (HJFET) (NE32584C [13]). Fig. 6 illustrates the simplified passive HEMT small-signal model [4] of NE32584C. When the gate voltage changes from 3 to 0 V, the passive HEMT is switched from the off to the on ). Fig. 7 shows state and has 0.52-S channel conductance the complete circuit schematic of the 1-GHz SPDT switch. This switch is realized on an FR4 printed circuit board (PCB) board. From (8), it can be observed that the insertion loss is . Therefore, the dominated by the turn-on conductance of three-paralleled devices are selected as and to reduce the turn-on conductance from 0.52 to 0.17 S. The SPDT switch is designed for 1-GHz center frequency, 30% FBW, and 0.01-dB equal ripple frequency response. In the filter synthesis, the parameters of the filter prototype in Fig. 2(c) and the parameter are determined. Based on the design equations (1)–(3) and (5)–(7), the admittance and frequency and ) have to derivatives of each shunt resonator ( be equal to those of a third-order quarter-wavelength bandpass and ) and, thus, the parameters filter ( and can be determined. Fig. 8 shows the admittance of the and ) and those of the switches resonators ( ( and ) versus frequency. It can be observed that admittance of the resonators of the switch are close to those of the filter when the frequency is lower than 1.3 GHz so that the conditions in (1)–(3) and (5)–(7) are satisfied.

In this 1-GHz circuit design, the parasites in the package and layout discontinuities are taken into consideration. Fig. 9 shows the frequency responses of the ideal third-order bandpass filter and those of the simulated through path of the switch (Port 1 V and V). The frequency to Port 2 with responses of the switch are almost the same. The simulated results show 1-dB insertion loss at the center frequency and return losses better than 23 dB. Fig. 10 shows a chip photograph of the SPDT switch. Fig. 11(a) shows the measured and simulated results of the through path of the switch. At 1-GHz center frequency, the measured insertion loss is 1.5 dB and the return loss is better than 25 dB. Fig. 11(b) shows the performance of the V isolation path of the switch (Port 1 to Port 3 with V). The measured isolation is better than 27 dB at and the center frequency. Fig. 11(c) is the return loss of Port 1, which shows better than 25-dB measured return loss at the center frequency. The measured results are close to the simulated results when the frequency is below 1.5 GHz. The discrepancies between the simulation and measurement above 1.5 GHz may be due to the inaccuracy of the parasitics of the packaged devices at higher frequencies. The power compression of the SPDT switch in the center frequency (1 GHz) is analyzed. The device parameters of the piecewise linear approximation are obtained and listed in Table I. In

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Fig. 15. Output power versus input power for insertion loss and isolation of the 1-GHz hybrid SPDT switch.

Fig. 14. (a) Compression of isolation from Port 3 to Port 1 isolation versus nonlinear conductances of 1-GHz SPDT hybrid switch. (b) Calculated nonlinear conductance versus input power (P ) at Port 3 of 1-GHz SPDT hybrid switch.

Fig. 16. Circuit schematic of the 60-GHz MMIC SPDT switch.

TABLE II SIMPLIFIED PASSIVE HEMT SMALL-SIGNAL MODEL AND THE PARAMETERS OBTAINED FROM PIECEWISE LINEAR APPROXIMATION PARAMETERS FOR A TWO-FINGER 150-mm pHEMT DEVICE

Fig. 17. 60-GHz MMIC SPDT switch with chip size of (2 mm

the schematic for power analysis, shown in Fig. 12, and are off-state transistors and and are on-state transistors. The nonlinear channel conductance ( – ) of each transistors are analyzed to predict the power compression. To find the relations between the insertion loss from Port 2 to Port 1 and each nonlinear conductance, the insertion losses are simu– , and are illustrated in Fig. 13(a). lated with sweeping and are possible nonIt can be observed that (they are 5, 5, and 73 mS, linear elements to affect input respectively).

2 1 mm).

The next step is to find an input power level such that the nonlinear conductance satisfies the condition of 1-dB compression. The branch voltage of different power levels can be simulated by the linear simulation tool with the conductance satisfying the condition of 1-dB compression. Using (15) and (17), the nonlinear conductance under different input power – ] are estimated. Fig. 13(b) shows [ the nonlinear conductance versus input power. It is observed is 19.9 and 22.7 dBm by only considering and that , respectively. The nonlinear conductance of still does not decrease to 73 mS for 24-dBm input power. Thus, the of the SPDT switch is 19.9 dBm (lower one actual input . Therefore, between 19.9–22.7 dBm), and is determined by . the 1-dB compression of the insertion loss is dominated by

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Fig. 19. Output power versus input power for insertion loss and isolation of the 60-GHz MMIC SPDT switch.

TABLE III COMPARISON BETWEEN THE SIMULATED AND MEASURED RESULTS

(1 GHz). The measured insertion loss is 20.5 dBm and isois 187 dBm. Both of them are close to the predicted lation 19.9-dBm insertion loss and 19-dBm isolation , respectively. The input third order intercept point (IIP ) at 1 GHz with two tones of 100-kHz offset was measured to be 25 dBm. B. 60-GHz MMIC SPDT Switch Fig. 18. Simulated and measured results of the 60-GHz MMIC SPDT switch. (a) Insertion loss and output return loss of the on-state switch. (b) Isolation and output return loss of the off-state switch. (c) Input return of the switch.

Regarding the isolation from Port 3 to Port 1, the 1-dB comand are 6.6, 166 and 57 mS, pression occurs when respectively, and are shown in Fig. 14(a). The relation between the input power and nonlinear conductances are illustrated in limits the isolaFig. 14(b). It can be observed that only tion when the input power is below 24 dBm and the predicted is 19 dBm. Fig. 15 is the measured output power versus input power for insertion loss and isolation at center frequency

The process used in this design is WIN’s 0.15- m-high linearity AlGaAs/InGaAs/GaAs pseudomorphic HEMT (pHEMT) MMIC process [5]. In this design, the HEMT device of two fingers with total periphery of 150 m (2f150) was used. Following the simplified passive HEMT small-signal model in Fig. 6 and the piecewise linear approximation, the parameters of the HEMT device are listed in Table II. The target frequency response of this switch is 60-GHz center frequency, 30% FBW, and 0.01-dB third-order equal ripple bandpass filter responses. With the target frequency response, the systematic synthesis approach in Section II is applied to determine the characteristic impedance and length of the transmission lines for the device with a total gatewidth of

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TABLE IV PREVIOUSLY PUBLISHED HEMT SPDT SWITCHES AROUND 60 GHz

150 m. Fig. 16 shows the complete schematic diagram of the 60-GHz MMIC SPDT switch. The discontinuous junctions and parasitic effects of the transmission lines are all characterized with the Sonnet electromagnetic simulation tool [6]. Fig. 17 shows a chip photograph of the SPDT switch with a chip size of 2 mm 1 mm. The chip was measured via an on-wafer test. Since that two RF ground–signal–ground (GSG) probes cannot be placed on the same side of the chip for an on-wafer test, one of the output ports is terminated with an on-chip 50- load. In order to measure the complete frequency-response shape of this SPDT switch, three different test sets for different bands (dc to 50 GHz, -band, and -band) are used. Fig. 18(a) illustrates the measured and simulated results of the on-state switch. At the center frequency of 60 GHz, the measured insertion loss is 2.5 dB and the return loss is better than 10 dB. Fig. 18(b) shows the off-state switch performance. The measured isolation is better than 30 dB at center frequency. Fig. 18(c) shows the return loss of the input port, which shows better than 10-dB measured return loss at center frequency. From 45 to 65 GHz, the MMIC SPDT switch has an insertion loss, isolation, and return loss better than 2.2, 30, and 8 dB, respectively. They all reasonably agree with the simulated results. Fig. 19 illustrates the measured output power versus input power for insertion loss and isolation. Due to the power limitation of the 60-GHz signal source, we can only measure 0.5-dB compression of the insertion loss when the input power is 15.3 dBm. For the isolation, the power compression does not occur when the input power is 15.3 dBm. Using the same analysis procedure in the 1-GHz hybrid SPDT switch, the conductances for the 0.5-dB compression of insertion loss are , 1.35 mS for , or 43 mS for by sweeping 1.1 mS for the nonlinear conductance, and then the nonlinear conductance can be evaluated by (15) and (17) with the swing voltage of the input power at port 2. It is also observed that the dominant at input power of 16 dBm. Regarding isolacomponent is

mS, mS, or mS are the tion, conductances for the 0.5-dB compression point of isolation. Thus, the 0.5-dB compression of isolation is estimated to be , although we cannot 19.2 dBm and the dominate element is measure this power level. Table III lists the comparison between the simulated and measured results of the 1-GHz hybrid and the 60-GHz MMIC SPDT switches. The dominate components for the power compression are determined. The simulated results are close to the measured results; therefore, the design approach of the FIS and the predictions of the power performance are verified. Table IV lists the published HEMT switch around 60 GHz and compares it with our SPDT FIS. The 2.5-dB insertion loss in the passband is the best result, except for [11], which shows that no tradeoffs is required when integrating the filter function into a switch. V. CONCLUSION In this paper, the concept of the FIS, which integrates the quarter-wavelength bandpass filter into an SPDT, has been expanded from the SPST switch to the SPDT switch. The systematic design approach is presented so that the design parameters of the SPDT switch can be determined by applying the filter synthesis. A more general analysis to predict the power compression has also been described. The experimental circuits of hybrid 1- and 60-GHz MMIC SPDT switches both have 30% FBW, and a 0.1-dB equal ripple filter frequency response was built and measured. The 1-GHz switch exhibits 1.5-dB insertion loss and 27-dB isolation at center frequency, while the -band switch has 2-dB insertion loss and 32-dB isolation at center frefor the quency. For the 1-GHz SPDT switch, the measured is 17 dBm. For insertion loss is 20.5 dBm and the isolation is 15.3 dBm. the 60-GHz SPDT switch, the insertion loss The measured data are close to the predicted results. The method also provides a simple and precise way to obtain the power compression point and the dominant components without generating

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nonlinear models and nonlinear simulation tools, which is useful to design for the power performance of SPDT switches. ACKNOWLEDGMENT The authors would like to thank to Prof. G. D. Vendelin, National Central University, Chung Li, Taiwan, R.O.C., for his advice and review. REFERENCES [1] T.-S. Martin, F. Wang, and K. Chang, “Theoretical and experimental Investigation of novel varactor-tuned switchable microstrip ring resonator circuits,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1733–1739, Dec. 1988. [2] J. Lee, Z.-M. Tsai, and H. Wang, “A bandpass filter-integrated switch using field-effect transistors and its power analysis,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, 2006, pp. 768–771. [3] S. F. Chang, W.-L. Chen, J.-L. Chen, H.-W. Kung, and H.-Z. Hsu, “New millimeter-wave MMIC switch design using the image-filter synthesis method,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp. 103–105, Mar. 2004. [4] K.-Y. Lin, W.-H. Tu, P.-Y. Chen, H.-Y. Chang, H. Wang, and R.-B. Wu, “Millimeter-wave MMIC passive HEMT switches using travelingwave concept,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1798–1808, Aug. 2004. [5] “WIN Semiconductors GaAs 0.15 m pHEMT Model Handbook,” WIN Inc., Taipei, Taiwan, R.O.C., 2003. [6] “Sonnet User’s Manual,” Sonnet Softw. Inc., Liverpool, NY, 1998. [7] Z.-M. Tsai, M.-C. Yeh, H.-Y. Chang, M.-F. Lei, K.-Y. Lin, C.-S. Lin, and H. Wang, “FET-integrated CPW and the application in filter synthesis design method on traveling-wave switch above 100 GHz,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2090–2097, May 2006. [8] K.-Y. Lin, Y.-J. Wang, D.-C. Niu, and H. Wang, “Millimeter-wave MMIC single-pole–double-throw passive HEMT switches using impedance-transformation networks,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1076–1085, Apr. 2003. [9] M. Madihian, L. Desclos, K. Maruhashi, K. Onda, and M. Kuzuhara, “A sub-nanosecond resonant-type monolithic T/R switch for millimeter-wave systems applications,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 7, pp. 1016–1019, Jul. 1998. [10] G. L. Lan, D. L. Dunn, J. C. Chen, C. K. Pao, and D. C. Wang, “A high performance V -band monolithic FET transmit-receive switch,” in IEEE Microw. Millimeter-Wave Monolithic Circuits Symp. Dig., New York, NY, Jun. 1988, pp. 99–101. [11] J. Kim, W. Ko, S.-H. Kim, J. Jeong, and Y. Kwon, “A high-performance 40–85 GHz MMIC SPDT switch using FET-integrated transmission line structure,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 12, pp. 505–507, Dec. 2003. [12] S. F. Chao, C.-H. Wu, Z.-M. Tsai, H. Wang, and C.-H. Chen, “Electronically switchable bandpass filters using loaded stepped-impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4193–4201, Dec. 2006. [13] “NEC hetero junction field effect transistors,” NEC Inc., Kanagawa, Japan, NE32584C Data Sheet, 1997.

Zuo-Min Tsai (S’01–M’06) was born in Mailo, Taiwan, R.O.C., in 1979. He received the B.S. degree in electronic engineering and Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2001 and 2006, respectively. He is currently a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University. His research interests are the theory of microwave or millimeterwave circuits.

Yu-Sian Jiang was born in Kaoshiung, Taiwan, R.O.C., in 1984. She received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2006, and is currently working toward the M.S. degree at National Taiwan University. She is currently with the Graduate Institute of Communication Engineering, National Taiwan University. Her research interests include monolithic microwave/millimeter-wave circuit design.

Jeffrey Lee (S’07) was born in Buffalo, NY, in 1984. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2006, and is currently working toward the Master degree at National Taiwan University. His research interests include the design of MMICs and the theory of microwave circuits.

Kun-You Lin (S’00–M’04) was born in Taipei, Taiwan, R.O.C., in 1975. He received the B.S. degree in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1998, and the Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003. From August 2003 to March 2005, he was a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University. From May 2005 to July 2006, he was an Advanced Engineer with the Sunplus Technology Company Ltd., Hsin-Chu, Taiwan, R.O.C. In July 2006, he joined the faculty of the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, as an Assistant Professor. His research interests include the design and analysis of microwave/RF circuits. Dr. Lin is a member of Phi Tau Phi.

Huei Wang (S’83–M’87–SM’95–F’06) was born in Tainan, Taiwan, R.O.C., on March 9, 1958. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1980, and the M.S. and Ph.D. degrees in electrical engineering from Michigan State University, East Lansing, in 1984 and 1987, respectively. During his graduate study, he was engaged in research on theoretical and numerical analysis of electromagnetic radiation and scattering problems. He was also involved in the development of microwave remote detecting/sensing systems. In 1987, he joined the Electronic Systems and Technology Division, TRW Inc. He has been an MTS and Staff Engineer responsible for MMIC modeling of computer-aided design (CAD) tools, MMIC testing evaluation, and design and became the Senior Section Manager of the Millimeter-Wave (MMW) Sensor Product Section, RF Product Center. In 1993, he visited the Institute of Electronics, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., to teach MMIC related topics. In 1994, he returned to TRW Inc. In February 1998, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, as a Professor. He is currently the Director of the Graduate Institute of Communication Engineering, National Taiwan University. Dr. Wang is a member of Phi Kappa Phi and Tau Beta Pi. He was the recipient of the Distinguished Research Award of National Science Council, R.O.C. (2003–2006). In 2005, he was elected as the first Richard M. Hong Endowed Chair Professor of National Taiwan University. He has been appointed an IEEE Distinguished Microwave Lecturer for the 2007–2009 term.

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Printed and Integrated CMOS Positive/Negative Refractive-Index Phase Shifters Using Tunable Active Inductors Mohamed A. Y. Abdalla, Student Member, IEEE, Khoman Phang, Senior Member, IEEE, and George V. Eleftheriades, Fellow, IEEE

Abstract—This paper presents a printed and an integrated bi-directional tunable positive/negative refractive-index phase shifter utilizing CMOS tunable active inductors (TAIs). The printed phase shifter is comprised of a microstrip transmission line (TL), loaded with series varactors and a shunt monolithic microwave integrated circuit (MMIC) synthesizing the TAI. Using the TAI extends the phase tuning range and results in a low return loss across the entire tuning range. The integrated circuit (IC) phase shifter replaces the TLs with suitable lumped L–C sections. This enables integrating the entire phase shifter on a single MMIC, resulting in a compact implementation. The TAI used for both phase shifters is based on a modified gyrator-C architecture, employing a variable resistance to independently control the inductance and quality factor. The TAI is fabricated in the 0.13- m CMOS process and operates from a 1.5-V supply. The TAI chip is used to implement the TL phase shifter, which achieves a phase of 40 to 34 at 2.5 GHz with less than 19-dB return loss from a single stage occupying 10.8 mm 10.4 mm. The IC phase shifter is fabricated in the same process and achieves a phase from 35 to 59 at 2.6 GHz with less than 19-dB return loss from a single stage occupying 550 m 1300 m.

+

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Index Terms—Active inductors, CMOS, metamaterials, monolithic microwave integrated circuit (MMIC), negative refractive index (NRI), phase shifters, transmission lines (TLs).

I. INTRODUCTION ECENTLY there has been a strong interest in building phase shifters by cascading negative-refractive-index (NRI) metamaterial lines with positive-refractive-index (PRI) transmission lines (TLs) [1]. This allows building broadband compact phase shifters with linear frequency response. The PRI/NRI phase shifters are centered around the 0 mark and are capable of achieving positive and negative phase shifts. Centering the phase shift on 0 is desirable, for example, for scanning about the broadside direction in series-fed steerable antenna arrays. These composite 1-D metamaterial lines are realized by loading a host TL with series capacitors and shunt inductors [2], [3]. Several designs [4]–[6] have recently been published, which use a single variable loading element to tune

R

Manuscript received November 8, 2006; revised April 2, 2007. This work was supported by the Canadian Microelectronics Corporation, by the Center for Excellence in Communication and Information Technology Ontario, by the Natural Sciences and Engineering Research Council of Canada, and by Nortel Networks. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 3G4 (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.901076

the phase characteristics of NRI lines. This results in only positive phase shifts. Furthermore, tuning a single loading element (varactors in [4] and [5], and active inductors in [6]) results in high return losses. A composite PRI/NRI TL phase shifter is presented in [7] using two tunable loading elements: series and shunt ferroelectric varactors, which, however, require high control voltages (15 V). Furthermore, the design in [7] uses a fixed shunt inductor, which makes it impossible to satisfy the matching condition across the entire phase tuning range. Ferroelectric varactors also result in a modest phase tuning range of 12.5 /unit stage. In [8] and [9], the authors presented a tunable active PRI/NRI phase-shifter implementation based on loading a host microstrip TL with series varactors and a shunt monolithic microwave integrated circuit (MMIC) tunable inductor fabricated in a 0.13- m CMOS process. It was shown that using both varactors and tunable inductors extends the phase tuning range and maintains the input and output matching across the entire phase tuning range. In this paper, we substantially extend the previous work in [8] and [9] by introducing a single MMIC solution to implement the tunable active PRI/NRI phase shifter, eliminating the need for any printed or off-chip components. This is achieved by replacing the TL sections by their lumped L–C equivalent circuit in a suitable form, allowing for their integration with the tunable active inductor (TAI) on the same MMIC; hence, resulting in a large area saving. Furthermore, it will be theoretically and experimentally demonstrated that using varactors to implement the shunt capacitors of the lumped PRI TL sections extends the phase tuning range as compared to the TL phase shifter, while still maintaining input and output matching. Furthermore, this paper presents the underlying theory behind the design and implementation of the TAI used in both the TL phase shifter of [8] and [9] and the integrated circuit (IC) phase shifter. The TAI is based on a modified gyrator-C architecture, which employs a variable feedback resistance to independently control the inductance and quality ( ) factor. Tuning the TAI inductance without affecting the is a key feature to overcome the degradation of the phase-shifter insertion loss and return loss due to the dewhen the phase is being tuned via the crease of the TAI’s TAI inductance. Moreover, it is important to have control over the TAI’s without affecting the inductance since this allows controlling the phase-shifter insertion loss without affecting its phase response. In order to achieve the independent inductance and quality factor tuning capability, a variable feedback resistance is added to the gyrator-C architecture. This enhancement technique is applicable to any TAI based on the gyrator-C architecture, and it provides a general framework that can be applied to existing designs [10], [13], [14].

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Fig. 1. PRI/NRI metamaterial phase-shifter unit cell.

This paper begins by describing the theory and design equations for both the TL and IC phase shifters. Section III presents the TAI design and implementation, followed by the presentation of a MMIC prototype, which is fabricated and experimentally characterized. In Section IV, a packaged version of the MMIC TAI is used to implement the TL phase shifter, which is also experimentally characterized. The design and implementation of the IC phase shifter is presented in Section V together with its experimental characterization. Finally, in Section VI, the advantages and drawbacks of the two implementations are discussed and both implementations are compared. II. THEORY A. TL PRI/NRI Phase Shifter Fig. 1 shows the unit cell of the TL PRI/NRI phase shifter [1]. It is composed of a regular microstrip line with a characteristic , where and are the TL inimpedance ductance and capacitance per unit length, respectively. The microstrip line is loaded with two series capacitors and a shunt inductor . Cascading the PRI TL with the NRI section compensates the phase shift incurred by the propagating signal. The phase-shifter unit cell is analyzed herein using periodic analysis for terminated periodic structures. This technique can be applied to a finite number of unit cells when terminated with the corresponding Bloch impedance [15]. Using this technique simplifies the analysis and offers good design insight. Hence, one can show that the insertion phase of the unit cell is given by [1], [2]

(1) where is the phase lag due to one section of the PRI microstrip line given by . By investigating the zero-phase frequencies of the periodic structure, one can show that the condition to close the stopband in the region around the zero-phase point is (2) It is also important to investigate the return loss of the phaseshifter unit cell. When a unit cell of the TL phase shifter is terminated with an impedance , one can show that the reflection at the zero-phase frequency is coefficient expressed as

(3)

This indicates that, by satisfying the stopband closure condition of (2), the TL phase shifter becomes perfectly matched at . However, when the comthe zero-phase frequency ponent values are varied to tune the phase shift, the location of the zero-phase frequency changes. Hence, the frequency at which the phase shifter is matched changes. Nevertheless, the TL phase shifter still achieves a low return loss as long as the stopband closure condition of (2) is satisfied. This is mainly due to the nature of the PRI TL sections, which determines the characteristic impedance of the loaded TLs. Under the stopband closure condition, the phase shift can be approximated as (4) Equation (4), which was originally derived in [1], indicates that positive and negative phase shifts can be realized by a single unit cell without having to go through a complete phase rotation as in a traditional left-handed (high-pass) or right-handed (lowpass) architecture. The phase can be tuned by simultaneously changing the values of both the loading elements and . If and are varied from their nominal value to and , the phase tuning range can be expressed as (5) This results in at least doubling the phase tuning range compared to varying only. Furthermore, changing and simultanewill result in a low return loss across the entire ously phase tuning range. B. IC PRI/NRI Phase Shifter To reduce the size of the PRI/NRI phase-shifter unit cell, it is desirable to get rid of the TL sections. The TL sections are important to compensate the phase incurred by the signal due to the loading elements. By replacing each TL with a lumped L–C section and carefully selecting the values of the series inductance and parallel capacitance, a similar phase response can be achieved while occupying a much smaller area. This implementation will eliminate the need for bulky TLs and will allow integrating the PRI/NRI phase shifter on a single IC. The proposed IC PRI/NRI phase-shifter unit cell is shown in Fig. 2. It is similar to that presented in [2] to model a TL loaded with discrete series capacitors and shunt inductors for the sake of analyzing it. However, a complete -model with two shunt capacitors and a series inductor is used here to synthesize the TL sections. This creates two zeros in the reflection coefficient transfer function, resulting in two frequencies at which the phase shifter is perfectly matched. This extends the IC phase-shifter bandwidth as opposed to a lumped element phase shifter based on the unit cell presented in [2]. Furthermore, this topology makes the phase shifter more suitable to the implementation in IC form since the discrete components are replaced with on-chip components fabricated on a silicon substrate. More specifically, the explanation lies with the implementation of the series capacitors , which will be replaced with MOS capacitors, as will be

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implementation of (2) since the TL section, with a characteristic impedance , can be modeled by the two shunt capacitors and the series inductor , given that . Based on the stopband closure condition of (9), the phase shift per unit cell can then be rewritten as Fig. 2. Proposed IC PRI/NRI metamaterial phase-shifter unit cell.

(10) described in Section V. The MOS series capacitors are associated with large parasitic gate and drain/source diffusion capacitance to the substrate. Therefore, these parasitic capacitors can be naturally lumped with the shunt capacitor , and accounted for as a contributor to the phase of the PRI section. Similarly, the effect of the parasitic capacitance associated with the series . inductors can also be lumped within the shunt capacitors This makes the proposed unit cell in Fig. 2 well suited for the MMIC phase-shifter implementation. Using periodic analysis, one can show that the phase shift of the unit cell can be expressed as

(6)

(11) Similar to (4), the phase expression of (11) has two terms. The first term results in a phase lead and is caused by the NRI section (high pass), whereas the second results in a phase lag and is caused by the PRI section (low pass). Hence, similar to the TL phase shifter, a 0 phase shift can be realized by a single unit cell without having to go through a complete 360 phase rotation. Furthermore, positive and negative phase shifts can be realized depending on which of the two terms dominate. To center the phase shift around the 0 mark, the lumped component values should be chosen such that the phase contributions of the PRI and NRI sections cancel out. It is also important to investigate the return loss of the phaseshifter unit cell. When a unit cell of the IC phase shifter is terminated with an impedance , one can show that the reflection coefficient at the zero-phase frequency is expressed as

A simpler and intuitive expression for the phase shift can be obtained by assuming that the signal incurs a small phase shift ; . This is used to simplify (6), resulting hence, in the following expression: (12) (7) By equating the phase shift frequencies

to 0, one can find the zero-phase

and (8) To close the stopband of the periodic structure, should coincide, i.e., lower frequency of results in the following stopband closure condition:

and the . This

(9) This approximation is based on the assumption that the shunt is smaller than the series loading capacitor capacitor , which will guarantee that the cutoff frequency of the right-handed (low-pass) section is higher than the cutoff frequency of the left-handed (high-pass) section. This stopband closure condition is similar to the condition obtained for the TL

Similar to (3), this indicates that, by satisfying the stopband closure condition of (9), the IC phase shifter becomes perfectly . However, when matched at the zero-phase frequency the component values are varied to tune the phase shift, the location of the zero-phase frequency changes according to (8). Hence, the frequency at which the phase shifter is matched changes. Since it is desired to achieve a wide bandwidth, it is important to investigate the phase-shifter matching at frequencies different from the zero-phase frequency. One can derive to zero at frequencies the matching condition by equating . This results different from the zero-phase frequency in the following matching condition: (13) Equation (13) indicates that the proposed IC phase shifter has a second frequency at which it is perfectly matched. This is a result of an additional zero in the reflection coefficient transfer function. Using (13), one can show that this second frequency dips, i.e., , is expressed as where

(14)

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Fig. 3. Gyrator-C architecture and its equivalent circuit.

Having two frequencies at which dips, i.e., and , helps extending the bandwidth of the IC phase shifter compared to a lumped element phase shifter based on the unit cell of [2], dips only at the zero-phase frequency. In addition, where compared to the TL phase shifter, the IC phase shifter provides a significant area reduction. As indicated by (11), the IC implementation also allows varying the phase contribution of both the PRI and NRI sections, as opposed to the TL implementation, which only allows varying the phase of the NRI section. In the IC implementation, the phase of the NRI section is tuned via and , while that of the PRI section is tuned via . To demonand are varied from their nominal values to strate this, become and respectively. This results in the following phase tuning range:

(15) , and are chosen in order where the tuning ratios , to satisfy the stopband closure condition of (9), as well as the matching condition of (13). By comparing (5) and (15), one can see that the IC phase shifter has an extra term, which further extends its phase tuning range compared to the TL phase shifter, while still satisfying the matching condition. Furthermore, integrating the phase shifter on a single MMIC eliminates the parasitics associated with the individual component packages, which, in turn, extends the tuning range even more. The performance limitations due to the integrated capacitors and inductors will be discussed in Section V-A. Nonetheless, it will be shown that the proposed IC phase shifter, utilizing active circuits to implement the shunt inductor, results in a reasonably low insertion loss across the entire phase tuning range. III. TAIS In both the TL and IC phase shifters, tunable capacitors are implemented using varactors either discrete or integrated, respectively, where the voltage across the varactor controls its capacitance. On the other hand, electronically TAIs are not readily available and need to be synthesized. A. Analysis and Design Most TAI implementations are based on the gyrator-C architecture [16], which is shown in Fig. 3. It consists of two and ). If transconductors connected back-to-back ( the output resistance and capacitance of a transconductor (where ) are modeled by and and its input

Fig. 4.

f (R

) versus the negative series resistance

R

.

, then the input impedance capacitance is modeled by is equivalent to that of the circuit shown in Fig. 3. The series and parallel resistors in the equivalent circuit model the inductor losses, and the capacitance models the self-resonance of of the inductor. The inductance and series resistance the equivalent circuit are given by (16) and (17) The inductance of the TAI can be tuned by changing the and according to (16). To undertransconductances on the TAI’s , stand the effect of tuning the inductance one has to derive the expression for using it basic definition . The result of this analysis shows that is a function of frequency and it peaks at with a peak value of (18) . Equation (18) indicates where that as is tuned, the TAI will also change. This is not a desirable feature for the phase shifters, which require independent control of and . To overcome this shortcoming, we exploit the dependence of on . The function has two nega, as shown in Fig. 4. Therefore, to achieve tive real roots , one should pick a negative value for a high value for close to one of the two roots and within the stable region, which is defined as the region where or . Values in between the two roots will produce a negative (i.e., of ), therefore, this region is avoided a negative real part for during the characterization of the TAI. Uses for this region will be explored in Section V-B to compensate for the phase-shifter losses. Consequently, the TAI has two possible operating points and in regions 1 and 2, respectively. For the TAI circuit to operate at either point, it is required . Furthermore, if this negative can to achieve a negative be varied without affecting , the TAI circuit will achieve the independent and tuning capability. To generate a variable , a feedback resistance is added to the tradinegative tional gyrator-C architecture, as shown in Fig. 5. One can show

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Fig. 5. Modified gyrator-C loop and its equivalent circuit.

that the inductance and series resistance of the equivalent circuit are given by (19)

Fig. 6. Proposed TAI circuit with the tunable feedback resistance.

and (20) From (19) and (20), will be positive at low frequencies, but will turn negative at higher frequencies. Furthermore, the negcan be controlled via the feedback resisative resistance . Thus, a variable feedback resistance will allow one tance to control the TAI operating point on Fig. 4. This will guarantee a stable operation with a tunable high . It is important to note that the inductance is independent of the feedback resistance as long as it is much smaller than the output resistance of the first transconductor. This will allow one to independently tune the TAI’s and . B. Circuit Implementation Fig. 6 shows the proposed TAI circuit schematic, the first of the modified gyrator-C loop is replaced transconductor , and the second transconductor by a differential pair is replaced by a common-source amplifier . The tunis inserted between the output of the able feedback resistance and the input of the second transconfirst transconductor . and mirror a ratio of the reference current in ductor and to bias the circuit. Moreover, mirrors to generate the necessary current to half of the current in . To ensure that and mirror the desired current bias , a low voltage cascode current from the reference transistor mirror is used. Sizing and appropriately will guarantee , and sit at the same potential, and that the drains of this is achieved by setting (21) can be obAn approximate expression for the TAI and tained directly from (19) and (20) by replacing and with and , respectively. Thus, the inductance and the series resistance of the equivalent circuit are given by (22)

Fig. 7. Grounded active inductor equivalent circuit.

and

(23) where and are the output resistance and capacitance of the differential pair transconductor. Equation (22) shows that is independent of the feedback resistance and, hence, of , provided that is much smaller than the output resistance of the differential pair transconductor. More elaborate circuit analysis replaces each transistor by its , but results in small-signal equivalent model a fairly complicated expression for the input impedance. To obtain a simplified expression, which is necessary to gain insight in the circuit operation, the effect of the output resistance of the nMOS transistors is neglected. Hence, the input impedance of the TAI can be represented by the equivalent circuit shown in Fig. 7. The impedance of the inductive branch is given by

(24) This results in the same inductance and series resistance given by (22) and (23), respectively. This validates our previous analysis, which was based on the Gyrator-C block diagram. , binary To implement the tunable feedback resistance weighted nMOS transistors operating in the triode region are used as shown in Fig. 8. The tunable nMOS resistors are connected in parallel to a fixed resistance to make the overall feedback resistance, and hence , less sensitive to variations of the overdrive voltage of the transistors. This will make less sensitive to the input signal level and will improve the circuit stability. The nMOS resistors are switched ON and OFF using a

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Fig. 8. Digital/analog feedback resistance

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R

. Fig. 10. Measured inductance versus frequency for different bias voltages and .

V

V

Fig. 9. TAI die micrograph.

5-bit digital word, which is serially shifted into an on-chip shift register; this allows coarse tuning of . To enable fine tuning of , the digital word is applied through five CMOS inverters. , which is The inverters have a variable supply voltage used to set the required gate voltage for the ON nMOS resistors [17]. This arrangement is used to minimize the number of input pads required by the TAI circuit and, hence, reduce the circuit area. Combining digital and analog control to tune the resistor value allows a wider tuning range for the feedback resistance and, hence, the quality factor of the TAI. This makes the circuit more robust to process and temperature variations. C. Physical Realization and Experimental Characterization Fig. 9 shows the die micrograph of the fabricated TAI circuit, the chip was fabricated in a 1.5-V 0.13- m CMOS process. The chip area is 0.5 mm , of which the active inductor occupies 150 m 170 m. A ground–signal–ground (GSG) probe was used to probe the TAI, while two multicontact wedges with dc needles were used to provide the bias and control voltages. The TAI was characterized by measuring the reflection coeffithat was, in turn, used to extract the inductance and cient quality factor. The tuning characteristics of the TAI are demonstrated through tuning modes I and II described below. 1) Tuning Mode I: In this mode, the inductance is tuned . The TAI is tuned via while maintaining a fixed peak and according to (22), where the two transconductances are set by the two bias voltages and in Fig. 6, reaccording to (18). To comspectively. This will also change is tuned to bring the pensate for this, the feedback resistance

Fig. 11. Measured of Fig. 10.

Q versus frequency for the same bias voltages V

and

V

TAI back to the desired value. According to (22), changing should not have a significant effect on the TAI . Changing the bias point will also affect the TAI self-resonance frequency . This is due to the change in the TAI , which necessitates a change in the frequency at which resonates with the parasitic capacitance. Experimental results in Fig. 10 show that and , the cirby decreasing the transconductances cuit inductance can be tuned from 0.93 to 2.7 nH at 2.5 GHz. It also indicates that when the inductance increases, the circuit’s self-resonance frequency shifts to a lower value. Fig. 11 shows the corresponding TAI quality factors versus frequency for the of 100 is achieved across the entire insame bias points, a ductance tuning range. It is worth mentioning that, as the inducshifts to a lower tance increases, the peak frequency value, due to the decrease in . The measured and plots presented in Figs. 10 and 11 are a sample of the measured results, which are shown here to indicate the tunability of the TAI circuit around the phase shifters’ design

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Fig. 13. Measured Fig. 12. Measured

Q versus frequency for different feedback voltages V

S

of the TAI for different feedback voltages

V

.

.

frequencies. Full characterization of the TAI circuit indicate that it can provide inductances as low as 0.83 nH and as high as 11.7 nH at 2 GHz, while maintaining the peak quality factor in excess of 100. 2) Tuning Mode II: In this mode, the inductance is fixed and the TAI is tuned via the feedback resistance . Increasing will decrease according to (23), assuming the TAI is (Fig. 4), will decrease. Similarly, deoperating at point creasing will result in a higher . Since the bias point is and are unaffected. Hence, does not not changed, change since the feedback resistance is much smaller than the output resistance of the differential pair transconductor. Furthermore, both the self-resonance frequency and the peak frewill remain constant. Tuning the TAI using the quency feedback resistance does not require any additional power dissipation, as opposed to using a cross-coupled differential pair to generate a negative resistance [18]. Fig. 12 shows that the of a 1.7-nH TAI can be tuned from 10 to peak quality factor 200 by sweeping the gate voltage of the nMOS transistors imfrom 800 to 950 mV. Across the tuning plementing range, shown in Fig. 12, the measured variation at 2.4 GHz is less than 6.7% of the nominal value. The measured reflection coefficient of the TAI is plotted in Fig. 13 for the different feedback voltages. 3) Linearity Measurements: The linearity of the TAI circuit has been experimentally characterized by using an RF signal source to excite the grounded TAI (which is a one-port circuit) through a circulator. The circulator directs the reflected wave from the TAI circuit to its third port and the distortion components are analyzed using a spectrum analyzer. The TAI achieves dBm input compression point ( dB), which corresponds to approximately an 800-mV voltage swing at the TAI input from a 1.5-V supply voltage. It also achieves a third-order input inter-modulation product (IIP3) of 12.5 dBm. 4) Comparison of TAI Performance: Table I presents a detailed comparison between the proposed TAI and different TAIs presented in the literature [10]–[14]. Among these recently published TAIs, this study provides the largest inductance tuning

range and the highest resonance frequencies (with the exception of the BiCMOS design) in spite of the low-voltage CMOS process. Furthermore, the proposed design provides a mechanism to control the TAI quality factor with small variations in the circuit inductance. Compared to the other CMOS and BiCMOS implementations, the proposed TAI dissipates more dc power in order to achieve better linearity. Unfortunately, linearity was not reported in most of the previous publications to allow for an adequate comparison. IV. TL PRI/NRI PHASE SHIFTER A. Design and Implementation To allow for a comparison between the TL and IC designs, some of the material in [8] is summarized here. The TAI chip described in Section III is packaged using a 4 mm 4 mm high-speed quad flat-pack no lead (QFN) package to minimize the package parasitics, and is used to implement the TL PRI/NRI phase shifter. Fig. 14 shows the unit cell of the tunable are replaced by TL phase shifter. The series capacitors discrete varactors, where the reverse voltage across the varactor controls the capacitance. The TL sections are implemented using microstrip lines on a low-loss 10-mil Rogers RT/Duroid 5880 substrate. A photograph of the TL phase shifter is given in Fig. 15, the bias and control lines going to the inductor chip are supplied from the lower side of the board, whereas the right and left connectors are the input and output ports of the TL phase shifter, which also supply the bias voltages to the series varactors. A low substrate permittivity of 2.2 was chosen to reduce the phase shift incurred by the signal, hence making the phase shifter more wideband. The varactors used are 1.7 mm 0.9 mm plastic packaged silicon hyper-abrupt junction varactor diodes from Skyworks, Irvine, CA (SMV1232). The TL phase-shifter unit-cell size is 10.8 mm 10.4 mm; the TL and series varactors occupy 10.8 mm 1 mm, while the chip and bias lines roughly occupy 10.8 mm 9.4 mm. The unit-cell width is mainly set by the MMIC inductor package, which can be easily reduced by more than 22% by using a smaller package size (3 mm 3 mm). The smaller package was

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TABLE I COMPARISON BETWEEN DIFFERENT TAI IMPLEMENTATIONS

Fig. 14. TL PRI/NRI metamaterial phase-shifter unit cell.

Fig. 16. Measured and theoretical phase responses versus frequency for different bias conditions. The phase expression of (1) is used for comparison.

Fig. 15. Tunable PRI/NRI phase-shifter unit cell.

not used here since the chip contained other test circuits that needed to be packaged. B. Experimental Results Fig. 16 shows the measured and theoretical phase responses when both the MMIC inductance and varactor capacitance are varied. The theoretical response is predicted using the exact phase expression of (1). This figure shows good agreement between the measurements and theory. Using the approximate expression of (4) results in an error of less than 6.5 , hence, (4) can still serve as a good starting point for initial hand calculations, and at the same time, it gives good design insight. The good

agreement between the theoretical and experimental results is achieved by extracting the values of the different circuit components using accurate simulations (electromagnetic (EM)/circuit simulations). The component values obtained are then used to predict the phase response based on the theoretical equations. At the design frequency of 2.5 GHz, the phase can be varied from 40 to 34 passing through the zero-phase point. The phase-shifter unit cell is capable of achieving both positive and negative phase shifts at the design frequency without going through an entire 360 rotation (88-mm microstrip line). This corresponds to a 73% area saving compared to meandering the microstrip line. Furthermore, over the entire phase tuning range, is maintained below the matching condition is satisfied, and 19 dB at the design frequency, as shown by Fig. 17. As the increases, its capacitance varactor reverse bias voltage decreases, and the MMIC has to decrease to satisfy (2). When

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Fig. 19. Proposed IC PRI/NRI metamaterial phase-shifter unit cell.

V. INTEGRATED PRI/NRI PHASE SHIFTER A. Design and Physical Implementation

Fig. 17. Measured S

versus frequency for different bias conditions.

Fig. 18. Measured S

versus frequency for different bias conditions.

approaches 4.2 V, the matching condition is increasingly difficult to satisfy since the package adds a fixed inductance to the MMIC inductance, thereby setting a minimum achievable . Nevertheless, the phase shifter achieves a bandwidth of 2.6 GHz over which is less than 10 dB (see Fig. 17). The is presented in Fig. 18, and is set by the measured varactor losses and the MMIC inductor . The insertion loss at 2.5 GHz varies from 0.55 to 1.1 dB over the entire phase tuning range. Across the entire 2.6-GHz bandwidth, the insertion loss varies from 0.25 to 4.6 dB. The TL phase shifter dissipates an average dc current of approximately 32.9 mA from a 1.5-V supply, which corresponds to 49.4 mW across the entire phase tuning range. This power is required to bias the TAI circuit in order to generate the required inductance. The average noise figure of the TL phase shifter is predicted from simulations to change from 6.1 to 9.3 dB at 2.5 GHz across the phase tuning range with an average value of 7.6 dB. From simulations, the main noise contributor to the phase shifter’s noise figure is the TAI circuit.

The schematic diagram of the fully integrated PRI/NRI tunable phase shifter is shown in Fig. 19. Since there is no need for any printed or off-chip components, the phase shifter was implemented using a single MMIC. The 0.13- m CMOS process was chosen to fabricate the phase shifter since the TAI has already been characterized in that process. The same TAI circuit described in Section III is used to implement the phase shifter’s are implemented using shunt inductor. The series capacitors on-chip MOS varactors; the MOS capacitors used can be tuned from 0.38 to 1.4 pF. The of the varactors has a strong impact on the phase-shifter insertion loss given that its effect can be modeled as a series resistance in the signal path. To achieve the large capacitance value required to make the design frequency 2.6 GHz, a larger series capacitor is required. To this end, a fixed 0.67-pF on-chip high- metal–insulator–metal (MIM) capacitor, i.e., , is connected in parallel to achieve the required capacitance without reducing . The MOS capacitance, i.e., , is set via the gate to drain/source voltage. The gate voltage of the MOS capacitors is set by the dc voltage ap. On plied at the input and output ports of the phase shifter the other hand, the drain/source voltages are set by the voltage generated by the TAI ( V). The shunt capacitors of the PRI sections are implemented using on-chip hyperabrupt junction varactors, which provide a wide tuning range. The varactor capacitance can be tuned from 90 to 270 fF by changing the varactor cathode voltage from 3.8 to V. The varactor anodes are biased by the voltage generated by the are impleTAI. The series inductors of the PRI sections mented using on-chip 1.7-nH spiral inductors with 2.5 turns and an outer diameter of 200 m. However, the spiral inductors have a low- at the design frequency, which is one of the main contributors to the phase-shifter insertion loss. The die micrograph of the IC phase shifter is shown in Fig. 20, and it occupies an area of 550 m 1300 m; the core circuit without the pads occupies 380 m 960 m. To the authors’ knowledge, this is the smallest tunable PRI/NRI metamaterial phase shifter reported in the literature that operates in this frequency band. The TAI occupies 150 m 170 m from the overall area, and is located in the middle section of the layout. The spiral inductors to the left and right of the TAI are the series inductors of the PRI sections, i.e., . They occupy a larger area than the TAI, and are surrounded by ground shields to minimize the coupling between them. The series MIM capacitors, occupy a very small area, and can be seen in the die micrograph. On the other hand, the series MOS capacitors and

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Fig. 20. IC PRI/NRI metamaterial phase-shifter die micrograph.

shunt varactors are not visible in the die micrograph because they are covered by the metal fill introduced by the foundry to achieve certain layer densities. The bias and control voltages are provided to the circuit from the bottom pads. Large on-chip de-coupling capacitors are used to stabilize the bias and control voltages by providing a low-impedance path to ground. The right and left pads correspond to the input and output ports of the phase shifter, which also provide the bias voltage to the series MOS capacitors.

Fig. 21. Measured S voltage V .

and phase shift at 2.6 GHz versus the TAI feedback

Fig. 22. Measured S voltage V .

and phase shift at 2.6 GHz versus the TAI feedback

B. Experimental Results The IC phase shifter is characterized by probing the dies and measuring the -parameters. Two GSG probes are used for the RF ports while a multicontact wedge with 8 dc needles is used to probe the dc pads. Fig. 21 shows the measured and theoretical phase responses for different bias conditions. The theoretical response is predicted here using the approximate phase expression of (10), which results in good matching between the measurements and theory. Unlike the TL phase shifter, the exact phase expression of (6) and the approximate phase expression of (10) yield very accurate results. Comparing the measured phase with the approximate phase expression of (11) results in a 12.2 phase error; this is mainly due to the approximations made in the derivation of (11), which utilizes the approximate matching condition of (9). Equation (11) can still serve as a good starting point for initial hand calculations, and at the same time, it gives good design insight. To tune the phase shift , the series loading capacitance , the shunt TAI, as well as the shunt capacitance are varied using , the TAI bias point ( ), and , respectively. The voltage is swept from 0.3 to 2.05 V, and for each value, the appropriate inductance is generated using and to satisfy the matching condition given by (9). In addition to that, is swept from 0.1 to 3.8 V to extend the phase tuning range. The phase shift at 2.6 GHz can be tuned from 35 to 59 passing through the zero-phase mark without the need for an entire 360 rotation. This represents a 50% increase in the phase tuning range compared to the TL phase shifter. As explained in Section II, this is due to the ability to control the shunt capacitance . Furthermore, the IC phase shifter eliminates the parasitics of the TAI package, which limited the inductance tuning range in the TL phase shifter. Fig. 22 shows the measured input reflection coefficient for the same bias conditions used to sweep the phase. The

worst case at the design frequency is 19 dB. The phase shifter has a bandwidth of 1.9 GHz across which is less than 10 dB. The IC phase shifter has a smaller bandwidth compared to the TL phase shifter, which is expected, and is mainly due to the frequency dependence of the matching condition of (13). Fig. 23 shows the measured insertion loss for the same bias points. As indicated, varies from 2.8 to 3.8 dB at the design frequency. Across the entire phase-shifter bandwidth, the insertion loss varies from 2.8 dB to a worst case of 7.2 dB. The phase-shifter insertion loss is mainly due to the losses associated with the spiral inductors and the series MOS varactors, and here the negative resistance associated with the TAI is used to minimize these losses. It is also important to note that, as the shunt capacitance increases, the cutoff frequency of the PRI section decreases, which increases the phase-shifter insertion loss. Hence, the worst case results when and are set to 2.05 and 0 V, respectively. To demonstrate the effect of the negative resistance generated by the TAI circuit on the phase-shifter performance, the

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from simulations to change from 8.4 to 12.8 dB at 2.6 GHz across the phase tuning range with an average value of 10.3 dB. Similar to the TL phase shifter, the main noise contributor is the TAI circuit. VI. DISCUSSION AND COMPARISON

Fig. 23. Measured S voltage V .

and phase shift at 2.6 GHz versus the TAI feedback

Fig. 24. Measured S voltage V .

and phase shift at 2.6 GHz versus the TAI feedback

amount of negative resistance is varied by sweeping and the measured is plotted in Fig. 24. The plot shows that, at 2.6 GHz, the phase-shifter insertion loss can be enhanced by 0.6 dB with less than 7.4 phase variation, which corresponds to a change of 7.8% of the phase tuning range. The effect of on can be explained by the variation in the TAI series resis. Using the derivation outlined in Section III, one can tance show that the TAI series resistance is directly proportional to the function from (2). Hence, as increases, the feedback resistance decreases, resulting in an increase in the series resistance of the TAI equivalent circuit . This will lead to a decrease in , until reaches its minimum value. After that, will start to increase. This is the reason behind the decrease of in Fig. 24 after it reaches the maximum loss compensation point. The IC phase shifter dissipates an average dc current of approximately 21 mA from a 1.5-V supply, which corresponds to 31.5 mW across the entire phase tuning range. This power is required to bias the TAI circuit in order to generate the required inductance. The noise figure of the IC phase shifter is predicted

Using the PRI/NRI phase-shifter topology of [1] to build the electronically tunable phase shifters allows building compact phase shifters with phase centered around the 0 mark. Compactness is an important feature for many applications such as antenna arrays, where it is necessary to have compact phase shifters to allow the antennas to be spaced close enough to avoid capturing grating lobes in the radiation patterns as the beam is scanned. Moreover, passing through the 0 mark is desirable in series fed antenna arrays for scanning through the broadside direction. In addition, the proposed approach of using TAIs maintains the bi-directionality of the phase shifters, thus allowing the same antenna array to operate as a transmitter and receiver. We have experimentally demonstrated that the proposed approach results in a wider phase tuning range while maintaining the matching of the phase shifter, as opposed to other implementations using a single tuning element [4]–[6], or using two tunable elements, but in the form of series and shunt varactors only [7]. On the other hand, using TAIs will impose limitations on the phase shifter’s linearity, especially since it operates from a low-voltage supply (1.5 V). This will be critical when an antenna array using these phase shifters operates in the transmit mode. To this end, the TAI circuit has been designed to achieve good linearity by selecting appropriate transistor sizes and bias points. This resulted in a 2.16-dBm input compression point, which corresponds to approximately an 800-mV voltage swing from a 1.5-V supply voltage, and a 12.5-dBm IIP3. However, this comes at the expense of power consumption for an active phase-shifter design. The TL and IC phase shifters consume an average dc power of 49.4 and 31.5 mW across the entire phase tuning range, respectively. The higher average power consumption of the TL phase shifter is mainly due to the parasitic package inductance, which adds a fixed series inductance to the TAI. Hence, in order to satisfy the matching condition, lower inductance values are required by the TAI in the TL design, which, in turn, requires higher bias currents for the TAI circuit. Also, designing the phase shifters utilizing active inductors generally results in higher noise figures compared to passive designs: 7.6 and 10.3 dB for the TL and IC phase shifter, respectively. Nevertheless, in a practical application, the noise figure of the phase shifter can be enhanced by preceding the phase shifter with a low-noise amplifier when operating within a receiver [19]. Both the TL and IC phase shifters presented in this paper are prototypes fabricated to prove the concept and to experimentally characterize them. When using these phase shifters within a practical system, a lookup table together with multiple DACs can be implemented to set the different bias voltages according to a single control input. All of the biasing circuitry can be easily integrated on the same die with the TAI for the case of the TL phase shifter and on the same die with the IC phase shifter. This is one of the main advantages of using a standard CMOS technology to implement the phase shifters as opposed to using other

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TABLE II COMPARISON BETWEEN DIFFERENT PRI/NRI PHASE-SHIFTER IMPLEMENTATIONS

high (transistor unity-gain frequency) technologies such as GaAs. Furthermore, this should not result in a significant increase in the die sizes since removing the dc/bias pads would result in some area saving. Moreover, generating the bias voltages on-chip will reduce the number of pins required from the TAI IC package for the case of the TL phase shifter and from the IC package of the fully integrated phase shifter, making it possible to move to a smaller package size for both designs. This will allow us to further shrink the dimensions of the TL phase shifter, as well as the packaged IC phase shifter. Table II presents a detailed comparison between the hereby presented TL and IC PRI/NRI phase shifters along with related PRI/NRI phase shifters reported in the literature. Note that although the phase shifters presented in [7], [20], and [21] utilize the PRI/NRI structure, they do not achieve a phase centered around the 0 mark at their design frequencies. Tuning the inductance in both the TL and IC phase shifters results in a very wide phase tuning range compared to the other implementations. To this end, the TL phase shifter achieves the highest /dB), which is defined as the figure-of-merit ( phase tuning range per decibels of insertion loss. In contrast, the IC phase shifter has the largest phase tuning range, but it achieves a figure-of-merit of only 34 /dB. This is attributed to the higher losses of the IC phase shifter due to the low of the on-chip series spiral inductors and MOS varactors. In spite of this, the IC phase-shifter implementation occupies a very small area compared to the other implementations, and has the potential of integration with RF and digital circuitry in a standard low-voltage and low-cost CMOS process. Furthermore, the fractional bandwidth of both the TL and IC phase shifters is much wider than those of other implementations reported in the literature. This is mainly due to the ability to tune the shunt inductance, which allows one to maintain the matching condition across the entire phase tuning range.

TAIs: the first synthesizes the PRI section using microstrip TLs, whereas the second replaces the TL sections with lumped L–C sections. Thus, the latter can be implemented on a single MMIC. Both phase shifters are capable of achieving positive, negative, and zero phase shifts without going through the entire 360 . Compared to other implementations having only one tunable element, using tunable MMIC active inductors and varactors extended the phase tuning range and at the same time maintained the input and output matching of the phase shifters. The TL phase shifter achieved an electronically tunable phase from 40 to 34 at 2.5 GHz from a single stage, with better than 19-dB return loss across the entire phase range. On the other hand, the IC phase shifter achieved an electronically tunable phase from 35 to 59 at 2.6 GHz from a single stage, with better than 19-dB return loss. The MMIC inductor of the TL phase shifter and the IC phase shifter were implemented in a low-cost low-voltage (1.5 V) standard CMOS process. The design of the TAIs was based on a modified gyrator-C architecture utilizing a feedback resistance, which allows independent control over the inductance and quality factor. The negative resistance generated by the active inductors is used to partially compensate for the phase-shifter losses. The bi-directional PRI/NRI phase shifters can be used in series-fed steerable antenna arrays or as the unit cell for 2-D tunable metamaterial structures. For the latter application, the small size of these tunable devices will ensure that the resulting tunable metamaterials operate in the homogeneous limit, where the unit cell is much smaller than the wavelength. Furthermore, more than one stage can be cascaded to extend the phase tuning range further according to the desired application.

VII. CONCLUSION

The authors would like to thank the Canadian Microelectronics Corporation (CMC), Kingston, ON, Canada, for providing the fabrication facilities.

In this paper, we have presented two bi-directional implementations for the PRI/NRI phase-shifter unit cell utilizing

ACKNOWLEDGMENT

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REFERENCES [1] M. A. Antoniades and G. V. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 103–106, 2003. [2] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [3] A. Lai, T. Itoh, and C. Caloz, “Composite right/left-handed transmission line metamaterials,” IEEE Micro, vol. 5, no. 3, pp. 34–50, Sep. 2004. [4] H. Kim, A. Kozyrev, A. Karbassi, and D. W. van der Weide, “Linear tunable phase shifter using a left-handed transmission line,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 5, pp. 366–368, May 2005. [5] C. Damm, M. Schussler, M. Oertel, and R. Jakoby, “Compact tunable periodically LC loaded microstrip line for phase shifting applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 2003–2006. [6] L.-H. Lu and Y.-T. Liao, “A 4-GHz phase shifter MMIC in 0.18 m CMOS,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 694–696, Oct. 2005. [7] D. Kuylenstierna, A. Vorobiev, P. Linner, and S. Gevorgian, “Composite right/left handed transmission line phase shifter using ferroelectric varactors,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 167–169, Apr. 2006. [8] M. A. Y. Abdalla, K. Phang, and G. V. Eleftheriades, “A 0.13 m CMOS phase shifter using tunable positive/negative refractive index transmission line,” IEEE Microw. Wireless Compon. Lett., 2006, accepted for publication. [9] M. A. Y. Abdalla, K. Phang, and G. V. Eleftheriades, “A tunable metamaterial phase-shifter structure based on a 0.13 m CMOS active inductor,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 325–328. [10] C. Yong-Ho, H. Song-Cheol, and K. Young-Se, “A novel active inductor and its application to inductance-controlled oscillator,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 8, pp. 1208–1213, Aug. 1997. [11] C. Leifso, J. W. Haslett, and J. G. McRory, “Monolithic tunable active inductor with independent Q control,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 6, pp. 1024–1029, Jun. 2000. [12] C. Leifso and J. W. Haslett, “A fully integrated active inductor with independent voltage tunable inductance and series-loss resistance,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 4, pp. 671–676, Apr. 2001. [13] R. Mukhopadhyay, Y. Park, P. Sen, N. Srirattana, J. Lee, C. Lee, S. Nuttinck, A. Joseph, J. D. Cressler, and J. Laskar, “Reconfigurable RFICs in Si-based technologies for a compact intelligent RF front-end,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 81–93, Jan. 2005. [14] H. Chao-Chih, K. Chin-Wei, H. Chien-Chih, and C. Yi-Jen, “Improved quality-factor of 0.18 m CMOS active inductor by a feedback resistance design,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 12, pp. 467–469, Dec. 2002. [15] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005, pp. 372–375. [16] B. D. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep., vol. 3, pp. 81–101, Apr. 1948. [17] E. Alarcon, H. Martinez, E. Vidal, J. Madrenas, and A. Poveda, “D-MRC: Digitally programmable MOS resistive circuit,” in Midwest Circuits Syst. Symp., Aug. 2001, vol. 1, pp. 215–218. [18] A. Thanachayanont and A. Payne, “CMOS floating active inductor and its applications to bandpass filter and oscillator designs,” Proc. Inst. Elect. Eng.—Circuits, Devices, Syst., vol. 147, no. 1, pp. 42–48, Feb. 2000. [19] T. M. Hancock and G. M. Rebeiz, “A 12-GHz SiGe phase shifter with integrated LNA,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 977–983, Mar. 2005. [20] C. Damm, M. SchuBler, J. Freese, and R. Jakoby, “Artificial line phase shifter with separately tunable phase and line impedance,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 423–426. [21] D. Kuylenstierna, E. Ash, A. Vorobiev, T. Itoh, and S. Gevorgian, Sr TiO “X -band left handed phase shifter using thin film Ba ferroelectric varactors,” in Proc. 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006, pp. 847–850.

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Mohamed A. Y. Abdalla (S’98) was born in Cairo, Egypt, in 1978. He received the M.Sc. and B.Sc. degrees in electronics and electrical communications engineering from Cairo University, Giza, Egypt, in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree in electrical and computer engineering at the University of Toronto, Toronto, ON, Canada. From 2000 to 2002, he was with the Electronics and Electrical Communications Engineering Department, Cairo University, as a Teaching Assistant, and then as an Assistant Lecturer. Since Fall 2002, he has been a Research and Teaching Assistant with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto. His research interests include analog ICs such as transconductors and current-mode circuits, and RF/microwave circuits such as active inductors, phase shifters, directional couplers, and tunable metamaterial structures, as well as electronically steerable antenna arrays. Mr. Abdalla assisted in the organization of the 2005 International Solid State Circuits Conference (ISSCC). He has also assisted with the organization of the distinguished lecture series for the IEEE Solid-State Toronto Section. In December 2005, his paper entitled “A novel CMOS realization of the differential input balanced output current operational amplifier and its applications” published in July 2005 in Analog Integrated Circuits and Signal Processing was ranked among the top five most viewed papers. He was the recipient of a three-year University of Toronto Ph.D. Fellowship, as well as the Edward S. Rogers Sr. Ontario Graduate Scholarship (2004/2005). He was also the recipient of the 2003 Best Teaching Assistant Award.

Khoman Phang (S’90–M’00–SM’05) received the B.A.Sc., M.A.Sc., and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1990, 1992, and 2001, respectively. In 1993, he was a Visiting Researcher with the Sony Corporation, Tokyo, Japan. In 1994, he joined IBM Microelectronics Ltd., Toronto, ON, Canada, where he was involved in the development of infrared wireless networking products. In 2000, he joined the University of Toronto, where he is currently a Senior Lecturer with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering. His research interests include analog ICs, optical communication systems, and ICs for biomedical applications.

George V. Eleftheriades (S’86–M’88–SM’02– F’06) received the Diploma (with distinction) from the National Technical University of Athens, Athens, Greece, in 1988, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1993 and 1989, respectively, all in electrical engineering. From 1994 to 1997, he was with the Swiss Federal Institute of Technology, Lausanne, Switzerland, where he developed millimeter- and sub-millimeter-wave receiver technology for the European Space Agency and created fast computer-aided-design tools for planar packaged microwave circuits. In 1997, he joined the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada, where he is currently a Professor and a Canada Research Chair (Tier 1). He leads a group of 15 graduate students in the areas of negative-refraction metamaterials and their microwave and optical applications, integrated antennas and components for broadband wireless telecommunications, novel antenna beam-steering techniques, low-loss silicon micromachined components, sub-millimeter-wave radiometric receivers, and EM design for high-speed digital circuits. He coedited/coauthored Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley/IEEE Press, 2005). Prof. Eleftheriades currently serves as an IEEE Distinguished Lecturer for the Antennas and Propagation Society (IEEE AP-S). He was the recipient of the 2001 Gordon Slemon Award (teaching of design) presented by the University of Toronto and the 2001 Ontario Premier’s Research Excellence Award. He was the recipient of a 2004 E.W.R. Steacie Memorial Fellowship presented by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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A 20-Gb/s 1 : 2 Demultiplexer With Capacitive-Splitting Current-Mode-Logic Latches Jun-Chau Chien, Student Member, IEEE, and Liang-Hung Lu, Member, IEEE

Abstract—This paper presents a high-speed 1 : 2 demultiplexer (DEMUX) implemented in a 0.18- m CMOS process. By employing a capacitive-splitting architecture for the current-mode-logic latches, a significant speed improvement is achieved in the proposed DEMUX. Provided a 223 1 pseudorandom bit sequence from the pattern generator, the fabricated circuit operates at an input data rate up to 20 Gb/s. The fully integrated DEMUX consumes a dc power of 150 mW from a 2-V supply voltage. Index Terms—Bandwidth enhancement, capacitive-splitting latches, current-mode-logic (CML), demultiplexers (DEMUX), high-speed flip-flops, inductive peaking, optical-fiber communications.

I. INTRODUCTION N THE receiver frontend of an optical-fiber communication system, the demultiplexer (DEMUX) is an essential building block, which deserializes a high-speed data stream into several lower speed parallel data outputs. Owing to the increasing demands on the bandwidth for data transmissions, DEMUX integrated circuits operating at very high speed are required. In conventional circuit implementations, the maximum operation of speed of such circuits is limited by the cutoff frequency the transistors. Therefore, III–V compound semiconductors and SiGe technologies were preferred for applications at a data rate beyond 10 Gb/s [1], [2]. With recent advances in the fabrication technologies, CMOS devices demonstrate a great potential for the high-speed circuit designs. Due to the reduced parasitics and enhanced transconductance from the device scaling, deep-submicrometer CMOS transistors with a cutoff frequency over 100 GHz were reported [3]. Based on state-of-the-art process technologies, a 1 : 2 DEMUX in 120-nm CMOS [4] and a 1 : 4 DEMUX in 90-nm CMOS [5] have been fabricated to operate at a data rate of 40 Gb/s. However, for a cost-efficient process such as the 0.18- m CMOS, the highest data rate demonstrated for the DEMUX circuits is limited within 10 Gb/s [6], [7]. In this paper, capacitive-splitting architectures, which were originally developed for broadband amplifiers, are proposed for the high-speed latches to alleviate the speed limitations caused

I

Manuscript received November 30, 2006; revised May 7, 2007. This work was supported in part by the National Science Council under Grant 94-2220-E002-026 and Grant 94-2220-E-002-009. The authors are with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, 10617 Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.902071

Fig. 1. Cascaded stages with shunt-peaking inductance for bandwidth enhancement.

by the finite cutoff frequencies of the transistors. By employing the proposed technique, a fully integrated DEMUX operating at a data rate of 20 Gb/s is presented using a standard 0.18- m CMOS process. The remainder of this paper is organized as follows. Section II describes the proposed capacitive-splitting architectures for high-speed latches. The design and experimental results of the 1 : 2 DEMUX are presented in Sections III and IV, respectively. Finally, a conclusion is given in Section V. II. CAPACITIVE-SPLITTING CML LATCHES For high-speed DEMUX circuits, the highest input data rate is typically limited by the maximum operation speed of the master–slave flip-flops (MS-FFs), especially in CMOS designs. Conventionally, an MS-FF is composed of two cascaded static latches, and current-mode-logic (CML) stages with a seriesgated topology [4] are widely used for the implementation of the high-speed latches. In order to overcome the speed limitations, bandwidth enhancement techniques, which were originally developed for broadband amplifiers, are investigated for the CMOS DEMUX circuits. A. Circuit Architectures for Bandwidth Enhancement In the design of conventional cascaded amplifiers with resistime tive loads, the bandwidth is primarily limited by the constant of the individual stages. A widely used technique to enhance the bandwidth of such circuits is to employ the shuntpeaking inductors [8] in series connection with the resistive loads, as shown in Fig. 1. As the peaking inductance resonates with the nodal capacitance in the vicinity of the rolloff frequency, an extended bandwidth is demonstrated. For ultra-wideband applications, another effective technique is proposed to further boost the amplifier bandwidth by the capacitive-splitting

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(2) where

Fig. 2. Capacitive-splitting architectures. (a) NE termination topology. (b) FE termination topology.

architecture [9]–[11]. With the series inductance inserted in between the cascaded stages to separate the capacitive loadings, wideband characteristics can be achieved. In order to perform a systematic analysis on the bandwidth enhancement techniques, two simplified circuit models derived from the capacitive-splitting amplifiers are illustrated in Fig. 2, and represent the output capacitance of the prewhere vious stage and the input capacitance of the following stage, is the series inductance, which separates respectively, and and . Note that the major distincthe capacitive loadings tion between these two capacitive-splitting architectures lies in where the resistive-load branch is located. In the circuit model, as depicted in Fig. 2(a), the resistive-load branch is placed at the same node as , referred to as the near-end (NE) termination network. On the other hand, the far-end (FE) termination network, as shown in Fig. 2(b), places the loading branch on the side of . Even with the similarity in the capacitive-splitting architectures, very different circuit characteristics in terms of the frequency and step responses are demonstrated, especially and are not evenly separated. Therefore, both cawhen pacitive-splitting architectures are investigated for the design of the high-speed CML latches. B. Frequency and Step Response Analysis Based on the simplified circuit models in Fig. 2, the transfer functions of the capacitive-splitting architectures are derived to evaluate the frequency and step response of the systems. For the NE and FE networks, the transfer functions can be expressed as

(1)

From (1) and (2), both networks can be characterized as fourthorder systems, which only differ from each other by one coefficient in the denominator. Theoretically, the poles of the systems can be analytically derived, but the expressions become too complicated to provide any useful insight for circuit design. Consequently, numerical and conceptual analyses are instead performed. Before the analysis, it is noted that the transfer functions and are identical when . Nevertheless, this is generally not the case in practical designs since the is smaller than the gate capacitance drain capacitance in typical CMOS cascaded stages. For simplicity, the assumpis employed as a special case. The first analysis is tion to illustrate the impact of the inductance ratio, which is defined , on the frequency responses. In both networks, as the shunt inductance is selected to provide a maximum flat frequency response as is the case in conventional shunt-peaking topology [8] (3) is the load resistance, and repwhere resents the total capacitance. With realistic inductor models, Fig. 3(a) and (b) shows the frequency responses of the NE and FE networks, respectively, with various . It is observed from Fig. 3(a) and (b) that an additional high-frequency resonance . As and gain peaking take place due to the existence of increases, the resonant frequency moves toward the lower frequency band, while the midband gain recession diminishes. Therefore, with a proper selection of , the 3-dB bandwidth can be pushed beyond the high-frequency resonance, leading to an enhanced bandwidth with reasonable gain flatness in the passband. Fig. 4 depicts the frequency responses of the proposed capacitive-splitting architectures with the maximum achievable bandwidth. It is evident that the FE network outperforms its NE counterpart in terms of the bandwidth at the expense of excessive gain peaking in the vicinity of the rolloff frequency. In order to have a better understanding of the distinction in the frequency responses between these two networks, behavior illustrations, as presented in [10], are provided. In the NE netand work, as shown in Fig. 5(a), the series connection of first resonates at , resulting in gain roll-up in the frequency response. The gain reaches a maximum value when the

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forced to flow through mated by

and

, and the gain peaking is esti-

(4) At frequencies beyond , the -network can be treated as . Consequently, the third resan equivalent capacitance is determined by the inductance and onant frequency the capacitance . As for the FE network, similar illustrations are presented in Fig. 5(b). The first resonance, which and , takes place at . The high is determined by impedance provided by the parallel resonator forces the curto charge the capacitance . At frequencies higher than rent , the parallel resonator becomes capacitive and is modeled by a shunt resistance and a shunt capacitance as (5) (6) and act as a At the second resonant frequency short circuit to absorb the current , leading to a gain roll-up in the frequency response. The gain roll-up peaks when the -netresonates. From (5) and (6), the resowork nant frequency and associated gain peaking can be expressed as

Fig. 3. Simulated frequency responses with various inductance ratio m for the: (a) NE topology and (b) FE topology. (! = (R C ) ).

(7)

(8)

Fig. 4. Simulated frequency responses of the proposed capacitive-splitting architectures with the maximum achievable bandwidth. (! = (R C ) ).

-network resonates at . Due to the high is impedance demonstrated by the -network, the current

From the above analysis, two major differences between the proposed capacitive-splitting architectures are observed. First, the gain peaking occurs at the second resonance frequency in the NE network, while it takes place at the third resonant frequency in the FE network. Secondly, as predicted in (4) and (8), the FE network exhibits more severe gain peaking than the NE network does. As a result, the FE network can extend the 3-dB bandwidth more effectively, which shows good agreement with the simulation results in Fig. 4. Thus far, the analysis of the frequency responses are based on , which is a valid approximation for the assumption most cases in cascaded amplifiers. However, in the designs of high-speed digital circuits such as CML latches, the behavior several times larger than must of both networks with , simube investigated. Provided a fixed total capacitance lated frequency responses for both circuit topologies with varare illustrated in Fig. 6. In ious capacitance ratio the NE network, the third resonant frequency increases with a

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Fig. 5. Behavior illustrations of the capacitive-splitting architectures. (a) NE topology. (b) FE topology.

Fig. 6. Simulated frequency responses with various capacitance ratio (C

=C

diminishing gain peaking as increases. On the other hand, the gain peaking and rolloff frequency of the FE network increase with . From the numerical analysis performed above, a

) for the: (a) NE topology and (b) FE topology (! = (R

C

)

).

conclusion can be made that, though the capacitive-splitting architecture with the FE topology is advantageous for bandwidth enhancement, the excessive gain peaking in the frequency re-

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Fig. 7. Absolute root locus for the capacitive-splitting architectures as the capacitance ratio k increases.

Fig. 9. Circuit schematic of the capacitive-splitting CML latches with the: (a) NE topology and (b) FE topology.

Fig. 8. Simulated step responses of the NE and FE capacitive-splitting architectures for: (a) k = 2 and (b) k = 4.

sponse is undesirable in practical circuit designs, especially for high-speed latches with a large capacitance ratio. To further evaluate the influence of the capacitance ratio on the system behavior, the root locus and step response of both capacitive-splitting architectures are characterized. Provided that and the inductance ratio are fixed, the total capacitance the absolute root locus for both proposed topologies with an in, the creasing capacitance ratio are shown in Fig. 7. For total capacitance is evenly distributed between and , leading to an identical characteristic equation for both architectures. As increases, two poles move with decreasing , while

Fig. 10. Block diagram of the 1 : 2 DEMUX.

the other two poles move with increasing . Note that, in the FE network, the poles with smaller locate in close proximity to the axis, especially when becomes large, resulting in a reduced damping factor and an enhanced tendency of oscillation. In order to verify the observations, simulated step responses of

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Fig. 11. Simulated: (a) output jitter and (b) eye-opening improvement factor of the capacitive-splitting CML latch with various L and m.

Fig. 12. Simulated output eye diagrams of the high-speed latches with: (a) conventional CML stage, (b) shunt-peaking inductance, (c) NE capacitive-splitting topology, and (d) FE capacitive-splitting topology.

the capacitive-splitting architectures with various are shown in Fig. 8. As indicated in the plots, the FE network exhibits a faster transition due to the extended bandwidth. However, the output ringing and settling time become unacceptable when a large capacitance ratio is required. C. Capacitive-Splitting High-Speed CML Latches Based on the concept of the capacitive-splitting technique for bandwidth enhancement, two CML latches are proposed. Fig. 9(a) and (b) illustrates the circuit schematics with the NE

and FE topology, respectively. Due to the existence of the crosscoupled pair for data storing and the input capacitance of the is typically larger than by a factor of following stage, 4 in practical CML latch circuits. Although it is more effective to achieve an extended bandwidth by using the FE topology, the severe gain peaking near the rolloff frequency and the reduced damping factor are not desirable for high-speed latch operations. Even if the oscillation is prevented by selecting the circuit parameters carefully or deliberately adding a large series resistance along with , larger ringing and long settling time

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Fig. 15. Measurement setup for the DEMUX circuit.

Fig. 13. Circuit schematic of the differential output buffer.

Fig. 16. Re-timed singled-ended eye diagram at 13 Gb/s with 2 input. (horizontal scale: 20 ps/div., vertical scale: 50 mV/div.).

0 1 PRBS

Fig. 14. Microphotograph of the fabricated 1 : 2 DEMUX.

would introduce excess intersymbol interference (ISI), leading to a significant degradation in the quality of the output eye diagram for high-speed data transmission. As a result, the NE network is adopted for the circuit implementation. III. DESIGN OF 1 : 2 DEMUX Fig. 10 shows the block diagram of the 1 : 2 DEMUX, which is composed of two MS-FFs and two output buffers. In order to sample the data stream provided at the input, a half-rate clock is applied to both of the MS-FFs with inverted phases. The outputs of the MS-FFs are buffered by a two-stage differential amplifier to drive the external 50- load of the testing instruments. The design of the capacitive-splitting MS-FF starts with the circuit parameters of the CML latches. In consideration of the specifications including the voltage gain, output swing, and gate delay, a load resistance of 100 is employed and realized by polysilicon resistors in this particular design. The tail current is selected to provide an output voltage swing of 600 mV for sufficient signal-to-noise ratio. Once the tail current is determined, – is optimized for the transistor size of the tracking stage the switching speed. In order to achieve high-speed operations,

Fig. 17. Output singled-ended eye diagram with an input data rate of 20 Gb/s (horizontal scale: 50 ps/div., vertical scale: 50 mV/div.

the capacitance contributed by the cross-coupled pair – has to be minimized. Therefore, the transistor size of and is chosen such that the loop gain of the latch stage is slightly larger than unity to ensure the latch function. Finally, the clock – with a large transistor size is used for switching pair better sensitivity. After determining the circuit parameters of the CML latches, the capacitive-splitting architecture is constructed by incorporating the inductors into the design. Since both frequency- and time-domain characteristics are strongly influenced by the capacitive-splitting architecture, careful attention is required in the and . In order to optimize the circuit perforselection of mance, simulated output jitter and eye-opening improvement and are demonstrated in factors with various values of

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TABLE I PERFORMANCE SUMMARY OF THE HIGH-SPEED CMOS DEMUX

Fig. 11(a) and (b), respectively. Note that the simulation re1 pseudorandom bit sequence sults are based on a 20-Gb/s 2 (PRBS), while the eye-opening improvement factor is the vertical eye opening normalized to that of a conventional CML latch without any inductors. In this design, a 0.45-nH shuntis employed. As indicated in Fig. 11(a), peaking inductance output jitter reaches a minimum value of 0.01 unit interval (UI) , and increases as . This is mainly due with to the nonconstant group delay, which introduces excess ISI. However, in consideration of the rise and fall time at the output, is used, leading to an eye-opening an inductance ratio improvement factor of 6. Fig. 12 shows the simulated output eye diagrams of the high-speed latches based on various circuit topologies for comparison. It is noted that the capacitive-splitting CML latch with the NE topology exhibits the best quality in the output characteristics, making it very attractive for highspeed applications. Fig. 13 illustrates the schematic of the two-stage differential amplifiers, which provide the required buffering for the deserialized output data. To prevent the degradation in the bandwidth of the latches for high-speed operations, the first amplifier stage is designed to minimize the capacitive loading of the MS-FFs, while maintaining moderate signal amplification. On the other hand, the second amplifier stage is employed to steer a large output current for sufficient voltage swing across the external load of 50 . IV. EXPERIMENTAL RESULTS The proposed DEMUX is implemented in a standard 1P6M 0.18- m CMOS process. Fig. 14 shows the microphotograph of the fabricated circuit with a chip area of 0.92 0.82 mm including the pads. In order to evaluate the high-speed characteristics of the proposed technique, the fabricated circuit was tested by on-wafer probing. Fig. 15 shows the measurement setup, where the differential input data are generated by a pattern generator and a 4 : 1 multiplexer, and a single-ended clock at half of the data rate is provided by a synchronized signal generator with an output phase shifter. The single-ended voltage swings for the input data and the clock signal are 0.6 and 1 V , respectively. With a 2-V supply voltage, the DEMUX consumes a dc power of 150 mW including the output buffers. The fabricated circuit was first tested in the re-time mode to evaluate the highest operating speed of the MS-FFs with the capacitive-splitting CML latches. Fig. 16 shows the re-timed single-ended eye diagram

at the MS-FF output for a 13-Gb/s data rate and a 13-GHz clock, exhibiting a voltage swing of 250 mV . The measured peak-to-peak and rms jitter are 27.14 and 4.14 ps, respectively. 1 PRBS at a data rate of 20 Gb/s and a 10-GHz With a 2 half-rate clock, the measured output single-ended eye diagram of the DEMUX is shown in Fig. 17. The obtained eye opening is 150 mV, while the measured peak-to-peak and rms jitter are 21.2 and 3.6 ps, respectively. Note that, due to the limitations of the testing instruments, the measurement results are based on a single-ended clock signal. According to the simulation results, a higher data rate at 25 Gb/s can be achieved with differential clock signal provided. A comparison with the state-of-the-art results from previously published studies is tabulated in Table I. V. CONCLUSION In this paper, two different types of capacitive-splitting architectures, which are proposed to enhance the bandwidth and the operation speed of CML latches, have been analyzed and discussed. Based on the proposed circuit technique, a high-speed 1 : 2 DEMUX circuit is presented in a 0.18- m CMOS process. 1 PRBS input, the fabricated circuit operates at a With a 2 maximum data rate of 20 Gb/s, exhibiting a significant performance improvement compared with conventional topology with static CML latches. ACKNOWLEDGMENT The authors would like to thank the National Chip Implementation Center (CIC), Hsinchu, Taiwan, R.O.C., for chip fabrication and technical support. REFERENCES [1] K. Murata, K. Sano, H. Kitabayashi, S. Sugitani, H. Sugahara, and T. Enoki, “100-Gb/s multiplexing and demultiplexing IC operations in InP HEMT technology,” IEEE J. Solid-State Circuits, vol. 39, no. 1, pp. 207–213, Jan. 2004. [2] M. Meghelli, A. V. Rylyakov, and L. Shan, “50-Gb/s SiGe BiCMOS 4 : 1 multiplexer and 1 : 4 demultiplexer for serial communication systems,” IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1790–1794, Dec. 2002. [3] H. S. Bennett, R. Brederlow, J. C. Costa, P. E. Cottrell, W. M. Huang, A. A. Immorlica, Jr., J.-E. Mueller, M. Racanelli, H. Shichijo, C. E. Weitzel, and B. Zhao, “Device and technology evolution for Si-based RF integrated circuits,” IEEE Trans. Electron Devices, vol. 52, no. 7, pp. 1235–1258, Jul. 2005. [4] D. Kehrer, H.-D. Wohlmuth, H. Knapp, M. Wurzer, and A. L. Scholtz, “40-Gb/s 2 : 1 multiplexer and 1 : 2 demultiplexer in 120-nm standard CMOS,” IEEE J. Solid-State Circuits, vol. 38, no. 11, pp. 1830–1837, Nov. 2003.

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[5] K. Kanada, D. Yamazaki, T. Yamamoto, M. Horinaka, J. Ogawa, H. Tamura, and H. Onodera, “40 Gb/s 4 : 1 MUX/1 : 4 DEMUX in 90 nm standard CMOS,” IEEE Int. Solid-State Circuits Conf. Tech. Dig., pp. 152–153, Feb. 2005. [6] A. Tanabe, M. Umetani, I. Fujiwara, T. Ogura, K. Kataoka, M. Okihara, H. Sakuraba, T. Endoh, and F. Masuoka, “0.18-m CMOS 10-Gb/s multiplexer/demultiplexer ICs using current mode logic with tolerance to threshold voltage fluctuation,” IEEE J. Solid-State Circuits, vol. 36, no. 6, pp. 988–996, Jun. 2001. [7] J. Cao, M. Green, A. Momtaz, K. Vakilian, D. Chung, K.-C. Jen, M. Caresosa, X. Wang, W.-G. Tan, Y. Cai, I. Fujimori, and A. Hairapetian, “OC-192 transmitter and receiver in standard 0.18-m CMOS,” IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1768–1780, Dec. 2002. [8] S. S. Mohan, M. M. Hershenson, S. P. Boyd, and T. H. Lee, “Bandwidth extension in CMOS with optimized on-chip inductors,” IEEE J. SolidState Circuits, vol. 35, no. 3, pp. 346–355, Mar. 2000. [9] B. Analui and A. Hajimiri, “Bandwidth enhancement for transimpedance amplifiers,” IEEE J. Solid-State Circuits, vol. 39, no. 8, pp. 1263–1270, Aug. 2004. [10] S. Galal and B. Razavi, “40-Gb/s amplifier and ESD protection circuit in 0.18-m CMOS technology,” IEEE J. Solid-State Circuits, vol. 39, no. 12, pp. 2389–2396, Dec. 2004. [11] B. Y. Banyamin and M. Berwick, “Analysis of the performance of four-cascaded single-stage distributed amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2657–2663, Dec. 2000. [12] A. Rylyakov, S. Rylov, H. Ainspan, and S. Gowda, “A 30 Gb/s 1 : 4 demultiplexer in 0.12 m CMOS,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2003, pp. 176–177. [13] B.-G. Kim, L.-S. Kim, S. Byun, and H.-K. Yu, “A 20 Gb/s 1 : 4 DEMUX without Inductors in 0.13 m CMOS,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2006, pp. 528–529.

Jun-Chau Chien (S’05) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. and M.S. degrees in electronics engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004 and 2006, respectively. He is currently with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University. His research interests focus on integrated circuit designs for high-speed communication systems.

Liang-Hung Lu (M’02) was born in Taipei, Taiwan, R.O.C., in 1968. He received the B.S. and M.S. degrees in electronics engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 1991 and 1993, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2001. During his graduate study, he was involved in SiGe HBT technology and MMIC designs. From 2001 to 2002, he was with IBM, where he was involved with low-power and RF integrated circuits for silicon-oninsulator (SOI) technology. In August 2002, he joined the faculty of the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor. His research interests include CMOS/BiCMOS RF and mixed-signal integrated-circuit designs.

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Multitone Fast Frequency-Hopping Synthesizer for UWB Radio Kari Stadius, Member, IEEE, Tapio Rapinoja, Jouni Kaukovuori, Student Member, IEEE, Jussi Ryynänen, Member, IEEE, and Kari A. I. Halonen, Member, IEEE

Abstract—A fast frequency-hopping six-band local oscillator signal generator is described in this paper. Targeted for a WiMedia ultra-wideband radio transceiver, it offers operation in mandatory band group 1 and in extensional band group 3. The circuit entity consists of three parallel phase-locked loops (PLLs), each including two voltage-controlled oscillators, one per band group, and a signal multiplexer for fast frequency selection. A broadband poly-phase RC filter is used for in-phase/quadrature generation. Furthermore, the synthesizer generates the clock signal for analog-to-digital converters by mixing the signals from the first and third PLL. The circuit was fabricated in a 0.13- m CMOS process and it consumes 32 mA from a 1.2-V supply. It achieves 2-ns frequency settling time with a 3-MHz hopping rate. Index Terms—Frequency synthesizer, phase-locked loop (PLL), ultra-wideband (UWB).

I. INTRODUCTION

U

LTRA-WIDEBAND (UWB) communication offers great potential for future very high bit-rate short-range wireless networks [1]. As a part of recent standardization within the IEEE Working Group 802.15.3a, a multiband orthogonal frequency-division multiplexing (OFDM) system has been proposed, and based on this, the WiMedia UWB standard [2], formerly known as the MBOA standard, has been established. The 3.1–10.6-GHz frequency band has been divided into 528-MHz sub-bands, and these are arranged into band groups, as depicted in Fig. 1. Operation in the first band group is defined mandatory and the remainder of the band groups are optional extensions. Band group 2 overlaps with the UNII band, which potentially includes large interferers that can corrupt the UWB reception. Thus, a more attractive solution for future applications is to design a high-end UWB radio for operation in band groups 1 and 3. The UWB standard utilizes fast frequency hopping inside one band group at a time. The symbol length is 312 ns and hopping may take place after each symbol, resulting in a 3-MHz hopping speed, a rate significantly faster than in any previous frequency-hopping system. The allowed switching time between symbols is only 9.47 ns, and this sets high demands on the applicable frequency synthesizer for UWB radio. Three basic methods exist for fast frequency-hopping frequency synthesis. These are the single-sideband (SSB) mixing method, dual-PLL method, and local oscillator (LO) signal multiplexing.

Manuscript received September 27, 2006; revised January 29, 2007. The authors are with the Electronic Circuit Design Laboratory, Helsinki University of Technology, Espoo 02015 TKK, Finland (e-mail: [email protected] tkk.fi). Digital Object Identifier 10.1109/TMTT.2007.901131

Fig. 1. WiMedia UWB band allocation. The LO tones for the dark shaded band groups 1 and 3 are generated with the developed circuit.

We will give details and compare these methods in Section II. In Section III, we will present circuit design details for a frequency synthesizer based on signal multiplexing. Section IV is devoted to experimental results. The research presented in this paper is part of UWB transceiver development, although this paper will solely focus on LO generation. II. FAST FREQUENCY-HOPPING FREQUENCY SYNTHESIZER ALTERNATIVES A conventional phase-locked loop (PLL) frequency synthesizer takes hundreds of reference cycles to settle into the desired frequency. Even the fastest implementations take hundreds of nanoseconds for settling and, therefore, it is clear that a more advanced structure is needed for achieving less than 10-ns settling time. Three potential solutions are known, and they will be compared here. The major criteria are settling time, number of operation bands, spectral purity, power consumption, and die area. A. SSB Mixing Method Originally, in the IEEE Working Group 802.15.TG3a, an SSB mixing-method-based synthesizer was proposed by Batra et al. as a potential solution for fast frequency hopping, and since then, several such circuit implementations based on this method have been reported [3]–[14]. They do vary on the actual structure and implementation details, but fundamentally all rely on the SSB mixing method. For convenience, a simple conceptual structure is shown in Fig. 2. Actually, this structure was used as such in [6]. In the SSB mixing method, a PLL frequency synthesizer provides a fixed frequency tone, and by feeding the tone through the frequency divider, another mixing component is generated. The SSB mixer is thereafter used for generating the desired output tone. For instance, in Fig. 2, the PLL produces a tone at 3960 MHz and a divide-by-7.5 frequency divider is used for generating a tone at 528 MHz. Since the switching from one band to another requires only changing the operation mode of the SSB mixer, which inherently has small inertia, this method provides very fast frequency switching. The method consists of only one PLL including a voltage-controlled oscillator (VCO), and a set of transistor-only circuits in dividers

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target must be less than 0.2 s. According to [17], the frequency settling time of an ordinary integer- PLL is related by

(1)

Fig. 2. Conceptual structure of the SSB mixing method for frequency-hopping LO generation. Here, the SSB mixer is a three-mode device with upper/lower sideband rejection or bypass mode. The abbreviation “fr” refers to raster frequency of 528 MHz.

and an SSB mixer. Thus, it consumes a relatively small die area and has moderate power consumption. Unfortunately, there does exist a severe drawback. The SSB mixing is not an ideal operation. Instead, the resulting output spectrum includes a variety of undesired tones. These spurious tones do interfere with other wireless systems such as GSM900, GSM1800, WCDMA, Bluetooth, WLAN a/b/g, etc. resulting in performance degradation. The acceptable level of far-off spurs depends on use scenario: in a friendly environment, the UWB system is operating with only weak interfering systems in the vicinity, in a hostile environment, strong interfering systems are present, and in the worst case scenario, we have a multisystem terminal where an UWB radio is operating concurrently with other systems, and thus, strong interferences are in close proximity. Estimation for accepted level of spurs at WLAN and Bluetooth bands is given in [15], but it does not cover cellular systems. In [16], a detailed discussion on the spur generation in SSB mixing method is also given. In a friendly environment, spectral purity of approximately 40 dBc can be accepted. However, according to our system simulations, a spectral purity level of roughly 60 dBc is needed for a multisystem terminal. For being able to achieve such a low level for these far-off spurs, the quality of the SSB mixing (gain mismatches, LO and RF leakage, image rejection, linearity) must be very good and harmonics of the mixing tones generated directly from the PLL and from the divider chain must be low, thus requiring some filtering. According to our simulations, it is hard to achieve spectral purity better than 40 dBc, particularly if production tolerances are kept in mind. As a conclusion, the research carried out by several groups, as well as by us on this topic, is justified and the SSB mixing method is a feasible solution, but for some applications, we need to find an alternative solution, which is fundamentally able to provide better spectral purity.

B. Dual PLL In a dual-PLL concept, also called a ping-pong PLL, two similar PLLs are in parallel, and while one is in the active mode, providing the correct LO frequency, the inactive one is locking toward the next frequency in the hopping plan. Thus, the requirement for locking time is equal to one symbol time. Since we have to provide some safety margin, in the UWB case, the

in (1) refers to the final desired frequency accuracy, and is the amplitude of the frequency shift. it is 1 kHz [2]. For being able to lock extremely fast, a very broad loop bandwidth (BW) is needed and, hence, the reference frequency has to be high as well. Even then, fast operation of a PLL requires very detailed design since the above equation is a linear approximation and it does not take into account cycle slips. A very broad loop BW relaxes VCO phase noise requirements, but on the other hand, poses challenges for a low-noise design of phase-frequency detector (PFD) and charge pump. A very high reference frequency is a problem by itself. It must have a very low phase noise spectrum, and if it is chosen to be very high, yet another PLL and VCO are needed to generate it. Direct frequency multiplication may be a neat solution, but it deteriorates phase noise of the reference signal even further. Finally, to cover all defined bands (here, UWB band groups 1 and 3), a VCO with a very large tuning range or more probably a set of parallel VCOs is needed. Two parallel PLLs will thus occupy a large die area. Since only one PLL output is active at a time, this method potentially offers good spectral purity. As a conclusion, the dual-PLL method is feasible, but sets high demands on circuit design and implementation. A very fast-settling PLL design for this type of approach has been presented in [18] and it provides us a good reference for comparison. They have selected to use a reference frequency of 528 MHz and a loop BW of 26 MHz. The reference frequency is directly fed into the PLL and the implementation is a plain PLL, not a complete UWB frequency-hopping LO generator. As such, the design is very elaborate, and achieves 150-ns settling time. The PLL consumes 60 mW, and since in an actual UWB LO generator two of these are needed with some additional circuitry for multiplexing the outputs (and furthermore a PLL for reference signal generation is required), the complete entity will consume a significant amount of power and die area. C. LO Signal Multiplexing In the third method, we simply have a set of parallel constant frequency sources followed by a multiplexer (MUX), which selects one of the output frequencies at a time. Since the latency is only caused by the switching action of the MUX, this method offers very high-speed hopping capability. Since only one frequency output is active at a time, good spectral purity is possible. Each PLL can be designed independently, giving opportunity to optimize the circuits without compromise with challenging performance demands. Here, the PLL settling time is irrelevant since they boot-up with the complete radio and thereafter just maintain the correct frequency. A set of parallel PLLs has potential drawbacks mainly in power consumption and die area. We have selected to study this approach further and circuit design details are given in Section III. Therefore, here this short introduction is satisfactory. Finally, it is worth mentioning that muxing principle is used successfully in [19].

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Fig. 3. Structure of the LO generator. All structures are differential although drawn here as single-ended for clarity.

Fig. 5. VCO and buffer (only one side drawn).

transfer function of a charge-pump-based PLL can be expressed by (3)

Fig. 4. Block diagram of the PLL.

III. LO GENERATOR CIRCUIT DESCRIPTION The structure of the implemented LO generator is depicted in Fig. 3. It includes three parallel PLL frequency synthesizers, a MUX for selecting the active signal, and a mixer for generating a clock signal for analog-to-digital converters (ADCs) operating at twice the raster frequency of 528 MHz. The in-phase/ quadrature (I/Q) generation is accomplished with a three-stage poly-phase RC filter followed by buffers for loss compensation. Each PLL includes two VCOs, one for band group 1 and another one for band group 3. Furthermore, the frequency divider is programmable for operation in either of the band groups. The three PLLs differ in VCO device sizing and in frequency divider division ratios only. A. PLL The desired six output tones can be expressed with

MHz

(2)

Here, is 13, 15, 17, 25, 27 or 29. Thus, all three PLLs operating in both band groups 1 and 3 can be constructed from a generic integer- PLL structure. Such a PLL includes a divide-by-24 prescaler and a programmable divider, and the reference frequency is 11 MHz. The structure of such a PLL is depicted in Fig. 4. Band group selection bit is used to activate either bg1-VCO or bg3-VCO, and to set the corresponding division ratio into the counter. Although each VCO has slightly difand the division ratios ( ) differ, the ferent tuning gain impacts of these variations on PLL behavior are small enough so that no additional tuning is needed. For instance, the open-loop

is the charge-pump current and is the where of an impedance of the loop filter [17]. Thus, higher upper band VCO is compensated by increased division ratio . In this type of LO generation arrangement, the requirements for PLL performance are quite relaxed. The level of spurious tones should be below 50 dBc and, according to analysis presented in [16], phase noise of 90 dBc/Hz at 1-MHz offset is sufficient. Since in this application each VCO will only operate at one precise frequency, a very small tuning range could be, theoretically speaking, possible. However, due to process, supply voltage, and temperature (PVT) variations, a sufficient tuning range is needed. With extensive simulations, we found out that an 8% tuning range is appropriate. VCO: The circuit includes six oscillators—two in each PLL. They all have the same topology, depicted in Fig. 5, and just device sizes are tailored for best performance. The outputs of the two parallel oscillators are fed into a common buffer, which combines the signal routes. Only one VCO in each PLL is active at any time. The circuit includes a cross-coupled NMOS pair for generation of negative conductance and an LC resonator consisting of a differential coil and accumulation-mode nMOS -noise convaractors. Resistors are used for biasing to avoid is the tribution of a current source/sink. The upper resistor primary bias resistor. Furthermore, another small resistor at the bottom is used to improve phase noise characteristics and to provide some isolation to ground rail. Thus, the VCO floats from both rails and has some immunity for low-frequency disturbances. A pMOS switch MP is used for enabling power-down action. It has no significant impact on VCO characteristics. The in parallel with the upper resistor provides a capacitor low-impedance path for the second harmonic, yet it is so small that low-frequency disturbances in the supply rail do not have impact on the VCO through this device. As a result, the first VCO tunes in 3290–3590 MHz within a tuning voltage range of 0.2–1.0 V, draws 2.2 mA from a 1.2-V supply, and has phase noise of 118 dBc/Hz at a 1-MHz offset. Correspondingly, the

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Fig. 7. PFD.

Fig. 6. (a) Structure of the prescaler. (b) Low-voltage SCL D-latch. (c) Structure of the counter. Control logic is tailored for each PLL for providing proper division ratios. Control bit is used for bg1/bg3 selection.

last VCO covers 7380–7950 MHz, draws 3.5 mA, and the phase noise level is 113 dBc/Hz. Divider Chain: The frequency divider consists of a divide-by-24 prescaler and a counter. The counter provides division according to the band group and PLL operation band. Thus, in PLL1, the division ratios are 13 or 25, in PLL2, 15 or 27, and in PLL3, 17 or 29. The prescaler includes a chain , of dividers with division ratios of 2, 4, and 3 as depicted in Fig. 6(a). Furthermore, it includes an additional D-flip-flop (DFF) at the end of the divider chain used for reducing cumulative jitter [20]. All dividers are based on DFFs, and each flip-flop includes two D-latches. The structure of such a latch is shown in Fig. 6(b). A conventional current sink is omitted due to the low supply voltage of 1.2 V. As such, removing the current sink increases the maximum toggle frequency, but also increases power consumption and sensitivity

for process spread. A small bias resistor is now used for reducing the sensitivity for variations on the dc level of the input signal and transistor characteristics. Here, the broad input band poses a design challenge. A lot of power is spent on being able to cope with an input range almost two octaves broad. In the final realization, the prescaler consumes 50% of the total power in each PLL. A simple level shifter is used between the prescaler and the counter since the logical levels differs. The counter has been implemented using static CMOS logic and is shown in Fig. 6(c). It is a 5-bit counter [21], where the outputs of the first five DFFs change state according to the amount of pulses fed into the counter. The control logic realizes the wanted division ratios. The counter is able to operate up to an input frequency of 1.5 GHz with nominal device parameters, thus providing a good margin for the required maximum operation frequency of approximately 400 MHz. The counter consumes only 0.1 mW. PFD, Charge Pump, and Loop Filter: The PFD, depicted in Fig. 7, consists of two DFFs and a delayed feedback containing an OR-gate, two inverters, and a 1-pF capacitor. The two inverters and the capacitor create a delay, which prevents the dead-zone problem, when the PLL is in locked state. The buffers and inverters at the output minimize the timing error between the UP and UPX/DN and DNX signals. This timing error incur ripple to the VCO tuning voltage. This voltage ripple is the reason for frequency inaccuracy and close-in spurious tones at the PLL output and, therefore, has to be minimized. The second source of ripple is mismatches in the charge pump. When the PLL is in the locked state, the average current generated by the charge pump is zero. The sink and source current mismatch in the charge pump causes that the charge pump output currents and do not have exactly the same waveform and amplitude. The charge pump current matching depends on the voltage level at the output, which also defines the wanted VCO frequency in steady state. Ideally this voltage is designed to be 0.6 V, but it can vary due to PVT variations. This means that, in steady state, the mismatch has to be low despite the VCO tuning voltage. An error correction feedback, depicted in Fig. 8, is used to solve this problem. The designed charge pump itself is a “switch-in-gate” type for being able to operate with a low dimensioning is supply voltage. The charge-pump current a compromise between the power dissipation, loop filter size, produces a high-impedance and noise contribution. Low

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Fig. 10. Principle of MUX and output buffer. Real circuit is differential and includes biasing circuitry. Active g -stage draws 1.2 mA and the buffer-stage draws 1.6 mA. Fig. 8. Switch-in-gate type charge pump with error correction feedback.

Fig. 9. Loop filter schematic.

level to the loop filter, which implies smaller capacitor values and die area. On the other hand, the current has to be high enough so that the loop filter resistors noise contribution does value of 100 A not reach an unacceptable level [17]. An was chosen for achieving low power dissipation and a reasonable high-impedance level to the loop filter. Through simulations of the PFD, charge pump, and loop filter, it can be found that a loop BW of 280 kHz is a good compromise between loop filter noise contribution, size, and attenuation of the VCO tuning voltage ripple. The design of the loop filter is a compromise between the area consumed in the components and the attenuation to the ripple and phase noise. As a result of comparing the filter order and total loop filter capacitance, which equals the consumed area, we can note that the impact of the loop filter order to the consumed area is quite small. Thus, it is reasonable to use a high-order loop filter to accomplish the maximum attenuation for the VCO tuning voltage ripple. The drawback is that higher order loop filters create more thermal noise through the large resistors, which are used to create extra poles. In this design, we have chosen to use a fourth-order passive loop filter, depicted in Fig. 9. The loop filter consumes only approximately 10% of the total PLL area. B. MUX The role of an analog 3-to-1 MUX is to select one out of the three PLL outputs at a time and feed it to the poly-phase filter. It has to provide fast switching action, good isolation, and broadband gain. Since strong input signals are continuously present in all three inputs, the isolation is an important parameter for avoiding the mixing of these three signals in the subsequent blocks. This mixing will generate similar far-off spurs, as discussed in Section II-A. A simplified structure of the MUX and succeeding buffer are shown in Fig. 10. The use of the cascode

Fig. 11. (a) Three-stage poly-phase filter. All capacitors are 160 fF, 1 = 120 2 = 200 , and 3 = 340 . (b) I/Q accuracy, lines with symbols R

;R

R

represent worst case process spread. Inside band groups 1 and 3, over 35-dB IRR is achieved.

transistor enhances both isolation and BW. The buffer is a combination of a voltage follower and a common-source stage with an RC-feedback mechanism. C. I/Q Generation and Buffer Quadrature signal generation is accomplished with a three-stage poly-phase RC filter. It offers adequate performance over both band groups 1 and 3 with a sufficient margin for process variations as well. The structure is illustrated in Fig. 11(a), and achieved accuracy of the quadrature signals is shown in Fig. 11(b). IRR denotes the image-rejection ratio and

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Fig. 12. Microphotograph of the LO generator.

shows how the inaccuracy of phase and amplitude balance of a nonideal LO signal affects the reception in an otherwise perfect receiver. Here, we have slightly modified the conventional circuit topology. Nodes N1 and N2, depicted in Fig. 11(a), are usually connected to signal ground, whereas here they are connected to signal inputs. The structure provides exactly the same IRR as the conventional method, but offers 2-dB lower loss. More precisely, here the phase balance deteriorates and the amplitude balance improves compared to conventional topology. The 10-dB loss of the poly-phase filter has to be compensated for being able to drive the mixers of the actual receiver chain with adequate LO amplitude. The buffer-amplifier consists of a resistively loaded common-source stage with a cascode transistor followed by a voltage follower for driving the relatively large input capacitance of the mixer. D. Clock Generation The LO generator has to provide a clock signal for the rest of the modules in the IC chip, particularly for the ADCs. In our receiver, ADCs are clocked with the double raster frequency, i.e., at 1056 MHz [22]. Since the PLL output frequencies are not integer multiple of raster frequency, it is not possible to use a simple integer- frequency divider for generating the clock signal. Instead, here we are mixing the outputs of the first and third PLL. The down-converting signal will be at 1056 MHz. The mixer is a low-power Gilbert-cell type mixer with a resistively degenerated input stage. An amplifier with low-pass filtering for leakage tones from the mixer feeds the signal into ADCs, where it is further amplified for generating a proper square-wave clock signal. IV. EXPERIMENTAL RESULTS The circuit was fabricated in a 0.13- m CMOS process, which offers accurate high-quality capacitors and resistors for analog circuits. The LO generator occupies 1.9-mm excluding bonding pads. The microphotograph is shown in Fig. 12. The LO generator had lots of bonding pads and a digital control bus, thus offering us a possibility to measure various building blocks as standalone circuits.

Fig. 13. Measured tuning curves of the six VCOs. Tuning voltage measurement step is here 60 mV.

A. PLL and VCO The measured tuning curves for free-running VCOs are depicted in Fig. 13. Linearity in the actual tuning range (0.2–1.0 V) is very good and does not generate any problems for PLL dynamics. However, it appeared that the oscillation frequency ranges have been shifted downwards by 6%–8%. Since the tuning ranges are still as large as simulated, this shift in frequency cannot be explained by significantly larger parasitic capacitances. During the design, we did take into account the additional feed lines from the inductor to rest of the VCO, and yet the frequency is too low. Furthermore, in the actual receiver chain, there was an LC notch filter, and there the resonance was also shifted similarly. Thus, it seems that there is a problem with inductor models in the design kit. By using 10.3-MHz reference frequency instead of the intended 11.0 MHz, PLLs do lock correctly and, thus, despite this frequency shift, we are able to test the complete LO generator. Fig. 14 shows a typical phase noise plot for the free-running VCO. The first VCO (for band #1) has phase noise of 115 dBc/Hz at 1-MHz offset and the last VCO (for band #9) has 101 dBc/Hz at 1-MHz offset, respectively. The characteristics of other four VCOs are between these two extremes. Fig. 15 also includes the phase noise of the complete PLL. The in-band noise level agrees well with the simulations. The close-in spurious tones, one result depicted in Fig. 15, are higher than in simulations by approximately 10 dB, yet they remain at an acceptable level. When only a single PLL is powered up, and obviously the MUX is also in the active mode, the output spectrum should only include the oscillation tone and its harmonics. However, signal is also from Fig. 16, we can observe that the present. We do not have any buffer in front of the first divider and, therefore, the divided signal present inside the first stage of the prescaler leaks backwards into the VCO output node, which is directly coupled to MUX. B. System Level Results Table I summarizes the power budget of the circuit. Below 40-mW total power consumption compares very well to other

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TABLE I POWER BUDGET OF THE LO GENERATOR

Fig. 14. Phase noise for free-running VCO and for locked PLL.

Fig. 17. Output spectrum in bg3-mode, MUX is set to select the PLL2 output.

Fig. 15. Close-in spurious tones of PLL3 in bg1-mode.

Fig. 18. Measured LO hopping from band 3 to band 1. Settling time is approximately 2 ns.

Fig. 16. Output spectrum of a single PLL. F

divided-by-two leaks through.

published UWB LO generators. Due to the leakage of previously discussed divide-by-2 signals into the MUX, and thereafter to the output, the spectral purity is not as good as we were targeting. Unwanted signals create interferences, and although

the levels of these unwanted tones are quite low, they do appear in the output spectrum. One measurement result is depicted in Fig. 17. In this case, operation is in band group 3 and the MUX output is set to select the PLL2 output. The adjacent channels leakage is approximately 30 dBc. The WiMedia UWB standard does not define a maximum level for the adjacent UWB channels, but generally 30 dBc is considered acceptable. The double clock signal from the clock-generating mixer also leaks into the output. In the layout, the MUX and mixer are very close to each other. All other possible interference tones at bands of interest are below the measurement noise floor showing over 60-dB purity. Frequency hopping was measured with a 20-Gsa/s digitizing oscilloscope. Fig. 18 shows a hopping from band 3 to

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band 1 and the signal recovers within 2 ns. Transitions between other bands are very similar. The LO hopping was also tested with the complete receiver, and a potentially severe problem, dc-offset saturation at the baseband output, did not occur. V. CONCLUSION In this paper, we have represented an LO generator design for a WiMedia UWB radio operating in band groups 1 and 3. The principle of using parallel PLLs and a MUX shows good potential for UWB radio operating in basic mode in band group 1 and in advanced mode operation in band groups 1 and 3. It offers fast switching and good spectral purity. Despite some imperfections in the experimental results, the implemented circuit was able to demonstrate the feasibility of this approach. It has very fast switching speed and it offers quite low power consumption. It also brought forth a potential problem: the leakage of unwanted signals into the output via parasitic paths. This issue must be emphasized in future research by detailed layout designing and coupling modeling. The remaining major problem is a relatively large die area, mainly caused by the six coils in this design. Our future research work will address this problem. Currently, the mostly likely solution is to implement VCOs with a large tuning range for band group 3 and derive band group 1 via a frequency divider. This will significantly reduce the die area without a major impact on power consumption. REFERENCES [1] A. Batra, J. Balakrishnan, G. R. Aiello, J. R. Foerster, and A. Dabak, “Design of a multiband OFDM system for realistic UWB channel environments,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2123–2138, Sep. 2004. [2] High Rate Ultra Wideband PHY and MAC Standard, ECMA Standard 368, Dec. 2005. [Online]. Available: http://www.ecma-international. org/publications/standards/Ecma-368.htm [3] C. Sandner and A. Wiesbauer, “A 3 GHz to 7 GHz fast-hopping frequency synthesizer for UWB,” in Proc. Int. Ultra Wideband Syst. Workshop, 2004, pp. 405–409. [4] C. Sandner, S. Derksen, D. Draxelmayr, S. Ek, V. Fillmon, G. Leach, S. Marsili, D. Matveev, K. Mertens, F. Michl, H. Paule, M. Punzenberger, C. Reindl, R. Salerno, M. Tiebout, A. Wiesbauer, I. Winter, and Z. Zhang, “A WiMedia/MBOA-compliant CMOS RF transceiver for UWB,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig. , 2006, pp. 122–123. [5] W.-K. Lee, W. Kim, D. Meacham, H. S. Kim, and Y. S. Kim, “Implementation of a multi-tone signal generator for ultra wideband transceiver,” in Proc. Int. Ultra Wideband Syst. Workshop, 2004, pp. 263–267. [6] C.-C. Lin and C.-K. Wang, “A regenerative semi-dynamic frequency divider for mode-1 MB-OFDM UWB hopping carrier generation,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., 2005, pp. 206–207. [7] A. Ismail and A. A. Abidi, “A 3.1- to 8.2-GHz zero-IF receiver and direct frequency synthesizer in 0.18-m SiGe BiCMOS for mode-2 MB-OFDM UWB communication,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2573–2582, Dec. 2005. [8] J. Lee, “A 3-to-8-GHz fast-hopping frequency synthesizer in 0.18-m CMOS technology,” IEEE J. Solid-State Circuits, vol. 41, no. 3, pp. 566–573, Mar. 2006. [9] R. van de Beek, D. Leenaerts, and G. van der Weide, “A fast-hopping single-PLL 3-band UWB synthesizer in 0.25 m SiGe BiCMOS,” in Proc. Eur. Solid-State Circuit Conf., 2005, pp. 173–176. [10] R. Roovers, D. M. W. Leenaerts, J. Bergervoet, K. S. Harish, R. C. H. van de Beek, G. van der Weide, H. Waite, Y. Zhang, S. Aggarwal, and C. Razzell, “An interference-robust receiver for ultra-wideband radio in SiGe BiCMOS technology,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2563–2572, Dec. 2005.

[11] S. Lo, I. Sever, S.-P. Ma, P. Jang, A. Zou, C. Arnott, K. Ghatak, A. Schwartz, L. Huynh, and T. Nguyen, “A dual-antenna phased-array UWB transceiver in 0.18 m CMOS,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., 2006, pp. 118–119. [12] A. Tanaka, H. Okada, H. Kodama, and H. Ishikawa, “A 1.1 V 3.1-to-9.5 GHz MB-OFDM transceiver in 90 nm CMOS,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., 2006, pp. 120–121. [13] C.-F. Liang, S.-I. Liu, Y.-H. Chen, T.-Y. Yang, and G.-K. Ma, “A 14-band frequency synthesizer for MB-OFDM UWB application,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., 2006, pp. 126–127. [14] C. Mishra, A. Valdes-Garcia, E. Sanchez-Sinencio, and J. Silva-Martinez, “A carrier frequency generator for multi-band UWB radios,” in Proc. IEEE Radio Freq. Integrated Circuit Symp., 2006, pp. 221–224. [15] D. M. W. Leenaerts, “Transceiver design for multiband OFDM UWB,” EURASIP J. Wireless Commun. Network., vol. 2006, pp. 1–8, Article ID 43917. [16] C. Mishra, A. Valdes-Garcia, F. Bahmani, A. Batra, E. SanchezSinencio, and J. Silva-Martinez, “Frequency planning and synthesizer architectures for multiband OFDM UWB radios,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3744–3756, Dec. 2005. [17] C. S. Vaucher, Architectures for RF Frequency Synthesizers. Boston, MA: Kluwer, 2002. [18] G.-Y. Tak, S.-B. Hyun, T. Y. Kang, B. G. Choi, and S. S. Park, “A 6.3-9-GHz CMOS fast settling PLL for MB-OFDM UWB applications,” IEEE J. Solid-State Circuits, vol. 40, no. 8, pp. 1617–1679, Aug. 2005. [19] B. Razavi, T. Aytur, C. Lam, F.-R. Yang, K.-Y. Li, R.-H. Yan, H.-C. Kang, C.-C. Hsu, and C.-C. Lee, “A UWB CMOS transceiver,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2555–2562, Dec. 2005. [20] S. Levantino, L. Romano, S. Pellerano, C. Samori, and A. L. Lacaita, “Phase noise in digital frequency dividers,” IEEE J. Solid-State Circuits, vol. 39, no. 5, pp. 775–783, May 2004. [21] S. Brown and Z. Vranesic, Fundamentals of Digital Logic with VHDL Design. New York: McGraw-Hill, 2000. [22] O. Viitala, S. Lindfors, and K. Halonen, “A 5-bit 1-GS/s flash-ADC in 0.13-m CMOS using active interpolation,” in Proc. Eur. Solid-State Circuit Conf., 2006, pp. 412–415. Kari Stadius (S’95–M’03) received the M.Sc. degree in electrical engineering and Licentiate of Technology degree from the Helsinki University of Technology, Espoo, Finland, in 1994 and 1997, respectively. He is currently a Research Scientist with the Electronic Circuit Design Laboratory, Helsinki University of Technology. His research interests include the design and analysis of RF transceiver blocks with special emphasis on frequency synthesizers, RF oscillators, and modeling of passive components.

Tapio Rapinoja was born in Oulu, Finland, 1981. He received the Master of Science degree in electrical engineering from the Helsinki University of Technology, Espoo, Finland, in 2006. He is currently a Research Engineer with the Electronic Circuit Design Laboratory, Helsinki University of Technology. His main research interests are PLLs.

Jouni Kaukovuori (S’05) was born in 1977. He received the M.Sc. degree in electrical engineering from the Helsinki University of Technology (HUT), Espoo, Finland, in 2002, and is currently working toward the D.Sc. degree at HUT. Since 2001, he has been a Research Engineer with the Electronic Circuit Design Laboratory, HUT. His main research interests are CMOS RF circuits for wireless applications.

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Jussi Ryynänen (S’99–M’04) was born in Ilmajoki, Finland, in 1973. He received the Master of Science, Licentiate of Science, and Doctor of Science degrees in electrical engineering from the Helsinki University of Technology (HUT), Espoo, Finland, in 1998, 2001, and 2004, respectively. He is currently a Senior Research Engineer with the Electronic Circuit Design Laboratory, HUT. His main research interests are RF circuits, low-noise amplifiers, and mixers in direct-conversion receivers.

Kari A. I. Halonen (M’02) was born in Helsinki, Finland, on May 23, 1958. He received the M.Sc. degree in electrical engineering from the Helsinki University of Technology, Espoo, Finland, in 1982, and the Ph.D. degree in electrical engineering from the Katholieke Universiteit Leuven, Heverlee, Belgium, in 1987. From 1982 to 1984, he was an Assistant with the Helsinki University of Technology, and as Research Assistant with the Technical Research Center of Finland. From 1984 to 1987, he was a Research Assis-

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tant with the Electronics, Systems, Automation, and Technology (ESAT) Laboratory, Katholieke Universiteit Leuven. Since 1988, he has been with the Electronic Circuit Design Laboratory, Helsinki University of Technology, as Senior Assistant (1988–1990), and the Director of the Integrated Circuit Design Unit, Microelectronics Center (1990–1993). From 1992 to 1993, he was on a leave of absence during the academic year, during which time he was a Research and Development Manager with Fincitec Inc., Helsinki, Finland. From 1993 to 1996, he was an Associate Professor, and since 1997, he has been a Full Professor of the Faculty of Electrical Engineering and Telecommunications, Helsinki University of Technology. In 1998, he became the Head of Electronic Circuit Design Laboratory, Helsinki University of Technology. He has authored or coauthored over 150 international and national conference and journal publications on analog ICs. He holds several patents on analog ICs. He specializes in CMOS and BiCMOS analog ICs, particularly for telecommunication applications. Dr. Halonen was an associate editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART I: FUNDAMENTAL THEORY AND APPLICATIONS from 1997 to 1999. He has been a guest editor for the IEEE JOURNAL OF SOLID-STATE CIRCUITS and the Technical Program Committee chairman for the 2000 European Solid-State Circuits Conference. He was the recipient of the Beatrice Winner Award presented at the 2002 International Solid-State Circuits Conference. He was also a recipient of a temporary grant presented by the Academy of Finland.

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A New -Band Low Phase-Noise Multiple-Device Oscillator Based on the Extended-Resonance Technique Jonghoon Choi, Student Member, IEEE, and Amir Mortazawi, Fellow, IEEE

Abstract—A novel multiple-device oscillator based on the extended-resonance technique is proposed. This oscillator achieves low phase noise through optimizing the circuit’s group delay. Based on this technique, an -band four-device SiGe HBT oscillator has been designed, fabricated, and tested. The measured phase noise is 119 dBc at 100-kHz offset frequency. To the authors’ best knowledge, this oscillator shows the lowest phase-noise performance among other reported -band microwave planar hybrid free-running oscillators. Index Terms—Extended resonance, microwave oscillators, oscillator noise, phase noise.

I. INTRODUCTION

T

HE PHASE noise of oscillators fundamentally degrades the performance of communication and radar systems [1]. Many techniques to reduce phase noise of oscillators have, therefore, been investigated. The most commonly used technique to achieve low phase noise in microwave oscillators is to employ high- resonators such as high-permittivity dielectric resonators. This approach has several drawbacks, however, including their large size, the requirement for post-production tuning, and integration complexity [2]. In this paper, a new low phase-noise multiple-device planar oscillator circuit is proposed. Previous work in multiple-device oscillators has shown that, for an -device oscillator, the phase noise drops as ; this is because only the carrier powers add constructively, while the noise powers do not add coherently phase-noise reduction is not significant [3]. However, the enough to address the low phase-noise requirements of modern communication systems. The oscillator proposed here is based on an extended-resonance technique [5]. The extended-resonance technique is a power-dividing and power-combining approach based on a ladder circuit structure resembling a filter. This circuit can be optimized to achieve a high group delay corresponding to a high oscillator loaded quality factor . Additionally, the phase-noise improvement can be obtained from the usual Manuscript received October 2, 2006; revised May 4, 2007. J. Choi was with the Radiation Laboratory, Electrical Engineering and Computer Science Department, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 USA. He is now with Qualcomm Incorporated, Campbell, CA 95008 USA (e-mail: [email protected]). A. Mortazawi is with the Radiation Laboratory, Electrical Engineering and Computer Science Department, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]) Digital Object Identifier 10.1109/TMTT.2007.901612

power combining property of such circuits. In this manner, the extended-resonance multiple-device oscillator can improve rate relative to the number of the phase noise beyond the devices employed. In [4], a low phase-noise design approach for the extended-resonance oscillator was presented and several GaAs pseudomorphic HEMT (pHEMT) extended-resonance oscillators were demonstrated. In this study, a more complete analysis of a power-dividing and power-combining extended-resonance devices is given. Furthermore, low circuit incorporating phase-noise design methodologies are proposed in order to optimize the circuit parameters for maximizing the oscillator loaded . Based on the proposed technique, a 9-GHz four-device extended-resonance oscillator using packaged SiGe HBT devices is demonstrated. II. EXTENDED-RESONANCE OSCILLATOR DESIGN The basic concept of the extended-resonance technique is given in [4]. This type of circuit forms an -way power divider and combiner. Fig. 1 shows a schematic of an extended-resonance oscillator, which consists of an extended-resonance amplifier and a feedback circuit. The extended-resonance amplifier is superficially similar to well-known distributed amplifier circuits, though it is quite different in operation [5]. The extended-resonance amplifier is a power-combining circuit where all the devices are uniformly excited, thus contributing equally to the output power. This type of circuit is generally a narrowband resonant structure, while a distributed amplifier is a broadband circuit. Finally, there are no input and output terminations in the extended-resonance circuit. In the design of the extended-resonance amplifier, the input and , are and output admittances, namely, first determined through large-signal simulations (Fig. 1). The input and output circuits can then be designed for equal power dividing and combining. The input power-dividing circuit design follows the extended-resonance procedure, starting from the th device and ending at the first device [5]. The condition for coherent power combination requires the voltage phase to be difference between successive device output ports equal to the phase delay between the corresponding device input . ports An extended-resonance oscillator is constructed by employing feedback around the amplifier. The circuit’s design is based on the amplifier’s large-signal operation according to the substitution theory [7], [8]. The voltages and currents at the input and output terminals of the amplifier can be determined

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Fig. 1. Circuit schematic of the extended-resonance oscillator.

at a specific RF input power level through circuit simulation, as shown in Fig. 1. A feedback circuit that provides the same large-signal condition as the amplifier can be designed by determining the terminal voltages and currents. Here, a parallel -shaped feedback network consisting of one conductive and three susceptive elements ( , , element ) is employed as the external feedback network. The susceptive elements and output load in the -feedback network are calculated based on the input and output terminal voltages and currents [7] (1) (2)

(3) (4) (5)

Fig. 2. Extended-resonance power-dividing circuit with vices.

N two-terminal de-

III. EXTENDED-RESONANCE DESIGN FOR LOW PHASE NOISE The main design objective for the extended-resonance oscillator is to minimize its phase noise while maximizing power combining from multiple devices. In the extended-resonance oscillator, low phase noise can be achieved through large group delay and power combining. It is well known that an important mechanism for lowering phase noise in an oscillator is through increasing the oscillator’s loaded . The oscillator’s loaded is defined [10] by

where (6)

(11)

(7)

is the phase of the oscillator’s open-loop transfer where function at a steady state. It must be noted that the oscillator’s loaded is proportional to the absolute value of the group delay in (11) [10]. In other words, the increase in the group delay leads to the increase in the frequency selectivity of the oscillator circuit, thus reducing the phase noise. The extended-resonance circuit is a ladder network, composed of multiple devices and their interconnecting transmission lines, which behaves like a filter. This can be explained by considering an extended-resonance circuit incorporating devices, as shown in Fig. 2. A detailed explanation on the circuit operation is given in [4]. At the th stage, the admittance looking

(8) (9) (10) and are the input terminal voltage and current, and Here, and are the output terminal voltage and current in the amplifier.

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circuit in Fig. 3(a),

,

, and

can be expressed as (12) (13) (14)

where is the characteristic admittance of the interconnecting transmission lines, and (15) where is the complex propagation constant and is the physical length of the interconnecting transmission line. and Using the above reflection coefficients, the voltages at the input terminals of the first and second devices can be written as (16) (17)

Fig. 3. (a) Two-device extended-resonance circuit. (b) Equivalent circuit for the expanded device part marked by a dashed line box in Fig. 3(a).

into the th device is transformed to its conjugate , resonating out the susceptance of the admittance next device. Therefore, multiple resonant circuits are cascaded in the extended resonant circuit, allowing the circuit to achieve high group delays. In order to investigate how to achieve maximum group delay, an extended-resonance power-dividing and power-combining circuit incorporating devices is analyzed. First, a two-device extended-resonance circuit is analyzed. Subsequently, this analysis is extended to devices. The two-device circuit shown in Fig. 3(a) is one solution among many possible extended-resois the large-signal transconducnance circuits [5], where tance of each device. The large-signal device input and output and , readmittances are denoted by and are transformed to spectively. and through transmission lines ( and ) and open stubs ( and ) respectively, as shown in Fig. 3(a). An equivalent circuit for a unit cell of this structure is shown incorporates the in Fig. 3(b), where the transconductance effect of the gate and drain transmission lines and open stubs - . An expression for the large-signal group delay in the extended-resonance circuit can be found in terms of the device and are given, is chosen and circuit parameters. If , and is equal to so that the phase difference between and for coherent power the phase difference between combination. To simplify the analysis, and are transformed to an equal value . The steady-state voltages at each device’s input and output ports are determined when an input is applied. Referring to the input dividing current source

For the output combining circuit in Fig. 3, the voltages and at the output terminals of the first and second devices can be represented by

(18)

(19) This analysis can be expanded to an -device extended-resonance circuit using a similar approach. A MATLAB code was written to investigate the effect of the circuit parameters on the overall group delay. The group delay of the -device extended-resonance circuit can be expressed as (20) where (21) and are the input and output currents and is the voltage at the output terminal of the th device. In the numerical analysis, the center frequency and the frequency deviation are assumed to be 10 GHz and 10 MHz, respectively. To account for the losses, the parameters for a Rogers TMM3 substrate are used , mil, and ). (

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Fig. 5. Effect of the characteristic impedance of the interconnecting transmission line on (a) group delay, insertion loss, and (b) phase-noise figure-of-merit for the four-device extended-resonance circuit with various device admittances.

Fig. 4. (a) Group delay, (b) insertion loss, and (c) phase-noise figure-of-merit versus the device conductance and susceptance in the four-device extendedresonance circuits.

Based on the numerical analysis, the effect of the device admittance on the group delay and insertion loss is examined for the four-device extended-resonance circuit. Fig. 4(a) and (b) shows the plot of group delay and insertion loss versus the values of and . It can be seen that the group delay and insertion loss are proportional to the absolute value of device susceptance . Since phase noise is inversely proportional to the square of the group delay (in seconds) and proportional to the insertion loss [11], a phase-noise figure-of-merit F is defined as dB

(22)

Fig. 4(c) shows the plot of the phase-noise figure-of-merit versus device admittance and indicates that large device susceptance is advantageous for low phase noise. Next to be considered is the effect of the characteristic impedance of the interconnecting transmission lines on the group delay and insertion loss. In Fig. 5(a) and (b), the plot of the group delay, insertion loss, and phase-noise figure-of-merit versus the characteristic impedance of the interconnecting transmission lines is given in the four-device extended-resonance circuit. The simulation result shows that phase noise is inversely proportional to the characteristic impedance of the interconnecting transmission lines. Fig. 6 shows the circuit schematic of the extended-resonance oscillator designed for the low phase-noise performance. Based on the above-described circuit analysis, the low phase-noise design methodologies for this oscillator are outlined as follows. and 1) By using admittance transforming circuits ( in Fig. 6), the input and output device admittance values are transformed to new admittance values with the highest achievable susceptance. 2) The highest achievable characteristic impedance value is chosen for the interconnecting transmission lines.

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Fig. 6. Circuit schematic of the extended-resonance oscillator designed for the low phase-noise performance. AT and AT represent the input and output admittance transforming networks, respectively. The admittance transforming networks transform the input and output device admittances G + jB and G + j B into the new admittances G + j B and G + j B , respectively.

3) The shunt susceptive stubs inserted halfway between the and in Fig. 6) are selected in order to devices ( increase the group delay. Further, the susceptive stubs can reduce the overall circuit size. In general, as the number of devices increases, more interconnecting stages with resonance characteristics are cascaded, thereby increasing the overall oscillator . However, it should be mentioned that the circuit insertion loss also increases with the number of devices, thus impeding the phase-noise improvement rate. Fig. 7 shows the group delay and insertion loss of the extended-resonance circuit with the number of devices. Further, the estimated phase-noise improvement of the -device extended-resonance oscillator compared to the two-device extended-resonance oscillator is calculated using the following equation and as shown in Fig. 7:

(dB)

(23)

where and represent the group delay and insertion loss of the -device extended-resonance circuit, respectively. The above equation incorporates the effects of group delay, insertion loss, and power combining on phase noise. Fig. 7 shows that the four- and eight-device oscillators are expected to achieve approximately 7.5- and 13-dB phase-noise improvements over the two-device oscillator, respectively. IV.

-BAND FOUR-DEVICE SIGE HBT OSCILLATOR

A. Circuit Design In our previous study [4], 6-GHz pHEMT one-, two-, and four-device oscillators were demonstrated. The phase-noise measurement result in Fig. 8 shows that the two- and four-device extended-resonance oscillators achieve 13- and 22-dB phase-noise improvements at 1-MHz offset frequency, as compared to the single-device oscillator, respectively. The measured phase noise of the four-device pHEMT extended-resonance oscillator is 128 dBc/Hz at 1-MHz offset frequency.

Fig. 7. Group delay, insertion loss, and phase-noise improvement of the extended-resonance circuit with the number of devices. Input device admittance of 0:04 j 0:1 and transmission line characteristic impedance of 80 are used in the numerical analysis.

0

In this study, an -band four-device extended-resonance oscillator is designed employing the design methodologies described in Section III. To further reduce the phase noise,

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Fig. 8. Phase-noise measurement result for the pHEMT single-, two-, and fourdevice oscillators.

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Fig. 10. Phase-noise measurement and simulation result for the SiGe HBT -band extended-resonance oscillators. The black line shows the measured phase noise for the four-device extended-resonance oscillator. The gray lines show the simulated phase noises for the four-, six-, eight-, and ten-device extended-resonance oscillators.

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TABLE I COMPARISON WITH OTHER REPORTED MICROWAVE PLANAR HYBRID OSCILLATORS

Fig. 9. Circuit layout of the SiGe HBT nance oscillator.

X -band four-device extended-reso-

packaged SiGe HBT devices (NEC NESG2030M05) with low noise are used in the extended-resonance oscillator design. The oscillator is designed at 9.1 GHz and constructed on a Rogers TMM3 substrate. Fig. 9 shows the circuit layout of the four-device SiGe HBT extended-resonance oscillator. Each of 2 V device is biased at a collector–emitter voltage with a collector current of 8 mA. The device input and output admittances are determined through large-signal simulations and transformed to input and output device admittances of and . The characteristic impedance of 80 is chosen for the interconnecting transmission lines. The above device admittances and characteristic impedance are selected by taking into account the tolerance to fabrication error. The simulation results show that the oscillator of 90 is achieved in contrast to the oscillator of 55 for the pHMET four-device extended oscillator shown in [4]. In the circuit simulation, a parasitic oscillation around 1 GHz was observed. To eliminate a possible low-frequency parasitic oscillation, a coupled line with an insertion loss of 0.2 dB is inserted in the feedback network. The use of the coupled line suppresses the parasitic oscillation without affecting the output power level.

B. Comparison Between Simulation and Measurement Results The measured output power of the four-device SiGe HBT oscillator is 9.7 dBm at 9.1 GHz. The total consumed dc power is 66 mW corresponding to a dc–RF efficiency of 14%. The oscillator phase noise is measured based on an FM discriminator technique using the Agilent E5504A phase-noise measurement system. The phase-noise measurement results are shown in Fig. 10. The measured phase noise is 119 and 138 dBc/Hz at 100-kHz and 1-MHz offset frequencies, respectively. Since the transistor nonlinear model does not include the device noise data, the phase noise is simulated only in the region using the Agilent ADS simulator. The measured phase noise shows a good agreement with the simulations in the region, as shown in Fig. 10. The simulated phase noises the for six-, eight-, and ten-device extended-resonance oscillators are also shown in Fig. 10. The predicted phase-noise improvements for six-, eight-, and ten-device oscillators are 4.4, 7.1, and 8.3 dB, as compared to the four-device oscillator, respectively Table I compares the performance of the four-device extended-resonance oscillator with other reported microwave

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planar free-running oscillators at - and -band. To the authors’ best knowledge, this oscillator shows the lowest phase-noise performance among other published -band microwave planar hybrid free-running oscillators.

V. CONCLUSION This paper has presented a new multiple-device extended-resonance oscillator capable of improving the phase noise beyond rate relative to the number of devices employed. A the complete analysis of -device extended-resonance circuits has been presented for the low phase-noise design. The analysis has shown that the oscillator loaded is increased by transforming large-signal device admittances to admittance values with large susceptance and choosing a large characteristic impedance value for interconnecting transmission lines. A low phase-noise four-device SiGe HBT oscillator has been demonstrated at 9.1 GHz. The oscillator has shown an excellent phase noise of 119 and 138 dBc/Hz at 100-kHz and 1-MHz offset frequencies, respectively. The proposed technique is also expected to achieve lower phase noise by incorporating more devices. Our goal is to design a monolithic integrated low phase-noise extended-resonance oscillator at millimeter-wave frequencies.

REFERENCES [1] J. Choi and A. Mortazawi, “Microwave oscillators,” in Encyclopedia of RF and Microwave Engineering. Hoboken, NJ: Wiley, 2005, vol. 3, pp. 2818–2827. [2] S. Qi, K. Wu, and Z. Ou, “Hybrid integrate HEMT oscillator with a multiple-ring nonradiative dielectric resonator feedback circuit,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 1552–1558, Oct. 1998. [3] K. Kurokawa, “The single-cavity multiple-device oscillator,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 10, pp. 793–801, Oct. 1971. [4] J. Choi and A. Mortazawi, “A novel multiple-device low phase noise oscillator based on the extended resonance technique,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, 2006, pp. 577–580. [5] A. Martin, A. Mortazawi, and B. C. De Loach, Jr., “An eight-device extended-resonance power-combining amplifier,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 6, pp. 844–850, Jun. 1998. [6] A. Mortazawi and B. C. De Loach, Jr., “Multiple element oscillators utilizing a new power combining technique,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2397–2402, Dec. 1992. [7] K. L. Kotzebue, “A technique for the design of microwave transistor oscillator,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 7, pp. 719–721, Jul. 1984. [8] M. Lee, S. Yi, S. Nam, Y. Kwon, and K. Yeom, “High-efficiency harmonic loaded oscillator with low bias using a nonlinear design approach,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1670–1679, Sep. 1999. [9] M. M. Driscoll, “Low noise oscillator design and performance,” presented at the IEEE Freq. Control Symp., New Orleans, LA, Jun. 2002.

[10] J. Nallatamby, M. Prigent, M. Camiade, and J. J. Obregon, “Extension of the Leeson formula to phase noise calculation in transistor oscillators with complex tank,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 690–696, Mar. 2003. [11] T. E. Parker, “Current developments in SAW oscillator stability,” in Proc. 31st Annu. Freq. Control Symp., Jun. 1977, pp. 359–364. [12] Y. Cassivi and K. Wu, “Low cost microwave oscillator using substrate integrated waveguide cavity,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 2, pp. 48–50, Feb. 2003. [13] L. Dussopt, D. Guillois, and G. M. Rebeiz, “A low phase noise silicon 9 GHz VCO and 18 GHz push–push oscillator,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 2, pp. 695–698. [14] A. P. S. Khanna, E. Topacio, E. Gane, and D. Elad, “Low jitter silicon bipolar based VCO’s for applications in high speed optical communication systems,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, vol. 3, pp. 1567–1570. [15] L.-H. Hsieh and K. Chang, “High-efficiency piezoelectric-transducer tuned feedback microstrip ring-resonator oscillators operating at high resonant frequencies,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1141–1145, Apr. 2003. [16] Y.-T. Lee, J. Lee, and S. Nam, “High- active resonators using amplifiers and their applications to low phase noise free-running and voltagecontrolled oscillators,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2621–2626, Nov. 2004. [17] E. Park and C. Seo, “Low phase noise oscillator using microstrip square open loop resonator,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, 2006, pp. 585–588.

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Jonghoon Choi (S’03) received the B.S. and M.S. degrees from Seoul National University, Seoul, Korea in 1997 and 1999, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2007. Upon graduation, he joined Qualcomm Incorporated, Campbell, CA, as a Senior RFIC Design Engineer. His research interests include low phase-noise RF/microwave oscillators and RF integrated circuits. Dr. Choi was a recipient of the Information Technology Fellowship administered by the Ministry of Information and Communication, Seoul, Korea, and the Honorable Mention Award of the Student Paper Competition presented at the 2005 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), Long Beach, CA.

Amir Mortazawi (M’90–SM’05–F’05) received the Ph.D. degree in electrical engineering from the University of Texas at Austin, in 1990. He is currently a Professor of electrical engineering with The University of Michigan at Ann Arbor. His research interests include millimeter-wave phased arrays, power amplifiers, power-combining techniques, and frequency-agile microwave circuits. Prof. Mortazawi is co-editor-in-chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Administrative Committee (AdCom). He is the co-chair of the IEEE MTT-16 Committee on Phased Arrays. He was an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (1998–2001), an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (2005), and was the guest editor for the December 1995 “Special Issue on the IEEE MTT-S Microwave Symposium” of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was a secretary to the IEEE MTT-S AdCom.

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Design and Analysis of a Millimeter-Wave Direct Injection-Locked Frequency Divider With Large Frequency Locking Range Chung-Yu Wu, Fellow, IEEE, and Chi-Yao Yu Abstract—In this paper, direct injection-locked frequency dividers (ILFDs), which operate in the millimeter-wave (MMW) band, are analyzed. An analytically equivalent model of the direct ILFDs is developed, and important design guidelines for a large frequency locking range are obtained from it. These guidelines are: 1) maximize the quality factor of the passive load; 2) maintain low output amplitude; and 3) increase the dc overdrive voltage of the input device. A direct ILFD without varactors is designed and fabricated using a 0.13- m bulk CMOS process to verify the developed model and design guidelines. A pMOS current source is used to restrict the output amplitude and to increase the dc overdrive voltage of the input device to achieve a large frequency locking range. The size of the input device is only 3.6 m/0.12 m and the measured frequency locking range is 13.6% at 70 GHz with a power consumption of 4.4 mW from a supply voltage of 1 V. In short, the proposed divider has the potential to be integrated into an MMW phase-locked loop system. Index Terms—Frequency locking range, injection-locked frequency divider (ILFD), millimeter-wave (MMW) integrated CMOS circuit, phase-locked loop (PLL), 0.13- m bulk CMOS technology.

I. INTRODUCTION

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ITH RAPID advances in CMOS technology, the CMOS circuit operating in the millimeter-wave (MMW) band has attracted increasing interest and research [1]–[16]. Since frequencies around 60 GHz have been opened for unlicensed use in the U.S. and Japan, it seems possible that such circuits can be used in the front-end systems of gigabits/s point-to-point links, wireless local area networks, high data-rate wireless personal area networks, and radars. In general, phase-locked loops (PLLs) are extensively used in CMOS RF front-end systems as frequency synthesizers or clock sources to generate local oscillating signals. In an MMW PLL, the main blocks with the highest operating frequency are typically the voltage-controlled oscillator (VCO) and the frequency divider. More specifically, the main design issues of an MMW VCO concern the oscillating frequency tuning range, phase noise, power consumption, and output power level [7]–[10]. Most of these degrade as the input capacitance of

Manuscript received February 20, 2007; revised May 22, 2007. This work was supported in part by National Nano Device Laboratories, the Ansoft Corporation, the United Microelectronic Corporation, and the National Science Council, Taiwan, R.O.C., under Grant NSC95-2215-E-009-023. The authors are with the Institute of Electronic Engineering, National Chiao-Tung University, Hsinchu, 300 Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.902067

the next stage, which may be a frequency divider, increases. Therefore, the reduction of the input capacitance of the divider becomes very important as the operating frequency to the MMW band increases. In addition, the wide operating frequency range of the divider is also important in the MMW band in order to cover the inevitable shift of the center operating frequency caused by the process variations in the small values of integrated spiral inductance or parasitic capacitance. A small operating frequency range will severely reduce the reliability of the MMW PLL. Therefore, the main design challenge facing the MMW divider designers is to reduce input capacitance while maintaining a wide operating frequency range. As in other integrated CMOS RF circuits, power consumption and noise performance are also important in divider design. In comparison with flip-flop-based static frequency dividers [11], injection-locked frequency dividers (ILFDs) [12]–[14] generally have lower power consumption and higher frequency capability in bulk CMOS technologies. However, ILFDs usually suffer from narrow locking ranges. This limitation can be significantly improved by proper structure selection and correct design methodology, as will be shown in this paper. As the scaling down of CMOS technology toward a 90- or 65-nm node, ILFDs provide a good low-power design choice besides the static dividers in the MMW PLL integration [16]. A conventional LC-based ILFD is shown in Fig. 1(a). The input stage is used to provide both an input signal path and a dc bias path. Thus, is typically large, resulting in a large input capacitance. Moreover, the input signal is significantly dein Fig. 1(a). By using graded by the parasitic capacitor and the a peaking inductor between the drain terminal of ground, this problem can be reduced [18]; however, this strategy requires a greater chip area. Moreover, the Miller divider proposed in [15] faces the same problems of a large input capacitance and the need for a peaking inductor. The 50-GHz direct ILFD in [12], as shown in Fig. 1(b), provides a solution for MMW operation with a low input capacitance, but it suffers from a narrow frequency locking range. Therefore, the passive loads of a direct ILFD in another study [14] are optimized to increase the frequency locking range. However, the power consumed by the resulting ILFD is relatively large, probably because the devices involved are also large and a current-limiting device is absent. Varactors are used at the output nodes in another ILFD [13] to increase the locking range. However, with a PLL system design, the need to synchronize the controlling voltages between the varactors in both the VCO and divider significantly increases design complexity.

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Fig. 2. General block diagram of a differential direct ILFD.

Fig. 1. Schematic diagrams of: (a) a conventional ILFD and (b) a direct ILFD [12].

In this study, an analytical model and design guidelines of a direct ILFD are presented. Based on the developed model and guidelines, a direct ILFD without a varactor is designed and fabricated using 0.13- m bulk CMOS technology. With a simple pMOS current source and a suitable design, the proposed direct ILFD has a low input capacitance, large frequency locking range, low power consumption, and favorable noise suppression capability. Therefore, the proposed direct ILFD can be integrated with an MMW VCO into an MMW PLL system. The frequency locking range of a conventional ILFD [19] is given by , where and are the resonant frequency and the quality factor of the LC resonator, respectively, and is the injection ratio. Accordingly, the locking range is inversely proportional to the factor of the LC resonator. However, the proposed analytical model herein reveals that for a direct ILFD, increasing the factor can reduce the power consumption without reducing the locking range. This result differs from the conventional one. For verification, the other direct is fabricated and ILFD with an LC resonator with a lower comparative measurements are made. The measurements show that the center frequency of the proposed direct ILFD is around 70 GHz. The operating frequency range is 13.6% at 70 GHz with 4.4 mW from a supply voltage of 1 V and an input device size of only 3.6 m/0.12 m, which is smaller than that used in another study [12]. This paper is organized as follows. Section II proposes the analytical model and design guidelines of a direct ILFD. Based on these design guidelines, the circuit design considerations of the proposed direct ILFD are presented in Section III. The phasenoise analysis is presented in Section IV. The measured results are described and discussed in Section V. Finally, Section VI draws some conclusions. II. ANALYTICAL MODEL AND DESIGN GUIDELINES The general block diagram of a differential direct ILFD is cell with positive feedback shown in Fig. 2. The active is designed to provide a negative resistance to compensate for the power loss from the resistive load per oscillating cycle for

the stable output oscillating signals. and represent the cell. The input stage equivalent passive loads of the active only. The input voltage is implemented by using an nMOS ) is applied to the gate node of , where is the input phase. For the sake of convenience, it is , as shown in Fig. 2. If the input frequency assumed that falls into the divided-by-2 locking range, then the differenat the drain and the source nodes of tial output voltages are given by , where is the output , then can be denoted as , which reprephase. If sents the phase difference between the input and output signals. can be regarded as a mixing device and In this situation, the mixing channel current of is denoted by . is In most cases, the input voltage is a large signal so operated in the on–off mode. Fig. 3(a) and (b) shows the two and , as is equal to sample waveforms of and , respectively. As shown in Fig. 3, the time interval beis . Since tween the two neighboring turn-on periods of the frequency of the differential output voltages at the drain is exactly half of that of the input and source nodes of the voltage, the resulting in the two neighboring turn-on periods displays the same shapes, but opposite polarities as those shown in Fig. 3(a) and (b). Therefore, the fundamental frequency of is and the fundamental component of is denoted by . To develop the desired analytical model, is decomposed into in-phase and quadrature components (1) strongly deAs shown in Fig. 3(a) and (b), the shape of pends on . Therefore, the amplitudes of both components in (1) should also be the functions of . In fact, is determined by the input frequency Fig. 4(a)–(c) plots the HSPICE simulated waveforms of and when is equal to, larger than, or smaller , where is the resonant frequency of the equivalent than passive load in Fig. 2. The waveforms of and calculated from are also shown in each. . In this Fig. 4(a) plots the waveforms in the case of and so the phase of is case, equals the same as the output voltage signal. Therefore, can be with the value of . modeled as a single resistor The equivalent model in this case is also shown in Fig. 4(a) exceeds , as the waveforms When the input frequency plotted in Fig. 4(b), becomes slightly smaller than so

WU AND YU: DESIGN AND ANALYSIS OF MMW DIRECT ILFD WITH LARGE FREQUENCY LOCKING RANGE

Fig. 3. Two waveforms of V (b) =4.

;V

6; and I

, as ' is equal to: (a) =2 and

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Fig. 4. Simulated waveforms of V ; V ; I ; I cos(!t + '); I sin(!t + ') and the equivalent model as: (a) 2! = 2! , (b) 2! > 2! , and (c) 2! < 2! .

lags behind the output voltage signal. Therefore, is larger than 0 and can be modeled as in parallel with an and are calculated as inductor

smaller than 0 and can be modeled as in parallel with , whose capacitance is given by a capacitor

(2)

(6)

and

The output frequency

can be easily calculated as

(3) The equivalent model in this case is also presented in Fig. 4(b). The output frequency can be easily calculated as (7)

Therefore, the maximum available value of mined by the maximum available value of denoted by , and is given by

(4)

is deterTherefore, the minimum available value of mined by the minimum available , which is denoted , and can be expressed as by

is deter, which is

(8)

(5)

From (5) and (8), the input frequency locking range denoted by can be calculated as

The waveforms and equivalent model of the final case in which the input frequency is less than are shown in such Fig. 4(c). In this case, becomes slightly larger than that leads to the output voltage signal. Therefore, is

(9) Given the symmetric differential structure in Fig. 2, for a parequals , and ticular output voltage amplitude

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(9) can be further simplified as (10) should be designed as large as posAccording to (10), for fixed values of sible to maximize the locking range and . However, since all voltage signals that are applied to are large signals, no analytical equation exists for . in Therefore, HSPICE is adopted to find the values of the variously biased cases. Fig. 5(a)–(d) shows contour maps of for various dc overdrive voltages of and output voltage amplitudes with different input voltage amplitudes . In all these cases, increases with for a fixed and decreases as increases in the highregion. According to the proposed model, shown in Fig. 4, and the derived locking range equation (10), for a fixed , the quality of the passive load in Fig. 4 does not directly factor influence the locking range. More accurately, the value of only indirectly influences the locking range through a change or , which changes , as shown in Fig. 5. For in cell, a low of the passive load results example, for a given and the locking range in a smaller and, thus, a larger that is given by (10). However, in low and high cases, the locking ranges can more fairly be compared with a fixed and . In this situation, is fixed, as shown in Fig. 5, such that the locking ranges in low and high cases are the same for a fixed and , as determined by (10). Since a lower passive load has a lower , the cell needs to consume more power in order to compensate for to maintain the same at resonance. Therefore, for any output voltage amplitude required , using a higher passive load can reduce the power requirement without any reduction in the locking range. From the above analysis, some design guidelines for a direct of the input device should ILFD can be inferred. Firstly, be designed as large as possible to maximize the and frequency locking range. Secondly, a tradeoff exists between the and the frequency locking range. output voltage amplitude should be set at its minimum tolerant value to Therefore, maximize the frequency locking range. Finally, the factor of the passive load should be as large as possible to reduce the required dc power consumption without reducing the frequency locking range. III. CIRCUIT DESIGN A. Circuit Structure Based on the design guidelines in Section II, the proposed ILFD circuit for high-speed operation is shown in Fig. 6. The circuit structure is simple in that it has no varactor, but it still provides a large frequency locking range. is used In order to reduce the input capacitance, nMOS as the only input stage to generate the injected current . Furthermore, instead of a complementary cross-coupled pair [12], cell an nMOS cross-coupled pair is used to implement the in Fig. 2. Since the frequency locking range is inversely proportional to the total capacitance value at the output node, as in (10), the absence of a pMOS cross-coupled pair can significantly increase the frequency locking range. Adding a pMOS , as shown in Fig. 6, provides two advantages current source

Fig. 5. Contour maps of g = 0:3 V.

0:4 V, and (d) v

as: (a) v

= 0:6 V, (b) v = 0:5 V, (c) v =

over an ILFD presented in an earlier study [14], increasing the locking range. Firstly, since a tradeoff exists between the output

WU AND YU: DESIGN AND ANALYSIS OF MMW DIRECT ILFD WITH LARGE FREQUENCY LOCKING RANGE

Fig. 7. Simulated frequency locking ranges and g of the inductor.

Fig. 6. Circuit structure of the proposed direct ILFD.

voltage amplitude and the frequency locking range, the output voltage amplitude can be set to its minimum value by designing to maximize the locking range. an appropriate dc current of Secondly, the dc voltage at the output node can be set much . lower than the VDD because the dc current is limited by can be biased in the high overdrive voltage reTherefore, gion. Additionally, through the resistor , the dc voltage at the can be equal to those at the drain and substrate node of can be kept source nodes such that the threshold voltage of low to increase overdrive voltage. B. Size of Input Stage As the operating frequency increases to the MMW band, the shown in Fig. 6 is restricted by size of the input nMOS the input capacitance since a large input capacitance makes the integration of an ILFD with an MMW VCO difficult. In the design, the target of the operating frequency is 70 GHz, and the is designed as 3.6 m, with the width of the input nMOS minimum length. From the simulation, the input capacitance of is less than 10 fF, which is an acceptable load for an on-chip 70-GHz VCO. C. Design of pMOS Current Source The dc current of the pMOS current source denoted by directly influences the output voltage amplitude . According [20]. to the model in Fig. 4, can be estimated as Notably, a tradeoff exists between and the frequency locking range. Therefore, should be designed appropriately such just equals the required value at the edges of the frethat quency locking range. D. Design of Integrated Spiral Inductor and Cross-Coupled Pair constrains the value of , Since the small size of careful design of an integrated spiral inductor and cross-coupled pair to achieve a large frequency locking range is important. It can be seen from (10) that the frequency locking range

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with different values

is proportional to the inductor value . Initially, the frequency locking range increases with an increase in inductance. However, as increases over an optimum value, the locking range begins to drop for the following two reasons. Firstly, the output can be expressed as center frequency

where is the width (length) of and in is the overlap capacitance per unit width, is the Fig. 6, gate–oxide capacitance per unit area, and is capacitance must be reduced from the next stage. Thus, as increases, . At a fixed dc current, this drop to maintain the required and and, thus, reduces increases the dc gate voltages of the overdrive voltage of and and, thus, the locking is too small to maintain enough , range. Secondly, if such that the power loss per oscillating cycle from and in Fig. 4 cannot be compensated for when the input frequency falls in the range specified in (10), then the frequency locking range rapidly declines. Therefore, in this design, iterative simulations are required to find the optimum inductance of the spiral inductor for the maximum frequency locking range. factor of the passive As mentioned in Section III-C, the load should be designed as large as possible to reduce the power . Accordingly, no extra resistor is connected consumption or in parallel to the inductor in the proposed circuit. The results of Ansoft’s Nexxim simulation involving the frequency locking ranges with various inductances are shown in Fig. 7. In the simulation, the center output frequency is around 70 GHz, the input amplitude is 0.6 V, the input nMOS size is 3.6 m/0.12 m, and the minimum required output voltage amvalue in each case is obtained plitude is 250 mV. The from Fig. 5 so the locking range can be given by (10). Fig. 7 value and the locking range given by (10), also plots the which are consistent with the simulation results. In order to consider the frequency shift resulting from the process and temperature variation [17], the corner models provided by the foundry are used to simulate the performance of the proposed divider. The shift of the center frequency is 6.98% from 0 C FF corner to 100 C SS corner. The simulated input sensitivity curves with an inductance of 400 pH in these two

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Fig. 8. Simulated input sensitivity curves in the corner cases.

extreme cases are shown in Fig. 8. The locking range of the divider is sufficient to cover the frequency shift as the input power is larger than 0 dBm. IV. PHASE-NOISE ANALYSIS Here, the noise model in an earlier study [19] is modified and used to analysis the phase noise of a direct ILFD. The block diagram of a direct ILFD is redrawn in Fig. 9(a) with the active cell replaced by a negative resistor . is now given by a single sinusoidal function

(11) where is the amplitude of and is the phase of , which can be decomposed to and the extra phase . Here, is related to the phase difference between the input and output voltage signal and, thus, it can be given as a function of . Fig. 9(b) presents the linear loop for the phase noise analysis, and are the random variables that represent where the small phase fluctuations of the input and output voltage sigrepresents the small phase response of the nals. Here, equivalent load in Fig. 9(a) and is given by (12) is the quality factor of the equivalent load and is the offset frequency. The values of the partial differentiations in Fig. 9(b) can be easily calculated using where

(13) and

Fig. 9. (a) Block diagram of the direct ILFD. (b) Linear loop for the phase-noise analysis.

where (16) The calculation of the transfer function of the free-running and and output phase-noise spectral densities ( ) is as in an earlier cited study [19]; only the result is shown here as follows: (17) From (15), the input phase noise appears at the output with a 6-dB reduction and low-pass shaping, dominating the output . When phase noise when the offset frequency is less than , then from (17), the output the offset frequency exceeds phase noise is dominated by the phase noise of the divider in free run. This result is similar to that of a conventional ILFD. with various The simulated curves of and at the central frequency are plotted in Fig. 10. From Fig. 10, increases with and generally exceeds GHz and . Therefore, 1 GHz when with respect to noise, this structure is also suitable for MMW becomes large, its internal noise can operations because as be suppressed even at a large offset frequency. V. MEASUREMENT RESULTS

(14) where is the derivative of . From (12)–(14), the transfer function of the input and output phase-noise spectral densities and , respectively, is given by

(15)

The proposed ILFD shown in Fig. 6 is designed and fabricated using 0.13- m bulk CMOS technology with a supply voltage of 1 V. The size of the is only 3.6 m/0.12 m. A low- ILFD with a resistor that is connected in parallel with to reduce the factor is also fabricated on the same chip to observe the relationship between the locking range and the factor. The chip micrographs of both fabricated ILFDs are shown in Fig. 11.

WU AND YU: DESIGN AND ANALYSIS OF MMW DIRECT ILFD WITH LARGE FREQUENCY LOCKING RANGE

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Fig. 12. (a) Measured output amplitude versus input frequency. (b) Measured and calculated/simulated locking range and the minimum output amplitude versus I .

Fig. 10. Simulated curves of S and (c) V at the central frequency.

=S

with different: (a) ! , (b)

V

,

Fig. 11. Micrographs of ILFDs.

After the losses from the cable and buffer have been deembedded, the measured output amplitudes versus the input are presented in frequencies for the various values of Fig. 12(a). The locking range can be determined by the difference between the frequencies at the two ends of each curve in Fig. 12(a). Fig. 12(b) plots the curves of the locking range and the minimum output amplitude in throughout the locking

range versus . The simulated and calculated curves are also shown for comparison. The locking range can be increased at the cost significantly by choosing a suitable value for of a reduced output voltage amplitude. This result is consistent should be kept larger with those of the analysis. Notably, than the specific current to maintain a sufficient to compensate for the power loss form the equivalent resistive load per oscillating cycle. Otherwise, the stable output oscillating signals cannot be maintained. Thus, the locking range declines rapidly, as shown in the long broken-line regions of the measured curves in Fig. 12(b). The maximum measured locking of 4.4 mA from range is 13.6% (66.4–76 GHz) with an , the calculated locking a 1-V supply. Except at the low ranges from (10) are consistent with the measurement results. The measured frequency locking ranges as the supply voltage decreases to 0.8 V are plotted in Fig. 13. The locking ranges are considerably smaller than those in the 1-V case because the drop and in the supply voltage reduces the overdrive voltage of . This result is also consistent with analytic results. also The measured locking ranges versus the output voltage amplitudes of the proposed and low- ILFDs are plotted

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Fig. 15. Measured input sensitivities of both ILFDs. Fig. 13. Locking range as the supply voltages are 0.8 and 1 V.

Fig. 14. Measured locking ranges versus output voltage amplitudes of both proposed and low- ILFDs.

Q

in Fig. 14. The value of in each case is marked on the measured curves. For any required output voltage amplitude, , but reducing the factor not only increases the required also reduces the frequency locking range. The locking range reduces the overdrive declines because an increase in . The measured input sensitivivoltage and thereby also ties of both dividers are plotted in Fig. 15. The proposed ILFD also has a greater input sensitivity than the low- ILFD. The measured output phase noise and phase noise of the input signal from the Agilent MMW Source Module E8257DS15 [22] are both plotted in Fig. 16(a). This figure reveals that the output phase noise is determined by the input phase noise below the 300-kHz offset frequency. Beyond the 300-kHz offset, the output phase noise is corrupted by a flat noise floor of approximately 120 dBc/Hz. The waveform of this extra noise is flat and shapeless so its source is not within the closed loop that is shown in Fig. 9(b). Since only the single-ended output signal is measured, this noise floor may be from the common-mode noise from the pMOS current source, supply voltage, and ground, or the instrument itself. The output phase noise and the phase noise in free run are both plotted in Fig. 16(b). Although

Fig. 16. (a) Measured output phase noise and the phase noise of input signal from Agilent MMW Source Module E8257DS15 [22]. (b) Measured output phase noise and the free-run phase noise.

the output signal in free run is noisy, the output phase noise after locking is almost independent of the phase noise in free run below the 10-MHz offset frequency. Beyond the 10-MHz offset frequency, the phase noise in free run is also corrupted by a flat noise floor at around 120 dBc/Hz. Therefore, the internal noise in the loop in Fig. 9(b) from the ILFD is observably suppressed before the 10-MHz offset frequency at the very least. The performances of the proposed divider and other CMOS frequency dividers at above 40 GHz are compared in Table I. Without a varactor, the locking range of the proposed divider is 13.6% at 70 GHz. Finally, the device size of the input stage is 3.6 m/0.12 m, which is smaller than that presented elsewhere [12].

WU AND YU: DESIGN AND ANALYSIS OF MMW DIRECT ILFD WITH LARGE FREQUENCY LOCKING RANGE

TABLE I PERFORMANCE COMPARISON BETWEEN THE PROPOSED CMOS ILFD AND OTHER CMOS FREQUENCY DIVIDERS

VI. SUMMARY In this paper, an analytical model for a direct ILFD is presented. From the proposed model, important design guidelines have been developed. Based on the design guidelines, a 70-GHz direct ILFD has been designed and fabricated using 0.13- m bulk CMOS technology, where a pMOS current source was used to restrict the output voltage amplitude and to increase the overdrive voltage of the input device to improve the frequency locking range. For a direct ILFD, a higher passive load can release the power required without decreasing the frequency locking range. Even if the input device size is small and the varactor is not used, the frequency locking range is large. Therefore, the proposed direct ILFD can be integrated with an MMW VCO easily and is a favorable choice for use in a CMOS MMW PLL system.

REFERENCES [1] C. H. Doan, S. Emami, A. M. Niknejad, and R. W. Brodersen, “Millimeter-wave CMOS design,” IEEE J. Solid-State Circuits, vol. 40, no. 1, pp. 144–155, Jan. 2005. [2] B. Razavi, “A 60-GHz CMOS receiver front-end,” IEEE J. Solid-State Circuits, vol. 41, no. 1, pp. 17–22, Jan. 2006. [3] B. Razavi, “CMOS transceivers for the 60-GHz band,” in IEEE Radio Freq. Integrated Circuit Symp. Dig., San Francisco, CA, Jun. 2006, pp. 11–13. [4] T. Yao, M. Gordon, K. Yau, M. T. Yang, and S. P. Voinigescu, “60-GHz PA and LNA in 90-nm RF-CMOS,” in IEEE Radio Freq. Integrated Circuits Symp. Dig., San Francisco, CA, Jun. 2006, pp. 147–150. [5] S. Emami, C. H. Doan, A. M. Niknejad, and R. W. Bronderson, “A 60-GHz down-converting CMOS single-gate mixer,” in IEEE Radio Freq. Integrated Circuits Symp. Dig., Long Beach, CA, Jun. 2005, pp. 163–166. [6] C. S. Lin, P. S. Wu, H. Y. Chang, and H. Wang, “A 9–50-GHz Gilbert-cell down-conversion mixer in 0.13-m CMOS technology,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 5, pp. 293–295, May 2006. [7] C. Cao and K. K. O, “A 90-GHz voltage controlled oscillator with a 2.2-GHz tuning range in a 130-nm CMOS technology,” in VLSI Circuits Symp., Kyoto, Japan, Jun. 2005, pp. 242–243.

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[8] P. C. Huang, M. D. Tsai, H. Wang, C. H. Chen, and C. S. Chang, “A 114 GHz VCO in 0.13 m CMOS technology,” in IEEE Int. Solid-State Circuits Conf. Dig., San Francisco, CA, Feb. 2005, pp. 404–405. [9] R. C. Liu, H. Y. Chang, C. H. Wang, and H. Wang, “A 63 GHz VCO using a standard 0.25 m CMOS process,” in IEEE Int. Solid-State Circuits Conf. Dig., San Francisco, CA, Feb. 2004, pp. 446–447. [10] C. Cao and K. K. O, “Millimeter-wave voltage-controlled oscillators in 0.13-m CMOS technology,” IEEE J. Solid-State Circuits, vol. 41, no. 6, pp. 1297–1304, Jun. 2006. [11] K. J. Wang, A. Rylyakov, and C. K. Yang, “A broadband 44-GHz frequency divider in 90 nm CMOS,” in IEEE Compound Semicond. Integrated Circuit Symp., Oct. 2005, pp. 196–199. [12] M. Tiebout, “A CMOS direct injection-locked oscillator topology as high-frequency low-power frequency divider,” IEEE J. Solid-State Circuits, vol. 39, no. 7, pp. 1170–1174, Jul. 2004. [13] K. Yamamoto and M. Fujishima, “70 GHz CMOS harmonic injection-locked divider,” in IEEE Int. Solid-State Circuits Conf. Dig., San Francisco, CA, Feb. 2006, pp. 600–601. [14] K. Yamamoto and M. Fujishima, “55 GHz CMOS frequency divider with 3.2 GHz locking range,” in Proc. Eur. Solid-State Circuits Conf., Leuven, Belgium, Sep. 2004, pp. 135–138. [15] J. Lee and B. Razavi, “A 40 GHz frequency divider in 0.18-m CMOS technology,” IEEE J. Solid-State Circuits, vol. 39, no. 4, pp. 594–601, Apr. 2004. [16] J. Lee, “A 75-GHz PLL in 90 nm technology,” in IEEE Int. Solid-State Circuits Conf. Dig., San Francisco, CA, Feb. 2007, pp. 432–433. [17] D. Lin, J. Kim, J. O. Plouchart, C. Cho, D. Kim, R. Trzconski, and D. Boning, “Performance variability of a 90 GHz static CML frequency divider in 65 nm SOI CMOS,” in IEEE Int. Solid-State Circuits Conf. Dig., San Francisco, CA, Feb. 2007, pp. 542–543. [18] H. Wu and A. Hajimiri, “A 19 GHz 0.5 mW 0.35 m CMOS frequency divider with shunt-peaking locking-range enhancement,” in IEEE Int. Solid-State Circuits Conf. Dig., San Francisco, CA, Feb. 2001, pp. 412–413. [19] A. Mazzanti, P. Uggetti, and F. Svelto, “Analysis and design of injection-locked LC dividers for quadrature generation,” IEEE J. Solid-State Circuits, vol. 39, no. 9, pp. 1425–1433, Sep. 2004. [20] A. Hajimiri and T. H. Lee, “Describing function analysis of oscillators,” in The Design of Low Noise Oscillator. Norwell, MA: Kluwer, 1999, pp. 179–186, Appendix F. [21] H. R. Rategh and T. H. Lee, “Superharmonic injection-locked frequency dividers,” IEEE J. Solid-State Circuits, vol. 34, no. 6, pp. 813–821, Jun. 1999. [22] “Agilent Technologies mm-wave source modules from OML, Inc. for the Agilent PSG signal generators,” Agilent Technol., Santa Clara, CA, 2005, p. 3.

Chung-Yu Wu S’76–M’76–SM’96–F’98) was born in 1950. He received the M.S. and Ph.D. degrees in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1976 and 1980, respectively. Since 1980, he has been a consultant to high-tech industry and research organizations and has built up strong research collaborations with high-tech industries. From 1980 to 1983, he was an Associate Professor with National Chiao Tung University. From 1984 to 1986, he was a Visiting Associate Professor with the Department of Electrical Engineering, Portland State University, Portland, OR. Since 1987, he has been a Professor with National Chiao Tung University. From 1991 to 1995, he served as the Director of the Division of Engineering and Applied Science, National Science Council, Taiwan, R.O.C. From 1996 to 1998, he was honored as the Centennial Honorary Chair Professor with National Chiao Tung University. He is currently the President and Chair Professor of National Chiao Tung University. In Summer 2002, he conducted post-doctoral research with the University of California at Berkeley. He has authored or coauthored over 250 technical papers in international journals and conferences. He hold 19 patents, including nine U.S. patents. His research interests are nanoelectronics, biochips, neural vision sensors, RF circuits, and computer-aided design (CAD) analysis. Dr. Wu is a member of Eta Kappa Nu and Phi Tau Phi. He was a recipient of 1998 IEEE Fellow Award and a 2000 Third Millennium Medal. He has also been the recipient of numerous research awards presented by the Ministry of Education, National Science Council (NSC), and professional foundations in Taiwan, R.O.C.

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Chi-Yao Yu was born in Taipei, Taiwan, R.O.C., in 1978. He received the Master’s degree in communication engineering from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 2002, and is currently working toward the Ph.D. degree at National Chiao Tung University, Hsinchu, Taiwan, R.O.C. His current research is focused on communication systems, mixed-signal integrated circuit design, CMOS MMW circuits, and CMOS RF front-end circuit design.

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W -Band Waveguide Impedance Tuner Utilizing

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Dielectric-Based Backshorts

Tero Kiuru, Student Member, IEEE, Ville S. Möttönen, Member, IEEE, and Antti V. Räisänen, Fellow, IEEE

Abstract—We present the design, simulations, fabrication, and -band waveguide impedance tuner. The measurements of a design consists of a WR-10 waveguide with two -plane arms for tunable dielectric-based backshorts. The impedance tuner is fabricated using a simple split-block technique and is easily scalable to submillimeter wavelengths. Our design shows an excellent input impedance range, tuning accuracy, repeatability, and a high attainable maximum reflection coefficient over the frequency band of 75–110 GHz. Our tuner applies dielectric-based backshorts, which provide a tuning accuracy better than that of other waveguide backshorts. Index Terms—Dielectric-based backshort, impedance tuner, -plane, waveguide, -band.

I. INTRODUCTION

A

T MILLIMETER and submillimeter wavelengths, the input and output powers of devices are usually low. Consequently, accurate and reliable source and load impedance determination is important, e.g., in a mixer or multiplier optimization and in a low-noise amplifier (LNA) design. In addition, an accurate and reliable reflection coefficient setting is important for noise parameter measurements at microwave and millimeter-wave regions [1]–[3]. For example, at least four noise-figure measurements with different source reflection coefficients have to be made in order to determine the noise parameters of a linear two-port. The noise figure is then extracted with mathematical fitting methods [2]. The accuracy of the calculated noise parameters depends on the accuracy of the source reflection coefficients. An impedance tuner is a key element in a source or load impedance determination. Commercially, wideband waveguide impedance tuners are available up to 170 GHz. One operating principle of these tuners is based on a needle probe inserted inside the waveguide. The place and depth of the probe determines the phase and magnitude of the reflection coefficient [4], [5].1 A

Manuscript received November 30, 2006; revised March 23, 2007. This work was supported in part by the Academy of Finland and Tekes–Finnish Founding Agency for Technology and Innovation under the Centre of Excellence Program. T. Kiuru and A. V. Räisänen are with the TKK Helsinki University of Technology, SMARAD Centre of Excellence/Radio Laboratory, Millimetre Wave Laboratory of Finland (MilliLab), Espoo FI-02015 TKK, Finland (e-mail: tero. [email protected]). V. S. Möttönen was with the TKK Helsinki University of Technology, SMARAD Centre of Excellence/Radio Laboratory, Millimetre Wave Laboratory of Finland (MilliLab), Espoo FI-02015 TKK, Finland. He is now with the National Board of Patents and Registration of Finland, FI-00101 Helsinki, Finland. Digital Object Identifier 10.1109/TMTT.2007.901112 1Maury Microwave, Ontario, CA. Device characterization, electro-mechanical tuners (ATS), automated tuners. [Online]. Available: http://www.maurymw. com/

design with a waveguide magic-T using tuning arms is commercially available2 and impedance tuners employing traditional waveguide backshorts as tuning elements have been in use for some time, e.g., in klystron matching [6] and in noise parameter measurements [7]. The performance of these designs at submillimeter wavelengths degrades due to the complex structures and scaling difficulties of the traditional backshorts. A waveguide transition in [8] applies for the impedance tuning a new low-loss noncontacting waveguide backshort. Both the measured performance at the -band and simulated performance at 1.6 THz are good. However, according to measurements and simulations, this design is prone to resonances. Recently, research on the impedance tuners based on microelectromechanical systems (MEMS) has been vigorous, e.g., [9]–[11]. MEMS-based tuners are compact and electrically tunable. However, at higher millimeter-wave frequencies, the maximum attainable reflection coefficient of MEMS-based backshorts is less than that of the waveguide backshorts, and, to the best of the authors’ knowledge, electrically tunable MEMS-based backshorts have not been demonstrated at submillimeter wavelengths. Reference [12] describes a micromechanical tuning element, called a sliding planar backshort, which can be fabricated with photolithographic micromachining techniques. Tests in a coplanar waveguide environment indicate an excellent return loss at 620 GHz. The mechanical tuning of this backshort is very challenging. In this paper, we present the design, simulations, fabrication, and measurements of a novel waveguide impedance tuner for the -band (75–110 GHz). The waveguide impedance tuner consists of a main waveguide (WR-10) and -plane arms (WR-10) with dielectric-based backshorts. This tuner shows an excellent impedance range, tuning accuracy, repeatability, and a high attainable maximum reflection coefficient [high voltage standingwave ratio (VSWR)] over the entire -band. Moreover, our design is easily scalable up to submillimeter wavelengths. II. DESIGN OF WAVEGUIDE IMPEDANCE TUNER A. Basic Structure We use two series, i.e., -plane, tuning elements in our -band (frequency range: 75–110 GHz) waveguide impedance tuner. The tuner has WR-10 waveguides with the width mm and height mm. The use of series tuning elements enables easier fabrication and assembly in the case of split-waveguide block techniques. Two elements provide a very large impedance range and, in addition, the tuner structure does 2Millitech Inc., Northampton, MA. [Online]. Available: http://www.millitech.com/pdfs/specsheets/IS000060-EHT-HBT.pdf

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Fig. 2. Dielectric-based tunable waveguide backshort. One waveguide block half is shown with a waveguide end, guide channel, and a dielectric slab. [14].

Fig. 1. (a) E -plane T-junctions of the waveguide impedance tuner with equal height waveguides. (b) Equivalent circuit of the tuner in a circuit simulator ADS. Impedances Z and Z correspond to the characteristic impedances of an empty waveguide and a waveguide with quartz slab, respectively. Lengths l and l correspond to the lengths of these waveguides, respectively. Total length l l is constant. The S -parameter matrix of the double-T junction simulated in HFSS is imported into ADS.

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not become complex. The tuning elements are formed by a tunable backshort [13], [14] in the -plane arm of the T-junction of equal height waveguides. Fig. 1(a) shows a simplified tuner structure. Different equivalent circuits exist for the waveguide T-junction depending on the T-junction reference planes, e.g., [15, pp. 336–350]. An equivalent circuit can be applied in a circuit simulator for the tuner design. However, we have used a finite-element method-based electromagnetic structure simulator [Ansoft’s High Frequency Structure Simulator (HFSS)] to obtain an accurate model for the double-T junction. This model, embedded in a circuit simulator [Agilent’s Advanced Design System (ADS)] [see Fig. 1(b)] provides a better design accuracy. B. Dielectric-Based Backshort as a Tuning Element Our waveguide tuner uses a dielectric-based backshort [13], [14] as a tuning element. Fig. 2 shows an overview of the backshort. Basically, a dielectric slab, moved inside the waveguide through a hole in a fixed waveguide end, changes the effective electrical length of the structure and, thus, the phase of the reflected wave. A rigid slab aligns well with the guide channel and does not require guide grooves in the waveguide. We have

Fig. 3. Simulated input impedance coverage of the lossless tuner on the Smith diagram at 92.5 GHz. The impedance spots correspond to backshort positions with 100-m steps.

TABLE I DIMENSIONS AND PARAMETERS OF THE TUNER

used a 127- m-thick and 1.2-mm-wide quartz slab (dielectric ) in a 200- m-high and 1.27-mm-wide guide constant channel. In [13], we have described the structure, design, and testing of a -band backshort, and have also compiled the design features. The measured return loss for the -band back. Refshort with a quartz slab is less than 0.21 dB erence [14] studies the effects of different dielectric materials, material thicknesses, a material tapering, and frequency scaling on the backshort performance. In comparison to other existing waveguide backshorts [8], [16]–[26], the one presented here has several important advantages: a simple design and fabrication, easy tuning (physical tuning range greater than with other

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Fig. 4. Simulated input impedance coverage on the Smith diagram when losses are taken into account. (a) 75 GHz. (b) 92.5 GHz. (c) 110 GHz. The impedance spots correspond to backshort positions with 100-m steps.

backshorts), the most accurate adjustment, low losses, high reliability, insensitivity to alignment errors, not prone to resonances in the operational frequency band like noncontacting backshorts, and readiness for scaling. Features of different backshort designs are further discussed in [14] and [27]. C. Spacing of Tuning Elements Theoretically, two tuning elements enable matching of almost all the impedances when the spacing between the elements is close to zero or of the half-wavelength in the wave. However, the tuning is then very sensitive to the guide at 75 GHz is twice at 110 GHz, has to frequency. Since have an appropriate value in order for the tuner to operate well over the full waveguide frequency band. Since the -plane arms of the T-junctions are close to each other, we have simulated the double-T junction as one four-port structure in HFSS, and then studied the impedance coverage in ADS with an imported four-port -parameter matrix and ideal tunable backshorts [see Fig. 1(b)]. By changing , we have optimized the operation over the -band. At this point, when optimizing the impedance coverage, we have used lossless structures. The optimization remm ( – , 75–110 GHz). Fig. 3 sulted in shows the simulated input impedance at Port 1 (load impedance in Port 2 is equal to ) (see Fig. 1) for the lossless tuner with 100- m backshort position steps at 92.5 GHz. The Smith diagram coverage is 96% at 75 GHz, 92% at 92.5 GHz, and 93% at 110 GHz. Table I shows the main dimensions and parameters of the tuner. D. Effect of Losses Losses of the waveguide impedance tuner comprise the conductive losses of the waveguide walls and dielectric losses of the quartz slabs. Due to the losses, it is impossible to obtain reflection coefficients with magnitudes very close to unity and, therefore, the Smith diagram coverage is receded from the edges of the diagram. Fig. 4 illustrates the simulated input impedance for the lossy tuner structure with 100- m backshort position steps. At 75 GHz, the Smith diagram coverage is 81%, 77% at 92.5 GHz and 74% at 110 GHz. By choosing output waveguides of different lengths on each side of the double-T junction, we

can change the position of the unmatched region on the Smith diagram by turning the tuner around. This will further increase the achievable Smith diagram coverage. In our design, the length difference between the input and output waveguides is 770 m, corresponding to a 130 phase difference in the reflection coefficient at 92.5 GHz if the tuner is turned around. The simulated combined Smith diagram coverage from Port 1 and Port 2 is 91% at 75 GHz, 94% at 92.5 GHz, and 94% at 110 GHz. The already low losses of our design can be further reduced by decreasing the length of the waveguides. Measurements and simulations show that the minimum length where the tuning elements still provide a phase shift of 360 over the entire -band is 10.6 mm. Therefore, the tuning element waveguides could be 1.9 mm shorter. The length of the main waveguide in our design is 35 mm. Theoretically, only the correct function of the double-T junction limits the main waveguide length. This limit is only a few millimeters at the -band. In reality, however, the limit is set by the size of the micrometers used for tuning and the space required for the attachment of the waveguide block halves. Dielectric losses could be reduced by using a material with lower losses. Suitable materials are unfortunately scarce. We use fused quartz due to its rigidity, low loss, and isotropic permittivity. III. FABRICATION AND ASSEMBLY Fig. 5 shows the fabricated tuner. The dielectric slab moves inside the waveguide through the guide channel. The slab is fixed by pressure to a slide, which has a grooved shape to reduce friction. A magnet placed in the other end of the slide connects the slide to the micrometer head with a nonrotating spindle. By using the magnet, misalignment in the micrometer fastening does not affect the slab alignment. The metal elements of the tuner are made of aluminium alloy using the split-block technique and the dielectric slabs are made of fused quartz. The following fabrication and assembly steps are taken. Step 1) Waveguide Milling: The slide channels and the WR-10 main waveguide together with the waveguides for the tuning elements are milled using the split-block technique. After milling, the split-waveguide block halves are gold-coated for optimal electrical operation.

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Fig. 5. Waveguide impedance tuner with one split-waveguide block half removed. One micrometer head with a slide and dielectric slab is shown separately.

Step 2) Fabrication of the Slides: The slides for the dielectric slabs are milled to the required size and shaped to minimize the friction. Strong permanent magnets are embedded in the slides for the attachment of the slides to micrometer spindles. Step 3) Milling of the Guide Channels for the Dielectric Slabs: The slides are fixed to one split-waveguide block half for the maximum alignment precision of the dielectric slabs. The guide channels are milled simultaneously to the split-waveguide block half and to slides. Step 4) Attaching the Dielectric Slabs to the Slides: The dielectric slabs are positioned in the center of the guide channel so that part of the slab rests on the slide and part on the split-waveguide block half. A pressure piece is attached to the slide in order to clamp the dielectric slab in place. Step 5) Assembly of the Impedance Tuner: After the dielectric slabs and micrometer heads are connected to the slides and the slides are mounted in one of the split-waveguide block halves, the other waveguide block half is installed on top of the first. IV. TUNER PERFORMANCE A. Scattering-Parameter Measurements The scattering parameters of the waveguide impedance tuner were measured with an HP-8510 vector network analyzer. The frequency range of the measurements was 75–114 GHz and the number of frequency points was 101. The averaging factor was 64. The measured input impedances are shown in Fig. 6 at 75, 92.5, and 110 GHz. The results were obtained by keeping one dielectric slab in place at the end of the waveguide and by moving the other slab in 2-mm steps from the 0 mm (dielectric slab completely drawn in the guide channel) to the 12-mm position. The first slab was then moved 2 mm inside the waveguide. The same procedure was repeated until the both slabs were 12 mm inside the waveguides. Some additional measurements were made in

order to better illustrate the impedance coverage on the Smith diagram. The circles in Fig. 6 correspond to the obtainable reflection coefficients with one of the dielectric slabs fixed in place. They are obtained by fitting an equation of a circle using three data points. As a result of fitting with three data points and the fact that the magnitude of the reflection coefficient from the dielectric-based backshort changes slightly as a function of the position of the dielectric slab [13], some measured data points do not lie exactly on the circle. In the simulations of the impedance tuner, the steps for the dielectric slab positions were 100 m apart. This means that the number of obtained reflection coefficients in the simulations (Figs. 3 and 4) is 400 times greater than in the performed measurements. This, however, is not the limit even for the manually tunable impedance tuner. A careful measurer can easily move the dielectric slabs with 10- m steps by turning the micrometer heads. With this modest tuning accuracy, the number of obtainable reflection coefficients is 40 000 times greater than that demonstrated here (Fig. 6). With the help of commercial automated linear motors, the achievable step size is in the order of 0.1 m [4]. This indicates that with a programmable waveguide impedance tuner, virtually any reflection coefficient inside the achievable range could be obtained with outstanding accuracy. The impedance coverage on the Smith diagram calculated from measured data is 70% at 75 GHz, 76% at 92.5 GHz, and 71% at 110 GHz. The maximum attainable reat 75 GHz, 0.96 flection coefficient is 0.935 at 92.5 GHz, and 0.952 at 110 GHz. Table II comprises the results of the tuner measurements. The achieved Smith diagram coverage correlates well with the simulations at 92.5 GHz (77%) and at 110 GHz (74%) and moderately well with that at 75 GHz (81%). According to simulations, the Smith diagram coverage is very sensitive to the tuning element spacing (Fig. 1). If in the fabricated impedance tuner was less than the designed 500 m, the largest effect would be seen at the lower frequencies of the -band, as the electrical length of is closer to zero and the matching becomes very sensitive to backshort positions. Indeed, delicate m. measurements under the microscope show that However, this inaccuracy only partly explains the small differences in simulations and measurements and it is assumed that some error comes from the four-port simulations of the double-T junction with HFSS. It is also possible that misalignment between the split-waveguide block halves occurred during the milling or in the assembly. However, this is very difficult to detect, especially near the double-T junction. The measured insertion loss of the waveguide impedance tuner over the -band with two different backshort positions is shown in Fig. 7. The average insertion losses are 0.30 and 0.31 dB for the different backshort positions, respectively. B. Repeatability of Reflection Coefficient As discussed above, the tuning accuracy of the waveguide impedance tuner is very good even when it is tuned manually with micrometer heads. The repeatability of the impedance tuner depends on the accuracy of tuning and rigidity of the tuner structure. The latter is assumed to be excellent because the only moving parts inside the structure are the slides that the dielectric slabs are attached to. The machined slides have a very tight fit with the guide channels and can move only in wanted directions. We have studied the repeatability of the impedance tuner by measuring the same randomly chosen

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Fig. 6. Measured input impedance coverage (points) of the tuner on the Smith diagram. (a) 75 GHz. (b) 92.5 GHz. (c) 110 GHz. Large circles represent the obtainable input impedances with one of the dielectric slabs fixed in place.

TABLE II TUNER CHARACTERISTICS AT 75, 92.5, AND 110 GHz

repeatability would be considerably better, due to the two decade improvement in positioning accuracy. The nonintuitive result, that the magnitude of the reflection coefficient is the most sensitive at 75 GHz, where the wavelength is the longest, originates from the fact that the randomly chosen reflection coefficient happens to be at the sensitive region of the Smith diagram, as can be seen in Fig. 6(a). A disadvantage of this type of impedance tuner is that the change in the impedance is different for constant steps of the slab position. C. Scalability of Waveguide Impedance Tuner

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Fig. 7. Insertion loss of the waveguide impedance tuner over the -band. The circles denote the backshort positions of 2 and 10 mm and the squares the positions 8 and 4 mm, respectively.

reflection coefficient (same backshort position) manually 30 times (only ten measurements are shown in Fig. 8 for clarity). Between measurements, the backshort positions were changed arbitrarily and then returned to original settings. The mean reflection coefficients obtained with this randomly chosen tuner position were 0.764 31.06 at 75 GHz, 0.062 51.53 at 92.5 GHz, and 0.0064 77.47 at 110 GHz. The corresponding standard deviations in the magnitude and phase of the reflection coefficient are presented in Table II. As can be seen in Fig. 8, the differences in the magnitude and phase of consecutively measured reflection coefficients are very small even when the tuner is operated manually with the micrometer heads. With the previously mentioned automated linear motors, the

According to the authors’ knowledge, there are currently no commercial standalone impedance tuners above 170 GHz. The noise figure measurements are already being performed at -band [3] and the need for sensitive receivers for astronomical and other applications is driving the development of noise measurements towards submillimeter wavelengths. Moreover, an accurate, large-range, and low-loss impedance tuner could be used to optimize mixers and doublers operating at submillimeter wavelengths. In one type of commercial impedance tuner [4], [5], a needle probe is moved inside a slotted waveguide. The place and depth of the probe determines the phase and magnitude of the reflection coefficient. These impedance tuners are accurate and exhibit low losses and good Smith -band and below, but as frequency diagram coverage at increases, some problems start to arise. The smallest achievable phase step is directly proportional to the probe positioning accuracy. For example, a 10- m tuning distance corresponds to a 1.7 phase shift at 92.5 GHz with the traditional backshorts, but only to a 0.37 phase shift in our dielectric-based backshort [14]. In our design, the phase shift is caused by the movement of the dielectric slab inside the waveguide (change in the electrical distance of the waveguide fixed short) and is, therefore, controllable by the choice of the material and dimensions of the slab. This enables even more accurate tuning, e.g., by decreasing the thickness or width of the dielectric slab. This feature is a great advantage at submillimeter wavelengths (300 GHz). At these frequencies, the fabrication of waveguide impedance tuners based on probes or traditional backshorts becomes very difficult. Suitability of different backshorts at submillimeter wavelengths is discussed in [27]. As frequency increases, the size of the waveguides decreases. For example, a WR-3 waveguide designed to operate at the frequency band

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the rectangular waveguides, the metal losses per a wavelength are proportional to the square root of the frequency and the dielectric losses per wavelength remain constant [15, pp. 18 and 57]. The surface accuracy limits are of course more severe, but as discussed above, the conventionally machined rectangular waveguides can be well used at submillimeter wavelengths. As a design example, a waveguide impedance tuner with WR-2 (325–500 GHz) waveguides could be realized as follows. The waveguide structure is milled using the split-waveguide block technique. The 50- m-thick and 150- m-wide quartz slabs are fabricated with the diamond cutting. The guide channel for the quartz slab is 254- m wide and 70- m high. The cutoff frequency for the waveguide mode is 1.1 THz in this quartz filled channel. High-precision (with the nonrotating spindle) micrometer heads with 2.5-mm tuning range and 1- m tuning accuracy are connected to the slides in which the quartz slabs are attached. With this design, a 10- m tuning distance corresponds to a 1.5 phase shift at 412.5 GHz, whereas with the traditional backshorts, the phase shift would be 7.0 . Preliminary simulations show that the return loss for this backshort design would be below 0.3 dB. In addition, in [14], the performance of a dielectric-based backshorts is studied at frequency bands of 140–220 and 220–325 GHz, and it is concluded that they exhibit low losses at both frequency bands and are suitable for the submillimeter wave operation. Due to the use of the dielectric-based backshorts, our waveguide impedance tuner design can be easily reproduced at submillimeter wavelengths with reliable, accurate, and low-loss operation. V. CONCLUSION

Fig. 8. Difference in the reflection coefficient between consecutive measurements of the same backshort position. (a) Magnitude. (b) Phase angle. Circles denote the frequency: 75 GHz, crosses: 92.5 GHz, and squares: 110 GHz.

of 220–325 GHz has a width of 0.86 mm. Machining a slot wide enough for the probe, but narrow enough to avoid the disturbance of the surface current in the waveguide walls and, hence, causing losses, is very demanding. The fabrication of magic-T-based impedance tuners at submillimeter wavelengths is difficult due to the complex structure and the fact that the correct alignment of the traditional contacting and noncontacting backshorts in a very small waveguide for reliable operation is very demanding. In our design, the waveguide is intact, except for the very narrow guide channel for the dielectric slab at the end of the waveguide. The fabrication techniques used in our tuner design are relatively simple. The split-waveguide block halves are milled with traditional machining techniques and the rectangular quartz slabs are fabricated with diamond cutting. These techniques are mature and split-block mixers and multipliers with quartz substrates have been demonstrated, e.g., at 220 GHz [28] and 640 GHz [29], [30]. With the state-of-the-art high precision equipment, rectangular waveguides have been directly machined up to 2.5 THz [31]. If the wanted structures cannot be fabricated using conventional machining, micromachining techniques [32]–[34] can be used instead. The metallic and dielectric losses are not of a great concern. In

A waveguide impedance tuner for -band applications has been designed, fabricated, and measured. The impedance tuner consists of two dielectric-based backshorts forming a double-T junction with a WR-10 waveguide. Simulations and measurements show that it exhibits excellent impedance coverage, tuning accuracy, repeatability, low losses, and a high VSWR over the frequency band of 75–110 GHz. Due to the length difference in input and output waveguides, the impedance coverage can be further increased by turning the tuner around. The measured impedance coverage on the Smith diagram is more than 70% at the -band. By turning the tuner around, the estimated impedance coverage for the fabricated tuner is over 90%. The tuner is easily scalable to submillimeter wavelengths, where, to the authors’ knowledge, there are no commercial standalone impedance tuners currently available. ACKNOWLEDGMENT The authors acknowledge E. Kahra, Radio Laboratory, TKK Helsinki University of Technology, Espoo, Finland, and H. Rönnberg, Metsähovi Radio Observatory, TKK Helsinki University of Technology, for the fabrication of the impedance tuner. REFERENCES [1] C. E. McIntosh, R. D. Pollard, and R. E. Miles, “Novel MMIC sourceimpedance tuners for on-wafer microwave noise-parameter measurements,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 2, pp. 125–131, Feb. 1999. [2] M. Kantanen, M. Lahdes, T. Vähä-Heikkilä, and J. Tuovinen, “A wideband on-wafer noise parameter measurement system at 50–75 GHz,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1489–1495, May 2003.

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[31] P. H. Siegel, R. P. Smith, and M. C. Gaidis, “2.5-THz GaAs monolithic membrane-diode mixer,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 596–604, May 1999. [32] C. M. Mann, “Fabrication technologies for terahertz waveguide,” in Proc. Terahertz Electron., 1998, pp. 46–49. [33] P. L. Kirby, D. Pukala, H. Manohara, I. Mehdi, and J. Papapolymerou, “Characterization of micromachined silicon rectangular waveguide at 400 GHz,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 6, pp. 366–368, Jun. 2006. [34] V. M. Lubecke, K. Mizuno, and G. M. Rebeiz, “Micromachining for terahertz applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-46, no. 11, pp. 1821–1831, Nov. 1998.

Tero Kiuru (S’06) was born in Tampere, Finland, on June 11, 1980. He received the M.Sc. degree in electrical engineering from the TKK Helsinki University of Technology, Espoo, Finland, in 2006, and is currently working toward the Ph.D. degree at the TKK Helsinki University of Technology. Since 2003, he has been a Research Assistant and Researcher with the Radio Laboratory, TKK Helsinki University of Technology. He is a member of the TKK Graduate School of Electrical and Communications Engineering. His current research interests include millimeter- and submillimeter-wave sources, characterization and modeling of Schottky diodes, and on-wafer measurement and calibration methods.

Ville S. Möttönen (S’00–M’05) was born in Oulu, Finland, in 1972. He received the Master of Science (Tech.), Licentiate of Science (Tech.), and Doctor of Science (Tech.) degrees in electrical engineering from the TKK Helsinki University of Technology, Espoo, Finland, in 1996, 1999, and 2005, respectively. From 1996 to 2007, he was with the Radio Laboratory (and its Millimeter Wave Group), TKK Helsinki University of Technology, as a Research Assistant and Research Associate. He is currently with the National Board of Patents and Registration of Finland, Helsinki, Finland. His current research interests include antennas for mobile devices and the development and design of millimeter- and submillimeter-wave receivers.

Antti V. Räisänen (S’76–M’81–SM’85–F’94) received the D.Sc degree (Tech.) in electrical engineering from the TKK Helsinki University of Technology, Espoo, Finland, in 1981. In 1989, he became the Professor Chair of Radio Engineering with TKK Helsinki University of Technology. In 1997, he became the Vice-Rector of TKK Helsinki University of Technology (1997–2000). He has been a Visiting Scientist and Professor with the University of Massachusetts, Amherst, Chalmers University of Technology, Göteborg, Sweden, University of California at Berkeley, California Institute of Technology, Jet Propulsion Laboratory (JPL), Pasadena, and Paris Observatory, Paris, France, and the University of Paris 6, Paris, France. He currently supervises research in millimeter-wave components, antennas, receivers, microwave measurements, etc. with the TKK Radio Laboratory and Millimetre Wave Laboratory of Finland—ESA External Laboratory (MilliLab). He leads The Centre of Smart Radios and Wireless Research (SMARAD), which has obtained the national status of Centre of Excellence (CoE) in Research for 2002–2007 and 2008–2013. He has authored or coauthored approximately 400 scientific papers and six books, including Radio Engineering for Wireless Communication and Sensor Applications (Artech House, 2003). Dr. Räisänen has been conference chairman for four international microwave and millimeter wave conferences. He was an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (2002–2005). In 2006, he became a member of the Board of Directors of the European Microwave Association (EuMA).

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Variable Antenna Load for Transmitter Efficiency Improvement Ville Kaajakari, Ari Alastalo, Kaarle Jaakkola, and Heikki Seppä

Abstract—A wireless radio transmitter with an integrated variable-impedance antenna is demonstrated. The transmit signal is connected via a switch to the antenna at one of several feed points that correspond to different load impedances that are optimal for the power amplifier at different transmit power levels. It is shown that this can be used to improve the efficiency of the transmission at low and medium transmit powers. The variable antenna load is demonstrated with a planar inverted-F antenna, but the method can be applied to any antenna with suitable impedances. Index Terms—Antennas, impedance matching, microwave communication, power amplifiers (PAs).

I. INTRODUCTION

T

HE RF front ends of cellular phones have gone through rapid and continuing integration. This has been driven in part by the change from heterodyne architectures to direct down conversion that allows much of the receiver functionality to be integrated into the baseband signal processor. Another driver has been the integration of discrete components into modules. For example, the front-end switches and surface acoustic wave (SAW) filters are now integrated in the same low-temperature co-fired ceramic (LTCC) package. The benefits of modularization are: 1) reduced packaging costs; 2) reduced design time for a cell phone manufacturer; and 3) smaller final circuitry area. Despite the progress in integration, the antenna essentially remains as an unintegrated industry-standard 50- “black box” to the transceiver designer. In this paper, the efficiency benefits of a variable-impedance antenna integrated as a part of the transmitter are demonstrated. The power amplifier (PA) consumes a significant amount of the cell phone power budget and considerable effort has been spent in increasing the PA efficiency. Depending on transmission conditions, the output power level of the PA varies from 5 to 33 dBm for global system for mobile communications (GSM) handsets [1]. The PAs are optimized for efficiency at the maximum power level. However, the efficiency falls dramatically when operating at the lower power levels, well below 30 dBm, where the phones are used most of the time [2]. For example, Manuscript received January 9, 2007; revised May 8, 2007. This work was supported by Perlos Oyj and Asperation Oy. V. Kaajakari was with the VTT Technical Research Center of Finland, Espoo FIN-02044 VTT, Finland. He is now with the Institute for Micromanufacturing, Louisiana Tech University, Ruston, LA 71270 USA. A. Alastalo, K. Jaakkola, and H. Seppä are with the VTT Technical Research Center of Finland, Espoo FIN-02044 VTT, Finland. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.902093

a typical GSM PA efficiency for 20-dBm output power is only approximately 10%. Consequently, there is significant potential for power savings if the amplifier efficiency can be increased at the typically used power levels. There has been considerable interest in improving the PA efficiency, as this would directly translate to an increase in the operation time of battery-powered portable devices [3]. In general, the amplifiers are classified by their operation point (biasing point). The “class-A” amplifiers conduct current all the time and are thus very linear. The “class-B” and the higher class amplifiers conduct current only during a part of the input cycle and are, therefore, more efficient, but also less linear. There is considerable effort in utilizing schemes such as predistortion to use nonlinear amplifiers in communication systems that require linear signal paths. While these efforts address the demand for the high PA efficiency, they do not directly address the issue of the lowered efficiency at the low transmit power levels. The PAs reach their maximum efficiency when they operate at the full output voltage swing that normally correspond to the maximum output power. One way to increase the PA efficiency at lower power levels is, therefore, to adjust the dc supply voltage [4]–[6]. This approach increases the efficiency at the low power levels, but the method has two disadvantages, which are: 1) the variable supply requires additional circuit components and 2) the supply voltage conversion suffers from its own conversion losses that lower the overall efficiency also at the maximum power levels. Another approach to increase the transmitter efficiency at low power levels is to make the matching network between the PA output and antenna adaptive [7][8]. In principle, high efficiencies can be maintained by increasing the matching network impedance. For full benefit, however, the impedance change requirements are quite high. For example, tuning the output by 10 dB requires an impedance adjustment by a factor of ten. In addition, the tunable components need to have low losses and be highly linear to satisfy the spectral purity requirements for wireless communication. These factors have made implementation of tunable matching networks impractical for portable wireless devices. Methods to increase the transmission efficiency also include an approach to bypass the final stages of the transmitter amplifier for low output powers [9]. Fig. 1(a) shows the solution considered in this paper to increase the PA efficiency at the lower power levels with a minimal increase in the component count. The amplifier is connected through a switch to an antenna that has two different feed points with different impedances. At the lower power levels, the load impedance is increased to maintain full swing of the PA output

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Fig. 1. (a) PA integrated with an antenna that has two different impedance points. (b) By switching between high and low antenna resistances, the power efficiency curve can be adjusted to stay at good efficiencies for all output powers.

voltage that results in good efficiency. Thus, depending on the transmit power level, the power-efficiency curve is adjusted for an overall efficiency increase, as illustrated in Fig. 1(b). To further enhance the control of the efficiency, one can increase the number of feed points, but in this paper, we only consider two feeds for simplicity and because this already captures most of the available power saving. This paper is organized as follows. In Section II, the concept of using the variable load to increase efficiency at the lower output power levels is verified. This is followed by the demonstrator design and the associated tradeoffs in Sections III and IV. Section V presents the measurement results. Finally, this paper presents discussions and conclusions in Section VI. II. SIMULATED EFFICIENCY IMPROVEMENT The typical handset PAs are designed to reach their maximum efficiencies of approximately 50% when operated at the full output voltage swing of 3–5 V that correspond to the output , where is the output rms voltage and power of is the load resistance. As the supply voltage is limited by the battery, the cell phone PAs are designed to drive a rather low load resistance typically of 1–4 . This translates approximately to 2 W of maximum RF power, as required by the GSM specifications. For operation at the lower power levels, the output voltage swing is reduced, which, unfortunately, also reduces the amplifier efficiency. An alternative way to lower the power is to , but continuous adjustment is increase the load impedance not practical. In this paper, an amplifier topology is proposed that has two discrete load impedances for rough power tuning, while the fine power adjustment is still carried out by adjusting the voltage swing. To quantify the power savings that are achievable with the variable-load method, PA simulations were done. Fig. 2(a) shows a PA schematic that is based on a typical MESFET transistor [3]. The circuit simulations with different input drive levels and load impedances were carried out using APLAC RF simulation software [10]. The results for two load impedances are shown in Fig. 2(b). As expected, the power efficiency at the low drive levels is better for the higher load impedance. The peak efficiency for the higher impedance is optimized to be at a power level where a significant amount of power is still consumed, but the efficiency of a fixed-load amplifier has

Fig. 2. Simulated RF output power and supply efficiency for two different load impedances. The higher load impedance results in an increased efficiency at the lower output power levels, but also reduces the maximum output power. Shading indicates the region where power savings can be achieved with the variable load impedance method. (a) MESFET PA circuit with ideal load (after [3]). (b) Simulated output power and efficiency.

significantly dropped. The maximum efficiency obtained in the simulations is slightly higher than found for commercial amplifiers due to the idealized tank circuit used in the simulations. The maximum power saving occurs here at approximately dBm where the efficiency is increased from 28% to 60%. III. DEMONSTRATOR DESIGN Fig. 3 shows the components of the variable load amplifier module. The transmitter consists of: 1) the amplifier; 2) impedance matching element; 3) switch; and 4) antenna with two impedance points. These four elements are covered in Sections III-A–D. The demonstrator was designed for the 869-MHz industrial–scientific–medical (ISM) frequency, which is close to the GSM band. A. Substrate The module was implemented on a 3-mm-thick thermoplast substrate that has similar RF properties than polytetrafluoroethylene (PTFE). Fig. 4 shows the cross section of the module. The three-layer module has the ground plane in the middle, electronics on the front surface, and the planar antenna 10 cm, on the back side. The module size was 3.5 cm approximating the dimensions of a cell phone.

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on a direct mechanical contact and, thus, has the advantage of low series resistance leading to small power losses in the switch. However, with the antenna impedance levels of 40 and 200 , a solid-state switch would also have been an alternative. Unfortunately, the MEMS switch has proven unreliable and failed after a few minutes of testing. The best measurement results were obtained by removing the MEMS switch and directly wiring to the different antenna impedances. E. Antenna Fig. 3. Components of the variable load amplifier module. Not accounting for matching network and switch losses, the expected shift in the power efficiency curve is 7 dB.

Fig. 4. Cross-sectional view of the demonstrator module that integrates a variable load antenna, switch, and PA.

B. PA For the PA, the RFMD 2173 from RF Microdevices1 was chosen. The amplifier is designed for the GSM 900 band and does not contain any impedance-matching elements to 50 in the amplifier chip/module. Thus, the amplifier is well suited for designing the variable-load modules. For the maximum power load output, the amplifier should be connected to a impedance. C. Impedance Matching Although the amplifier could, in principle, be directly connected to a 1.5– antenna load, the overall efficiency is increased if higher antenna impedances are used, as this reduces the negative effect of switch losses. For the demonstrator, a simple -match circuit is used to lower the load impedance seen by the PA. For the -match shown in Fig. 3, the transformation ratio is (1) where is the resistance seen by the amplifier, is the (antenna) load resistance, and is the -matching capacitance . As (1) [11]. The approximation in (1) is accurate if lowers shows, for fixed , increasing the load resistance . the PA load D. Switch For switching between the antenna loads, a microelectromechanical systems (MEMS) switch (M1C06-CDK2 from Magfusion)2 was planned to be used. The MEMS switch is based 1RF Micro Devices Inc., Greensboro, NC. [Online]. Available: www. rfmd.com 2Magfusion

Inc., Chandler, AZ. [Online]. Available: www.magfusion.com

The antenna that is used in the demonstrator module is a rectangular planar inverted-F antenna (PIFA) [see Fig. 5(a)] implemented on a 3-mm-thick thermoplast substrate. The dimensions of the PIFA are approximately 54.5 mm 20 mm. The short circuit needed between the actual antenna patch and the ground layer has been realized by a line of vias. The two feed points required to provide the two different feed impedances are easily implemented in the PIFA structure by locating the feed points at different distances from the short-circuiting end of the antenna. As the electronics and radiating antenna element are placed on opposite sides of the ground plane of the module, a signal is fed to the antenna through vias, as shown in Fig. 4. The feed impedance of a (narrow and thin) PIFA has a sine-squared dependency on the distance of the feed point from the short-circuiting end. As the desired feed impedances, i.e., 40 and 200 , are both resistive, the feed points are located symmetrically on both sides of the center line of the antenna patch. This is due to keeping the current distribution around the both feed points as similar as possible. The total distance between the feed points is determined by the distance between the contact pads of the MEMS switch and, therefore, the feed points cannot lie on the center line of the antenna, as would be the ideal case. Tuning of only the real part of the antenna impedance by varying the feed position has also been demonstrated for conventional patch antennas [12]. Fig. 5 shows the simulated radiation pattern. Simulated values for the radiation efficiency and the maximum gain are 62% and 1.9 dBi, respectively. IV. DESIGN TRADEOFFS The electrical equivalent of the matching and switching circuits is shown in Fig. 6. The matching network losses are that determines the unloaded modeled with the resistance -match quality factor . To obtain guidelines for optimum design, it is useful to analyze the overall power efficiency from the PA to the antenna. Including the matching and switch losses, the efficiency is (2) is the switch efficiency and is the matching efficiency. Thus, to lower , high load impedances are rethe switch losses due to quired. This, however, translates to a high transformation ratio , which will lower the matching efficiency. Therefore, there is an optimal load impedance that will minimize the combined switch and matching-network losses. As (2) shows, when a single -match is used for the two different load impedances, optimal design cannot be obtained for where

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Fig. 7. Simulation of the matching network. (a) Circuit schematic for simulating the PA load impedance and matching network efficiency. (b) Simulated impedance curves (magnitude and phase) for the MEMS switch (R = 0:1 ). At the operating frequency, only the real part of the impedance changes when the load resistance is changed.

Fig. 5. PIFA antenna and its simulated radiation pattern. (a) PIFA antenna geometry. (b) Simulated radiation pattern in X Z -plane. (c) Simulated radiation pattern in Y Z -plane.

Fig. 7 shows the circuit schematic for the amplifier load and , as seen by the PA. The series inductance the simulated represents the antenna feed through inductance and the switch is modeled with a transmission line and a resistance. The series inductance from the switch and feed through effectively change the circuit topology from an - to a -match. The simulated matching-network efficiencies are 74% and 92% for the lowand high-impedance loads, respectively. Accounting for the matching-network losses, the simulated power-efficiency curve difference for the two loads is 5 dB. V. MEASUREMENTS

Fig. 6. L-matching network is realized with a transmission line inductance L and a surface mounted 0204 ceramic capacitor C . The switch losses are modeled with a series resistance R . The matching network losses are modeled with the 1. resistance R that determines the L-tank quality factor Q

both loads. This will limit the performance of the system if the load impedance is widely varied. For the demonstrator, antenna loads of 40 and 200 were chosen, as they were easily realized with the antenna. These impedances, however, are rather high and the matching network losses limit the efficiency performance. A more optimal design would have been obtained with lower antenna impedances that could be realized by having the antenna feeds closer to the short circuit. This, however, would have made the impedances much more location sensitive.

Fig. 8 shows the fabricated demonstrator module. Unfortunately, the MEMS switch failed in the testing and the following measurements were done with the switch removed and manually wiring the two different antenna loads. The transmitter module efficiencies were measured in pulsed operation (20% duty cycle at 300 Hz), as continuous operation results in heating and loss of PA efficiency. The dc-power consumption was monitored by measuring the amplifier current and voltage during the RF-power-on cycle. In all measurements, the amplifier supply voltage was kept at 3 V. The transmitted RF signal was received with a Suhner antenna and captured using an Agilent Vector Analyzer (89600) in an RF power envelope detection mode. The laboratory measurements were calibrated in an anechoic room where the total radiated power was measured together with the radiation pattern of the antenna. In the antenna measurement, the radiation efficiency of 60% and the maximum gain of 1.7 dBi were recorded. These values correspond well to the antenna simulations. Fig. 9 shows the measured radiated power and the overall efficiency curves for the manual switch module. The rather low

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Fig. 8. Demonstrator for the variable-impedance antenna concept. (a) Front side of the demonstrator board showing the PA and switch. (b) Back side of the demonstrator board showing the antenna and feed through locations.

overall transmit efficiency is due to the antenna losses. The efficiency curve with the antenna losses deembedded is shown in Fig. 9(b). The efficiencies are close to 50% for the amplifier and matching network, which is in line with the manufacturer’s data sheet for the amplifier. At 25-dBm power output, the obtained efficiency improvement in Fig. 9 was 60% when switching from the low to high load impedance. This magnitude of improvement in PA efficiency can be considered significant. The obtained 3-dB shift in the efficiency curve correlates well with the simulated shift of 5 dB predicted for 40- and 200- antenna impedances. The 2-dB difference is likely due to a change in internal PA efficiency not accounted for in the simulations. It is well known that due to finite output resistance of PAs, the PA efficiency decreases with decreasing load impedance. Unfortunately, accounting for the change in PA efficiency is not possible, as the load–pull characteristics for the used amplifier are not available.

VI. DISCUSSIONS The variable-load antenna was demonstrated to be an effective method for adjusting the PA load impedance. The demonstrated shift of the power-efficiency curve, however, was only 3 dB and the overall efficiency was not optimal. To obtain larger shifts of the power-efficiency curve, the high-to-low impedance ratio should be increased. To obtain a better efficiency, lower antenna impedances should be used. Both goals could be realized by choosing the antenna feed points closer to the PIFA short. As -law, low impedances can be the PIFA impedance follows a realized. For example, impedances of 10 and 80 would result in a 9-dB shift. The practical realization of variable antenna with low impedance poses the following challenges.

Fig. 9. Measured transmitter efficiencies. (a) Measured overall transmitter efficiency (efficiency from the dc to the transmitted power). The rather low efficiency is due to the antenna losses. (b) Measured efficiency with antenna losses deembedded. The PA efficiency is increased by 60% at 25 dBm when switching from low to high load.

1) Close to the short, the antenna impedance varies rapidly with position and, thus, a good manufacturing precision is needed. 2) Antenna impedance may vary due to external disturbances. 3) The low load impedance requires a low-loss (preferably ohmic) switch. These challenges are discussed in the following. The modern circuit board manufacturing enables metal patterns smaller than 100 m and manufacturing tolerances are of the same order of magnitude. Thus, the first challenge of obtaining sufficient manufacturing tolerances is probably achievable with current technology, although practical realization may require a few design iterations. The second challenge of antenna impedance variations due to external disturbances is more serious as the cell phone antenna impedance significantly changes due to proximity effects [13]. With the cell phone antenna analyzed in [13], the 40- and and 200- antenna impedances would change to

KAAJAKARI et al.: VARIABLE ANTENNA LOAD FOR TRANSMITTER EFFICIENCY IMPROVEMENT

, respectively, due to a hand being placed close to the antenna. As a result, the simulated impedances seeing by the and PA would change from the ideal to detuned and . The significant reduction in the real part of the PA impedance load would reduce the PA efficiency and the radiated power. On a positive note, the proposed variable antenna load is no more susceptible to proximity effects than a single load antenna as both the antenna impedance and matching network follow the traditional design. To battle the large reactive load, two solutions are possible: the variable antenna load may be combined with variable matching/tuning or new additional antenna impedance points may be chosen to provide the desired reactance to counter the environmental variations. The third challenge of low-loss ohmic switches is both technical and economical. In the demonstrator, the use of a micromechanical switch was planned, but in the end, manual wiring was used due to the reliability problems of the switches. Thus, although all the other components (PA, matching, and antenna) were integrated into a realistic demonstrator, the switching was unrealistic for practical applications. Since the design of the prototype, alternative sources for the micromechanical switches have become available3 4. However, it remains to be seen whether the micromechanical switches can be competitive in RF transmitter applications. The solid-state switches are a possible alternative, but due to their higher series resistance, solid-state switches would require higher load impedances to obtain an optimum efficiency. We, therefore, believe that the full benefits of the proposed switchable-antenna load are obtained with micromechanical switches. VII. CONCLUSION A wireless transmitter with an integrated variable-impedance antenna has been demonstrated. By switching between different impedance points of a PIFA antenna, different PA loads have been realized. The PA efficiency has been increased by switching to high load at low transmit power levels. Future work is needed on optimizing the design and improving the switch reliability. ACKNOWLEDGMENT The results of this study are based on a collaborative project between the VTT Technical Research Center of Finland, Espoo, Finland, and Perlos Oyj/Asperation Oy, Perlos, Vantaa, Finland.

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[5] D. Dening, “Automatic VEE control for optimum power amplifier efficiency,” U.S. Patent 6 624 702, 2003, RF Micro Devices Inc., Greensboro, NC. [6] B. Sahu and G. A. Rincón-Mora, “A high-efficiency linear RF power amplifier with a power-tracking dynamically adaptive buck-boost supply,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 112–120, Jan. 2004. [7] A. Klomsdorf, L. E. Winkelmann, and M. Landherr, “Memory-based amplifier load adjust system,” U.S. Patent 6 556 814, 2003, Motorola Inc., Schaumburg, IL. [8] W. C. E. Neo, Y. Lin, X. D. Liu, L. C. N. de Vreede, L. E. Larson, M. Spirito, M. J. Pelk, K. Buisman, A. Akhnoukh, A. de Graauw, and L. K. Nanver, “Adaptive multi-band multi-mode power amplifier using integrated varactor-based tunable matching networks,” IEEE J. SolidState Circuits, vol. 41, no. 9, pp. 2166–2176, Sep. 2006. [9] J. B. Sung, S. W. Kang, C. H. Hyoung, J. H. Hwang, and Y. T. Kim, “Dual antenna diversity transmitter and system with improved power amplifier efficiency,” U.S. Patent Applicat. Publication U.S. 2005/0143024, 2005. [10] RF Design Tool. APLAC, An AWR Company, Espoo, Finland [Online]. Available: http:www.aplac.co [11] T. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge, U.K.: Cambridge Univ. Press, 1998. [12] L. I. Basilio, M. A. Khayat, J. T. Williams, and S. A. Long, “The dependence of the input impedance on feed position of probe and microstrip line-fed patch antennas,” IEEE Trans. Antennas Propag., vol. 49, no. 1, pp. 45–47, Jan. 2001. [13] K. R. Boyle, Y. Yuan, and L. P. Ligthart, “Analysis of mobile phone antenna impedance variations with user proximity,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 364–372, Feb. 2007.

Ville Kaajakari received the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Wisconsin–Madison. in 2001 and 2002, respectively. From 2002 to 2006, he was with the VTT Technical Research Center of Finland, Espoo, Finland, the largest nonuniversity research organization in Northern Europe. While with the VTT Technical Research Center of Finland, he was a Project Manager involved with development of micromechanical reference oscillators. Since 2006, he has been an Assistant Professor with the Institute for Micromanufacturing (IfM), Louisiana Tech University, Ruston.

Ari Alastalo received the M.Sc. and D.Sc. (Tech.) degrees in technical physics from the Helsinki University of Technology, Espoo, Finland, in 1997 and 2006, respectively. Until 1998, he was involved with magnetic impurities in superconductors. From 1998 to 2002, he was with the Nokia Research Center, where he focused on adaptive-antenna systems. Since 2002, he has been with the VTT Technical Research Center of Finland, Espoo, Finland, where he is currently a Senior Research Scientist leading a team of printable sensors. His research concerns RF components, MEMS, and printable electronics.

REFERENCES [1] “GSM global system for mobile communications, 3GPP TS 05.01/05. 05,” ETSI, Sophia-Antipolis, France, 2003. [2] J. Wiart, C. Dale, A. V. Bosisio, and A. L. Cornec, “Analysis of the influence of the power control and discontinuous transmission on RF exposure with GSM mobile phones,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 376–385, Nov. 2000. [3] S. C. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 1999. [4] “Application note 1205,” Maxim Integrated Products Inc., Sunnyvale, CA [Online]. Available: www.maxim-ic.com 3TeraVicta Technologies Inc., Austin, TX. [Online]. Available: http://www. teravicta.com/ 4WiSpry Inc., Irvine, CA. [Online]. Available: http://www.wispry.com/

Kaarle Jaakkola was born in Helsinki, Finland, in 1976. He received the Master of Science (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 2003. Since 2000, he has been with the VTT Technical Research Centre of Finland, Espoo, Finland, initially as a Research Trainee and, since 2003, as a Research Scientist. From 2000 to 2002, he participated in the Palomar (EC IST) Project, during which time he developed RF parts for a new RF identification (RFID) system. His current research interests include RFID systems, wireless and applied sensors, antennas, electromagnetic modeling, and RF electronics.

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Heikki Seppä received the M.Sc., Lic.Tech., and Dr.Tech. degrees in technology from the Helsinki University of Technology, Espoo, Finland, in 1977, 1979, and 1989, respectively. In 1979, he joined the VTT Technical Research Center of Finland, Espoo, Finland, where since 1989, he has been a Research Professor involved with information technology. In 1994, he became Head of the measurement technology field with VTT automation. His research has concerned electrical metrology, in general, and superconducting devices for measure-

ment applications, in particular. He is currently involved with research on dc superconducting quantum interference devices (SQUIDs), the quantized Hall effect, single electron transistor (SET) devices, RF instruments, printable electronics, and microelectromechanical devices.

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Rigorous Analysis of a Metallic Circular Post in a Rectangular Waveguide With Step Discontinuity of Sidewalls Constantine A. Valagiannopoulos and Nikolaos K. Uzunoglu, Fellow, IEEE

Abstract—The scattering of guided waves by a conductive post placed inside a rectangular waveguide with a step discontinuity on its sidewalls is analyzed through an integral-equation formulation. The interaction between post and discontinuity is examined with use of the Green’s function. Numerical computations are carried out based on the mode-matching technique, providing a stable and accurate solution. An equivalent-circuit model is developed from the electromagnetic analysis and the computed scattering -parameters. The developed solution accuracy is validated with simulation software. Index Terms—Green’s function, guided waves scattering, integral equations, mode matching, -parameters, waveguide discontinuities. Fig. 1. PEC circular inductive post in rectangular waveguide with step discontinuity. The left port is labeled 1 and the right one is labeled 2.

I. INTRODUCTION HE DESIGN of filters using rectangular waveguide structures and the analysis of the relevant scattering phenomena has attracted considerable attention over the past decades. The case of cylindrical posts inside waveguides has been extensively investigated because they are easy to be constructed. In [1, p. 257–262], the original Swinger’s solution is given for an electrically small post inside a rectangular waveguide, while Leviatan et al. [2] examine a nonsingular multifilament current representation also leading to results for large posts. The scattering parameters of a two-port device with multiple cylindrical obstacles are derived by Li et al. [3], contrary to [4], where the exact solution of the surface current is provided. A similar method is presented by Green [5] where the post can have arbitrary cross section. Step discontinuities in rectangular waveguides have also been studied in the past. A review of multiple step discontinuities for a shielded three-layered coplanar waveguide is given by Rahman and Nguyen [6], while a similar problem, but for microstrips, is examined in [7]. Finally, in [8], systems with step discontinuities are analyzed with respect to their scattering characteristics. It seems that the combination of step discontinuities in waveguide walls and posts has not been examined in the past. Bradshaw [9] investigates a post with a gap constituting the discontinuity, and in [10], the study is restricted to symmetrical

T

Manuscript received October 28, 2006; revised March 16, 2007. This work was supported in part by the Eugenides Foundation and by the Microwave and Fiber Optics Laboratory, National Technical University of Athens. The authors are with the School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece (e-mail: [email protected] ntua.gr; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.901597

multiport networks. In this paper, a nonsymmetrical two-port device is considered. The structure consists of two abruptly connected unequal rectangular waveguides of different height, while a perfectly conducting (PEC) circular post is placed near the wall discontinuity plane. The geometry of the device is shown in Fig. 1. The scattering -parameters of the two-port structure are determined and a lumped component model is developed based on the rigorously computed current, as in [11]. The waveguide is analyzed by applying a semianalytical method based on the solution of an integral equation on the cylindrical post surface current. The Green’s function, being the kernel function of this integral equation, is obtained by solving the waveguide structure with wall discontinuity. To this end, a mode-matching method is utilized assuming a double-sided excitation of the device. This approach leads to a decomposition of Green’s function into three terms corresponding to: 1) the response of sidewall discontinuity (derived similarly to [12]); 2) the influence of parallel plates in filament’s radiation (using methods of [13] and [14]); and 3) the primary field distribution due to the source singularity. The Green’s function has been written alternatively as a sum of modes in an expression computationally suitable for points far from the scatterer, as in [15]–[17]. The integration of the singular part is carried out using the analytical technique of [18]. Extensive numerical computations are executed to verify the validity of the boundary conditions and the consistency with previously published data [1], [2]. Convergence of numerical results and their sensitivity to truncation parameters are checked, assuring the accuracy of the proposed method. Finally, independent checks with simulation software are used to assert the validity of the developed solution.

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II. FORMULATION OF THE BOUNDARY VALUE PROBLEM The geometry of the problem under study is shown in Fig. 1. or a polar coIn the following, either a Cartesian ordinate system is used interchangeably. Throughout the analtime dependence is adopted and suppressed. ysis, an Consider two rectangular waveguides (labeled 1 and 2, as shown in Fig. 1) having the same width and different heights and . Only one of the sidewalls is discontinuous and the unequal walls are short circuited. A PEC circular post of radius , centered at the coordinate origin, is placed in the vicinity of the discontinuity between the two waveguides. The longitudinal distance between the axis of the cylinder and . The the discontinuity plane is denoted by two waveguide regions separated by the edge can be described for the in terms of the azimuthal parameter for the first one. The angle second one and is defined as

Substituting (4) into (3) and imposing the boundary condition on the PEC cylindrical boundary for the vanishing electric field yield to

(5) where is the primary field distribution and is the response quantity concerning the influence of the surrounding environment. Equation (5) denotes each azimuthal and leads to a linear system whose angle unknowns are the complex constants . In order to proceed, for and multiply both sides by integrate with respect to

(1) . The area inside waveguides is with assumed to have homogeneous electromagnetic properties and without losing taken to be free space. It is supposed generality and the discontinuity region is excited by the two modes) one at a time. These waves are dominant waves ( incident to the corresponding ports and are independent of the variable. It is, therefore, dictated that the secondary waves are also uniform along the -axis and, thus, the posed boundary problem is 2-D. This happens because both waveguides are . All the taken to support only the dominant mode field distributions are then described through the single electric field component (parallel to the -axis) defined as a scalar or depending on the used coordinates. function The transverse magnetic field component is computed by , employing the equation and are the wavenumber and intrinsic impedance where into the vacuum, respectively. In order to determine the scattered fields inside the waveguides, the scattering integral is utilized. As described in [3], the electric field produced by an infinite cylindrical PEC scatterer placed inside a 2-D region, assuming the relevant Green’s funcis known, can be defined by the following equation: tion (2) is the region wavenumber and is the intrinsic where is the scalar surface current on the PEC impedance. In (2), post, flowing parallel to the post axis on the scatterer’s perimeter curve . In the current case, (2) reduces to (3) The surface current can approximately be described in terms of a finite sum of azimuthal harmonics weighted by complex coefficients

(4)

(6) system of (6) is solved, Once the the surface current developed on the cylinder is obtained, and thereby every field quantity within the investigated area is computable [through (3)]. As the post-wall discontinuity is excited from both sides (two incident waves), a couple of correcurrent distributions denoted by sponding to different coefficients will be derived. The upper scripts in describing currents indicate the excitation port (left 1 and right 2). Accordingly, the field quantities and are written with a superscript within parentheses, indicating which waveguide (1 or 2) is excited. As the excitation field quantities are known, the current description can be achieved if one determines the and the total Green’s response fields . These two derivations are described in function Sections III and IV. III. IMPLEMENTATION OF MODE MATCHING Consider an infinite rectangular waveguide of height and width . It is well known [20, p. 367] that the supported by such a device have an -dependent normalized modes component given by (7) with with

. The modal wavenumbers are taken and , which are equal to (8)

The above model will be used to introduce the mode-matching procedure applied on the discontinuity plane of two unequal and and abruptly connected waveguides with heights . The method is based on the projection of the boundary conditions by using sets of mode functions of the two adjacent

VALAGIANNOPOULOS AND UZUNOGLU: RIGOROUS ANALYSIS OF METALLIC CIRCULAR POST IN RECTANGULAR WAVEGUIDE

waveguides [12]. The utilized cross products of the modes are defined as follows: (9) where are nonzero positive integers. If modes are orthogonal, i.e.,

, the two (10)

is the Kronecker’s delta. In case of unequal wavegwhere uides, the result is obtained as

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The vertical PEC edge occupies the region and, therefore, the tangential (equaling total) electric field should be zero across it. The continuity of the tangential magnetic field concerns only the common section of the two waveguides, as the surface current on the edge is linearly dependent to the aforementioned electric quantity. As (15) denotes , namely, the region of the first waveeach guide, the equation should be projected on its own mode set and integrated from to . For the same reason, (16) is multiplied by and integrated from to . After some algebraic manipulations, a linear system is formulated as follows:

(11) The purpose of the mode-matching technique is to determine the fields developed due to the step discontinuity. These fields can be written as infinite sums of modes of the waveguide they are referred to, advancing towards an outgoing direction from the discontinuity. If the sum is truncated, the discontinuity response field in the first waveguide has the form

(17)

(12) (18) as similarly does the quantity in the second waveguide

(13) . The number of terms where taken under consideration is common for both expressions for simplicity. Suppose a bilateral excitation, comprised of two and , denoting the field scalar quantities inside first and second waveguide, respectively, in case the effect of discontinuity is ignored. These quantities vanish on the PEC planes

(14) , continuity of tangential field components On the plane is enforced as follows:

The matrix of the system is inverted without numerical problems [22] and the unknown coefficients are determined. Equations (17) and (18), which constitute the linear system, -mode excitations. When the can be specialized to treat first waveguide is excited, the excitation quantities possess the expressions (19) (20) Adopting unilateral excitation (zero ) means that the should not only describe the effect sum of modes of the discontinuity, but the transmitted field from the first to second waveguide as well. In this case, the right-hand sides of (17) and (18) are simplified by use of (11). Through common numerical solvers, one can reach the approximate values of the constants referred as (21)

(15) A similar way is followed in case the excitation is the mode of the second waveguide

(16)

(22) (23)

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For each modal coefficient, a different notation is used as follows:

(24) Following the above analysis, the electric fields expressing the response of the waveguide wall discontinuity are defined by

(25)

(26) In (6), the fields are functions of the polar coordinates . Applying the conversion relations and , the alternative expressions are found to be

Fig. 2. Possible positions of the Green’s source across the surface of the scatterer. The sources are infinite filaments of electric current with a specific magnitude.

The singular one is chosen to be the following free-space Green’s function, which is well known [24]:

(31)

(27)

where is the Hankel function of zeroth order and second is defined by type. The smooth component different formulas for each side of the discontinuity, correand . sponding to different waveguide heights It is thus sensible to investigate the representative waveguide , whose Green’s function is denoted as with height

(28)

(32)

The Green’s function of the waveguide with wall discontinuity corresponds to the response of the structure when it is excited by a linear-source singularity. The scalar Green’s function is computed by assuming a line current parat a point placed at an arbitrary allel to the -axis with amplitude [23]. As shown in Fig. 2, the filamentary current point is placed on the axis . The desired Green’s function is comprised of two components: the first expressing the influence of the parallel plates on the filamentary excitation, and the second being the response of the step discontinuity (similar to the results in Section III)

In deriving , the perfect conductivity of the walls is exploited and the results of image theory are used. In case only one PEC half area is present, the symmetric image of the source would be sufficient for describing the whole situation. The second PEC plane dictates the placement of infinite images of Fig. 3. It is also noticeable that the -coordinate of all the images is common. After certain straightforward simplifications similar to [15], (33) is obtained, shown at the bottom of the following page. The separate Hankel term of (33) becomes , which is outside the investisingular at gated region , and consequently, cannot pose a numerical difficulty. Unfortunately, the series in (33) converges very slowly and is not convenient for computation. That happens due to the Hankel function’s slowly dumping behavior as its argument increases ([25]) as follows:

IV. SPLITTING OF GREEN’S FUNCTION

(29) The excitation part can be written as a sum of a singular and a normal (smooth) quantity (30)

(34)

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in [15]. The rapidness of convergence of this series increases as gets larger than as follows:

(36) The excitation part of the Green’s function can now be determined. Apart from the singular term, it arises from (36) with , proportionately to which waveguide the observation point belongs to, i.e.,

(37) Fig. 3. Infinite images constituting the influence of the parallel plates on the radiation of the filament. The central source is the primary one. The first-order images have opposite direction, the second-order singularities are the images with respect to the two infinite planes of the first order ones, etc.

To convert such a series to a rapidly converging integral, the Laplace transform and a geometric series formula are introduced. By following a similar method with [14], one concludes the result shown in (35) at the bottom of this page. The exponential of the denominator makes the numerical computation of . the integral very fast in all cases, except for those with Therefore, the implementation of such a formula will meet no problem for applications with observation points near discontinuity, such as the mode-matching technique. An alternative expression for the whole excitation part of the Green’s function, , is educed by employing computationally suitable for Poisson summation formula. The result is as follows and used

As stated above, the singular term is common, a fact that can be proved useful during the following steps. The part of the Green’s function expressing the impact of the will be derived by mode matching. discontinuity With reference to (17) and (18), the bilateral excitation is defined by the following:

(38) (39) By using excitation quantities not expressed in modal form, one can utilize mode matching for a great variety of sources. By supposing bilateral excitation, its expression is the same regardless of which waveguide contains the source. The filament in the inquired case equally belongs to both sides. For this reason, the in . derived formula will be valid for each

(33)

(35)

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The integrals in (17) and (18) involving the integrand funccan also be computed analytically tion of and, hence, the complexity is diminished while accuracy is increased. A numerical integration with respect to is demanded , but outside the singularity region because for of a neutralization. By solving the linear system for a unitary Green’s source placed at angle , the complex constants are as. signed. Refer to them as Thereby the form of the Green’s function’s component owed to the discontinuity is found, i.e.,

Applying the orthogonality of harmonic functions

(44) replacing the singular part by (43), and finally using twice (44) give

(45)

and both the response Once the Green’s function fields and are specified, the only remaining step for solving the integral equation (5) is to perform the double and single integrations of (6). As the integrand is irregular at containing the singular part , a numerical procedure cannot be used. Although the Green’s function is integrable by definition, the result must be found analytically. This is possible only if the integration intervals have an extent of . That remarks upon the significance of separating the singularity in order to be common for both waveguides. In order to carry out the analytical operation, an expansion of the Hankel function originating from the addition theorem is utilized [18] as follows:

As far as the normal part of the excitation Green’s function is concerned, the integrations should be carried out numerically. The expressions are complicated and a principal function for each case cannot be found. The double is executed by using an orthogintegration of points ( equally spaced points for onal grid of each of the variables ) and applying the simple trapezoidal rule in each dimension. With respect to , the expressions are uniform, but this in not the case for the integration with respect to , as different formulas correspond to each waveguide. The number of points used for each region’s formula equals the refor the second side’s expreslated fraction of , i.e., for the first side’s expressions. When it sions and , comes to the response part of the Green’s function the situation is similar to the above, but with one difference: this function can in no way be integrated analytically, as only its discrete values with respect to are available. It is thus imperative to carry out another numerical integration (with the same parameters as the aforementioned ones). The mode-matching procedure described in Section II is repeatedly performed times with different values (Green’s source positions) in order to reand each time. ceive the respective constants The 1-D integrations in (6) regarding expressions of and are also computed numerically. Once the currents are computed and the distributions are approximately determined, the expressions for the electric field in each side can be derived. It is meaningful to write the explicit forms of all the quantities for observation points relatively far from the scatterer, namely, for . When the first waveguide is excited,

(43)

(46)

(40) Unlike (37), the function in (40) is not analytic with respect to —just its values for different source positions are known (see Fig. 2). This is natural, as the derivation uses a numerical procedure throughout, of which the position of singularity is supposed to be invariant. The total Green’s function in polar coordinates is shown in (41) at the bottom of this page, where the singular part is defined equivalently as follows:

(42)

V. COMPUTATION OF CURRENT DISTRIBUTION ON THE POST

(41)

VALAGIANNOPOULOS AND UZUNOGLU: RIGOROUS ANALYSIS OF METALLIC CIRCULAR POST IN RECTANGULAR WAVEGUIDE

the total response field (excluding the primary excitation) inside the first waveguide is given by (47), shown at the bottom of this page. In the second one, the total response field is (48), shown at the bottom of this page. Similarly, when the second waveguide is excited, we get (49)–(51), shown at the bottom of this page. In order to compute (47)–(51), we make use of (36), where the

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absolute values are replaced by their magnitudes. As is apparent from the preceding analysis, the superscript in parentheses again corresponds to the type of the excitation of the discontinuity ( mode from the first or second waveguide), and the additional subscript to the area it is referred to (field into first or second waveguide).

(47)

(48)

(49)

(50)

(51)

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VI. COMPUTATION OF SCATTERING MATRIX Prior to proceeding to the computation of the -parameters, it should be made clear that two dominant modes with wavenumand are involved, one in each waveguide. bers and far from Consider also two reference planes the cylinder axis such that (52) It is assumed that the distance of each of the planes from the discontinuity is an integral multiple of each guide’s wavelength so that

Fig. 4. Typical equivalent circuit for the circular post. Due to the discontinuity, the network is not symmetric. It is, however, reciprocal.

(53)

Due to the scaling imposed by (54), the power conservation law can be verified [1, p. 108] as follows:

As stated from the beginning, the only dominant mode in both and, therefore, only the first term waveguides is the of the sums (47)–(51) will not vanish for points close to the remote reference planes. Due to (53), the dependence on is suppressed. The same should be done for the -dependence of the field functions. After straightforward computations utilizing the Poynting vector [20, p. 21], it is inferred that a transfers traveling mode such as this in (7) with towards the electromagnetic power equal to -direction. Thus, the unequal mode functions are replaced by their transmitting power

(59) It is of interest to find the impedance matrix terms of the scattering matrix [19, p. 192]

expressed in (60)

is the 2 2 unitary matrix. The elements of are where normalized with respect to the characteristic impedance of the propagating mode. Due to (54), the common impedance is used as follows:

(54) (61) The square root is used because mode functions express field, not power. After dropping the dependence on the variable, one can rewrite the total response quantities with mutual notation , shown in (55) at the bottom of this page, where the direcis defined as tion indicator (56)

To develop the equivalent circuit (Fig. 4) of a two-port discon, the well-known relations tinuity with impedance matrix are used between the actual impedances and the elements of [28]. It should be stressed that the system is reciprocal, namely, , and thus, the equivalent T network naturally arises. The circuit is comprised of two unequal series impedances and a shunt one, i.e., , as follows: (62)

The same procedure for the initial excitation fields gives

(63) (64) (57)

The scattering parameters of the two-port device (matrix can be defined as follows:

)

If the discontinuity is lossless, then the aforementioned impedances should be purely imaginary. This can be an extra check based on power conservation. VII. NUMERICAL RESULTS

(58)

The semianalytical method developed above has been used to produce numerical results. The normalized series and shunt

(55)

VALAGIANNOPOULOS AND UZUNOGLU: RIGOROUS ANALYSIS OF METALLIC CIRCULAR POST IN RECTANGULAR WAVEGUIDE

components of the equivalent circuit are calculated. Extensive numerical checks concerning the truncation limits of the sums defining the investigated quantities have been carried out. Checking the accuracy of the surface current in (4) has is shown that a number of azimuthal harmonics is not related with the cylinder’s sufficient. The value of diameter because single-mode waveguides are considered. Another convergence issue is the required number of modes in (12) and (13) expressing the response of the step discontinuity from both sides for any type of excitation used. Due to the moderate heights of the waveguides, a number of 20–25 modes is adequate. When it comes to the -parameter used because for the integrations of (45), it is proportional to the latter determines the rapidity of the oscillations of the is kept low, harmonic function. In the examined case where a sampling of 25–30 points for each integration variable is adequate. Contrary to the aforementioned ones, the integration of (35) is of an infinite interval. Therefore, the upper truncation (instead of infinite) must be selected in the first limit place. Given the fact that this formula is used exclusively to is small, the variation of the mode matching, where equal to 2 or 3 integrand is not abrupt and, therefore, a is sufficient. For the trapezoidal integration in (35), one can points per unity of take into account approximately . In the cases under consideration, the parameters referred above are chosen so that all the corresponding quantities are convergent. Additionally, for each point, the Green’s source is located, check of boundary conditions across the discontinuity is performed, and a similar test is also made for the vanishing electric field on the PEC cylinder. Finally, in each case, the satisfaction of power conservation laws of (59) is assured. The impedance parameters of (62)–(64) are ensured to be close to imaginary numbers, with excellent results. In the following cases, suppose a zero distance between the cylinder axis and discontinuity plane. Such a parameter makes little difference in electrically small problems such as the investigated ones. The heights of the waveguides do not take values very close to the limiting ones (half and one wavelength in vacuum ) in order to acquire more reliable results. The parameter is fixed throughout the numerical applications and . In the cases with absent discontinuity equals , the values have been checked against those of [1] and [2] by taking into account the necessary proportional constants. In other cases, the results are in agreement with the solutions obtained by the finite-elements method used in Ansoft’s commercial field High Frequency Structure Simulator (HFSS). In Fig. 5, the percentage differences between the -parameters derived via the proposed analytic procedure and those computed from the simulation program are presented as a function of the radius of the scatterer. The values for the waveguides’ heights are and . For all the -parameters, the percent error is relatively low (below 3%). In Figs. 6 and 7, the height of the second waveguide is again equal to , which is almost the same as , which corresponds to a centered scatterer into the second waveguide. Three different heights for the first one are considered, which are: , (solid line); 2) (dashed 1) (dotted line), while the indepenline); and 3)

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Fig. 5. Percentage of relative differences of the magnitudes of S -parameters computed through analytic formulas and simulation (Ansoft’s HFSS).

Fig. 6. Normalized series impedances. The two upper curves correspond to the first side’s quantities. The two lower curves correspond to the second side’s quantities. The solid curve is common for both sides, as in that case, the device becomes symmetrical. The height of the second waveguide is fixed at h =L = 3:01=3.

. The solid dent variable is the normalized cylinder’s radius lines describe the behavior of a continuous waveguide with a centered post. In Fig. 6, the imaginary parts of the normalized are series impedances presented. As the cylinder’s radius gets larger, there is a dispersion in the first side’s curves and a coincidence in the second side’s curves. One could also observe the change in the behavior , which for small scatterers, corresponds to inductor and of after a certain radius, inverses its sign indicating capacitive bealways corresponds to a cahavior. On the other hand, pacitor. In Fig. 7, the imaginary part of the normalized shunt appeared. The quantities are vanimpedance ishing for large scatterers. Due to the centralized position of the post, the dispersion is not significant. In Figs. 8 and 9, with constant height of the first waveguide , three different heights of the second one are

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Fig. 7. Normalized shunt impedances. The height of the second waveguide is fixed at h =L = 3:01=3.

Fig. 9. Normalized shunt impedances. The height of the first waveguide is fixed at h =L = 5:99=3.

Fig. 8. Normalized series impedances. The two upper curves correspond to the first side’s quantities. The two lower curves correspond to the second side’s quantities. The solid curve is common for both sides, as in that case, the device becomes symmetrical. The height of the first waveguide is fixed at h =L = 5:99=3.

Fig. 10. Surface current magnitude in polar plot. The graphs occupying the left portion correspond to the excitation from the same side. The same for the right side’s graphs. The two solid lines are symmetrical with respect to both axes. The height of the second waveguide is fixed at h =L = 3:01=3.

considered, i.e., (dotted line), (dashed line), and (solid line). The indepen. Solid lines correspond dent variable is again the parameter to a symmetric waveguide with an off-centered post. In addition, the dotted lines in the two cases concern exactly the same problem. In Fig. 8, the same quantities as in Fig. 6 are presented, but the conclusions are inversed. The dispersion is observed in the second side’s curves and the coincidence is seen in the first side’s curves. The capacitive behavior is also stronger in the than in the case of . case of Such a property is not proportionate to the discontinuity height, as shown in Fig. 6. In Fig. 9, the same quantities of Fig. 7 are demonstrated. One notices that the less discontinuous the system is constructed, the larger the shunt inductive impedances

become. In Fig. 10, the polar plots of the magnitudes of surface currents are presented as a function of the azimuthal angle . The rest of the parameters are the same as with Figs. 6 and 7. The currents when the second waveguide is excited are not affected by the size of discontinuity. The currents when the opposite port is excited are shifting their maxima toward the upper direction the larger the discontinuity gets. The symmetry of solid lines with respect to the vertical axis reflects the symmetry of a continuous waveguide as a two-port device. In Fig. 11, the same quantities as in Fig. 10 are presented for the cases described by Figs. 8 and 9. There is not a symmetry with respect to the horizontal axis for solid curves, as shown in Fig. 10, because the post in this case is off centered. One should notice that the current is suppressed when the height of the first

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REFERENCES

Fig. 11. Surface current magnitude in polar plot. The graphs occupying the left portion correspond to the excitation from the same side. The same for the right side’s graphs. The two solid lines are symmetrical with respect only to vertical axis. The height of the first waveguide is fixed at h =L = 5:99=3.

waveguide gets smaller, especially when the excitation concerns the same side. VIII. CONCLUSION This paper has presented a rigorous solution to the scattering of guided waves by a cylindrical post placed inside a stepped discontinuous rectangular waveguide. By expanding the surface current distribution on the cylinder as a sum of harmonic azimuthal functions, the integral equation of the scattering field has been reduced to a matrix equation. The Green’s function for a filamentary current in a discontinuous rectangular waveguide has been converted to a sum of three terms, i.e.: 1) a singular one; 2) a rapidly converging integral; and 3) an infinite sum emanating from the mode-matching technique. The separation of the free-space singular term and the derived unified expressions regardless of the position of the source contribute to the reliability of the proposed semianalytical method and the numerical computations. The current distribution on the post surface is derived by the method of moments. With use of this distribution and by utilizing an alternative representation of the Green’s function, the scattering matrix of the two-port network is then obtained. Extensive numerical checks have been carried out, including comparison with previously published results and a commercial simulator, convergence tests, and validity of the boundary conditions on the post and across waveguide discontinuity. The developed approach provides an equivalent-circuit model of the two-port network in terms of scattering ( ) parameters. Due to the high accuracy of the method, the obtained results could find useful application in designing microwave filters with high-selectivity transfer functions. The suggested technique could be easily extended to the case of multiple posts and discontinuities.

[1] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1951. [2] Y. Leviatan, P. G. Li, A. T. Adams, and J. Perini, “Single-post inductive obstacle in rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 10, pp. 806–812, Oct. 1983. [3] P. G. Li, A. T. Adams, Y. Leviatan, and J. Perini, “Multiple-post inductive obstacles in rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 4, pp. 365–373, Apr. 1984. [4] Y. Leviatan, D. H. Shau, and A. T. Adams, “Numerical study of the current distribution on a post in a rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 10, pp. 1411–1415, Oct. 1984. [5] R. B. Green, “A grating formulation for some problems involving cylindrical discontinuities in rectangular waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 10, pp. 760–763, Oct. 1969. [6] K. M. Rahman and C. Nguyen, “On the analysis of single- and multiple-step discontinuities for a shielded three-layer coplanar waveguide,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 9, pp. 1484–1488, Sep. 1993. [7] C. J. Railton and T. Rozzi, “The rigorous analysis of cascaded step discontinuities in microstrip,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 7, pp. 1177–1185, Jul. 1988. [8] R. N. Simons and G. E. Ponchak, “Modeling of some coplanar waveguide discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1796–1803, Dec. 1988. [9] J. A. Bradshaw, “Scattering from a medal post and gap,” IEEE Trans. Microw. Theory Tech., vol. MTT-21, no. 5, pp. 313–322, May 1973. [10] J. M. Reiter and F. Arndt, “Rigorous analysis of arbitrarily shaped H - and E -plane discontinuities in rectangular waveguides by a fullwave boundary contour mode-matching method,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 4, pp. 796–801, Apr. 1995. [11] B. Z. Steinberg and Y. Leviatan, “Multiresolution study of two-dimensional scattering by metallic cylinders,” IEEE Trans. Antennas Propag., vol. 44, no. 4, pp. 572–579, Apr. 1996. [12] S. F. Mahmoud and J. C. Real, “Scattering of surface waves at a dielectric discontinuity on a planar waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 2, pp. 193–198, Feb. 1975. [13] M. E. Veysoglu, H. A. Yueh, R. T. Shin, and J. A. Kong, “Polarimetric passive remote sensing of periodic surfaces,” J. Electromagn. Waves Applicat., vol. 5, pp. 267–280, 1991. [14] A. W. Mathis and A. F. Peterson, “A comparison of acceleration procedures for the two-dimensional periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 44, no. 4, pp. 567–571, Apr. 1996. [15] H. Rogier and D. De Zutter, “A fast converging series expansion for the 2-D periodic Green’s function based on perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1199–1206, Apr. 2004. [16] R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 8, pp. 734–736, Aug. 1985. [17] R. E. Jorgenson and R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 38, no. 5, pp. 633–642, May 1990. [18] A. Q. Howard, “The electromagnetic fields of a subterranean cylindrical inhomogeneity excited by a line source,” Geophysics, vol. 37, pp. 975–984, Dec. 1972. [19] R. E. Collin, Field Theory of Guided Waves. New York: McGrawHill, 1960. [20] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [21] E. W. Weisstein, Complete Orthogonal System. Champaign, IL: MathWorld—A Wolfram Web Resource, YEAR. [Online]. Available: http://mathworld.wolfram.com/CompleteOrthogonalSystem.html [22] D. S. Jones, Theory of Electromagnetism. Oxford, U.K.: Pergamon, 1964, pp. 269–271. [23] G. F. Roach, Green’s Functions. Cambridge, U.K.: Cambridge Univ. Press, 1982, p. 227. [24] A. D. Polianin, Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton, FL: CRC, 2002, p. 494. [25] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Washington, DC: Nat. Bureau Standards, 1971, p. 364. [26] R. E. Bellman and R. S. Roth, The Laplace Transform. Singapore: World Sci., 1984. [27] P. Morse and H. Fesbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. [28] J. Cathey, Schaum’s Outline of Electronic Devices and Circuits. New York: McGraw-Hill, 2002, pp. 8–12. [29] A. Sommerfeld, Partielle Differentialgleichungen der Physik. Frankfurt, Germany: Verlag Harri Deutsch, 1992, pp. 122–129.

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Constantine A. Valagiannopoulos was born in Athens, Greece, in 1982. He received the Diploma degree in electrical and computer engineering from the National Technical University of Athens, Athens, Greece, in 2004, and is currently working toward the Ph.D. degree in electrical and computer engineering at the National Technical University of Athens. He is currently with the Microwave and Fiber Optics Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens. His research interests include electromagnetics, microwaves, propagation, and applied mathematics. Mr. Valagiannopoulos was the recipient of the MMET*04 International Conference Award and a “Eugenides” Foundation Scholarship.

Nikolaos K. Uzunoglu (M’82–SM’97–F’06) was born in Constantinople, Turkey, in 1951. He received the B.Sc. degree in electronics from the Technical University of Istanbul, Istanbul, Turkey, in 1973, the M.Sc. and Ph.D. degrees from the University of Essex, Essex, U.K., in 1974 and 1976, respectively, and the D.Sc. degree from the National Technical University of Athens (NTUA), Athens, Greece, in 1981. From 1977 to 1984, he was a Research Scientist with the Office of Research and Technology, Hellenic Navy. In 1984, he became an Associate Professor with the Department of Electrical Engineering, NTUA, and in 1987, he became a Professor. He has authored or coauthored over 120 papers in refereed international journals, and has authored three books in Greek on microwaves, fiber-optics telecommunications, and radar systems. His research interests include electromagnetic scattering, propagation of electromagnetic waves, fiber-optic telecommunications, and high-speed circuits operating at gigabit/second rates. Since 1988, he has been the national representative of Greece to the Technical Telecommunication Committee, European Cooperation in the Field of Scientific and Technical Research (COST), and has actively participating in several COST telecommunications projects. He has also been the Project Manager of several RACE, ESPRIT, ACTS, and National Research and Development Projects in the fields of telecommunication and biomedical engineering applications. Dr. Uzunoglu was elected a Foreign Member of the National Academy of Sciences of Armenia in 1998. He was the recipient of the 1981 International G. Marconi Award in Telecommunications. In 1994, he was elected as an Honorary Professor of the State Engineering University of Armenia. He was also the recipient of an honorary Ph.D. diploma from the Universities of Bucharest, Cluj-Napoca, and Orade.

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Reactance of Posts in Circular Waveguide Qian C. Zhu, Student Member, IEEE, Allan G. Williamson, Senior Member, IEEE, and Michael J. Neve, Member, IEEE

Abstract—Experimental and numerical simulation results are presented for the reactance of centered variable-height single and double posts in a circular waveguide. The reactance is a function of the waveguide and post dimensions and frequency, and the results presented are useful in the design of a variety of circular waveguide devices involving these elements. Index Terms—Circular waveguide, filters, matching networks, tuning posts.

I. INTRODUCTION

Fig. 1. (a) Variable-height symmetrical double posts and (b) variable-height single post in circular waveguide.

BSTACLES such as irises and posts are frequently used as elements in waveguide matching networks and filters. These obstacles form discontinuities in the waveguide and their effects can be modeled as reactances. In practice, posts are often realized as screws so that the insertion depth in the waveguide can be easily adjusted. For this reason, they are sometimes preferred to other discontinuities that are less easily adjusted. Irises and posts in rectangular waveguides have been extensively investigated (e.g., [1]–[7]) and are used in many practical microwave systems in use today. By comparison, applications of circular waveguides are far less common. Nevertheless, circular waveguides are better suited to some applications, and, for example, have advantages in industrial microwave applications where the easier manufacturability afforded by the circular geometry reduces the costs of custom manufacture, and in food industry applications, permits easier internal cleaning in operation—a major hygiene consideration. A problem for the design engineer of such applications is the dearth of information about coupling to circular waveguides, and the characteristics of posts, and other obstacles, in circular waveguides. This is one of the motivations for the research reported here, which has yielded useful information on the behavior of posts in a circular waveguide. In comparison to rectangular waveguides, irises in a circular waveguide have received only limited attention to date [2], [8]–[12], and only two authors have considered posts [13], [14]. In [13], the mutual impedance between two arbitrarily located posts in a circular waveguide was derived, while in [14], a symmetrical double post in a circular waveguide was analyzed using a multistrip moment method where the multistrips were used to approximate the current on the post surfaces. To verify the susceptance results obtained, experimental data for one

particular case of a symmetrical double post in a circular waveguide was reported [14]. Other than this, no other published data has been found for posts in a circular waveguide. The theoretical analysis of obstacles in a circular waveguide is more complicated than that for similar situations in a rectangular waveguide—and even that is not trivial. However, given the aforementioned advantages of circular waveguides, a quantitative understanding of the reactance of posts, and some representative data would be advantageous in the design for some circular waveguide applications. As a first step to gain insight to the characteristics of such structures, experimental data together with numerical simulation results has been gathered for a range of centered variableheight symmetrical double-post [see Fig. 1(a)] and single-post [see Fig. 1(b)] geometries in circular waveguides. This paper can be viewed as a companion to [7] in which reactance results for posts in rectangular waveguides were presented. Measurements were made in a metal-walled circular wavemode was the guide at frequencies where the dominant only propagating mode. The measured results were then compared with numerical simulation results obtained using the software package CST Microwave Studio [15].

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Manuscript received September 22, 2006; revised March 1, 2007. The authors are with the Department of Electrical and Computer Engineering, The University of Auckland, Auckland 1142, New Zealand (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.901605

II. EXPERIMENTAL SETUP The experiments were conducted using a circular waveguide test jig comprising a central test section containing the post(s) and two coaxial line-to-circular waveguide transitions, as shown in Fig. 2. The circular waveguide diameter was 82.2 mm (giving cutoff frequency of 2.139 GHz), and measurements a were made in the frequency range from 2.30 to 2.76 GHz. An Agilent PNA-series network analyzer was used to obtain the -parameters. At the start of each measurement, a two-port thru-reflect-line (TRL) calibration was performed at the reference plane – between the test section and one of the coaxial line-to-waveguide transitions. Subsequently, a port extension was carried out to shift the reference plane by a distance to the plane –

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TABLE I REACTANCE FOR DOUBLE POSTS

Fig. 2. Cross-sectional view of the circular waveguide test jig.

Fig. 3. T-equivalent circuit.

located at the center of the variable length post(s). (This apmode is inproach assumes that the attenuation of the significant over the distance and that the shifted -parameters accurately model the post at plane – .) Ideally, and . However, in practice, due to imperfections in and differ by a small amount (as do and the system, ). Consequently, the means of and and that of and were used as the reflection coefficient and transmission coefficient , respectively. Similar to the rectangular waveguide situation [7], the reactive effect of the post in circular waveguide can be represented by a T-equivalent circuit, as shown in Fig. 3. The normalized and may be calculated from the reflection coreactances and transmission coefficient using [4] efficient (1) (2) III. RESULTS AND DISCUSSION Measurements were made of a symmetrical double-post configuration [see Fig. 1(a)] with post diameters of 4.0 and 10.0 and ) and a single-post configuration mm ( [see Fig. 1(b)] with post diameters of 3.1, 5.0, and 10.0 mm and ), respectively. An experi( mental data set for the normalized reactances and at three and , frequencies ( mode in where is the cutoff frequency of the dominant the circular waveguide) is given in Tables I and II for the doubleand single-post cases, respectively. Interpolation between the presented values should permit designers to estimate results for other cases. For the symmetrical double posts (in which the two posts are vertically aligned to each other), results were obtained for var. Fig. 4 shows the measured ious post insertion depths results from experiment, confirmed by CST’s Microwave Studio plotted against for simulation, for the shunt reactance

double posts with . Also shown in Fig. 4 for comparison is the reactance curve for a single post in a rectangular waveguide that has a broad wall width the same as the diameter, and the height the same as the radius, of the circular waveguide with is tested. Note that the nature of the variation of similar in both cases, which is capacitive for a large proportion of the insertion range. In comparison to the rectangular waveguide case, the available inductive range is slightly greater for the double posts in the circular waveguide. The other situation investigated was the centered variableheight single post. The experimental and Microwave Studio reare shown in Fig. 5. (The reactance curve for a post sults for in a square waveguide with width and height equal to diameter of the circular waveguide is also presented. It can be seen that there is a degree of similarity between the results.) A key point with respect to post insertion is to note that the variation of for a single post is quite different to that of double posts in the circular waveguide. The reactance of a single post can be considered in three re), the post is gions. For small insertion depth (i.e., is quite sensitive to insertion depth. When the capacitive and ), the post becomes insertion is increased (i.e.,

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TABLE II REACTANCE FOR SINGLE POST

Fig. 4. Reactance x plotted against post insertion (p =D) for centered symmetrical double posts. 2a=D = 0:049 and f =f = 1:26. : experimental results for double posts in circular waveguide. —: Microwave Studio results for double posts in circular waveguide. - - -: theoretical results for a single post in a rectangular waveguide (waveguide width = D; height = D=2) [5].

Fig. 5. Reactance x versus post insertion (p=D ) for a centered single post. 2a=D = 0:036 and f =f = 1:26. : experimental results for a single post in circular waveguide. —: Microwave Studio results for a single post in circular waveguide. - - -: theoretical results for a single post in a square waveguide (width and height = D ) [5].

changes from being highly inductive to highly capacitive as is increased, with the post being transparent at some particular value of . Note also the two resonances ( is zero, and the post is totally reflective), which is different to the single post in a rectangular waveguide and double posts in a circular waveguide, as discussed above. To the authors’ knowledge, these distinctive features have not previously been reported in the literature. The parameter , which is related to the post thickness [16], varies almost linearly with post insertion for the range of post diameters tested. The reactance is small since the posts tested here are electrically thin. The data set in Tables I and II could be interpolated to other intermediate post diameters, lengths, and frequencies of operation. It would be advantageous to have an interpolation formula and/or an analytical approach to the problem and this is currently being investigated. It should be noted, however, that initial attempts suggest that this is more complicated than the corresponding rectangular waveguide situation, which in itself, is not trivial. IV. CONCLUSION inductive where varies gradually with insertion and for some becomes insensitive to variation in . After range of this (i.e., ), there is a region where increases and

Experimental data and numerical simulation results for the reactance of centered variable-height symmetrical double posts and single post in a circular waveguide as a function of waveguide size, post diameter and length, and frequency has been

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presented. For the symmetrical double posts, the form of variwith respect to post insertion ation for the shunt reactance is similar to that of a single post in a rectangular waveguide, whereas the single-post case is similar to a single post in a square waveguide. ACKNOWLEDGMENT The authors would like to thank J. Roelvink, The University of Auckland, Auckland, New Zealand, for providing the rectangular waveguide results shown in Figs. 4 and 5, and to acknowledge useful discussions with R. Keam and J. Holdem, both with Keam Holdem Associates, Auckland, New Zealand, regarding the potential application of circular waveguides in industrial applications, and in particular, applications in the food industry. REFERENCES [1] L. Lewin, Advanced Theory of Waveguides. London, U.K.: Iliffe, 1951. [2] N. Marcuvitz, Waveguide Handbook, ser. MIT Rad. Lab.. New York: McGraw-Hill, 1951, vol. 10, pp. 218–229, 271–273, 243–246. [3] J. Schwinger and D. Saxon, Discontinuities in Waveguides. New York: Gordon and Breach, 1968. [4] Y. Leviatan, P. G. Li, A. T. Adams, and J. Perini, “Single-post inductive obstacle in rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 10, pp. 806–812, Oct. 1983. [5] A. G. Williamson, “Variable-length cylindrical post in a rectangular waveguide,” Proc. Inst. Elect. Eng., vol. 133, pt. H., pp. 1–9, Feb. 1986. [6] Y. Huang, N. Yang, S. Lin, and R. F. Harrington, “Analysis of a post with arbitrary cross section and height in a rectangular waveguide,” Proc. Inst. Elect. Eng., vol. 138, pt. H, pp. 475–480, Oct. 1991. [7] J. Roelvink and A. G. Williamson, “Reactance of hollow, solid, and hemispherical-cap cylindrical posts in rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 10, pp. 3156–3160, Oct. 2005. [8] L. Carin, K. J. Webb, and S. Weinreb, “Matched windows in circular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 9, pp. 1359–1362, Sep. 1988. [9] R. W. Scharstein and A. T. Adams, “Galerkin solution for the thin circular iris in a TE -mode circular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 1, pp. 106–113, Jan. 1988. [10] Z. Shen and R. H. MacPhie, “Scattering by a thick off-centered circular iris in circular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 11, pp. 2639–2642, Nov. 1995. [11] S. P. Yeo and S. G. Teo, “Thick eccentric circular iris in circular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 8, pp. 1177–1180, Aug. 1998. [12] G. Bertin, L. Piovano, L. Accatino, and M. Mongiardo, “Full-wave design and optimization of circular waveguide polarizers with elliptical irises,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1077–1083, Apr. 2002. [13] B.-S. Wang, “Mutual impedance between probes in a circular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 6, pp. 1006–1012, Jun. 1989. [14] X. H. Zhu, D. Z. Chen, and S. J. Wang, “A multistrip moment method technique and its application to the post problem in a circular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 10, pp. 1762–1766, Oct. 1991.

[15] Microwave Studio. CST , Darmstadt, Germany. [Online]. Available: http://www.cst.com [16] A. G. Williamson, “Analysis and modelling of a coaxial-line/rectangular waveguide junction,” Proc. Inst. Elect. Eng., vol. 129, pt. H, pp. 262–270, Oct. 1982. Qian C. Zhu (S’06) was born in Guangxi, China, on October 30, 1982. She received the B.E. (Hons.) degree in electrical and electronic engineering from The University of Auckland, Auckland, New Zealand, in 2004, and is currently working toward the Ph.D. degree at The University of Auckland. Her research interests are in the area of excitation and reactive obstacles in circular waveguide.

Allan G. Williamson (M’78–SM’83) received the B.E. and Ph.D. degrees in electrical engineering from The University of Auckland, Auckland, New Zealand. He then worked in industry. In 1975, he joined The University of Auckland, as a Lecturer, became a Professor of telecommunications in 1988, and is currently Department Head. His early research was concerned with microwave passive devices, while more recently, he has been involved with mobile radio research and issues related to cellular radio, personal communications, and indoor wireless systems. Prof. Williamson is a Fellow of the Institution of Engineering and Technology, U.K., and a Fellow of the Institution of Professional Engineers New Zealand. He has served as chairman of both the IEEE New Zealand North Section and the IEEE New Zealand Council.

Michael J. Neve (S’87–M’91) was born in Auckland, New Zealand, on October 29, 1966. He received the B.E. (Hons.) and the Ph.D. degrees from The University of Auckland, Auckland, New Zealand, in 1988 and 1993, respectively. From May 1993 to May 1994, he was Leverhulme Visiting Fellow with the University of Birmingham, Edgbaston, Birmingham, U.K., during which time he was involved with radio-wave propagation research using scaled environmental models. From May 1994 to May 1996, he was a New Zealand Science and Technology Post-Doctoral Fellow with the Department of Electrical and Electronic Engineering, The University of Auckland. From May 1996 to December 2000, he was a Part-Time Lecturer/Senior Research Engineer with the Department of Electrical and Electronic Engineering, The University of Auckland. In 2004, he was a Visiting Scientist with the Commonwealth Scientific and Industrial Research Organisation (CSIRO) Information and Communication Technologies (ICT) Centre, Sydney, Australia. He is currently a Senior Lecturer with the Department of Electrical and Computer Engineering, The University of Auckland. His current research interests include radio-wave propagation modeling in cellular/microcellular/indoor environments, the interaction of electromagnetic fields with man-made structures, cellular system performance optimization, and antennas. Dr. Neve is the Chair of the IEEE New Zealand North Section. He was a corecipient of a 1992/93 Institution of Electrical Engineers (IEE) Electronics Letters Premium for two publications resulting from his doctoral research.

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Fourier Decomposition Analysis of Anisotropic Inhomogeneous Dielectric Waveguide Structures Ramin Pashaie, Student Member, IEEE

Abstract—In this paper, we extend the Fourier decomposition method to compute both propagation constants and the corresponding electromagnetic field distributions of guided waves in millimeter-wave and integrated optical structures. Our approach is based on field Fourier expansions of a pair of wave equations, which have been derived to handle inhomogeneous mediums with diagonalized permittivity and permeability tensors. The tensors are represented either by a grid of homogeneous rectangles or by distribution functions defined over rectangular domains. Using the Fourier expansion, partial differential equations are converted to a matrix eigenvalue problem that correctly models this class of dielectric structures. Finally, numerical results are presented for various channel waveguides and are compared with those of other literature to validate the formulation. Index Terms—Anisotropic, dielectric waveguide, Fourier decomposition method, inhomogenous.

I. INTRODUCTION IELECTRIC waveguides are widely used for the transmission of electromagnetic energy and in the structure of optical devices such as directional couplers and modulators at optical frequencies. As the design criteria for these devices become tighter, results of approximate methods often do not have the desired accuracy. Examples of approximate methods are the effective index method [1], [2], the semivectorial finite-difference method [3], [4], and the variational methods [5]. Rigorous techniques include the vector finite-element method (FEM) [6], [7], the domain integral-equation method [8]–[11], and the method of lines [12], [13]. The FEM is an exact and general technique that can be utilized in the analysis of inhomogeneous anisotropic dielectric waveguides. Despite flexibilities of such a numerical method, the incorporation of unbounded regions outside the waveguide and the occurrence of spurious solutions (as a result of inexplicit satisfaction of the divergence equation) are two major problems one has to face using this method. An important advantage of the domain integral-equation method is that it avoids spurious solutions; nevertheless, this method is time consuming. The method of lines is a rigorous seminumerical technique. It is well known for its accuracy, speed of computation, and minimal memory requirements; however, it is relatively complicated.

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Manuscript received February 24, 2007; revised April 15, 2007. The author is with the Electrical and System Engineering Department, University of Pennsylvania, Philadelphia, PA 19104 USA (e-mail: [email protected] upenn.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.902616

In 1989, Henry and Verbeek proposed an interesting method for modal analysis of arbitrary shaped inhomogeneous dielectric waveguides with small changes in the refractive index function (weakly guiding) and negligible induced birefringence for modes that are far from their cutoff [14]. This analysis, which was further investigated by Marcuse [15], allows evaluation of propagation constants and field distributions within any desired precision. The method is based on expansion of the unknown field in a complete set of orthogonal functions within limited boundaries. which converts a linear partial differential equation into a matrix eigenvalue problem. By solving the eigenvalue problem, all the guided modes of the waveguide are clarified. Nevertheless, only those guided modes for which the field goes to zero before the boundary is reached are computed accurately. More recently, Hewlett and Ladoucer improved this method by introducing a preliminary mapping of the infinite transverse plane onto a unit square by applying a suitable analytic transformation function [16]. This modification allows computation of even near-to-cutoff modes by using relatively few waves. The main drawback is that both computation time and memory requirements can become exceedingly large before adequate accuracy is obtained; however, this problem can be solved by using suitable optimization parameters. Henry and Verbeek’s method can be used for modal analysis of inhomogeneous isotropic dielectric structures such as ion-exchange optical waveguides in the substrate of glass [17]. Nevertheless, extending the utility of this technique to applications that make use of anisotropic mediums (e.g., LiNbO ) requires reformulation of the method. In this paper, Henry and Verbeek’s method is extended and applied to the vector form of the wave equation. With minimal increase in computation time and memory requirements, the method is improved for modal analysis of anisotropic inhomogeneous dielectric structures. Recently, independent of this study, other researchers have studied full vector analysis of inhomogeneous dielectric waveguides [18]–[20]. However, to the author’s knowledge, the formulation in this paper covers the most general case, including anisotropic mediums. Basic equations are derived in Section II followed by the description of the eigenvalue problem. In Section III, numerical results are presented and discussed. Finally, concluding remarks are made in Section IV.

II. FORMULATION Here, wave and field equations that must be solved are derived directly from Maxwell’s equations. The Fourier expansion formulation is then applied to the developed equations so that an appropriate eigenvalue problem is obtained.

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where the wavenumber is defined as Gauss equations (5) and (6), we find that

. From the

(9) (10)

Fig. 1. Dielectric waveguiding structure with arbitrary shaped anisotropic inhomogeneous dielectric subregions ; ; . . . ; and embedded in

dielectric media.

A. Basic Equations Consider the case where an anisotropic inhomogeneous is embedded waveguiding region with arbitrary shape within the substrate , and this embedding consists of other . The cross section of dielectric subregions such a dielectric structure is illustrated in Fig. 1. The position is specified using a right-handed Cartesian reference frame. The -axis is chosen such that the material properties of the waveguide configuration is invariant in the -direction. Each sub-domain is assumed to be anisotropic and inhomogeneous, with diagonal permittivity and permeability tensors as follows:

(1)

(2)

The two vector wave equations (7) and (8) can be further expanded into six scalar wave equations. In an inhomogeneous anisotropic medium, each scalar wave equation is coupled with at least one other, and solving these equations is prohibitively difficult. Nevertheless, uncoupled scalar wave equations can and be developed by considering only the modes and using (9) and (10) accordingly. The superposition of the and the modes will then completely characterize propagating modes (including hybrid modes) in the structure

(11)

(12) and in all the regions and boundDue to the continuity of aries, in (11) and (12), one can change the order of differentiaand to and tion in terms , respectively. Consequently, the - and -oriented electric and magnetic components of a propagating mode in this - and -mode comdielectric structure are related to the ponents via the following set of equations: (13)

Both dielectric and ohmic losses are included since and are taken to be complex quantities defined as and , where are components of the conductivity tensor. In a source-free anisotropic inhomogeneous medium, Maxwell’s equations are written as follows: (3) (4) (5) (6) Maxwell’s curl equations (3) and (4) are coupled first-order differential equations. They become uncoupled by applying the (4) following sequence of mathematical operations, (3) into (4). Second-order vector wave into (3) and equations are then obtained for electric and magnetic fields (7) (8)

(14) (15) (16) (17) (18) Finding an analytical solution for (11) and (12) is complicated and numerical (or seminumerical) methods are often preferred. In the Fourier decomposition method, all dielectric sub-domains and are enclosed in a virtual box whose dimensions are and along - and -axes, respectively. and are large enough to ensure that the electromagnetic fields of the guided modes are zero on these artificial boundaries.

PASHAIE: FOURIER DECOMPOSITION ANALYSIS OF ANISOTROPIC INHOMOGENEOUS DIELECTRIC WAVEGUIDE STRUCTURES

The electric field of the mode and the magnetic field of mode are then expanded in terms of a complete set of the orthogonal basis functions (19)

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(27) (28) In these equations, and we expand the electric field using and the

are nonnegative integers. Next, and the magnetic field functions, respectively,

(20) In practice, infinite summations are approximated by considering finite terms in the expansions. The main restriction of the Fourier method appears as the modal cutoff is approached. Close to cutoff, more terms need to be included in the field expansion, and the bounding box should be chosen much larger to make the implicit assumption of zero field on the boundary more reasonable. An explicit solution to this difficulty is proposed in [16] in which the whole – -plane is mapped onto a unit square in the – -plane via analytical transformation functions such as (21)

(29)

(30) where the indices

and

are

(22) and are arbitrary scaling parameters. With this where mapping, boundary conditions for all bounded modes are satisand characfied automatically. One can develop the teristic wave equations in transformed coordinates by applying the mapping equations (21) and (22) to (11) and (12) as follows:

(23)

(31) (32) In the Fourier decomposition method, boundary conditions at the interfaces between the dielectrics are not applied directly. The field distribution functions are assumed to be piecewise smooth, continuous, and square integrable over the entire space. However, the jump discontinuity of the fields at the dielectric interfaces causes the appearances of the Gibbs phenomena. As a result, the solution of the vector wave equation is in good agreement with the exact solution, except on the dielectric interfaces. Nevertheless, the energy of this systematic error depresses with an increasing number of terms in the series expansion [15]. Substituting (29) and (30) into characteristic equations (23) and , respectively, and and (24), multiplying by integrating over the unit square, the matrix eigenvalue equations will be developed for both TE and TM modes. For the electric field,

(33)

(24) Solving these two Sturm–Liouville wave equations is the subject of Section II-B. B. Eigenvalue Problem Consider the following orthogonal functions

(34) (35) (36) (37)

: (38) (25) (26)

(39)

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while for the magnetic field,

(40) (41) (42) (43) (44) (45) (46) In these equations, is a normalization parameter that represents dimension of the dielectric waveguide. The variables and are the waveguide’s degree of guidance and and are the modal cladding parameters [16]. The expressions for the and are found in Appendix I. double integrals Equations (33)–(39) and (40)–(46) express the eigenvalue problems, which can be readily solved using standard nuand and the merical routines. The eigenvalues corresponding eigenvectors and of the system matrices and yield the unknown modal propagation constants and the associated Fourier expansion coefficients, respectively. The modal fields space via (19) can then be readily reconstructed in the space with the transforand (20) and mapped back to the mation functions (21) and (22). in size. ConThese system matrices are sequently, they possess eigenvalues, including both bounded and continuum modes of the waveguide. Eigenvalues and the maximum between the minimum of in the mode, and eigenvalues of between the minimum of and the maximum in the mode, are related to the of bounded modes. At modal cutoff, the propagation constant is equal to for the mode, and for the mode. Thus, at the cutoff, and (33) and (40) are simplified to the following forms: (47)

(48) In particular, the eigenvalues and and and the eigenvectors of the system matrices and yield the cutoff values of and and the associated Fourier expansion coefficients, respectively. As before, it is then straightforward to reconstruct the modal fields at the cutoff using (19) and (20) and the transformation functions (21) and (22).

and are the kernels of the Fourier Double integrals decomposition method. Speed and accuracy of results obtained are critically affected by the way these integrals are computed. Generally, it is more convenient to describe the cross section of dielectric waveguides (particularly the waveguides formed from etched layers or those fabricated by diffusion process) as a series of rectangular meshes where the electromagnetic permittivity and permeability tensors are approximately constant over and are zero and closed-form each mesh. In this case, equations can be written for other integrals. An efficient procedure for this purpose is presented in Appendix II. Rapid increase in the computation time, which is of order for each eigenvalue problem, is the main disadvantage of this method. Nevertheless, accurate results are often developed before a great increase in the computation time occurs. and and the Moreover, by adjusting scaling parameters normalization coefficient , accurate results can be obtained by using minimal terms in the field expansions. For a channel diand , optimum values electric waveguide with dimensions for and are and and the optimum value for norand malization parameter is the geometrical average of [16]. Quasi-optimum values of the mapping parameters can be chosen automatically by utilization of adaptive techniques [21]. Finally, it should be noticed that an appropriate selection of the orthogonal basis functions reduces the number of required terms in the expansions and considerably affects the precision of results and the computation time. Selection of the orthogonal functions depends on the geometry of the structure. As an example, for multilayer cylindrical dielectric structures, such as optical fibers, Fourier Bessel expansion would be the best choice. III. NUMERICAL RESULTS To validate the performance of the method, we defined three dielectric waveguides and analyzed them at optical frequencies. The results are obtained by employing the formulation given above. In each case, we compared our results with other available literatures. We adopt the modal identification format found in [22]. A. Anisotropic Dielectric Waveguide As a first example, an optical channel waveguide with rectangular cross section, embedded in a uniform media, is analyzed. The relative permittivities of the waveguide are , , and for the background. The scaling paand and the normalrameters are chosen to be ization parameter is . In each direction, 15 terms are considered in the expansions. Fig. 2 illustrates the dispersion curves for the first four guided modes of the waveguide. The results are compared with data obtained from the method of lines [12] and FEM [7]. As can be seen, close correspondence exists. The normalized field distrimode for is depicted in Fig. 3. bution of the B. LiNbO Optical Waveguide As a second example, an anisotropic LiNbO channel waveguide has been analyzed. The rectangular waveguide is surrounded by a uniform substrate with slightly smaller permeability tensors and homogeneous superstrate (air) that covers the

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Fig. 2. Dispersion cures for the first four modes of the illustrated channel waveguide. Relative permeability tensor of the channel is = = 2:31, = 2:19, and = 2:05 for background. = b, = b=2, and = . Expansions with 15 terms in each direction are used.

p

Fig. 3. Normalized field distribution of the E

mode in the anisotropic optical waveguide (the first numerical example) for

b = 4 :0 .

structure. Relative permeability tensor in the channel is , , and in the substrate, . In this case, physical dimensions of the channel is , m. The optimization parameters are chosen to where be , , and the normalization parameter is . Similar to the previous example, expansions with 15 terms in each direction are used. Fig. 4 illustrates the normaland of the ized dispersion curves of the guided modes waveguide. As before, we compare the results with the method of lines [12] and FEM [6]. The cutoff wavelengths of these two modes can be computed with the direct procedure studied before. For instance, the normalized propagation constant of the mode is , which is computed using 25 expansion. 25 C. Channel Waveguide With Permittivity and Permeability Tensors As a final example, a channel waveguide with permittivity and permeability tensors is analyzed three times for three dif-

Fig. 4. Dispersion cures for the E and E modes of the illustrated waveguide. Relative permeability tensor in the channel is = 2:22 , = = 2:3129 , and in the substrate, = = 2:29 , for background. In this case, b = 1:0 m, = 2:5b, = b=2, and = . Field expansions with 15 terms in each direction.

p

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and

are as follows:

Fig. 5. Permittivity and permeability tensors of the channel waveguide in the last example.

ferent permittivity and permeability tensors that are listed in the table in Fig. 5. Fig. 6 illustrates the dispersion curves of the and in each of the three cases first two guided modes and compares the results with results obtained via the method of lines [12]. In spite of small differences in dispersion curves at lower frequencies, the dispersion curves we obtained agree quite well with other literature. IV. SUMMARY In this paper, the Fourier decomposition method has been extended to determine both propagation constants and modal field distributions of the guided modes in anisotropic inhomogeneous dielectric structures with arbitrary cross sections. The method has been successfully applied in numerical computation of the propagation constants of channel waveguides with second rank electromagnetic permittivity and permeability tensors.

(50)

APPENDIX I are the following integrals: APPENDIX II Here, the closed-form equations are presented as follows for and when the permittivity and computation of permeability tensors are constant on a rectangular mesh (our standard rectangular mesh is depicted in Fig. 7):

(49)

PASHAIE: FOURIER DECOMPOSITION ANALYSIS OF ANISOTROPIC INHOMOGENEOUS DIELECTRIC WAVEGUIDE STRUCTURES

Fig. 6. Dispersion curves of illustrated channel waveguide with permittivity and permeability tensors. = 2:5b, = b=2, and = include 15 terms in each direction.

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p

. Field expansions

(51) Fig. 7. Standard rectangular mesh.

Here, , equations:

and

are defined with following

(52) It is apparent that these three integrals have simple analytical solutions. ACKNOWLEDGMENT The author would like to thank Dr. E. Mehrshahi, Iran Telecommunication Research Center, Tehran, Iran, and Dr. D. Jaggard, Electrical and Systems Engineering Department, University of Pennsylvania, Philadelphia, PA, for review of this paper’s manuscript and useful comments. REFERENCES [1] G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt., vol. 16, pp. 113–118, Jan. 1977.

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[2] K. Van D. Velde, H. Thienpont, and R. Van Geen, “Extending the effective index method for arbitrarily inhomogeneous optical waveguides,” J. Lightw. Technol., vol. 6, no. 6, pp. 1153–1159, Jun. 1988. [3] C. M. Kim and R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightw. Technol., vol. 7, no. 10, pp. 1581–1589, Oct. 1989. [4] M. S. Stern, “Semivectorial polarized field solutions for dielectric waveguides with arbitrary index profiles,” Proc. Inst. Elect. Eng., vol. 135, pt. J, pp. 333–338, Oct. 1988. [5] S. Akiba and H. A. Haus, “Variational analysis of optical waveguides with rectangular cross section,” Appl. Opt., vol. 21, pp. 804–808, Mar. 1982. [6] B. M. A. Rahman and J. B. Davies, “Finite-element solution of integrated optical waveguides,” J. Lightw. Technol., vol. LT-2, no. 10, pp. 682–688, Oct. 1984. [7] Y. Lu and F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 6, pp. 1215–1223, Jun./Jul. 1993. [8] E. W. Kolk, N. H. G. Baken, and H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 1, pp. 78–85, Jan. 1990. [9] N. H. G. Baken, J. M. van Splunter, M. B. J. Diemeer, and H. Blok, “Computational modeling of diffused channel waveguides using a domain-integral equation,” J. Lightw. Technol., vol. 8, no. 4, pp. 576–586, Apr. 1990. [10] H. J. M. Bastiaansen, N. H. G. Baken, and H. Blok, “Domain-integral analysis of channel waveguides in anisotropic multi-layered media,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 10, pp. 1918–1926, Oct. 1992. [11] H. P. Urbach and E. S. A. M. Lepelaars, “On the domain integral equation method for inhomogeneous waveguides,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 1, pp. 118–126, Jan. 1994. [12] P. Berini and K. Wu, “Modeling lossy anisotropic dielectric waveguides with the method of lines,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 5, pp. 749–759, May 1996. [13] U. Rogge and R. Pregla, “Method of lines for analysis of dielectric waveguides,” J. Lightw. Technol., vol. 11, no. 12, pp. 2015–2020, Dec. 1993. [14] C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightw. Technol., vol. 7, no. 2, pp. 308–313, Feb. 1989.

H

[15] D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by Galerkin method,” IEEE J. Quantum Electron., vol. 28, no. 2, pp. 459–465, Feb. 1992. [16] S. J. Hewlett and F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: Scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightw. Technol., vol. 13, no. 3, pp. 375–383, Mar. 1995. [17] R. V. Ramaswamy and R. Srivastava, “Ion-exchange glass waveguides: A review,” J. Lightw. Technol., vol. 6, no. 6, pp. 984–1000, Jun. 1988. [18] M. Koochakzadeh, R. M. Baghee, M. H. Neshati, and J. Rashed-Mohassel, “Solution of the vector wave equation for dielectric rod waveguides using the modified Fourier decomposition method,” in IRMMW-THz 2005, Sep. 2005, vol. 2, pp. 559–560. [19] A. Ortega-Monux, J. G. Wanguemert-Perez, L. Molina-Fernandez, E. Silvestre, and P. Andres, “Fast-Fourier based full-vectorial mode solver for arbitrarily shaped dielectric waveguides,” in MELCON, May 2006, pp. 218–221. [20] J. Xiao, M. Zhang, and X. Sun, “Solutions of the full- and quasi-vector wave equations based on -field for optical waveguides by using mapped Galerkin method,” Opt. Commun., vol. 259, pp. 115–122, 2006. [21] J. G. Wanguemert-Perez and I. Molina-Fernandez, “Analysis of dielectric waveguides by a modified Fourier decomposition method with adaptive mapping parameters,” J. Lightw. Technol., vol. 19, no. 10, pp. 1614–1627, Oct. 2001. [22] J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J., pp. 2133–2160, Sep. 1969.

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Ramin Pashaie (S’07) was born in Tehran, Iran. He received the B.S. degree in electrical engineering (with a major in microelectronics) from Shahid Beheshti University (SBU), Tehran, Iran, in 1998, the M.S. degree in fields and waves from Khaje Nasir Tousi University of Technology (KNTU), Tehran, Iran, in 2001, and is currently working toward the Ph.D. degree in electrical and system engineering at the University of Pennsylvania, Philadelphia.

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Analysis of a TE22;6 118-GHz Quasi-Optical Mode Converter Hansjörg Oliver Prinz, Student Member, IEEE, Andreas Arnold, Günter Dammertz, and Manfred Thumm, Fellow, IEEE

Abstract—A quasi-optical mode converter for the European 118-GHz gyrotron has been continuously improved for long pulse operation. Measurements at the last generation show a double peak structure at the output window plane, but simulations have not been able to explain this behavior in the past. The overall output system was analyzed carefully. Besides measurements, the first full 3-D electromagnetic analysis of a complete quasi-optical mode converter was performed. This gives the indication of the double-peak structure at the output window plane. An enhancement of the system for a single-peak Gaussian-like output beam will be presented.

TE22 6

Index Terms—Gaussian beam, gyrotron, high-power microwaves, mode converter, quasi-optical mirrors.

I. INTRODUCTION ODAY, modern high-power gyrotrons for fusion experiments deliver up to 1 MW of microwave power to the plasma [1]. To achieve these power levels, a large collector surface is needed. As this would require a highly overmoded waveguide for an axial output, a radial output of the microwave power is more feasible. This is realized with a quasi-optical mode converter. It separates the electron beam from the microwaves. The rotating high-order cavity mode is converted to a Gaussian freespace mode and directed towards the output window. The quasi-optical mode converter consists of a waveguide antenna with a helical cut (launcher), a quasi-elliptical, and two focusing mirrors. The volumetric cavity mode is converted into a mode mixture by perturbations on the launcher’s inner wall [2]. The superposition of modes forms a pencil-like microwave beam, which is radiated through the helical cut of the launcher (see [3], and references therein). The quasi-elliptical mirror forms a plane wave out of the launcher’s radiated field. A set of focusing mirrors directs the microwave beam towards the output window, where the beam diameter is limited by the window aperture. The linearly polarized Gaussian mode can be used directly for low-loss transmission to the application. The quasi-optical mode converter discussed here is installed -mode 118-GHz [4] gyrotron. A redesign of in the the mode converter to enhance the efficiency of the gyrotron shows two power peaks in the output beam. This was shown

T

Manuscript received January 3, 2007; revised April 9, 2007. This work was supported by the European Communities under the Association EURATOMForschungszentrum Karlsruhe Contract. H. O. Prinz and G. Dammertz are with the Institute of Pulse Power and Microwave Technology, Forschungszentrum Karlsruhe, D-76344 Eggenstein-Leopoldshafen, Germany (e-mail: [email protected]). A. Arnold and M. Thumm are with the Institut für Höchstfrequenztechnik und Elektronik, University Karlsruhe, 76131 Karlsruhe, Germany. Digital Object Identifier 10.1109/TMTT.2007.902779

Fig. 1. Power distribution in the window plane. (a) Simulation with diffraction integral. (b) Low-power measurement.

by [4], where the gyrotron was equipped with the quasi-optical system. This result could also be confirmed by low-power measurements. The low-power setup of an earlier version of the mode converter is stated in [5]. In Fig. 1, the low-power measurement is shown in comparison to the simulation with the Forschungszentrum Karlsruhe (FZK) code based on Kirchhoff’s diffraction integral. As can be seen from the power plots in linear scale, the simulation is not able to predict the behavior of the output beam. II. SETUP The quasi-optical output system consists of a launcher with wall perturbations and a set of three mirrors. The setup is shown in Fig. 2, where the mirror’s center points are indicated. The perturbation on the launcher’s wall causes preshaping of the launcher’s radiated field [6], [7]. The unrolled wall currents of the launcher with a radius taper of 2 mrad are shown in Fig. 3. The first of the three mirrors has a quasi-elliptical shape (first

0018-9480/$25.00 © 2007 IEEE

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Fig. 2. Cross section through the quasi-optical output system of the TE

Fig. 3. Calculated currents on the launcher’s wall in TE

118-GHz gyrotron.

mode at 118 GHz.

focus: 50 mm, second focus: 750 mm), which is used for azimuthal focusing of the radiated field from the launcher. The second mirror is a plane, and the third mirror is a parabolic one with an equivalent curvature radius of 1500 mm perpendicular to the gyrotron axis and 476 mm parallel to the gyrotron axis. The window aperture is 80 mm. III. ANALYSIS The analysis of the quasi-optical mode converter is split into two parts: the launcher—dealing with propagation in a waveguide—and the mirror system—dedicated to free-space propagation with focusing elements. A detailed comparison of measurement, simulation, and theory will be given.

Fig. 4. Launcher radius versus launcher length.

Always a top issue is the mechanical alignment of the components when handling millimeter-wave systems. This can be handled as shown below. A. Launcher A major concern is the mechanical accuracy at the fabrication of highly overmoded waveguides because every wall deformation results in mode conversion. As Fig. 4 demonstrates, with a measurement of the launcher’s inner radius, the deviations of the calculated curve are small compared to the measured points, which are shown as crosses. There is also the curve of the radius plotted for the case that the launcher would have been manufactured with the wrong rotation sense of the helical perturbation (dashed line), but this is certainly not the case, as can be seen

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118-GHz QUASI-OPTICAL MODE CONVERTER

Fig. 5. Calculated currents on the launcher’s wall in TE of the advanced launcher.

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mode at 118 GHz

from the plot. This measurement was done at different angles, all agree well. The design of the launcher itself can be still optimized, when analyzing the wall currents, as shown in Fig. 3. The bunching is well done, but the position of the cut, shown as a solid line, could be optimized to cross less currents by changing its tilt. Cutting any currents causes diffracted power, which could be the reason for the double-peak structure. By optimizing the wall perturbation, an advanced launcher was designed. Its wall currents are shown in Fig. 5. Here, currents lower than 18 dB were realized at the whole cut, but this has no influence on the output beam structure, as shown by a measurement. Even another launcher design with a radius taper of 4 mrad did not change the situation. Launcher simulations were carried out by the coupled mode equations as a very fast synthesis method, and by the electric field integral equation (EFIE) [8] as an analysis method with high accuracy. The measurements were performed at low power (cold measurements). A mode generator was used to excite the high-order cavity mode at 118 GHz, which is injected into the launcher [5]. The field plots were measured through a 2-D scan at the desired plane.

Fig. 6. Power distribution at x = ulation. (b) Measurement.

040 mm (plane of second mirror). (a) Sim-

B. Mirrors As no effects originating directly from the launcher could be demonstrated, further actions included the mirrors. In terms of simulations a new approach had to be undertaken because, as shown in Fig. 1, the calculation of the mirrors using Kirchhoff’s diffraction integral could not predict the behavior. The way to go was to extend the launcher’s calculation based on the EFIE for mirrors [9], which will be shown in the following. This demands high computation performance and a huge amount of RAM, but it showed excellent results, as will be presented in this paper. As the behavior of the microwave beam close to the launcher shows no abnormal effects, an analysis along the beam up to the position of the output window was carried out. In Fig. 6, a

Fig. 7. Power distribution at x = 140 mm (plane of third mirror). (a) Simulation. (b) Measurement.

measurement versus a simulation on the position of the second mirror is shown. They agree very well and show an astigmatic

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Fig. 8. Power distribution at x = 300 mm (plane behind third mirror). (a) Simulation. (b) Measurement.

Gaussian beam. There is a minor difference inside the maximum, but this finds an explanation because the plane is in the near field of the radiating aperture, thus the field is influenced by the measuring probe. Following the beam, the pattern shown in Fig. 7 is found at the position of the third mirror. Here again, an elliptical Gaussian beam was measured and simulated. The position, shape, and size are well predicted. Beside the main maximum, there is a mm, which is also shown in the simulasecond peak at tion. This plane is already far enough from the radiating aperture that no near-field effects can be detected any more. As known from Fig. 1, the double peak appears after the third mirror in the window plane, but the pattern looks fine on the third mirror, as depicted in Fig. 7. That is the reason why a measurement was carried out at a position behind the third mirror while it was removed. Fig. 8 shows the beam pattern in this plane at mm. As we would assume, the beam expands like a Gaussian beam after the waist. Also here, simulation and measurement are in excellent agreement. Analyzing the beam after its reflection at the third mirror leads to the patterns shown in Figs. 9 and 10. At a distance of 100 mm before the window plane, there is a nearly circular beam, which is nicely shown in measurement and simulation. Approaching the window plane, the measurement shows the double-peak structure. The simulation result in the same plane shows equal beam size and position, but the structure inside the maximum could not be predicted exactly, but as can be seen from Fig. 10(a), the peak is not centered any more, but down-shifted. This result indicates different beam properties in contrast to the one obtained by Kirchhoff’s diffraction integral.

Fig. 9. Power distribution at x =

0340 mm. (a) Simulation. (b) Measurement.

Fig. 10. Power distribution at x = with EFIE. (b) Measurement.

0440 mm (window plane). (a) Simulation

For further analysis, another tool that is available using the EFIE calculation was used. As the currents corresponding to the

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118-GHz QUASI-OPTICAL MODE CONVERTER

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Fig. 11. Cross section of the mode converter with electric field from 3-D simulation [9].

field of the whole structure are calculated in one, the field in any cross cross section can be calculated. For this problem, the mm, as plotted in Fig. 11, was the most useful. It section at shows the setup in the same plane as Fig. 2, but with the power distribution plotted. The launcher’s cross section is shown in Fig. 11 mm in the range of mm. As expected, up to the power inside the launcher is guided between the wall and the mode. At mm to mm caustic radius of the mm, the beam radiated from the launcher is reflected and by the first mirror. Even if this mirror has a strong curvature in -direction, it is the plane in the -direction, and thus, represented by a single straight line in this plot. The second mirror ranges from mm to mm. Since it is the plane, there is no other effect other than changing the beam’s direction. The beam mm then hits the third mirror, which is in the range of to 480 mm. After it is reflected by the third mirror, it propagates towards the window. The focusing characteristic can nicely be seen because the beam is rather broad at the mirror, at approximm, the beam radius has a minimum, and later mately it expands again. A detailed analysis of Fig. 11 shows that sidelobes on the third mirror are quite far apart from the main beam. They are focused to the center of the beam at the window position. As they are not in-phase with the main beam, the superposition of two beams may lead to an addition or subtraction of the field at different positions depending on the phase. This superposition is represented by the measured maximum–minimum structure in the window plane plotted in Fig. 10(b) at different -positions. At the upper peak, there is constructive superposition, between the two peaks, the wave superposes destructively, and at the lower peak, the superposition is again constructive.

Fig. 12. Power distribution at x = sorber.

0440 mm, third mirror covered with ab-

third mirror was carried out. To suppress the power on the outer parts of the third mirror, it was covered with an absorbing material. Only an area of 100 100 mm was kept free. This represents the reflection area of the main beam. The measurement result is shown in Fig. 12. As predicted, a circular Gaussian beam could be measured in the window plane. As some of the power is cut off there, approximately 8% less power is in this circular beam compared to the one having the double-peak structure. To prove the problem is related to the -direction, the absorbing material in the -direction was removed. The measurement result in Fig. 13 supports this argumentation. When removing all the absorbing material, the original pattern, as shown in Fig. 10(b), of course, appears again. V. SOLUTION

IV. EXPERIMENT To prove this theory about the appearance of the double-peak structure, an experiment with a smaller effective aperture of the

To avoid the double-peak structure, the third mirror has to be less focused. This way, the sidelobes shall not be focused to the inside of the beam in the window plane, but when focusing less,

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TABLE I CURVATURE RADII OF THE ADVANCED MODE CONVERTER

Fig. 13. Power distribution at x = sorber only in z -direction.

0440 mm, third mirror covered with ab-

Fig. 15. Power distribution at x = third mirror.

0440 mm, new design with second and

plane in Fig. 14. With a Gaussian content of 94% and a power transmission of 93% through the window, this is not optimum. If there were a way to introduce more changes to this mode converter, a well-focused output beam could be achieved. Since the second mirror is flat in the current design, adding a curvature to it would open the possibility of better adapting the mode converter to the input-beam properties. This is demonstrated by an advanced design, which optimizes the pattern in the window plane to match a profile of a Gaussian mode in the waist in the best way possible. The design parameters are documented in Table I. The calculation result for the window plane, shown in Fig. 15, demonstrates a well-focused beam. It has a beam radius of 20 mm with a Gaussian content of 95% and a power transmission of 98.2%. VI. CONCLUSION

0

Fig. 14. Power distribution at x = 440 mm (window plane) for a design with a new third mirror. (a) Simulation. (b) Measurement.

the beam size will, of course, increase. Here, a compromise has to be found that the beam still fits through the window, which has an aperture of 80 mm. A modification of the existing gyrotron is feasible only by changing the third mirror because an exchange of any other part would be very costly. Thus, a new design with a toroidal third mirror, which has the curvature radius of 909 mm in the -direction and 6250 mm perpendicular to this, was developed. The simulation results in the window plane are shown in Fig. 14. The output beam has a Gaussian content of 94%. To prove the theoretical investigations, this mirror was fabricated and cold measurements were carried out. The measurement is again in excellent agreement with the calculations, as can be seen from the comparison of the patterns in the window

The complex quasi-optical output system of the European 118-GHz gyrotron was analyzed. Full 3-D calculations of the dimpled-wall launcher and the three mirrors solving the EFIE were performed and show excellent agreement with the low-power measurements. The analysis implicated the focusing of sidelobes into the main beam as the cause of the double-peak structure. The over-focusing of the last mirror could be verified by a measurement with an effectively smaller third mirror and later with a third mirror with a larger curvature radius. The EFIE model could not exactly predict this behavior, but gives an indication. For a new advanced mode converter, the third mirror was redesigned. This avoids the double-peak structure in the window plane, but the beam radius increases. Keeping the original beam radius and omitting the double-peak structure, a new design of the last two mirrors was presented. ACKNOWLEDGMENT The authors would also like to thank C. Darbos, Commissariat à lénergie atomique (CEA), Cadarache, France, J.-P. Hogge, Centre de Recherche en Physique des Plasmas,

PRINZ et al.: ANALYSIS OF

118-GHz QUASI-OPTICAL MODE CONVERTER

Plasma Physics (CRPP), Lausanne, Switzerland, R. Magne, CEA, B. Sautic CEA, and M. Q. Tran, CRPP, for useful discussions and for the manufacturing of the modified third mirror. Furthermore, the authors would like to express their thanks to J. Neilson, Calabazas Creek Research (CCR), Saratoga, CA, for his stimulating suggestions and hints for enhancing the simulations. This study was carried out within the framework of the European Fusion Development Agreement.

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Andreas Arnold received the Dipl.-Ing. degree in electrical engineering from the University of Karlsruhe (TH), Karlsruhe, Germany, in 1997, and is currently working toward the Ph.D. degree at TH. Since 1994, he has been with the Gyrotron Group, Forschungszentrum Karlsruhe (FZK), Eggenstein-Leopoldshafen, Germany, where he is involved with millimeter-wave electronics, network analysis, and overmoded waveguides. Since 1997, he is also an Assistant Researcher with the Institute for Microwaves and Electronics, University of Karlsruhe (TH). His research topics are millimeter-wave measurement systems for testing high-power gyrotron assemblies.

REFERENCES [1] M. Thumm, “Progress in gyrotron development,” Fusion Eng. Design, vol. 66–68, pp. 69–90, 2003. [2] G. G. Denisov et al., “110 GHz gyrotron with a built-in high-efficiency converter,” Int. J. Electron., vol. 72, pp. 1079–1091, 1992. [3] A. Möbius and M. Thumm, , C. Edgcombe, Ed., Gyrotron Output Launchers and Output Tapers, Gyrotron Oscillators—Their Principles and Practice. London, U.K.: Taylor & Francis, 1993, pp. 179–222. [4] C. Darbos et al., “New design of the gyrotron used for ECRH experiments on Tore Supra,” in Proc. 13th Joint Electron Cyclotron Emission/Electron Cyclotron Resonance Heating Workshop, Nizhny Novgorod, Russia, 2004, pp. 409–414. [5] O. Braz, “Cold tests and high power measurements on an advanced quasi-optical mode converter for a 118 GHz gyrotron,” in 20th Int. Infrared Millim. Waves Conf., Orlando, FL, 1995, pp. 281–282. [6] A. A. Bogdashov and G. G. Denisov, “Asymptotic theory of high-efficiency converters of higher-order waveguide modes into eigenwaves of open mirror lines,” Radiophys. Quantum Electron., vol. 47, pp. 283–295, 2004. [7] M. Thumm, X. Yang, A. Arnold, G. Dammertz, G. Michel, J. Pretterebner, and D. Wagner, “A high-efficiency quasi-optical mode converter for a 140-GHz 1-MW CW gyrotron,” IEEE Trans. Electron Devices, vol. 52, no. 5, pp. 818–824, May 2005. [8] J. Neilson, “Surface integral equation analysis of quasi-optical launchers,” IEEE Trans. Plasma Sci., vol. 30, no. 3, pp. 794–799, Mar. 2002. [9] H. O. Prinz, J. Neilson, and M. Thumm, “3D-analysis of quasi-optical output systems for high power gyrotrons,” in Joint 31st Int. Infrared Millim. Waves Conf./14th Int. Terahertz Electron. Conf., Shanghai, China, 2006, p. 518.

Hansjörg Oliver Prinz (S’04) was born in Karlsruhe, Germany, in 1978. He received the Dipl.-Ing. degree in electrical engineering from the University of Karlsruhe (TH), Karlsruhe, Germany, in 2004. Since 2002, he has been with the Gyrotron Group, Research Center Karlsruhe, Forschungszentrum Karlsruhe (FZK), Eggenstein-Leopoldshafen, Germany, where he was initially involved with diagnostic systems for high-power gyrotrons. Since 2004, his research interests have been quasi-optical mode converters and multifrequency gyrotrons.

Günter Dammertz was born in Moers, Germany, on March 3, 1942. He received the Diploma degree in physics (for work on electron gamma angle correlation) from the Rheinische Friedrich Wilhelm University, Bonn, Germany, and the Dr. rer. nat. degree in physics (for work on superconducting RF structures) from the University of Karlsruhe (TH), Karlsruhe, Germany, in 1972. In 1968, he joined the Forschungszentrum Karlsruhe (FZK), Eggenstein-Leopoldshafen, Germany, where he had been involved in the development of superconducting RF accelerators and particle separators, the development of high current neutral beam sources, and the investigation of a spallation source. Since 1984, he has been involved with the development of high-power gyrotrons, especially in the experimental program and in technical investigations.

Manfred Thumm (SM’94–F’02) was born in Magdeburg, Germany, on August 5, 1943. He received the Dipl.Phys. and Dr. rer. nat. degrees in physics from the University of Tübingen, Tübingen, Germany, in 1972 and 1976, respectively. In June 1990, he became a Full Professor with the Institute for Microwaves and Electronics, University of Karlsruhe, Karlsruhe, Germany, and Head of the Gyrotron Development and Microwave Technology Division, Institute for Technical Physics, Research Center Karlsruhe, Forschungszentrum Karlsruhe (FZK). Since April 1999, he has been the Director of the Institute for Pulsed Power and Microwave Technology, FZK, where his current research projects concern the development of high power gyrotrons, dielectric vacuum windows, transmission lines and antennas for nuclear fusion plasma heating, and industrial material processing. He has authored or coauthored three books, nine book chapters, 195 research papers in scientific journals, and 850 conference proceedings articles. He holds ten patents on active and passive microwave devices. Dr. Thumm was the recipient of the 2000 Kenneth John Button Medal and Prize in recognition of outstanding contributions to the science of the electromagnetic spectrum. In 2002, he was awarded the title of Honorary Doctor, presented by the St. Petersburg State Technical University, for his outstanding contributions to the development and applications of vacuum electron devices.

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Mixed-Mode Chain Scattering Parameters: Theory and Verification Holger Erkens, Student Member, IEEE, and Holger Heuermann, Senior Member, IEEE

Abstract—Chain scattering parameters or -parameters are a useful tool for calculating cascaded two-ports. With the increasing importance of mixed-mode -parameters, a need for converting the -parameters from their unbalanced form into a balanced form emerges for suiting both common and differential mode waves, as well as the mode conversion. This paper presents the derivation of the equations for transformations between mixed-mode - and -parameters for a mixed-mode two-port. Although derived in a way very similar to monomode -parameters, no simplifications were necessary. Measurement results exemplify the quality of the -parameter transformation under real-life conditions. Index Terms—Cascaded two-ports, chain scattering parameters, dual-mode parameters, mixed-mode parameters, scattering parameters, -matrix, -matrix.

I. INTRODUCTION

T

HE HISTORY of chain scattering parameters, also known as -parameters, goes back to the definition of the scattering matrix. The introduction of the -parameters, describing reflection and transfer characteristic of -ports, in [1] was a great step to faster and easier developments of RF components. Later, -parameters derived from -parameters have proven very useful in handling cascaded two-ports. The book authors gave the -parameters different names and abbreviations, e.g., the name cascading parameters and the abbreviation R in [2]. The resulting equations of two-port -parameters are implemented in any field-simulation software, as well as in many manufacturing software packages, as the so-called deembedding process. The mixed-mode -parameters for common and differential modes have been introduced in [3] and [4] in more detail. Since then, several papers have been published that describe circuits with mixed-mode -parameters, [5]–[7]. The generalized mixed-mode -matrix including the common, the differential, as well as the unbalanced mode, is given in [8] in German. An English version has recently been published in [9]. A solution for calculating the -matrix of two cascaded networks has already been published in [10]. The solution for calculating the -parameters of cascaded two-ports of unlimited number will be presented in this paper. It begins with the fundamentals of

Manuscript received August 3, 2006; revised March 13, 2007. H. Erkens is with the Chair of Integrated Analog Circuits, RWTH Aachen University, 52074 Aachen, Germany (e-mail: [email protected] de). H. Heuermann is with the Department of RF Technology, Aachen University of Applied Sciences, 52074 Aachen, Germany. Digital Object Identifier 10.1109/TMTT.2007.902587

Fig. 1. Block diagram of a measurement setup. The DUT can be deembedded with T -parameters.

mixed-mode parameters. The main part of this paper gives the derivation of the mixed-mode -parameters, and is followed by experimental results, which conclude this paper. II. FUNDAMENTALS OF MONOMODE AND MIXED-MODE PARAMETERS is defined as a matrix The unbalanced scattering matrix that links vectors of incident waves and emergent waves, repre. Thus, for a two-port, one sented by and can write (1) Describing waves in this way, one can directly write transmission and reflection properties of an -port device. For the special case of a two-port device, the unbalanced chain scattering malinks the port-related waves trix (2) The transformation is done by solving the linear equations that form the -matrix for and and performing a comparison of the coefficients [11], [12]. When described with -parameters, the waves and form a joint vector. Thus, the complete -matrix for a cascade equals the matrix product of the single -matrices. Using -parameters, the -parameter data of a measured device can be extracted from the measurement equipment with perfect accuracy. For a so-called in-fixture measurement with a vector network analyzer (VNA), the test fixture is well characterized at times when an in-fixture calibration is not possible. The device-under-test (DUT) can only be measured afterwards, together with the known networks 1 and 3, as shown in Fig. 1. After the measurement, all -matrices are transformed into their -matrix equivalent. The transformation of the DUT is

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The mixed-mode -matrix, as in (4), is rearranged for the calculation of the mixed-mode -parameters

(6)

Fig. 2. Similar to monomode chain scattering parameters. A T -matrix can be formed for mixed-mode systems.

For a straightforward calculation, the re-sorted mixed-mode -matrix is divided into four 2 2 matrices

(7) with and being the two-ports incorporating the test fixcan be calculated from the scattering parameters ture. of the whole cascade. The deembedded DUT -parameters are then extracted by calculating

with

(3) A -matrix for mixed-mode scattering parameters is a convenient tool for enhancing the measurement accuracy of four-port VNAs in direct differential on-wafer measurements. The VNA measures all 16 unbalanced -parameters of a device and calculates the mixed-mode -parameters

(4)

with denoting the differential mode, denoting the , as well as denoting the convercommon mode, and ). The standard method sion between both modes ( , GSOLT for multiport VNA calibrations is given in [13].

Equation (7) can be written as two equations (8) (9) Solving (9) for , the result becomes the first equation of the mixed-mode -matrix

III. DERIVATION OF THE MIXED-MODE -PARAMETERS Fig. 2 illustrates mixed-mode two-ports by using the novel mixed-mode -parameters. Deembeding can be done by (3) just as in a monomode system. The mixed-mode chain matrix is defined as follows:

(10) The second equation is found by inserting (10) into (8) and solving it for as follows: (11)

(5)

Equations (10) and (11) are the matrix representation of the mixed-mode transmission matrix

(12) The index numbers in do not describe reflection and transmission, as in . The chosen indexing is used to keep -parameters consistent with mixed-mode -parameters and the unbalanced -matrix.

Some permutations are necessary to translate (12) into the form of (5), which is divided in four quadrants with one mode in each. Comparing the resulting coefficients of (12) with the ones

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from (5), the following scalar equations for the transformation can be derived:

(13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

Fig. 3. Simplified circuit diagram of the measured circuit cascade.

(35) (36)

(23)

(37) (38) (39)

(24)

(40) (41) (25)

with the factor (42)

(26) with the factor

IV. MEASUREMENT VERSUS CALCULATION (27)

The calculation from mixed-mode transmission parameters back to mixed-mode scattering parameters is done in analogy to the description in (5)–(12). For that reason, only the following equations for will be given: (28) (29) (30) (31)

(32)

(33)

(34)

For verifying the theory, circuit simulator results have been compared with calculation results and showed absolutely no deviation from each other. For a comparison with measurement results, three mixed-mode two-ports have been measured individually and as a cascade. Application of the proposed algorithm followed the measurement. Fig. 3 shows the cascade of mixed-mode two-ports. For the measurement, a differentialmode semiconductor DCS1800-filter [14] has been cascaded with a novel mixed-mode p-i-n diode switch in its ON state [15]. The printed circuit boards (PCBs) have been connected with male–male subminiature A (SMA) adaptors, which form the third mixed-mode two-port of the cascade. These adaptors have been modeled by adding a third mixed-mode two-port with perfect matching and a phase shift for transmission. For the verification of the algorithm, the complete cascade has been measured with a four-port VNA that provides mixedmode parameters as output. The measurement result has been compared to the result calculated by the introduced algorithm. This calculation was done by measuring both circuits individually with the VNA. Afterwards, their mixed-mode -matrices and the model of the SMA-adaptor have been transformed into mixed-mode -matrices. The result of the calculation has then been derived by transforming the matrix product back to its

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the SMA adaptor only being modeled with perfect return loss for both the differential and common mode. V. CONCLUSION With differential and mixed-mode circuits moving more and more into the focus of the RF designer, the need for adequate design and measurement tools is continuously growing. Finding a generalized solution for mixed-mode -parameters is a logical next step on the way to fully differential RF system solutions. A complete solution to mixed-mode -parameters has been presented in this paper. The equations work for common and differential mode -parameters, as well as for all conversion parameters. An experimental verification has been done and presented. The measurement results show that the transformation algorithm is precise under practical conditions, provided that the deembedding is done with high accuracy. Fig. 4. Measurement results of the differential mode reflection and transmission.

ACKNOWLEDGMENT The authors would like to thank Rohde&Schwarz, Munich, Germany, for providing their four-port VNA ZVB8. REFERENCES

Fig. 5. Measurement results of the mode-conversion reflection and transmission.

-parameter equivalent. A comparison that verifies the algorithm must show good agreement between the measured mixedmode -parameters and the calculated result. The differential-mode measurement results are presented in Fig. 4. Both the transmission and reflection show a certain amount of broadband ripple that comes from the mismatch between both measured circuits. However, the calculated and the measured results show a very high agreement that could be enhanced by a better model or a measurement of the adaptors connecting both PCBs. Fig. 5 shows that the mode-conversion results also sufficiently match. However, the difference between calculated and measured results is larger for the values below 30 dB, in general, and the values approaching the upper frequency limit of the measurement, in particular. This is obviously the result of

[1] K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 3, pp. 194–202, Mar. 1965. [2] G. F. Engen, Microwave Circuit Theory. London, U.K.: Peregrinus, 1992. [3] M. Moeller, H.-M. Rein, and W. Wernz, “15 Gbit/s high-gain limiting amplifier fabricated using Si-bipolar production technology,” Electron. Lett., vol. 30, pp. 1519–1520, Sep. 1994. [4] D. E. Bockelman and W. R. Eisenstadt, “Combined differential-mode and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995. [5] W. R. Eisenstadt, B. Stengel, and B. M. Thompson, Microwave Differential Circuit Design Using Mixed-Mode S -Parameters. Boston, MA: Artech House, 2006. [6] H. Erkens and H. Heuermann, “Blocking structures for mixed-mode systems,” in Proc. 34th Eur. Microw. Conf., Amsterdam, The Netherlands, 2004, pp. 297–300. [7] B. Stengel and B. Thompson, “Neutralized differential amplifiers using mixed-mode S -parameters,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, 2004, vol. 1, pp. A197–A200. [8] H. Heuermann, Hochfrequenztechnik. Wiesbaden, Germany: Vieweg Verlag, 2005. [9] A. Ferrero and M. Pirola, “Generalized mixed-mode S -parameters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 458–463, Jan. 2006. [10] H. Shi, W. T. Beyene, J. Feng, B. Chia, and X. Yuan, “Properties of mixed-mode parameters of cascaded balanced networks and their applications in modeling of differential interconnects,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 360–372, Jan. 2006. [11] P. Badharamik, L. Besser, and R. W. Newcomb, “Two scattering matrix programs for active circuit analysis,” IEEE Trans. Circuit Theory, vol. CT-18, pp. 610–619, Nov. 1971. [12] D. Wood, “Reappraisal of the unconditional stability criteria for active 2-port networks in terms of S parameters,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 2, pp. 73–81, Feb. 1976. [13] H. Heuermann, “GSOLT: The calibration procedure for all multi-port vector network analyzers,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, 2003, pp. 1815–1818. [14] H. Heuermann, “A synthesis technique for mono- and mixed-mode symmetrical filters,” in Proc. 34th Eur. Microw. Conf., Amsterdam, The Netherlands, 2004, pp. 309–312. [15] H. Erkens and H. Heuermann, “Novel RF switch concepts for differential wireless communications frontends,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2376–2382, Jun. 2006.

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Holger Erkens (S’07) was born in Stolberg, Germany, in 1977. He received the Dipl.-Ing. degree in electrical engineering from the Aachen University of Applied Sciences, Aachen, Germany, in 2004, and is currently working toward the Ph.D. degree at RWTH Aachen University. He is currently with the Chair of Integrated Analog Circuits, RWTH Aachen University. His current research interests include system- and circuit-level design techniques of wireless data transmission systems and phased-array radar transceivers.

Holger Heuermann (M’92–SM’06) received the Ph.D. degree in electrical engineering from the University of Bochum, Bochum, Germany, in 1995. From 1991 to 1995, he was a Research Assistant with the University of Bochum, where he was involved with RF measurement techniques. From 1995 to 1998, he was with Rosenberger Hochfrequenztechnik, Tittmoning, Germany, where he was engaged in the design of RF equipment for measurements with network analyzers. In 1998, he joined Infineon Technologies, Munich, Germany, where he led a development group for wireless front-end modules and integrated circuits. Since 2002, he has been with the Aachen University of Applied Sciences, Aachen, Germany, where he is currently a Professor heading the Department of RF Technology. He has authored or coauthored over 45 papers. He holds 25 patents. His current research interests include RF components, design of front-end circuits, as well as high-precision, multiport, and multimode scattering-parameter measurements. Dr. Heuermann is member of the Microwave Measurements Technical Committee MTT-11.

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A General Multigrid-Subgridding Formulation for the Transmission Line Matrix Method Luca Pierantoni, Member, IEEE, and Tullio Rozzi, Life Fellow, IEEE

Abstract—This paper introduces a general multigrid-subgridding formulation aimed at improving the efficiency of the transmission line matrix method. The rigorous algebraic procedure, based on Hilbert space formulation, leads to the definition of a general boundary scattering matrix, which interfaces 3-D regions of uniform computational mesh with different spatial resolution, thus allowing the modeling of objects of very disparate geometrical or electrical scale, e.g., such occurring in nanodevices. Computational acceleration and accuracy are measured by comparing the run time of simulations performed with subgridding to the run time of simulations without any subgridding. Index Terms—Hilbert space, scattering matrix, subgridding, transmission line matrix (TLM).

I. INTRODUCTION

T

HE transmission line matrix (TLM) is a time-domain space-discretizing method in which the dynamics of the electromagnetic (EM) field is described by applying Huygens’ principle [1]–[4]. This is a powerful method that allows the EM full-wave modeling of 3-D structures with nearly arbitrary geometry in a wide range of applications from EM compatibility to optics [5]–[7]. In TLM, as well as in the other numerical methods, e.g., finite-difference time-domain method or finite-element method, aimed at solving EM problems, a major difficulty is caused by dealing with physical objects of very different geometrical or electrical scale present in the same environment. This situation commonly arises, for example, in the modeling of integrated circuits for satellite communication operating above 50-GHz clock speed [8], microelectromechanical systems (MEMS) [9], photonic circuits in integrated optical architectures [7], and very recently, in the electrical/electronic modeling of nanotubes and nanodevices [10]. In these problems, small objects (e.g., metal wedges, thin wires), which significantly affect the EM field distribution, are embedded in regions that are orders of magnitude larger; the finest object imposes a grid size that is redundant for the rest of the region, thus sometimes leading to prohibitive memory and computation effort. For the finite-difference time-domain method, as well as for the TLM, considerable effort has been devoted to alleviating these problems, as reported in [11]–[15]; the general strategy is the use of different cell sizes over different portions of the computational domain. Manuscript received November 10, 2006. The authors are with the Dipartimento di Elettromagnetismo e Bioingegneria, Università Politecnica delle Marche, Ancona 60100, Italy (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.902581

The approach where two or more meshes of different spatial resolution co-exist in the same problem are referred to as “multigrid” or “subgridding” methods. More exactly, we refer to multigrid, when the interface between two grids develops on a 2-D surface and subgridding when 3-D subregions with finer grids are progressively nested inside regions with coarser grids. A fundamental attempt to solve the problem of multigridding in a 3-D TLM scheme is that of Herring and Christopolous in [11], where the basic ideas of the continuity of tangential electrical and magnetic fields were introduced, together with conditions on charge and energy conservation, transparency, and time synchronism. An elegant and accurate technique for interfacing segments of a uniform computational TLM mesh with different spatial resolution is described in [12]. In this technique, suited for a 2-D interface, multigrid rectangular TLM meshes are stitched together with triangular elements. A new definition of multigrid is introduced by Wlodarczyk in [13], where the interface between a coarse and fine mesh is defined as an electrical connection. The scheme presented in [13] differs from that presented in [11] in a number of aspects: it uses the same time step on both the coarse and fine meshes, it is defined as an electrical connection rather than an averaging process, and it is lossless. In order to adapt different grids, matching transformers must then be introduced. These transformers have turn ratios equal to the aspect ratios of the cell faces on the interface. The goal of this study is to introduce a general formulation of a multigrid-subgridding technique for 3-D TLM schemes [1]–[4]. The starting idea is using the continuity of tangential field components between fine and coarse grids, as in [11], but also generalizing the matching transformers concept introduced in [13]. With respect to [11], the novel aspect consists of a compact reformulation of the problem by means of spatial and temporal basis functions, simultaneously considered in each equation. By imposing the continuity of the tangential EM field at the interfaces of different grids, we derive a general boundary scatthat is real, lossless, and reciprocal, describing tering matrix an instantaneous scattering in the frequency domain, as well as in the time domain, thus ensuring the conservation of energy. The rigorous algebraic procedure is based on the Hilbert vector space formulation, according to the guidelines and formalism introduced by Krumpholz and Russer in [1], and used in [16] and [17]. operator The equivalent network representation of the describes EM coupling among transmission lines belonging to nodes of different grids.

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Fig. 1. TLM computational mesh with different spatial resolution. Boundary interface of coarse-to-fine grids.

This operator can also be seen as the generalization of the ideal transformer introduced in [13], matching grids of different characteristic impedances. boundary operator into a TLM By incorporating the solver, large regions are discretized by coarse grids and vice versa, without any loss of accuracy. Despite its apparent complexity, the definition of local boundary matrix operators is of great help in the practical implementation. The proposed subgridding method is applied to modeling: 1) the transmission characteristics of an iris loaded-rectangular waveguide and 2) a capacitive MEMS switch. The results obtained by the TLM with subgridding are compared to the ones by the TLM without subgridding, showing that the same accuracy is maintained, whereas the speed is considerably increased. II. THEORY Referring to Fig. 1, let us consider a boundary interface between grids of different sizes conformal to the rectangular coordinates. A boundary interface is a surface separating cells belonging to the coarser mesh from the ones of the finer mesh. On both sides, the size of the mesh can either be homogeneous or inhomogeneous. We assume that each boundary interface provides a mesh ratio of 1 : 2 over a rectangular direction, as depicted in Figs. 1 and 2. This assumption is accounted for sake of simplicity, but the approach can be generalized for 1 : ratios. By considering a 1 : 2 mesh ratio, it follows that each rectangular subdomain of the coarser mesh with surface is divided in four identical smaller rectangular ones over the finer mesh with surface . In the TLM method, the continuous space is segmented into cells by defining intersecting planes; ports are defined at the tangential planes between two neighboring cells and a scattering center is defined at the center of each cell with the ports [1]–[4]. This physical model is called the node and comprises the scattering center, which is connected via transmission lines to the ports at the tangential planes between neighboring cells, as depicted in Figs. 1 and 2. Pulses are scattered at the nodes and propagate on these transmission lines to the neighboring nodes where they are again scattered. In the Hilbert space formulation [1], the propagation and scattering of the wave amplitudes are expressed by an operator equation. A matrix equation of the

Fig. 2. Boundary interface on the x–y -plane. Incident and reflected pulses related to the (m; n)-node of the coarse mesh (left side, black line) are denoted with = (x; y ), whereas the pulses belonging to the (p; q )-node as a of the fine mesh (right side, grey line) are b .

form (1) is defined for each of these cells (or nodes), describing the of (1) is the scattering process [1], [2]. The matrix scattering matrix of the node . This node is located at , , ) in a Cartesian coordinate system, where ( , , and constitute the spatial increment and denotes the spatial indices in the -, -, and -direction of the grid constituted by the location of the nodes. The vectors and comprise the incident and reflected pulses that propagate on the transmission lines, and is the discrete time coordinate. On the tangential planes between neighboring cells, we and can define a correspondence between the vectors and the EM field components according to the field boundary mapping introduced in [1] and [16]. We now deal with a computational mesh with different spatial resolution, as depicted in Figs. 1 and 2, where a boundary interface from coarse-to-fine grid is conformal to the – -plane -plane. This tangential plane is placed and located at the between the and cells once is fixed. Our goal is to derive the boundary scattering matrix interfacing the two grids. To this purpose, on this boundary plane, we now consider the expansion of the tangential electric and magnetic fields with ), as in [16] respect to the coarser grid (left side, at

(2)

(3) where is the discrete time coordinate and and are indices for the discrete space – coordinates; for the sake of simplicity, we omit the -index for the -coordinate. The incident and scattered wave amplitudes at node are denoted by

PIERANTONI AND ROZZI: GENERAL MULTIGRID-SUBGRIDDING FORMULATION FOR TLM METHOD

and , respectively, while is the polarization unit in the boundary plane, as shown in vector of index Fig. 2. In this same figure, for clarity, only one node of the first fine mesh layer is drawn. and are the rectangular subdois the rectangular time-domain main base functions and . The quantities are the line expansion function impedances for the boundary cell of indices . fulThe rectangular subdomain base functions fill the relation

(4) is the surface of the elementary cell on the transverse where – plane, as shown in Fig. 2. For the time-domain expansion , the following relationships holds: functions (5)

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Therefore, each coarse cell overlaps four finer cells through the boundary plane and each event in the coarser cell lasts for . two in each finer cell This naturally leads to choosing the discrete time step of the finest mesh (by considering nested subgridded regions) as the driving time step of the scattering process in the entire computational domain. The dynamics of the computational process develops as follows. 1) Incident pulses transfer EM energy from a coarser grid to , a finer one (through the boundary) at a time and receives the scattered ones at the next time step . to 2) In the finer mesh, during the time from , the EM field components are sampled two times, from to and from to . 3) The EM field components are mapped into the scattered and transferred back to the coarser pulses at grid, again through the boundary. , the conOnce we have fixed the driving time step tinuity of the tangential EM field on the boundary plane is imposed, by setting (2) equal to (6) and (3) equal to (7) as follows:

in which is the time step in this grid. In the same way, we now consider the expansion of the tangential electric and magnetic fields with respect to the finer grid ) on the transverse boundary plane, as (right side, at shown in Figs. 1 and 2

(6)

(9)

(7) and , where the indices and refer to the coordinates is the discrete time coordinate and denotes the polarization at the boundary plane. and are the rectanThe functions is the rectangular gular subdomain base functions and , being the time step time-domain base function being the line impedances for the in this (finer) grid and boundary cell of indices . and , we set For the basis functions

(10) In order to ensure synchronism, in (9) and (10), the wave amand are sampled plitude vectors of the coarse grid , where is the integer according to the relation . This means that, for example, when the time-step part of variable, being fixed by the finer grid, is , , , , , , etc., , the corresponding field values in , , , , the coarser grid are sampled for , , etc., , as required. Now, by multiplying (9) and (10) by the fine grid time-domain , integrating them over and taking function into account (4), (5), and (8) yields

(8) Without loss of generality, we can consider an homogeneous (being mesh. A spatial mesh ratio of 1 : 2 implies and ).

(11)

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The wave amplitude vectors of the coarse and fine grids of (13) are defined in the Appendix. By referring to (13), is an (8 8) identity matrix, and and are of diagonal form, defined as follows: (14) with

being a diagonal (4

4) matrix (15)

is the line impedance of the coarse grid, and where ( ; ) are the line impedances of the fine grid. We now introduce the vectors (16) Fig. 3. Transmission line, wave amplitudes, and ports on the boundary interface between the (m; n) node of the coarse mesh and the (p; q ; p = 1; 2, q = 1; 2) nodes of the fine mesh. The grey line denotes x polarization and the black line denotes the y polarization.

We also define the extended square as in [16]

and

matrices,

(17) for the electric field and and matrices By taking into account the extended (17), the and matrices (14), we derive the final equations for the wave amplitudes vectors (16) (12) for the magnetic field. In (11) and (12), is the unit vector in the -direction. By multiplying both members of (11) and (12) by the basis , integrating over the transverse functions of the corresponding fine cell, and considcross section ering the orthogonality relation (7), we derive the appropriate set of coupling coefficients, which can be compacted in matrix form as

(13) The state vectors and belong to the coarser grid, whereas and belong to the finer grid. In order to highlight the structure of (13), let us consider a cell (or node), which is interfaced to four finer generic ( ; , ) cells (or nodes), as depicted in Fig. 3. In Fig. 3, we show the transmission lines propagating the wave pulses from the node to the boundary, where they form the tangential field components (9) and (10). We consider the boundary field mapping of the classical symmetrical condensed node reported in [1] and [17]. In the coarser grid, ports 10 and 12 belonging to the node lean on the boundary at ; in the finer grid, at , ports 9 and 11 lean on the subdomains , ; ) in which is divided. (

(18) with being a (8 8) null matrix. Finally, we obtain the boundary scattering equation (19) where (20) Since we denote waves incident into TLM cells with and waves scattered by TLM cells with , we shall denote waves incident on a boundary surface by and the waves scattered via the boundary surface by . The boundary scattering matrix (21) describes an instantaneous scattering and, therefore, is real and is identical in frequency-domain, as well as time-domain representation. Since the scattering is lossless and reciprocal, we obtain (21) The operator is real, symmetric, Hermitian, and unitary. Its structure is defined in the Appendix. boundary operator (20) and (21) provides the scatThe tered wave amplitudes from the boundary into the two grids. The scattered waves reflected by the boundary toward the finer grid are (22)

PIERANTONI AND ROZZI: GENERAL MULTIGRID-SUBGRIDDING FORMULATION FOR TLM METHOD

Fig. 4. Equivalent network representation of the S boundary operator providing the EM coupling among the transmission lines of the two different grids.

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Fig. 5. Analyzed rectangular waveguide (WR90) with a = 22:86 mm, b = 10:16 mm, and c = 76:2 mm, loaded by a inductive iris of thickness d = 2 mm, d = 6 mm, and d = 10 mm, respectively. The subgridding region size is = 3:81 mm.

whereas the scattered pulses toward the coarser grid are (23) or

The entire process has a simple interpretation in terms of the Hilbert ideal transformer [18] of Fig. 4, where the -ports on ports on the the left side (coarse grid) are coupled to the right side (fine grid). The above formulation can be extended in a straightforward manner to derive boundary scattering matrices for interfaces conformal to the – - and – -plane. III. EXAMPLES We first apply the proposed method to the analysis of a discontinuity in a hollow waveguide. We consider the rectangular WR90 waveguide of Fig. 5, loaded by an inductive iris of thickness . We proceed through the following steps. Step 1) We first calculate the transmission coefficient by considering a TLM discretization without subgridding and homogeneous mesh, mm. Step 2) We then discretize the guide by using a mesh of mm (coarse mesh) everywhere apart a subgridding region surrounding the iris where we mm. insert a finer mesh with We consider a guide length of mm; the excitation pulse is a sinusoidal electric field over the 8–12-GHz band, cenGHz, modulated by a Gaussian distribution and tered at transverse distribution at . In order to having the excite this modal configuration, we inject impulses into branch #11 of all the nodes along the excitation plane, reported as follows:

(24)

Fig. 6. Magnitude of the transmission coefficient for the inductive iris of Fig. 5. The iris is of thickness d and dimensions are expressed in millimeters. Results with subgridding and without subgridding are compared to the exact solution [19].

where

, is the th-time step, is the time position where the Gaussian is centered, , being GHz, GHz, and with GHz. The point is the central point of the transverse cell of indices and . plane The rectangular waveguide is terminated at the by exact modal absorbing boundary conditions reported in [16]. In the upper left-side sketch of Fig. 5, we depict the subgridding mm (ten cells), which extends region, with mm beyond the conducting walls of the iris, whereas in mm. For both the rest of the guide, the grid is simulations, the time step is ps. calculated by: In Fig. 6, we show the magnitude of 1) TLM without subgridding and 2) TLM with subgridding, compared to the accurate solution provided by the mode matching of [19]. We note excellent agreement among the three curves, thus highlighting that the same accuracy can be provided by using fewer cells with a speedup of 4.3, as reported in Table I, where the worst case mm for the subgridding is considered. Now, we load the same waveguide by the cascade of two mm , separated by a distance of 20 mm, inductive irises

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TABLE I GRID PARAMETERS AND COMPUTATIONAL TIME FOR THE IRIS-LOADED WR90 GUIDE OF FIG. 5

Fig. 8. Magnitude of S for the MEMS capacitive switch of [9] and [19]. Comparison among results by TLM with subgridding, TLM without subgridding, and measured data. Geometrical (in micrometers) and electrical details are shown in the inset.

Fig. 7. Magnitude of the transmission coefficient for two cascaded inductive irises, loading the same WR90 of Fig. 5 with d = 4 mm and separated by a distance of 20 mm.

and report the magnitude of the transmission coefficient in Fig. 7, showing excellent agreement with respect to the exact solution [19]. The computational data of Table I (single iris) can also be considered valid for the current case (cascade of irises) in which the finer mesh includes the two irises. The computational platform consists of a 450-MHz 500-MRAM PC. As a cogent example, we now analyze the MEMS capacitive switch reported in [20]. This is a challenging complex structure that exhibits several geometrical details, finite dielectric layers, lossy thick metal, 10 ). Details of the geand critical aspect-ratio (up to 10 ometry and dielectric parameters are shown in Fig. 8. m; the excitation We consider a MEMS length of is provided through a wideband Gaussian pulse. The discretization strategy is the following (see Fig. 9). 1) We first place a magnetic wall at the plane of longitudinal symmetry of the MEMS. m, 2) In a subregion comprising the underpath a homogenous grid with m is chosen fs . 3) This subregion is nested inside another subregion with m, comprising the metallic membrane m and the upper section of the bulk silicon region. This process continued over the rest of the structure: each ho, is progressively nested mogeneous “finer” grid with, say, in an homogeneous “coarser” one with . In

Fig. 9. Magnitude of S . Comparison among TLM, TLM with subgridding, and measured data of [19]. In the inset, details of the nested regions are reported.

and calFigs. 8 and 9, we compare the magnitude of culated by using the subgridding discretization outlined above to the ones obtained by using TLM with homogeneous mesh m and to measured data [9], [20]. We observe very good agreement among the three curves. This confirms that the same accuracy provided by the TLM without subgridding is achieved by TLM simulation with subgridding, which, on the other hand, shows a remarkably improved performance in terms of computational effort; in fact, for TLM with subgridding, we get a kilo-node grid, whereas for TLM without subgridding, we need a grid larger, by at least, one order of magnitude. IV. CONCLUSION This paper describes a general and compact formulation for the multigrid-subgridding techniques in the TLM method. The theoretical foundation is based on the Hilbert space representation of the field quantities. It starts from the conservation of energy and derives a general boundary scattering matrix. This boundary operator interfaces regions of different spatial resolutions, which can be progressively nested: each subregion with a fine mesh is embedded into another with a coarser one, and so on.

PIERANTONI AND ROZZI: GENERAL MULTIGRID-SUBGRIDDING FORMULATION FOR TLM METHOD

This technique is easily incorporated into a TLM solver and permits to model complex problems where objects of disparate geometrical or electrical scale are present in the same environment. The accuracy and the efficiency in terms of computational acceleration of TLM with subgridding is tested by analyzing, for examples, an iris loaded waveguide and a MEMS. The results show that by using the proposed subgridding method, the same accuracy is maintained, whereas a minor computational effort is required. APPENDIX The wave amplitude vectors of the coarse and fine grids introduced in (13) are defined as follows:

(25)

(26)

In the above equation (26), the definition of wave pulses , , , and , and the subscript ( , , ) indicates the localization in the subdomains , as shown in Fig. 3. boundary operator (20), (21) is a blocks partitioned The matrix (27) where

and

are (8

8) diagonal matrices

(28) In (28), the blocks defined as

and

are (4

4) diagonal matrices,

(29)

(30)

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REFERENCES [1] M. Krumpholz and P. Russer, “A field theoretical derivation of TLM,” IEEE Trans. Microw. Theory Tech., vol. 12, no. 9, pp. 1660–1668, Sep. 1994. [2] W. J. R. Hoefer, “The transmission-line-matrix method—Theory and applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 10, pp. 882–893, Oct. 1985. [3] H. Jin, R. Vahldieck, and J. S. Huang, “Direct derivation of the symmetrical condensed node and hybrid symmetrical condensed node from Maxwell’s equations using centered differencing and averaging,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2554–2561, Dec. 1994. [4] V. Trenkic, C. Christopoulus, and T. M. Benson, “Optimization of TLM schemes based on the general symmetrical condensed node,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 457–465, Mar. 1997. [5] S. Lindenmeier, L. Pierantoni, and P. Russer, “Hybrid space discretizing integral equation methods for numerical modeling of transient interference,” IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 425–430, Nov. 1999. [6] M. Righi, W. J. R. Hoefer, M. Mongiardo, and R. Sorrentino, “Efficient TLM diakoptics for separable structures,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 854–859, Apr. 1995. [7] A. Massaro, L. Pierantoni, and T. Rozzi, “Accurate analysis of wave propagation in negative uniaxial crystal,” IEEE J. Quantum Electron., vol. 40, no. 7, pp. 821–829, Jul. 2004. [8] M. Bhattacharya, P. Mazumder, and R. J. Lomax, “FD-TLM electromagnetic field simulation of high-speed III–V heterojunction bipolar transistor digital logic gates,” in 14th Int. VLSI Design Conf., 2001, pp. 470–474. [9] K. Strohm, C. Rheinfelder, and A. Schurr et al., “SIMMWIC capacitive RF switches,” in 29th Eur. Microw. Conf., Munich, Germany, Oct. 4–8, 1999, vol. 2, pp. 411–414. [10] T. Rozzi and D. Mencarelli, “Application of algebraic invariants to full-wave simulators—Rigorous analysis of the optical properties of nanowires,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 797–803, Feb. 2006. [11] J. H. Herring and C. Christopoulus, “Solving electromagnetic field problems using a multiple grid transmission-line modeling method,” IEEE Trans. Antennas Propag., vol. 42, no. 12, pp. 1654–1658, Dec. 1994. [12] P. Sewell, J. Wykes, A. Vukovic, T. M. Benson, and C. Christopoulos, “Multi-grid interface in computational electromagnetics,” Electron. Lett., vol. 40, no. 3, pp. 162–163, Feb. 2004. [13] J. Wlodarczyk, “New multigrid interface for the TLM method,” Electron. Lett., vol. 32, pp. 1111–1112, Jun. 1996. [14] M. Marrone and R. Mittra, “A new stable hybrid three-dimensional generalized finite difference time domain algorithm for analyzing complex structures,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1729–1737, May 2005. [15] B. Donderici and F. L. Teixeira, “Improved FDTD subgridding algorithms via digital filtering and domain overriding,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2938–2952, Sep. 2005. [16] P. Russer, B. Isele, M. Sobhy, and E. A. Hosny, “A general interface between TLM models and lumped element circuit models,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., San Diego, CA, May 23–27, 1994, pp. 891–894. [17] L. Pierantoni, C. Tomassoni, and T. Rozzi, “A new termination condition for the application of the TLM method to discontinuity problems in closed homogeneous waveguide,” IEEE Trans. Microw. Theory Tech, vol. 50, no. 11, pp. 2513–2518, Nov. 2002. [18] T. Rozzi and W. F. G. Mecklenbrauker, “Wide-band network modeling of interacting inductive irises and steps,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 2, pp. 235–245, Feb. 1975. [19] A. Weisshaar, M. Mongiardo, A. Tripathi, and K. Tripathi, “CAD-oriented full-wave equivalent circuit models for waveguide components and circuits,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2564–2570, Dec. 1996. [20] F. Coccetti, W. Dressel, P. Russer, L. Pierantoni, M. Farina, and T. Rozzi, “Accurate modeling of high frequency microelectromechanical systems (MEMS) switches in time- and frequency-domain,” Adv. Radio Sci., vol. 1, pp. 135–138, 2003.

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Luca Pierantoni (M’94) was born in Maiolati Spontini, Italy, in 1962. He received the Laurea (summa cum laude) degree in electronics engineering and Ph.D. degree from the University of Ancona, Ancona, Italy, in 1988 and 1993, respectively. From 1989 to 1995, he was a Research Fellow with the Department of Electronics and Automatics, University of Ancona. From 1996 to 1998, he was a Senior Research Scientist with the Institute of HighFrequency Engineering, Technical University of Munich, Munich, Germany. In 1999, he joined the Dipartimento di Elettromagnetismo e Bioingegneria, Università Politecnica delle Marche, Ancona, Italy, as an Assistant Professor. His current research interests are the development of analytical/numerical methods for the modeling of integrated optical circuits and nanodevices. Dr. Pierantoni has been a member of the Italian National Institute for the Physics of Matter (INFM) since 2001.

Tullio Rozzi (M’66–SM’74–F’90–LF’07) received the Dottore degree in physics from the University of Pisa, Pisa, Italy, in 1965, the Ph.D. degree in electronic engineering from The University of Leeds, Leeds, U.K., in 1968, and the D.Sc. degree from the University of Bath, Bath, U.K., in 1987. From 1968 to 1978, he was a Research Scientist with the Philips Research Laboratories, Eindhoven, The Netherlands. In 1975, he spent one year with the Antenna Laboratory, University of Illinois at UrbanaChampaign. In 1978, he became the Chair of Electrical Engineering with the University of Liverpool. In 1981, he became the Chair of Electronics and Head of the Electronics Group, University of Bath, where he was also Head of the School of Electrical Engineering on an alternate three-year basis. Since 1988, he has been a Professor with the Dipartimento di Elettromagnetismo e Bioingegneria, Università Politecnica delle Marche, Ancona, Italy, where he is also Head of the department. Dr. Rozzi was the recipient of the 1975 Microwave Prize presented by the IEEE Microwave Theory and Technique Society (IEEE MTT-S).

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 8, AUGUST 2007

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Resonance Absorption in Nonsymmetrical Lossy Dielectric Inserts in Rectangular Waveguides Leonid A. Rud, Senior Member, IEEE

Abstract—Features of resonance power absorption in a lossy dielectric resonator shaped as a rectangular trapezium in a rectangular waveguide are studied. It is found that such a resonator can provide strong absorption (or reflection) of the TE10 -mode input from the straight (or inclined) interface port. The maximum absorption is observed when the sum of the coupling coefficients equals unity. Simple formulas relating the scattering and absorption coefficients with the coupling coefficients at the resonance frequency are presented. The physical mechanism of resonance absorption is discussed. Index Terms—Lossy dielectrics, nonsymmetrical couplings, rectangular waveguides, resonance absorption, resonators.

I. INTRODUCTION

P

OWER absorption in partially filled and all-dielectric microwave resonators should be taken into account and reliably characterized when designing frequency-selective systems of various destination. Circuit-theory models of lossy transmission resonators, unequally coupled with input/output ports, have been proposed and the qualitative analysis of relationships between the resonator - and characteristic parameters (resonance frequency, coupling coefficients, loaded and unloaded -factors) has been performed by many authors (see, e.g., [1] and [2]). In this analysis, in [2], a condition on the coupling coefficients for the full matching of asymmetrically loaded resonators was been emphasized. Circuit-theory-based techniques for characterizing various one- and two-port resonators and measuring their parameters have been classified and further developed in [3] and [4]. Along with reflection and transmission resonators, two-port symmetric absorption ones have been specifically considered in [3] and [4]. Using the -parameter presentations from [1]–[4] and the energy conservation law, one can estimate what fraction of the input power is absorbed in a resonator due to its intrinsic loss. Other approaches to the analysis of symmetric two-port absorption resonators have been developed in [5]–[7]. They are based on the search for the resonator’s natural oscillations with complex eigenfrequencies providing the required ratio between the unloaded and external (or loaded) factors. Formulas for estimating the scattering and absorption coefficient values as the functions of the ratio of loaded and dielectric factors at the given frequency are presented in [5] and [6]. The formulas have been empirically derived in [5] from the Manuscript received December 17, 2006; revised April 2, 2007. The author is with the Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, 61085 Kharkov, Ukraine (e-mail: [email protected] kharkov.ua). Digital Object Identifier 10.1109/TMTT.2007.901614

Fig. 1. Longitudinal cross section of a rectangular-trapezium dielectric insert in a rectangular waveguide.

analysis of resonance characteristics of a transmission-type resonator shaped as a dielectric plate insert in a below-cutoff waveguide restriction. They are then used to explain the features of resonance absorption in a symmetric trapezium insert possessing properties of a reflection-type resonator. Qualitative models of symmetric two-port lossy resonators based on the complex eigenfrequency approach have been proposed in [7]. They allow one to predict magnitudes of the scattering and absorption power coefficients when changing the coupling or loss coefficient. The validity of developed models has been verified in [7] by the examples of rectangular dielectric posts and partially filled -plane stubs in a rectangular waveguide. From [3]–[7], it follows that the maximum absorption level in a symmetric resonator can reach half of the input power at the resonance frequency if the coupling (or loss) coefficient becomes critical. In unequally loaded resonators, this level has to already be determined by the two coupling coefficients, the relationship between them, and the chosen input port. In this paper, features of the resonance power absorption in a nonsymmetrical dielectric insert shaped as a rectangular trapezium in a rectangular waveguide (see Fig. 1) are studied. Such a resonator has been previously investigated as a notch filter in [8] and applied in [9] to create an idler frequency contour in a millimeter-wave parametric amplifier based on the Schottky-barrier diode. The numerical analysis showed that these resonators might have very different resonance values of the reflection and absorption coefficients depending on what port had been chosen as the input one. The study of this phenomenon nature and conditions of the maximum absorption at resonance is the principal goal of this paper. It should be noted that the considered resonator is a rejection-type one. It works on higher mode oscillations and does not belong to any class of resonators reported in [1]–[7]. Our studies are based on the full-wave numerical models of the scattering and natural oscillation problems. II. NUMERICAL MODELS AND OPTIMIZATION PROCEDURE The considered resonator structure is shown in Fig. 1. It is placed in the rectangular waveguide and characterized by

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the tilting angle and the length of the filled regular section. The material parameters are the relative permittivity and the loss tangent . The perfect electric conductivity of the waveguide walls is assumed. The structure uniformity along the waveguide narrow walls allows one to consider a scalar modes. As a normalboundary-valued problem for the is used below ized frequency parameter, the quantity where is the free-space wavelength. A. Scattering Problem The considered numerical model of the scattering problem is based on the generalized -matrix technique. When using this technique, inclined and straight interfaces in rectangular waveguides are chosen as the key discontinuities. Joining these discontinuities by the filled waveguide section of the length results in the insert configuration shown in Fig. 1. The blocks of the scattering matrix of the straight interface are diagonal ones and their elements are calculated by analytical formulas. The inclined interface relates to so-called noncoordinate structures. To calculate its generalized -matrix, we have used one of the methods of analytical regularization, namely, the semi-inversion method [10]. Solution of the scattering problem allows calculating the relmodes after the -matrix ative powers of the scattered , where are the port elements as numbers. Using the energy conservation law, the absorption coefficients can then be found as (1) (2)

B. Eigenfrequency Problem Natural oscillations in the insert as an open waveguide-type resonator are of our interest as well. We will consider them as those formed by a superposition of the direct and backward -type waveguide modes in the insert cavity. The numerical model of this problem is based on the same technique as that for the above-described scattering problem. Due to the dielectric loss and power radiated into the waveguide ports, all eigenfrequencies are complex-valued quantities , where corresponds to the resonance frequency of the th oscillation and determines its quality factor . For the given geometrical and material paramecan be found as complex roots of ters, the sought values of the determinant equation

modes in the filled regular section of the length . The mode complex amplitudes in the filled section and in the waveguide ports are determined by the solution of a homogeneous system of linear algebraic equations whose determinant fits (3). For high-quality oscillation, the value of , determined by (3), agrees well with the insert resonance frequency in the scatfactor tering problem, while the obtained natural oscillation is close to the loaded factor of a lossy insert or the external and , we one of a loss-free insert . With these values of using the known can estimate the unloaded quality factor . relationship

C. Optimization Procedure The main goal of our study is to obtain maximum absorption coefficients (1) or (2) for the lossy dielectric inserts excited by mode of ports 1 or 2 at the given frequency. A prelimthe inary analysis has showed that at small angles , the absorption practically over the whole coefficients are such that . By this reason, single-mode operation range we have chosen the cost function of the full-wave optimization procedure in the form (4) According to (2), such a cost function provides a maximum value of . In (4), is the vector of objective variables and corresponds to the given resonance frequency. The optimization procedure employed is based on the steepest descent method. Generally, the vector of objective variables in (4) can include two geometrical and two material parameters. However, this vector can be set with a smaller number of variables in accordance with peculiarities of the studied problem. An initial guess for the parameter can be borrowed from the dependences calculated by the equation similar to (3) . The natural corresponding to a simple dielectric plate oscillation formed mainly by the internal direct and backward modes is of our interest. Therefore, the initial propagating value of is chosen so that to provide the -mode propagation in the insert regular section at the or over a range of . Usually, given value of the choice of the loss tangent value is dictated by the desired factor of the natural oscillation and expected bandwidth of the resonance response in the scattering problem. However, the can be associated with those for a real parameters and dielectric and not considered as the variable ones. Initial tilting angle is set to a small value to avoid considerable shift of the resonance frequency from that for the simple dielectric plate.

(3) III. CIRCUIT-THEORY MODEL CONSIDERATIONS and are full and diagonal matrices for the where modes reflected inside the insert region from waveguide the inclined and straight interfaces, respectively, is the unit matrix, and is a diagonal matrix with exponential elements determining the phase shifts or attenuation coefficients of the

Before starting the discussion of original numerical results, we consider it necessary to set forth the circuit-theory models of lossy transmission resonators with nonsymmetrical couplings. In particular, using the results of [1] and [2], we can present

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the matrix of the reflection and transmission power coefficients and at resonance as

(5) where and are the coupling coefficients defined as the ratio of inherent loss power to power radiated in ports 1 and 2, respectively [2]. They link the loaded and unloaded factor . Using (1), (2), and (5), we can as express the absorption coefficients at resonance as (6) From (5) and (6), it follows that a two-port lossy resonator with unequal couplings has equal transmission coefficients and unequal reflection and absorption coefficients for ports 1 and 2. In the resonance, the resonator is a full-matched one from the side of port 1 (2) if the difference between the coupling coeffi. Just this case was of cients equals unity, i.e., special interest in [2]. We would like to emphasize the known fact that such a full matching mode does not necessarily entail the full transmission because a part of the input power can be absorbed in the resonator according to (6). Another “critical” combination of the coupling coefficients, related with unity and not considered previously, is (7) Under such a condition, it follows from (6) that the absorption and are identicoefficients at the resonance frequency fied directly with the coupling coefficients, namely, (8) From (7) and (8), we then obtain the conclusion that the identity (9) is valid. The reflection and transmission coefficients become

(10) (11) If one of the coupling coefficients vanishes, e.g., , then and , whereas all the other power coefficients tend to zero. IV. RESULTS OF NUMERICAL EXPERIMENTS A. Inserts With Varying Material Parameters The results of the inserts optimization under condition (4) of the maximum absorption from port 2 are presented in Fig. 2 for the case of the fixed . The optimization has been

Fig. 2. Frequency responses of the: absorption [see (a) and (b)], reflection [see (c) and (d)], and transmission [see (e)] power coefficients for the inserts optimized at different and D = 0:1 : 1 = 0:6, = 8:17 , " = 6:42, tan = 0:0046; 2 = 0:7, = 8:58 , " = 4:41, tan = 0:0060; 3 = 0:8, = 9:14 , " = 3:07, tan = 0:0062; 4 = 0:9, = 9:94 , " = 2:09, tan = 0:0049.

0

0

0

0

carried out for a sufficiently large set of frequency points from . The resulting extremum values of the range all the coefficients, calculated after finishing the optimization process at each , are plotted in Fig. 2 by the dotted curves.

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They are envelopes of the resonance spikes in the frequency responses for the inserts resonating at different . Frequency responses for four inserts are shown in Fig. 2 by the solid curves. and 2 Additional spikes for inserts 1 are caused by the resonances on the mode able to propagate in the filled section at . This is a consequence of the following regularity: the optimal permittivity -mode cutoff value increases when approaching the port at at and decreases when at . The calculations show that the sum of absorption coefficients at any . Neglecting is such that small dielectric losses, we can conclude that the calculated values of the coupling coefficients satisfying (7) and (8) are and . Due to the smoothness of the envelope (dotted) curve in Fig. 2(a), we have succeeded in its approximation with the function

Fig. 3. Extremum values of the scattering and absorption relative power coefficients for the ebonite inserts optimized at successive frequency points. The P , P , P , L , and L dependences are plotted by solid, dashed, dashed–dotted, dotted, and short–dashed lines, respectively.

(12) and then to establish that (13) and calculated by (12) and (13), as The values of well as the reflection and transmission coefficients calculated by (10)–(13) are marked in Fig. 2(a)–(e) with black circles. One can see that the approximate and numerical data are in good agreement, thereby confirming qualitative conclusions followed from the above-presented analysis of the known results under condition (7). It is a demonstrative fact that the considered structure with two open boundaries provides a really strong absorption of the input power from port 2 (straight interface) and a strong reflection from port 1 (inclined interface) at lower resonance fre, quencies [compare Fig. 2(b) and (c)]. At , , the absorption coefficients tend to be equal, , , in the same manner as in symwhereas metrical resonators operating under critical coupling [3]–[5]. Another interesting fact is the behavior of reflection coefficient responses. For the loss-free nonsymmetrical inserts at working on a higher mode, they have to be resonance and coincide over the whole frequency range (notice can be expected at the domthat the values of ). For the lossy dielectric inant mode resonances only if increases, while inserts considered here, only the value of decreases at resonance. This follows from the analysis of (10), (12), and (13). Outside of the resonance vicinity, the and are indeed close to each other responses [compare Fig. 2(c) and (d)]. It should be noted that the above-listed features and conclusions are valid for the inserts optimized in terms of all the vari) as well as of the combinaable parameters ( , , , and or . The differences are observed tion only in the resulting parameters and behavior of the envelope curves near cutoffs. If the optimization is carried out to reach a maximum absorption from port 1, the resulting geometry looks like a waveguide dielectric wedge with a large angle . In this case, the fre-

Fig. 4. Change of the geometrical parameters obtained during the ebonite inserts optimization at successive frequency points.

quency responses usually have a multiresonance character. The obtained coefficients provide the validity of relations (7)–(11) only at one given resonance frequency, but with a lower accuracy than for the above-discussed case, and those relations fail in other resonances. B. Inserts With Fixed Material Parameters The envelope curves of the obtainable values of scattering and absorption coefficients and geometrical parameters resulted in the insert optimization are shown in Figs. 3 and 4, respectively, and for the ebonite insert with the parameters . Similar to the previous case, the relation is valid. The values of the reflection and transmission coefficients calculated by (10) and (11) and the exact values of and are marked in Fig. 3 by different symbols. As one can see, the results of (10) and (11) are in a good agreement with -mode cutoff in the insert the exact ones. Approaching the regular section causes a sharp increase in the parameters and (see Fig. 4) that leads to the oscillating character of the curves in Fig. 3. The analysis of other insert samples shows that both the maximum absorption level and its location on the frequency axis depend on the chosen material parameters. This is predicted by the results presented in Fig. 2. To verify the theoretical results, an ebonite insert had been manufactured for the 23 10 mm rectangular waveguide. Its geometrical parameters were chosen using the results in Fig. 4 and taking into account capabilities of the available machines.

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Fig. 5. Comparison of the measured (symbols) and theoretical (lines) frequency responses for the ebonite insert with d = 2:14 mm and = 11:35 .

The initially chosen parameters were mm and . They are close to those in Fig. 4 for GHz . The actual dimensions of mm and the manufactured insert were . The measured and calculated scattering coefficients of this insert are shown in Fig. 5. The predicted and measured resonance frequencies are 10.79 and 10.87 GHz, respectively. The observed frequency shift ( 0.7%) and some differences in the measured magnitudes may be caused by manufacturing inaccuracy, departures of the chosen material parameters from the actual ones, and limited capabilities of the noncomputerized scalar network analyzer used for measurements. The analyzer provides the frequency MHz in the manual mode setup accuracy of and the measurement accuracy of 0.2 dB. Nevertheless, the experimental results confirm that nonsymmetrical inserts can have sufficiently different reflection coefficients from two ports and, therefore, sufficiently different loss coefficients near the resonance. C. On the Physical Mechanism of Resonance Absorption The above-presented results and considerations do not provide an answer to the question of what is the reason for the considerable difference in the resonance absorption levels for two ports. It is evident that this phenomenon is related with the difference in coupling coefficients. However, in reality, they cannot be changed independently. This is because only the tilting angle influences the couplings of the ports modes with the mode in the insert cavity. The distinction is that higher the mode of port 1 is coupled with the internal mode immediately by means of the inclined interface. As for port 2, the coupling is provided by the same interface, but after mode through the straight interthe passing of the input face and regular section of the length . Consequently, a change in the angle results in a simultaneous change in the coupling coefficients and . At small , these interrelated coefficients . This raises anare such that (9)–(11) are fulfilled with other question as to why the mentioned relation takes place in the studied resonator structure. The answers to the above-stated questions could be partially obtained if there were circuit-theory models of higher mode resonators with the interrelated loss coefficients like those proper to the considered inserts (the development of such models might

Fig. 6. Electrical field patterns for the natural oscillation (a) and for the TE mode scattering. (b) Incidence from port 1. (c) Incidence from port 2. Asterisks denote the field maximum locations.

turn out, in principle, to be problematical). That is why we have tried to get a relatively comprehensive idea of the absorption physical mechanism from the comparative analysis of the natural oscillation and in-resonance scattered fields. In this analysis, we have used the algorithm of the field calculation similar to that published in [11]. field pattern of the natural oscillation is shown The and . in Fig. 6(a) for the ebonite insert with The solution of the determinant equation (3) yields at for this insert. The field pattern demonmode in forming the natstrates the dominant role of the ural oscillation field with two almost equal maxima. We can call one. The distinctive feature is that such an oscillation the outside the insert cavity, the field is stronger in port 2. In addi-mode magnitude in the aperture of port 2 is altion, the most two times greater than that in port 1. Hence, the insert has mode incident from port 2 to be more susceptible to the at and thereby provides the condition in the scattering problem. In Fig. 6(b) and (c), the resonant scattered fields are shown -mode input from ports 1 and 2. The for the cases of the corresponding absorption coefficients are and . Such a difference in the absorption levels for two ports can be explained by the difference in the field strength levels, which, in turn, is a consequence of the inequality in coupling coefficients. In particular, from the numerical analysis, it , where is follows that the field maximum when exciting the insert from the th port. Since the absorption coefficients are in the ratio

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where the integrals are taken over the insert space , the validity becomes evident. of the inequality

V. CONCLUSIONS The presented results have shown that nonsymmetrical lossy dielectric waveguide inserts with one inclined interface possess unexpected absorption and scattering properties under the condition that the sum of port coupling coefficients or absorption coefficients equals unity. This condition is fulfilled at the certain combination of the insert geometrical and material parameters at the given resonance frequency. In such a condition, nonsymmetrical inserts provide strong absorption (reflection) of the mode input from the straight (inclined) interface port. The obtained theoretical results are confirmed by the results of real measurements. Similar properties can take place in the other resonant structures such as a waveguide insert with a stepwise interface, a free-space dielectric layer with one periodically perturbed interface, and so on. Cascading them allows creating narrowband absorption filters with a smaller number of resonators in comparison with the filters based on symmetrical resonators [7]. However, the revealed properties are unwanted in some applications, e.g., in vacuum windows with sealing dielectric sections. Nonparallelism of the section surfaces or presence of a discontinuity near one of them can provoke higher order mode parasitic resonances in the section and cause the resonant reflection or absorption of the input power that represents a potential danger of window failure.

ACKNOWLEDGMENT The author is grateful to his colleagues at the Institute for Radioplysics and Electronics (IRE), National Academy of Sciences (NAS) of Ukraine, Kharkov, Ukraine, E. Sverdlenko for the help in carrying out the experimental verification of obtained theoretical results and A. Nosich for his valuable comments on this paper’s manuscript.

REFERENCES [1] J. L. Altman, Microwave Circuits. New York: Van Nostrand, 1964, ch. 5. [2] G. Boudouris and P. Chenevier, Circuits Pour Ondes Guidées (Théorie, Réalizations et Applications). Paris, France: Dunod, 1975, ch. 5. [3] M. C. Sanchez, E. Martin, and J. M. Zamarro, “Unified and simplified treatment of techniques for characterising transmission, reflection or absorption resonators,” Proc. Inst. Elect. Eng., vol. 137, no. 4, pt. H, pp. 209–212, Aug. 1990. [4] J. R. Bray and L. Roy, “Measuring the unloaded, loaded, and external quality factors of one- and two-port resonators using scattering-parameter magnitudes at fractional power levels,” Proc. Inst. Elect. Eng., vol. 151, no. 4, pt. H, pp. 345–350, Aug. 2004. [5] L. A. Rud, “Characteristics of resonance absorption of microwave energy in dielectric waveguide resonators,” Radiophys. Quantum Electron., vol. 34, no. 9, pp. 840–842, 1991. [6] L. Minakova and L. Rud, “Eigenmode spectra and microwave power absorption resonances in waveguide dielectric resonators,” in Proc. 6th Math. Methods Electromagn. Theory Conf., Lviv, Ukraine, Sep. 10–13, 1996, pp. 215–218. [7] L. B. Minakova and L. A. Rud, “Resonance absorption in single and cascaded lossy waveguide-dielectric resonators,” Microw. Opt. Technol. Lett., vol. 36, no. 2, pp. 122–126, Jan. 2003. [8] V. V. Borschevsky, V. S. Kolesnikov, V. P. Modenov, and Y. A. Pirogov, “Resonance properties of a dielectric prism in a rectangular waveguide,” Radiotekhnika, no. 2, pp. 78–79, Feb. 1985. [9] O. N. Sukhoruchko, L. A. Rud, O. I. Belous, and A. I. Fisun, “Dielectric rejection filter in a millimeter-wave amplifier,” Prikl. Radioelektron., vol. 4, no. 2, pp. 243–245, Apr.–Jun. 2005. [10] A. A. Kirilenko and L. A. Rud, “Diffraction of waves on an inclined dividing boundary between dielectric materials in a rectangular waveguide,” Radio Eng. Electron. Phys., vol. 22, no. 10, pp. 41–50, Oct. 1977. [11] A. A. Kirilenko, L. A. Rud, and V. I. Tkachenko, “Nonsymmetrical -plane corners for TE –TE -mode conversion in rectangular waveguides,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2471–2477, Jun. 2006.

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Leonid A. Rud (M’99–SM’01) received the Radiophysics Engineering degree from the Kharkov National University of Radio Electronics, Kharkov, Ukraine, in 1964, and the Ph.D. and D.Sc. degrees in radiophysics from Kharkov National University, Kharkov, Ukraine, in 1976 and 1990, respectively. Since 1971, he has been with the Institute of Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkov, Ukraine, where he is currently a Senior Scientist with the Department of Computational Electromagnetics. His research interests are in the mathematical simulation and computer-aided design (CAD) of waveguide and antenna components and frequency-selective devices. Dr. Rud was a recipient of the 1989 State Prize of the Ukraine in Science and Technology.

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An Efficient Application of the Discrete Complex Image Method for Quasi-3-D Microwave Circuits in Layered Media Wee-Hua Tang, Member, IEEE, and Stephen D. Gedney, Fellow, IEEE

Abstract—An efficient means of evaluating reactions arising from the mixed-potential integral equation for layered media for quasi-3-D microwave circuits is presented. Analytical formulations for the -integration in the spectral domain are derived, thus avoiding expensive 2-D evaluation and interpolation of the layered Green’s function. In this paper, closed-form formulations for the resultant Sommerfeld integrals are evaluated via the robust two-step discrete complex image method. The overall computational time can consequently be greatly reduced when analyzing quasi-3-D circuits in layered media using the proposed method. Index Terms—Discrete complex image method (DCIM), layered Green’s function, method of moments (MoM), mixed potential integral equation.

I. INTRODUCTION ANY efficient computer-aided design (CAD) tools have been developed for the design and analysis of printed microwave circuits. Such tools are based on a number of techniques such as the finite-different time-domain [1], the finite-element method [2], the transmission line method [3], and the method of moments (MoM) [4]. A MoM formulation can be extremely efficient since the degrees of freedom are limited to metallic surfaces, which is a small percentage of the circuit volume. The layered Green’s function coupled with the mixed potential integral equation [5]–[7] is typically used for layered media. A significant cost of the MoM implementation is the calculation of the Sommerfeld integrals (SIs), which are computed numerically [8]. This is typically done by pre-computing and tabulating the Green’s function. The Green’s function is then computed via interpolation when evaluating the reaction integrals [9]. The radial variation of the layered Green’s function can be expressed in a closed form using a method such as the discrete complex image method (DCIM) [10]–[12]. However, the closed-form expression has been limited to circuits with fixed vertical source and field coordinates. As a consequence, a different DCIM expansion must be performed for every unique

M

Manuscript received January 6, 2007; revised May 2, 2007. W.-H. Tang is with Motorola Electronic Singapore, Singapore (e-mail: [email protected]). S. D. Gedney is with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY 40506-0046 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.901608

vertical source and field plane. This adds considerable complexity when modeling circuits with both vertical and horizontal conductors. When using interpolation, multiple 2-D tables must be precomputed and then efficient 2-D interpolation schemes must be used to evaluate the layered Green’s functions. For example, Ling and Jin [9] proposed to tabulate the Green’s function as pair in a sheet, where is the a function of for every transverse distance between the source and field points. Ling and Jin utilized a Chebyshev interpolation scheme to interpolate pair and the Lagrange interpothe Green’s function for a lation scheme for . The tabulated data requires fine sampling in , , and in the near field to integrate through singularities. Relatively fine sampling is also required to yield accurate solutions in the far field due to oscillatory behavior of the dyadic Green’s function (DGF). As a consequence, the setup time can be expensive. The 2-D interpolation of the Green’s function is also very time consuming during the fill process. Kinanyman and Aksun and more recently, Vrancken and Vandenbosch presented efficient approaches for analyzing printed circuits with a 2.5-D geometry (i.e., circuits limited to purely horizontal and purely vertical metal surfaces) [8], [13]–[15]. They proposed to perform the vertical -integrations in the spectral domain, and compute the resultant Sommerfeld integrations numerically. As a consequence, only a 1-D interpolation of the Green’s function becomes necessary. Vrancken and Vandenbosch applied the numerical inverse Fourier transform to evaluate the analytical forms of vertical source and field integrations in the spatial domain [14], while Kinanyman and Aksun (and more recently Onal et al.) applied numerical integration [13], [15]. Kinanyman and Aksun then applied a three-level DCIM to transform the integrations into the spatial domain in a closed form by the general pencil of function (GPOF) [16]. In this paper, it is shown that analytically derived SIs based on performing the vertical integrations in the spectral domain can be derived for an arbitrary multilayered media. It is also shown that the resulting SIs can be accurately approximated in a closed form by appropriately employing Aksun’s two-level DCIM [12]. By having the expressions in an analytical form, it is recognized that if the field radiated by a vertical source lies on a material interface boundary, an asymptotic extraction must be performed prior to the DCIM expansion. It is also shown that far-field DCIM models can also be accurately represented by extracting surface wave modes. This paper can be summarized as follows. Section II outlines the analytical formulations for the vertical integrations

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where primes indicate source coordinates, is the parametric coordinates, and can be expanded as an SI DGF.

Fig. 1. Vertical field and source cells in a layered media that support vertical half-rooftop basis functions.

in the spectral domain. Section III presents the appropriate methods for formulating a DCIM expansion of the resultant SIs. Section IV presents validating results for various reactions using the proposed methods. Section V provides results for an actual implementation of the proposed method in a MoM simulation and demonstrates the acceleration of the proposed method.

are the transverse projection of the

(2) is the “spectral-domain” representation of the where Green’s function. The order of integral can then be interchanged, leading to

II. ANALYTICAL FORMULATION OF -INTEGRATION IN THE SPECTRAL DOMAIN

(3)

In the proposed formulation, the following discretization can be assumed: horizontal metal surfaces can be discretized with an arbitrary fitted mesh consisting of triangles or general quadrilaterals. The vertical mesh is restricted to rectangular quadrilaterals that are purely vertical. However, vertical quadrilaterals are not limited to being aligned along any horizontal axis. The divergence conforming function space proposed by Graglia et al. [17] is employed for both basis and testing functions. Vertical patches are here restricted to zeroth-order basis. These reduce to classical rooftop basis functions on the vertical rectilinear quadrilaterals. Note that the vertical quadrilaterals support both vertical currents (due to basis functions sharing horizontal edges), as well as horizontal currents (due to basis functions sharing vertical edges). However, due to the restriction of the geometry, these currents are assumed to be either purely vertical or purely horizontal. A Galerkin discretization of the mixed-potential integral equation is assumed [18], and consequently, the test and basis function spaces are identical. Initially, consider the reaction integral involving a vertical field cell and a vertical source cell. The field cell is bounded along the vertical -direction , and the source cell has the bounds , as by illustrated in Fig. 1. Assume the reaction is computed between a vertical test function and a vertical basis function, the support of the functions over a single patch is one-half of a rooftop function, as illustrated in Fig. 1. The reaction integral involving the vector potential can be expressed in a general form as

is known in a closed form in the spectral domain, and the -integration can be performed analytically. For example, consider the case when both the source and field cells are embedded is expressed as [19] in the th layer. Here,

(1)

(7)

(4) where

(5) (6) , is the generalized reflection coefficient as defined in [20], and the th layer has a lower bound of , and an upper bound of , as illustrated in Fig. 1. The -dependence of is simply complex exponential functions and, thus, the -integrations in (3) can be derived analytically in a closed form. For example, from (4)–(6), one can derive

TANG AND GEDNEY: EFFICIENT APPLICATION OF DCIM FOR QUASI-3-D MICROWAVE CIRCUITS IN LAYERED MEDIA

where the values for are provided in the Appendix. Given the closed form in (7), a spatial Green’s function that is only a function of the transverse coordinates can be derived via a Sommerfeld integration (8) where . Combining (7) and (8) with (2), the reaction integral expressed in (1) can simply be reduced to (9)

for the vertical to vertical reaction. Thus, once the vertical integrations are computed analytically, the reaction integrals only require a transverse integration. Next, consider a vertical source reacting with a horizontal field. This reaction is expressed by the integral

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Thus, they can be applied to any source–field patch pair with these vertical bounds. An important observation is that the singularity of the transhas been reduced. However, the kernel verse kernel may still exhibit a logarithmic singularity (the kernel may not if the two be numerically singular, but its derivative is) at domains overlap. Thus, care must be taken when computing the integral numerically. To ensure high accuracy and rapid convergence, numerical integration based on the Gauss–Legendre quadrature rule is performed for the transverse field integration, while integration based on a Lin–log rule [21] can be appropriately employed for the transverse source integration to capture the singularity. Analytical forms of the -integrations for the remaining polarizations of the vertical rooftop functions can be similarly derived. Analytical forms for when the source and observation patches are in different layers can also be derived. Furthermore, analytical formulations for the -integrations including reaction , , and pairs involving the layered Green’s functions due to currents and charges can also be derived in a similar manner. These have been summarized in [22]. III. CLOSED-FORM REPRESENTATION VIA THE DCIM The spatial Green functions arising from this method [e.g., (8) and (11)] require the evaluation of an SI, which can be expressed in the general form

for (10) where

(11)

(12)

(15)

where is the th-order of the Bessel function, are coeffi, and is an integer . If is cients that are functions of . If is odd, the ineven, the integrand has a complex pole at tegrand then has a branch-point singularity. In general, Aksun’s two-level DCIM [12] can evaluate this integral to good accu. One exception is if any of the exponentials racy for all defining have a zero argument. This can occur when a vertical patch is reacting with a horizontal patch at a material layer. For example, consider the reaction due to a source current located on a vertical patch and an observation point located on a horizontal patch, as illustrated in Fig. 2(a). Observing (12)–(14), when the observation point is located on the interface layer, the situation can arise where or . in This will lead to a zero argument for the exponentials of (12) or in (14). This leads to a constant term, and can cause a singularity to occur when computing the inverse vector via singular value decomposition steps of the GPOF. Fortunately, this is easily rectified via an asymptotic extraction. Equation (11) can be rewritten as

(13)

(14) It is noted that and are specialized to the bounds of the source and field cells. However, they are trans. lational-invariant relative to the transverse coordinates

(16)

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denotes the zeroth-order Hankel function of the where second kind. The first integral in (16) can then be accurately approximated by employing Aksun’s two-level DCIM [12]. This approach has the advantage over the conventional DCIM in that it does not require a priori knowledge of the Green’s function behavior. With the extraction of the quasi-static terms from the Green’s function, the two-level DCIM yields a highly accurate solution in the near-field region. When employing the two-level DCIM in the far-field, the error becomes significant as becomes large. This is due to the fact the DCIM can not accurately approximate the asymptotic behavior inherent in the surface-wave modes that are dominant in the far-field. To resolve this problem, surface-wave modes can be extracted from the DCIM in the spectral domain and add the surface-wave modes back in the spatial domain [23], . [24]. The far field is typically in the regions when Hence, in the far-field, (16) can be rewritten as

(19) where

(20)

Fig. 2. Conducting strip embedded with three-layer media between two conducting planes. The vertical conductor is located at z -bounds from 0.508 to 0.762 mm. All dielectric layers have a loss tangent (tan ) of 0.001. and Q (a) Cross-sectional view of the stripline. (b) Magnitude of Q computed via the SI and the DCIM. (c) Zoom-in view of far-field region. (From [22].)

where and are derived from (5) in the limit , and are expressed as

is the surface-wave term in the spectral domain of the function in (11), is the surface-wave mode, is the residual, and is the number of surface-wave modes being extracted. A numerical contour-integral scheme [23], [24] based on the Cauchy residue theorem is employed to efficiently compute the surface-wave modes and residual in (20). This approach avoids the derivations of analytical forms for the surface-wave modes of the Green’s functions in the layered media, which are difficult to derive. Finally, applying Aksun’s two-level DCIM and substituting (18) and (20) into (19) leads to

(17) where denotes the relative permittivity in the second integral is evaluated via the identity

(21)

th layer. The and (18)

(22)

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where , denote the number of sample points for the two, and , denote the coefficients level DCIM, and exponents obtained from the GPOF method in the first and is the spatialsecond contours of the two-level DCIM, and . domain representation of IV. IMPLEMENTATION Once the closed-form expressions are obtained, the matrix fill can be rendered much more efficient since only the transverse integrals need to be performed. When using zeroth-order divergence conforming basis functions, such as the Rao–Wilton–Glisson (RWG) basis, these reactions can be computed with analytical acceleration, as outlined in [22]. The current implementation employs high-order divergence conforming basis on curvilinear cells for horizontal patches. Consequently, numerical integration was used. The 2.5-D Green’s function was computed via an efficient 1-D adaptive-window interpolation scheme (i.e., only a function of ). The Green’s functions are pre-computed via the DCIM and tabulated. An interpolation scheme is then utilized to approximate the Green’s function. In this paper, we employed the Silvester Lagrange interpolation [25]. The sample points of the interpolation scheme are adaptively sampled over variable ranges of providing controllable accuracy. When computing reactions between horizontal source and observation patches that are coincident, or nearly coincident, an efficient approach based on the Khayat–Wilton transform [26], [27] was used. Otherwise, standard Gauss-quadrature integration is employed. V. VALIDATION The proposed method is validated through the study of circuits with both vertical and horizontal conductors in a multilayered media. The results obtained from the two-level DCIM method are compared with those computed via a direct Sommerfeld integration using the weighted averages method [28]. The first example is chosen to validate the accuracy and efficiency of the DCIM. Consider a stripline circuit printed on three different dielectric layers embedded by two conducting planes S/m, as illustrated in Fig. 2(a). with conductivity mm, while The horizontal source patch is located at the vertical observation patch has -bounds from 0.508 to 0.762 mm. The 3-D DGF is used to account for the reaction between the transverse and vertical -integration. The closed-form -integration of the Green’s function is computed via the DCIM and the results are compared with a direct numerical Sommerfeld integration. Fig. 2(b) shows the magnitude of the two pairs of analytical formulations of -integrations reacting with the Green’s versus , where is the free-space functions wavenumber. For the far-field region, the surface-wave mode extraction is included in the DCIM. Fig. 2(c) illustrates the zoom-in view of the far-field region. The next example is the microstrip spiral inductor illustrated in Fig. 3. The microstrip circuit is printed on a conductor-backed RT/Duroid substrate with a permittivity of 9.6, a loss tangent , and a thickness of 2 mm. The microstrip conductor width and spacing is 2 mm. The spiral inductor has two air bridges across one of the feed lines (port 2). The height and

Fig. 3. Microstrip spiral inductor discretized with a quadrilateral mesh. (a) Cross-sectional view. (b) 3-D view of actual mesh layout. (c) Magnitude of S -parameters computed via a direct SI or the DCIM, and independently computed by Zeland Software Inc.’s IE3D. (From [22].)

span of the air bridges are 1.0 and 6.0 mm, respectively. The -parameters were computed using the DCIM, and the closedform -integration of Green’s functions for vertical basis reactions is utilized. These results are illustrated in Fig. 3(c). The computed results agreed quite well with Zeland Software Inc.’s IE3D [29], even at the resonant frequency. The computed results also agree very well with data published by Ling and Jin in [9]. The CPU times recorded on a 2-GHz Pentium IV processor are recorded in Table I for the direct Sommerfeld Integration and DCIM evaluations of the DGF with the double complex floating point arithmetic computation. It is observed that the DGF computation is accelerated by nearly a factor of 7 for this example when evaluated via the DCIM. As a last example, a microstrip inductor with an air bridge with a finite-metallization thickness illustrated in Fig. 4 is studied. The microstrip circuit is printed on a conductor backed RT/Duroid with a permittivity of 9.6, a loss tangent , and a thickness of 0.2 mm. The microstrip conductor width and spacing is 0.2 mm. The height and span of the air bridge is 0.1 and 1.1 mm, respectively. The magnitude of the computed -parameters is illustrated in Fig. 4(c) and is compared with results simulated via Zeland Software Inc.’s IE3D.

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TABLE I CPU TIMES FOR THE MICROSTRIP SPIRAL INDUCTOR IN DOUBLE COMPLEX FLOATING POINT COMPUTATION (FROM [22])

were performed analytically, thus eliminating the need for numerical integration along the vertical direction. It was shown that the DCIM can be used to express the resulting DGF in a closed form. However, prior to the DCIM expansion, it was found that one must perform an asymptotic extraction for potential constants. The resulting expression can be approximated very accurately via a two-level DCIM. Furthermore, it was shown that the far field can be accurately evaluated by extracting the surface-wave modes of the layered media. Through validation, it was shown that this method is computationally efficient, and has controllable accuracy. APPENDIX The coefficients in (7) are derived as [22] (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9)

Fig. 4. Layout of a microstrip spiral inductor with an air bridge with finite thickness. (a) Cross-sectional view. (b) 3-D view of layout (not the actual mesh). (c) Comparison of magnitude of S -parameters with computed via the DCIM with IE3D. (From [22].)

(A.10)

VI. CONCLUSION An efficient approach to compute the vertical current reaction of microwave circuits in layered media has been presented. The approach is limited to circuits with purely horizontal and purely vertical conductors. This is by no mean limiting since circuits fabricated with lithography, etching, or epitaxial growth-type processes grow metal layers along the vertical dimension. Practical problems of interests are layered circuits with vias, interconnects, air bridges, metals with thickness, etc. In this paper, the analytical formulation of the integration for the vertical current represented by a rooftop basis reacts with the Green’s function in the spectral domain is derived. Expressing the DGF as an SI, and swapping the order of integration, the vertical integrals

(A.11) (A.12) (A.13)

REFERENCES [1] A. Taflove, Advances in Computational Electrodynamics: The FiniteDifference Time-Domain. Norwood, MA: Artech House, 1998. [2] D. E. Livesay and K. M. Chen, “Electromagnetic fields induced inside arbitrary shaped biology bodies,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 12, pp. 1273–1280, Dec. 1974. [3] C. Christopoulous, The Transmission-Line Modeling Method. New York: IEEE Press, 1995.

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[4] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [5] K. A. Michalski and D. L. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. II. Implementation and results for contiguous half-space,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 345–352, Mar. 1990. [6] K. A. Michalski and D. L. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I. Theory,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 335–344, Mar. 1990. [7] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 508–519, Mar. 1997. [8] P. Gay-Balmazand and J. R. Mosig, “Three-dimensional planar radiating structures in stratified media,” Int. J. Microw. Millim.-Wave Comput.-Aided Eng., vol. 7, pp. 330–343, Sep. 1997. [9] F. Ling and J.-M. Jin, “Full-wave analysis of multilayer microstrip problems,” in Fast and Efficient Algorithms in Computational Electromagnetics, J. M. J. W. C. Chew, E. Michielssen, and J. Song, Eds. Boston, MA: Artech House, 2001, pp. 729–780. [10] J. D. G. Fang, J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipole in a multilayer medium,” Proc. Inst. Elect. Eng., vol. 135, pp. 297–303, Oct. 1988. [11] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Delisle, “A closed-form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 588–592, Mar. 1991. [12] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 5, pp. 651–658, May 1996. [13] N. Kinayman and M. I. Aksun, “Efficient use of closed-form Green’s functions for the analysis of planar geometries with vertical connections,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 593–603, May 1997. [14] M. Vrancken and G. A. E. Vandenbosch, “Hybrid dyadic-mixed-potential and combined spectral-space domain integral-equation analysis of quasi-3-D structures in stratified media,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 216–225, Jan. 2003. [15] T. Onal, M. I. Aksun, and N. Kinayman, “An efficient full-wave simulation algorithm for multiple vertical conductors in printed circuits,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3739–3745, Oct. 2006. [16] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, Feb. 1989. [17] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher-order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 329–342, Mar. 1997. [18] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press, 1998. [19] W. C. Chew, J. S. Zhao, and T. J. Cui, “The layered medium Green’s function—A new look,” Microw. Opt. Technol. Lett., vol. 31, no. 4, pp. 252–255, Nov. 2001. [20] W. C. Chew, Waves and Fields in Homogeneous Media. Piscataway, NJ: IEEE Press, 1995. [21] J. Ma, V. Rokhlin, and S. Wandzura, “Generalized Gaussian quadrature rules for systems of arbitrary functions,” J. Numer. Anal., vol. 33, no. 3, pp. 971–996, 1996.

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[22] W. H. Tang, “Efficient integral equation method for 2.5D microwave circuits in layered media,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Kentucky, Lexington, KY, 2005. [23] F. Ling and J. M. Jin, “Discrete complex image method for Green’s functions of general multilayer media,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 400–402, Oct. 2000. [24] T. C. Cui and W. C. Chew, “Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Geosci. Remote Sens., vol. 37, no. 3, pp. 887–900, Mar. 1999. [25] P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers. Cambridge, U.K.: Cambridge Univ. Press, 1986. [26] M. A. Khayat and D. R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3180–3190, Oct. 2005. [27] W.-H. Tang and S. D. Gedney, “An efficient evaluation of near singular surface integrals via the Khayat–Wilton transform,” Microw. Opt. Technol. Lett., vol. 48, no. 8, pp. 1583–1586, Aug. 2006. [28] K. A. Michalski, “Extrapolation methods for Sommerfeld integral tails,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1405–1418, Oct. 1998. [29] IE3D: 3D Full-Wave Electromagnetic Simulation and Optimization Package. ver. 9.3, Zeland Softw. Inc., Fremont, CA, 2003. Wee-Hua Tang (S’00–M’07) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Kentucky, Lexington, in 1999, 2001, and 2005, respectively. From December 2005 to May 2006, he was a Post-Doctoral Researcher with the University of Kentucky. He is currently with Motorola Electronic Singapore, Singapore. His research interests are computational electromagnetics and microwave circuit modeling.

Stephen D. Gedney (S’84–M’91–SM’97–F’04) received the B.Eng. degree (with honors) from McGill University, Montreal, QC, Canada, in 1985, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987 and 1991, respectively. He is currently a Professor with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, where he has been since 1991. In the summers of 1992 and 1993, he was a National Aeronautics and Space Administration (NASA)/American Society for Engineering Education (ASEE) Faculty Fellow with the Jet Propulsion Laboratory, Pasadena. In 1996, he was a Visiting Professor with Hughes Research Laboratories (now HRL laboratories), Malibu, CA. In 2002, he was named the Reese Terry Professor of Electrical and Computer Engineering at the University of Kentucky. His research interests are in the area of computational electromagnetics with applications in the areas of electromagnetic scattering, microwave circuit modeling and design, and the analysis of mixed-signal systems.

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New Series Expansions for the 3-D Green’s Function of Multilayered Media With 1-D Periodicity Based on Perfectly Matched Layers Hendrik Rogier, Senior Member, IEEE

Abstract—A new formalism based on perfectly matched layers (PMLs) is presented to derive new series expansions for the Green’s function of an infinite set of point sources with a 1-D periodicity embedded in a layered medium. Several PML-based series expansions, both in the spatial and spectral domains, combined with suitable convergence acceleration techniques such as the Shanks transform and Ewald transform, are proposed and their efficiency is evaluated. For each pair of excitation and observation locations, an optimal series expansion in terms of accuracy and CPU time is proposed, resulting in a significant speed-up compared to existing approaches. Index Terms—Green’s functions for multilayered media, integral-equation techniques, perfectly matched layers (PMLs), periodic structures.

I. INTRODUCTION ANY practical waveguiding, scattering, and radiating devices, such as gratings [1]–[3], arrays [4], metamaterials and electromagnetic (EM)-bandgap structures [5]–[7], , can be efficiently nonradiating dielectric waveguides [8], modeled as 2-D configurations [1]–[3], [5], [6], or 3-D configurations [4], [7], [8] with 1-D periodicity [1]–[3], 2-D periodicity [4], [6], or 3-D periodicity [7]. Moreover, several configurations relevant in the analysis of EM field propagation, such as EM shieldings [9] and reinforced concrete walls [10], [11], are well approximated by a periodic structure with infinite extent. By applying the Floquet–Bloch theorem, the analysis of a periodic structure with infinite extent is restricted to a representative unit cell. For the description of the fields by means of an integral-equation technique, the periodic Green’s function is required to account for the periodic character of the configuration, as discussed in Section II. The Green’s function approach is especially efficient for a configuration of scatterers embedded in a planar multilayered dielectric background medium since the effect of the background medium can be fully incorporated into the Green’s function. However, in order to determine the EM fields of a single point source located in the layered medium, a time-consuming Sommerfeld integration is required for the inverse Hankel transform that transforms the analytical solution from the spectral domain to spatial domain.

M

Manuscript received February 7, 2007; revised April 23, 2007. This work was supported by the Fund for Scientific Research-Flanders (FWO-V). The author is with the Information Technology Department, Ghent University, B-9000 Ghent, Belgium (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.902580

The periodic Green’s function is written as a spatial-domain series or as a spectral-domain series of the field solution for the single point source. For a 2-D periodic grid of point sources, the spectral-domain series can be expressed in an analytical way and Sommerfeld integrations are avoided. In this paper, however, we consider a 1-D periodic grid of point sources. In that case, even for the spectral-domain series, an inverse Fourier transform is required, as demonstrated in Section III-A. An additional complication results from the fact that both spectral- and spatial-domain series tend to be slowly converging for certain positions of the excitation and the observation point. Therefore, much attention has been devoted to derive series expansions that converge more rapidly, mainly by combining both the spatial- and spectral-domain series. Convergence acceleration techniques were proposed in literature for the 1-D periodic 2-D Green’s function [12]–[16] and for the 2-D periodic 3-D Green’s function. In [17], we presented a new efficient approach to calculate the 1-D periodic 2-D Green’s function based on the use of perfectly matched layers (PMLs). Up to now, little has been published about the 1-D periodic 3-D Green’s function. In [18] and [19], the 1-D periodic 3-D Green’s function for a periodic set of point sources located in free space is accelerated by means of the Ewald transform. However, an extension of this approach in order to include a stratified dielectric background medium is not straightforward. In [20], the 1-D periodic 3-D Green’s functions for a microstrip substrate are derived in the spectral domain first, and the corresponding spatial-domain quantities are obtained through an efficient sum of inverse Fourier transforms. In this paper, a new formalism based on PMLs is proposed to derive a fast converging series expansion for the 1-D periodic 3-D Green’s function of layered media. As in [21]–[25], PMLs [26]–[29] are used to transform the open layered medium into a closed waveguide configuration. An efficient expansion for the 3-D Green’s function of a point source in the stratified background medium in terms of a set of discrete modes of the closed waveguide containing the PML is then possible, while the PMLs mimic the open character. For a theoretical background about issues of completeness and convergence, the reader is referred to [30] and [31]. Based on the PML-based modal expansion, analytical series expansions are then derived in Section III-B. As both the spectral- and spatial-domain series suffer from slow convergence, special attention is devoted in Section III-C to accelerate convergence of the PML-based series. This acceleration is based on the use of the Shanks transform and the Ewald transform. Moreover, as in some cases, the accuracy and efficiency of the PML-based series is not ensured in the singularity

0018-9480/$25.00 © 2007 IEEE

ROGIER: NEW SERIES EXPANSIONS FOR 3-D GREEN’S FUNCTION OF MULTILAYERED MEDIA

region (when source and observation point are close to each other), a hybrid series is derived in Section III-C that combines the classic nonperiodic Green’s function, capturing the correct singularity, with a PML-based series for the remaining periodic part. In Section IV, we validate the new series and evaluate their accuracy and efficiency in terms of CPU time. For each combination of excitation and observation locations, an optimal series expansion in terms of accuracy and CPU time is proposed. II. INTEGRAL EQUATION FOR A 1-D PERIODIC CONFIGURATION OF CONDUCTORS IN A LAYERED MEDIUM dielectric layers Consider a planar stratified medium of with finite thickness along the -direction, but of infinite extent in the - and -directions. The stack of dielectric layers can be backed by a perfect electrically conducting ground plane. A 1-D periodic 3-D configuration of planar conductors, located in different layers parallel to the -plane and infinitely thin in the -direction, can be described by means of the mixed potential integral equation

functions the resulting series

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to the spatial domain. Moreover,

(2) is known to be slowly convergent so many spatial Green’s function evaluations are required in order to obtain an accurate result. In free space, (2) reduces to

(3) , and the Poisson transform can be applied to with obtain the spectral-domain analog on

(1) (4) provided that we are able to determine the kernel funcand that account for tions both the 1-D periodicity and the presence of the multireplayered background medium. The excitation -plane of the resents the component tangential to the electric field response of the background medium to an imposed electric field, which can, e.g., be an incident . plane wave of the form The solution of (1) yields the surface current distribution on all planar conductors. By considering only planar conductors, the Green’s function dyadic ponent

and . In this with case, both spectral- and spatial-domain series can be combined efficiently using the Ewald transform in order to obtain two fast converging series [18], [19]. In the general case of a multilayered background medium, however, no analytical expressions are available for the terms in the spectral-domain series [20]

was reduced to a single comin (1). (5)

III. 1-D PERIODIC 3-D GREEN’S FUNCTIONS AND

A. Classic Green’s Function Series Consider a planar multilayered dielectric background medium in which we place a 1-D grid of point sources. We assume that the problem is periodic in the -direction with the period given by ; two adjacent point source excitations differ by, at most, a phase factor . The conventional approach of finding the 3-D Green’s function for this 1-D periodic configuration proceeds by first determining the spatial-domain Green’s and for functions a single point source of elementary current and charge, respectively. In order to evaluate these Green’s functions in a multilayered medium, a time-consuming Sommerfeld integration is required to convert the spectral-domain Green’s

Now, a time-consuming inverse Fourier transform must be evaluated for each term in the spectral series. In [20], a fundamental speedup is obtained by extracting the singular behavior and by transforming that part into two fast converging series by means of the Kummer and Poisson transforms. B. Series Based on PMLs Assume that the background medium (Fig. 1) is translation invariant in the - and -direction (a planar stratified medium) and that all material variations in the -direction are located in a region that is bounded in that direction. The largest distance over which material variations extend in the -direction is denoted by (in Fig. 2, where a single-layered microstrip substrate is shown, this coresponds to the thickness of the substrate). For a faster evaluation of the 1-D periodic 3-D Green’s functions, we

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being the eigenvalues and with with being the excitation coefficients of the eigenmodes. Note that, in a free-space environment terminated by two PMLs forming a waveguide of total thickness , the eigenvalues are given by and the excitation coefficients are , given by is chosen at the bottom PEC where the origin for and plate (Fig. 1). For a microstrip substrate of thickness (Fig. 2), we distinguish between TE and TM eigenmodes [33]. The eigenvalues of the TE and TM modes are the solution of

Fig. 1. Pertinent to the calculation of the Green’s function.

(7)

Fig. 2. 1-D periodic configuration of point sources on a microstrip substrate terminated by a PML.

then construct a parallel-plate waveguide by terminating the free space with two perfect electrically conducting plates backed by and with material paan isotropic PML with thickness and [32], as shown in Fig. 1. As the constitutive rameters parameters are chosen the same as for the air layer, the isotropic PML can be combined with the air region to form a new air region with a complex thickness. A similar approach also applies to a planar stratified medium above a ground plane, as for the microstrip substrate in Fig. 2. As in [21] and [22], the 3-D Green’s function can be expanded into a series of discrete eigenmodes of the resulting parallelplate waveguide in the following way:

(6)

with , , and , and for with and the TE case, and for the TM case. For both TE and TM modes, the propagation constants of the leaky modes (which concentrate in the microstrip substrate) and of the Berenger modes (which concentrate in the PML) can be determined very rapidly based on analytical approximations derived in [34]. Given the initial estimates, the exact locations of the propagation constants of the TE and TM modes are found by performing a few iterations with Newton’s method, as described in [35]. For general multilayered substrates, this technique is applicable for the Berenger modes only. The leaky modes are then determined based on the complex root finders proposed in [36]. can be exIn (6), the Green’s function panded into the TE modes of the closed PML-waveguide, and the excitation coefficients of the TE eigenmodes in or on top of are given by (8), shown at the bottom the substrate of this page.1 No TM modes are needed. , (6) is comAs for the Green’s function posed of both TE and TM modes. The excitation coefficients of the TE eigenmodes in or on top of the substrate are given in (9), shown at the bottom of this page,1 whereas the excitation coefficients of the TM eigenmodes in or on top of the are given by (10), shown at the bottom substrate of the following page.1 1The

index n is omitted for simplicity of notation.

(8)

(9)

ROGIER: NEW SERIES EXPANSIONS FOR 3-D GREEN’S FUNCTION OF MULTILAYERED MEDIA

The 3-D Green’s function for a 1-D periodic grid of point sources can then be written as

(11) Application of the Poisson transform leads to the following equivalent series expansion

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For small mode orders , series (11) is slowly convergent as a function of . In [33], the Shanks transform is proposed to accelerate convergence of the PML-based mode expansion of the 2-D Green’s function for a line source on a microstrip substrate. In a similar way, we apply the Shanks procedure for both series (11) and (12) in order to accelerate convergence as a function of the PML-based mode index . Moreover, in this paper, for each index , the Shanks transform is applied to accelerate converfor both series gence as a function of the periodicity index (11) and (12). For a successful acceleration of convergence by means of the Shanks algorithm, the series must asymptotically behave as a geometric series. In general, this is not the case for (11) as a function of , when considering small values of the PML-based mode order and, hence, the Shanks algorithm is not very effective in accelerating the slow convergence. There. To fore, an Ewald transform is applied for mode orders this end, we rewrite the periodic Green’s functions as a sum of the modified spectral-domain series (12)

(12) with

.

C. Convergence Analysis and Acceleration By applying the PML formalism, we have replaced series (2), of a spatial Green’s which is a series over a single index function terms that require time-consuming Sommerfeld integration, by two equivalent series (11) and (12), over double indices and , but for which the terms are easy to evaluate. Let us first concentrate on the index , which runs over the different modes in the waveguide formed by the substrate together with the PMLs. In [33], the convergence of a 2-D Green’s function expansion for a line source on a microstrip substrate is analyzed and it is shown that exponential convergence is obtained prois not too small. In a similar way, vided the distance series (12) converges at a rate proportional to for large, being constant for fixed, thus exponentially as a function of , yielding a rapidly converging series provided is not too small. Series (11) converges at a the distance , for large, rate proportional to a constant, thus exponentially as a function of , yield a fast or converging series provided that either the distance is not too small. As a function of , on the distance the other hand, series (12) converges at a rate proportional to , for large and for arbitrary, but fixed , thus exponentially as a function of , resulting in a rapidly conis not too small. Yet, verging series provided the distance in order to obtain exponential convergence for series (11) as a function of , it is required that the index is sufficiently large.

(13) and the modified spatial-domain series (11)

(14) in which as

is the th-order exponential integral defined

(15) A suitable choice for the Ewald splitting parameter has to be made. Based on the theory developed in [37] for the periodic 2-D Green’s function series in free space, a suitable choice is

(16)

(10)

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where is the maximum exponent permitted in the spectral series (13), is the desired error, and is the number of terms necessary to achieve convergence in the spatial series , , and (14). Typical choices are [37]. In practice, only the two lowest order TE leaky modes, the two lowest order TE Berenger modes, the two lowest order TM leaky modes, and the two lowest order TM Berenger modes are used in the Ewald accelerated series (13) and (14) so we for and for choose . The two series are to be complemented with either the remaining spectral-domain series

Fig. 3. Classic versus PML series for 1-D periodic 3-D Green’s function jG (0; y; 10 mm; 5 mm; y ; 9:5 mm)j.

(17) or the remaining spatial-domain series

(18) such that the complete series expansion accelerated by the Ewald procedure is given by (19) All acceleration schemes presented up to now do not allow to calculate the 1-D periodic 3-D Green’s function accurately and efficiently when both distances and are very small. Indeed, in [38], it is shown for the 2-D case that the PML-based series does not capture the correct singular behavior at the interface of a nonmagnetic microstrip of substrate. Therefore, we combine part of the PML-based series (11) with one term of the classical series (2) to capture the correct singularity. The following series is proposed in order to for evaluate the 1-D periodic 3-D Green’s function very small distances and at the interface:

(20) The evaluation of the Green’s function for a single point source in the stratified medium is then performed by means of the classical Sommerfeld integration. IV. EXAMPLES In order to assess the accuracy and efficiency of the different series expansions for the 1-D periodic 3-D Green’s functions

, we consider a microstrip substrate with thickness mm, permittivity , and permeability . In order to obtain an expansion into PML-based modes, a closed waveguide is formed by adding a perfect electrically conducting mm and plate above the substrate such that mm. A strongly absorbing PML is obtained for and . The free-space wavelength at the operating cm. Based on the root-finding frequency is chosen to be approach described in [35], it takes 1.56 s to determine the first 1000 Berenger TE modes, 2.22 s for the first 1000 leaky TE modes, 1.20 s for the first 1000 Berenger TM modes, and 1.83 s for the first 1000 leaky TM modes. Let us determine the Green’s functions and for cm a 1-D periodic set of point sources with spacing (Fig. 2). The number of terms for each series evaluation is chosen in an adaptive manner in order to ensure that the relative error is smaller than 10 . Moreover, for the PML-based series, no more than 1000 Berenger TE modes, 1000 leaky TE modes, 1000 Berenger TM modes, and 1000 leaky TM modes are taken into account. In a first numerical experiment, we investigate whether the PML sufficiently mimics the open character of the microstrip substrate by placing both excitation and observation points in the air region. The excitation point is located 0.5 mm above the substrate–air interface, whereas the observation point is placed at a height of 1 mm above the substrate–air interface. Both mm. points are separated by a lateral distance of In Fig. 3, we compare different PML-based expansions for mm mm mm , choosing with the classic Sommerfeld integrated spectral series (2) accelerated by the method proposed in [20]. The agreement between different approaches is seen to be excellent. ranging from up to Specifically, for distances , the relative error between the hybrid series expansion (20) and the approach proposed in [20] remains below 0.14%. mm mm mm , it Concerning was found that the relative error between the hybrid series expansion (20) and the approach proposed in [20] remains even below 0.006%. Hence, it is illustrated that the PML modes yield a sufficiently accurate representation of the

ROGIER: NEW SERIES EXPANSIONS FOR 3-D GREEN’S FUNCTION OF MULTILAYERED MEDIA

Fig. 4. Classic versus PML series for 1-D periodic 3-D Green’s function jG (0; y; 9 mm; 0; y ; 9 mm)j.

evanescent spectrum in order to represent the open character of the structure. Let us now turn our attention to a more practical situation and place both excitation and observation points at the substrate–air mm mm interface. In order to evaluate with efficiently in the singularity region, i.e., for dissmaller than , we make use of the hytances brid series expansion (20). Comparing this result to the classic Sommerfeld integrated spectral series (2), accelerated by the method proposed in [20], a relative error smaller than 0.11% is found. In Fig. 4, the results of different series expansions mm mm are shown as a funcfor , excluding the singularity by evaluating the tion of . A small discrepancy series starting from distances of between the classic Sommerfeld evaluation and PML-based series expansions is noticed due to the fact that the first series had not fully converged. For the classic Sommerfeld integration (2), 400 terms and Shanks acceleration were used. Yet, the classical 1-D periodic Green’s function evaluation had not fully converged. However, the agreement between the PML-based series and classic Sommerfeld integration spectral series (5), accelerated by the method proposed in [20], is excellent. A more detailed view is presented in Fig. 5, where the relative error of the different series expansions is presented. The hybrid series expansion (20) is used as the reference solution since it is the only expansion that provides accurate results (convergence within a relative error smaller than 10 ) at all distances (the spectral series (5) is divergent for ) and it was found to be stable for a large range of PML parameters. It is found that the relative error between the hybrid series (20) and the Baccarelli approach . As [20] remains smaller than 0.14% for all values of for the spectral-domain expansion (12) with Shanks acceleraand , the relative error drops below tion for both indices larger than . As for the spa0.03% for distances tial-domain expansion (11) with Shanks acceleration for both indices and , the relative error drops below 1% for distances larger than . Concerning the different series expansions for mm mm , a higher accuracy than for is obtained for the Green’s function

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Fig. 5. Relative error of the different PML and classic expansions for (0; y; 9 mm; 0; y ; 9 mm)j. jG

j

G

TABLE I CPU TIMINGS FOR CALCULATING 200 POINTS OF (0; y; 9 mm; 0; y ; 9 mm)j FOR jy [ =60; 3:2 ] y

0 j2

, especially near the singularity (for small values of ). In agreement with the theory derived in [38], all PML-based series expansions for already exhibit an accuracy better than 0.004% at distances as small as . The discrepancy between the hybrid series (20) and the classic Sommerfeld integration spectral series (2), accelerated by the method proposed in [20], is smaller than 0.02%. In Table I, the total CPU timings are shown for calcuup to lating 200 points in the range from based on the different series expansions for mm mm with a relative conusing a Pentium T7400 vergence error smaller than Centrino Duo 2.16-GHz machine with 2-GB RAM. It is clear that the PML-based expansions are several orders of magnitude faster than the series based on 400 conventional Sommerfeld integrations, accelerated by the Shanks algorithm. Moreover, all PML-based series evaluations are clearly much faster than the acceleration method proposed in [20] based on the classic Sommerfeld integration spectral series (2). The spectral series (12) with Shanks acceleration only performed on index is the fastest and most accurate approach. Given the exponential , convergence rate proportional to and for all which leads to very fast convergence for large and , the convergence acceleration values of

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Fig. 6. Classic versus PML series for 1-D periodic 3-D Green’s function jG (0; y; 9 mm; 5 mm; y ; 9 mm)j.

Fig. 7. Relative error of the different PML and classic expansions for (x; 0; 9 mm; x ; 0; 9 mm)j. jG

obtained by performing Shanks acceleration also on index is clearly insignificant and the overhead of the algorithm results in a slower evaluation procedure. For the spatial series, however, or applying the Shanks transform on both indices and using the Ewald technique by combining series (13), (14), and (18), clearly helps in terms of CPU time. In this range, however, all spatial series perform slower than the spectral PML-based series. As for the singularity region, the hybrid series expansion (20) is the preferred approach in terms of accuracy and effimm mm ciency. The evaluation of in the range from to based on (20) takes 0.35 s so the optimal evaluation based on (20) for the singularity region and (12) with Shanks outside the singuacceleration performed only on index larity region takes in a total of 0.81 s for distances up to . Similar conclusions can be drawn for the calmm mm : the most culation of optimal evaluation based on (20) for the singularity region and (12) with Shanks acceleration performed only on index outside the singularity region takes in a total 1.20 s for up to (compared to 242 m 23 s for distances the classic series (2) with Sommerfeld integration). In Fig. 6, mm mm mm is shown as a funcfor a separation mm and for tion of . The convergence problems with the series of 400 terms based on classic Sommerfeld integration and Shanks acceleration are again obvious, whereas an accuracy better than 0.007% is achieved for both the spectral- and spatial-domain series. The discrepancy between the hybrid series (20) and the classic Sommerfeld integration spectral series (2), accelerated by the method proposed in [20], is now smaller than 0.004%. and Let us now choose a fixed distance study the relative error for the different series expansions for mm mm as a function of , . Although the spectral-domain seas plotted in Fig. 7 for , ries (12) does not exhibit an exponential decay for and performing the Shanks transform on both indices yields an accuracy better than 1% as long as . In order to obtain the same accuracy with the spatial-domain

series (11) or with the Ewald technique by combining series (13), (14), and (18), the separation in the -direction must at . Yet, even with convergence accelleast be eration by means of the Shanks algorithm, slow convergence is seen for the spectral-domain series (12) in the range from up to . Within that range, it is found that the hybrid spatial series (20) is more efficient to in terms of CPU time. In the range from , the spectral-domain series (12) is the most efficient technique, provided that, in this case, the Shanks and . The optimal algorithm is applied for both indices mm mm , evaluation technique for based on the hybrid spatial series (20) in the singularity region , takes and spectral-domain series (12) for 2.47 s of CPU time (compared to 159 m 53 s for the classic series (2) based on Sommerfeld integration). mm mm , the spectral-doAs for main series (12), spatial-domain series (11), or the Ewald technique by combining series (13), (14), and (18) yield an accuracy better than 0.02% for separations in the -direction of at least . Again, slow convergence is observed for the spectral domain series (12) in the range from up to . Within that range, it is found that the hybrid spatial series (20) is more efficient in terms of CPU time. In Table II, the total CPU timings are shown for calculating 150 points based on different series expansions for mm mm in the range from to on a Pentium T7400 Centrino Duo 2.16-GHz machine with 2-GB RAM. The optimal evaluamm mm , based tion technique for on the hybrid spatial series (20) in the singularity region and , takes 3.68 s spectral-domain series (12) for of CPU time. In Fig. 8, we show the results for the Green’s function series mm mm inside the dielectric subfor and strate, calculated as a function of . The theory derived in [38] for a fixed distance predicts that the Green’s function converges to the correct singularity when evaluated inside the substrate, but not on the

ROGIER: NEW SERIES EXPANSIONS FOR 3-D GREEN’S FUNCTION OF MULTILAYERED MEDIA

j

G

TABLE II CPU-TIMINGS FOR CALCULATING 150 POINTS (x; 0; 9 mm; x ; 0; 9 mm)j FOR jy [ =10; 3 y

0 j2

=8]

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. The application at distances as small as of the hybrid series (20) in the singularity region and the takes 0.66 s spectral-domain series (12) for and 0.88 s for . As a of CPU time for reference, the extension of the technique proposed in [20] and 11 min 35 s for takes 5 min 48 s for evaluating . evaluating V. CONCLUSIONS A new efficient and accurate formalism based on PMLs has been presented to derive new series expansions for the Green’s function for both the magnetic vector potential and scalar potential of a set of point sources with 1-D periodicity embedded in a layered medium. Several PML-based series expansions, both in the spatial and spectral domain, combined with suitable convergence acceleration techniques such as the Shanks transform and Ewald transform, were examined in terms of accuracy and CPU time. In the singularity region, i.e., for the excitation and observation point locations for which (21)

Fig. 8. Classic versus PML series for 1-D periodic 3-D Green’s function (0; y; 7 mm; 0; y ; 7 mm) . G

j

j

interface. The resulting function near the singularity is, however, highly oscillating, thus the application of the Shanks transform is required to ensure convergence for all PML-based se, resulting in relative accuracies ries expansions for . better than 0.08% at distances as small as The accuracies obtained with the PML-based series expansions are of the same order of magnitude with a relafor tive error smaller than 0.1% at distances as small as . The optimal periodic series solution in terms of accuracy and CPU time consists of applying the hybrid series (20) in the singularity region and the spectral-domain series (12) for . This approach takes 0.58 s of CPU time for and 0.96 s for . For comparison, an extension of the technique proposed in [20] takes 9 min 9 s for evaland 18 min 7 s for evaluating . uating Finally, we study the Green’s function series mm mm for the excitation at the substrate–air interface and the observation point inside the dielectric substrate. In agreement with the theory derived in [38], the application of the Shanks transform is again able to ensure convergence for all PML-based series expansions for . The spectral PML-based series (12) is again found to be the most accurate with a relative accuracy better than . The accuracies 0.2% at distances as small as obtained with the PML-based series expansions for are of the same order of magnitude with a relative error smaller than 0.2% for both the spectral and spatial PML-based series

the most efficient approach is to combine the classic nonperiodic Green’s function based on Sommerfeld integration with a PML-based spatial-domain series for the remaining periodic part. For all other observation/excitation pairs, the spectral-domain PML-based series with Shanks acceleration for both the and indices is the preferred approach. Future research consists of applying the new periodic series expansion to a variety of scattering and microwave applications. REFERENCES [1] J. Moore, H. Ling, and C. S. Liang, “The scattering and absorption characteristics of material-coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag., vol. 41, no. 9, pp. 1281–1288, Sep. 1993. [2] P. Petre, M. Swaminathan, G. Veszely, and T. K. Sarkar, “Integral equation solution for analyzing scattering from one-dimensional periodic coated strips,” IEEE Trans. Antennas Propag., vol. 41, no. 8, pp. 1069–1080, Aug. 1993. [3] H.-C. Chu, S.-K. Jeng, and C. H. Chen, “Reflection and transmission characteristics of single-layer periodic composite structures for the TE case,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1065–1070, Jul. 1997. [4] C. Craeye, A. B. Smolders, D. H. Schaubert, and A. G. Tijhuis, “An efficient computation scheme for the free space Green’s function of a two-dimensional semiinfinite phased array,” IEEE Trans. Antennas Propag., vol. 51, no. 4, pp. 766–771, Apr. 2003. [5] A. Della Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 331–334, 2006. [6] G. Lovat, P. Burghignoli, F. Capolino, D. Jackson, and D. Wilton, “Analysis of directive radiation from a line source in a metamaterial slab with low permittivity,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 1017–1030, Mar. 2006. [7] M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1418–1430, Apr. 2005. [8] M. Bozzi, D. Li, S. Germani, L. Perregrini, and K. Wu, “Analysis of NRD components via the order-reduced volume-integral-equation method combined with the tracking of the matrix eigenvalues,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 339–347, Jan. 2006.

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[9] H. Rogier, B. Baekelandt, F. Olyslager, and D. De Zutter, “The FE-BIE technique applied to some 2-D problems relevant to electromagnetic compatibility: Optimal choice of mechanisms to take into account periodicity,” IEEE Trans. Electromagn. Compat., vol. 42, no. 3, pp. 246–256, Aug. 2000. [10] E. Richalot, M. Bonilla, M. F. Wong, V. Fouad-Hanna, H. Baudrand, and J. Wiart, “Electromagnetic propagation into reinforced-concrete walls,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 3, pp. 357–366, Mar. 2000. [11] D. Peña, R. Feick, H. D. Hristov, and W. Grote, “Measurement and modeling of propagation losses in brick and concrete walls for the 900-MHz band,” IEEE Trans. Antennas Propag., vol. 51, no. 1, pp. 31–39, Jan. 2003. [12] A. W. Mathis and A. F. Peterson, “A comparison of acceleration procedures for the two-dimensional Green’s function,” IEEE Trans. Antennas Propag., vol. 44, no. 4, pp. 567–571, Apr. 1996. [13] S. Singh and R. Singh, “Application of transforms to accelerate the summation of periodic free-space Green’s function,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 11, pp. 1746–1748, Nov. 1990. [14] R. M. Shubair and Y. L. Chow, “Efficient computation of the periodic Green’s function in layered dielectric media,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 3, pp. 498–502, Mar. 1993. [15] R. Lampe, P. Klock, and P. Mayes, “Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 8, pp. 734–736, Aug. 1985. [16] R. E. Jorgenson and R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag., vol. 38, no. 5, pp. 633–642, May 1990. [17] H. Rogier and D. De Zutter, “A fast converging series expansion for the 2-D periodic Green’s function based on perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1199–1206, Apr. 2004. [18] F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 3-D Green’s function with one dimensional periodicity using the Ewald method,” in IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 2006, pp. 2847–2850. [19] F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 3-D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method,” J. Comput. Phys., vol. 223, pp. 250–261, Apr. 2007. [20] P. Baccarelli, C. D. Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1350–1362, Apr. 2006. [21] F. Olyslager and H. Derruder, “Series representation of Green dyadics for layered media using PMLs,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2319–2326, Sep. 2003. [22] D. Vande Ginste, E. Michielssen, D. De Zutter, and F. Olyslager, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1538–1548, May 2006. [23] P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 349–354, Feb. 2001. [24] H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas Propag., vol. 49, no. 2, pp. 185–195, Feb. 2001. [25] H. Rogier and D. De Zutter, “A fast technique based on perfectly matched layers for the full-wave space domain solution of 2-D dispersive microstrip lines,” IEEE Trans. Computer-Aided Design Integr. Circuits Syst., vol. 22, no. 12, pp. 1650–1656, Dec. 2003. [26] J. P. Bérenger, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas Propag., vol. 44, no. 1, pp. 110–117, Jan. 1996.

[27] W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations in stretched coordinates,” Microw. Opt. Technol. Lett., vol. 7, pp. 599–604, Sep. 1994. [28] S. D. Gedney, “An anisotropic PML absorbing media for the FDTD simulation of fields in lossy and dispersive media,” Electromagnetics, vol. 16, pp. 399–415, 1996. [29] L. Knockaert and D. De Zutter, “On the stretching of Maxwell’s equations in general orthogonal coordinate systems and the perfectly matched layer,” Microw. Opt. Technol. Lett., vol. 24, pp. 31–34, Jan. 2000. [30] L. F. Knockaert and D. De Zutter, “On the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1650–1653, Nov. 2002. [31] F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math., vol. 64, no. 4, pp. 1408–1433, 2004. [32] H. Derudder, F. Olyslager, and D. De Zutter, “An efficient series expansion for the 2-D Green’s function of a microstrip substrate using perfectly matched layers,” IEEE Microw. Guided Wave Lett., vol. 9, no. 12, pp. 505–507, Dec. 1999. [33] H. Rogier and D. De Zutter, “Convergence behavior and acceleration of the Berenger and leaky modes series composing the 2-D Green’s function for the microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1696–1704, Jul. 2002. [34] H. Rogier and D. De Zutter, “Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 4, pp. 712–715, Apr. 2001. [35] H. Rogier, L. Knockaert, and D. De Zutter, “Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminated by a perfectly matched layer,” Microw. Opt. Technol. Lett., vol. 37, pp. 167–171, May 2003. [36] L. Knockaert and H. Rogier, “An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media,” AEU—Int. J. Electron. Commun., vol. 59, pp. 230–238, Jun. 2005. [37] F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2977–2984, Sep. 2005. [38] H. Rogier and D. De Zutter, “Singular behavior of the Berenger and leaky modes series composing the 2-D Green’s function for the microstrip substrate,” Microw. Opt. Technol. Lett., vol. 33, pp. 87–93, Apr. 2002.

Hendrik Rogier (S’96–A’99–M’00–SM’06) was born in 1971. He received the Electrical Engineering and Ph.D. degrees from Ghent University, Gent, Belgium, in 1994 and in 1999, respectively. He is currently a Post-Doctoral Research Fellow of the Fund for Scientific Research-Flanders (FWO-V), Department of Information Technology, Ghent University, where he is also a Part-Time Professor with the Department of Information Technology. From October 2003 to April 2004, he was a Visiting Scientist with the Mobile Communications Group, Vienna University of Technology. He has authored or coauthored approximately 35 papers in international journals and approximately 50 papers in conference proceedings. His current research interests are the analysis of EM waveguides, EM simulation techniques applied to electromagnetic compatibility (EMC), and signal integrity problems, as well as to indoor propagation and antenna design, and in smart antenna systems for wireless networks. Dr. Rogier was a two-time recipient of the URSI Young Scientist Award presented at the 2001 URSI Symposium on Electromagnetic Theory and at the 2002 URSI General Assembly.

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Interpolated Coarse Models for Microwave Design Optimization With Space Mapping Slawomir Koziel, Senior Member, IEEE, and John W. Bandler, Life Fellow, IEEE

Abstract—The efficiency of space-mapping optimization depends on the quality of the underlying coarse model, which should be sufficiently close to the fine model and cheap to evaluate. In practice, available coarse models are often cheap, but inaccurate (e.g., a circuit equivalent of the microwave structure) or accurate, but too expensive (e.g., a coarse-mesh model). In either case, the space-mapping optimization process exhibits substantial computational overhead due to the excessive fine model evaluations necessary to find a good solution if the coarse model is inaccurate, or due to the cost of the parameter extraction and surrogate optimization sub-problems if the coarse model is too expensive. In this paper, we use an interpolation technique, which allows us to create coarse models that are both accurate and cheap. This overcomes the accuracy/cost dilemma described above, permitting significant reduction of the space-mapping optimization time. Examples verify the performance of our approach. Index Terms—Coarse model, engineering optimization, microwave design, space mapping, space-mapping optimization.

I. INTRODUCTION PACE MAPPING [1]–[5] is a methodology that allows efficient optimization of expensive or “fine” models by means of the iterative optimization and updating of so-called “coarse” models, which are less accurate, but cheaper to evaluate. Provided that the misalignment between the fine and coarse models is not significant, space-mapping-based algorithms typically provide excellent results after only a few evaluations of the fine model. A similar idea is shared by other surrogate-model-based methods [6]–[12], however, many of them do not use a coarse model: the surrogate model is created by direct approximation of the available fine model data. Space mapping is widely used in the optimization of microwave devices [1]–[3], [13]–[17], where fine models are often based on full-wave electromagnetic simulations, whereas coarse models may be physically based circuit models. Recently, space-mapping techniques have been applied to design problems in a growing number of areas (e.g., [18]–[20]).

S

Manuscript received February 2, 2007. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007239 and Grant STPGP336760, and by Bandler Corporation. S. Koziel is with the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 (e-mail: [email protected]). J. W. Bandler is with the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1, and also with Bandler Corporation, Dundas, ON, Canada L9H 5E7 (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.902618

A number of papers cover different aspects of space mapping, including the development of new algorithms ([2], [3], [21], [22]), space-mapping-based modeling [23]–[28], theoretical foundations ([21], [29], [30]), etc. It is well known that the performance of a space-mapping optimization algorithm depends on the quality of the underlying coarse model, which should be as good a representation of the fine model to be optimized as possible. On the other hand, the coarse model should also be easy to optimize and significantly less expensive than the fine model. Under these conditions, a space-mapping algorithm can reach a satisfactory solution after a few fine model evaluations. Moreover, the total cost related to the parameter extraction and surrogate optimization sub-problems, involving multiple coarse model evaluations, is negligible in comparison with the total cost of fine model evaluation. In practice, however, available coarse models are either cheap, but inaccurate, e.g., a circuit equivalent of the microwave structure, or accurate, but too expensive, e.g., a microwave structure evaluated using the same simulator as the fine model, but with a coarser mesh. In the first case, the space-mapping optimization process exhibits substantial computational overhead due to the excessive fine model evaluations necessary to find a good solution (i.e., the number of space-mapping iterations is larger than it could be if the accurate model were used). In the latter case, the total cost of solving the parameter extraction and surrogate optimization sub-problems may be comparable with the total cost of fine model evaluation or may even determine the total cost of the space-mapping optimization process. In this paper, we utilize an interpolation technique, which allows us to create coarse models that are both accurate and, at the same time, sufficiently cheap. In particular, the original coarse model is evaluated on a relatively coarse simulation grid and the modified model is obtained by interpolating this data using a suitable methodology. In this way, the original coarse model (which is typically assumed to exhibit sufficient accuracy, but is too expensive to make space-mapping optimization efficient) is evaluated at a limited number of points, which allows us to reduce the total space-mapping optimization time.

II. MOTIVATION Let us consider the following optimization problem: a second-order tapped-line microstrip filter [31] shown in Fig. 1. . The fine model The design parameters are is simulated in FEKO [32]. The number of meshes for the fine model is 360, which ensures mesh convergence for the

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Fig. 1. Geometry of the second-order tapped-line microstrip filter [31].

Fig. 3. Second-order tapped-line filter: initial fine model response (solid line) and coarse model R response (dashed line) at x .

Fig. 2. Coarse model R

of the second-order tapped-line filter (Agilent ADS).

structure. Simulation time for the fine model is 290 s. The design specifications are dB for dB for dB for

GHz GHz GHz

GHz GHz GHz

We also use to denote the response vector of the fine model. In this case, the model response is the evaluation of at 33 frequency points uniformly distributed in the interval from 3 to 7 GHz. is the structure We consider two coarse models. Model in Fig. 1, also simulated in FEKO; however, the number of and meshes is only 48. The number of meshes for are measured at the optimal solution of , which is mm. The simulation time for is approxis the circuit model implemented in imately 11 s. Model Agilent ADS [33] shown in Fig. 2. Evaluation time for is and are also approximately 1.2 s. As before, symbols used to denote the response vectors of the respective models. For this problem, we used input space mapping and output space mapping [21]. In particular, the space-mapping surrogate , where vector model is defined as is found using parameter extraction [21], after which is . the residual vector evaluated by We perform space-mapping optimization twice: using model with its optimal solution as a starting point, and then with its corresponding optimal solution using model mm as a starting point. Figs. 3 and 4 show the responses of the fine and coarse model at , as well as the fine and coarse model at , reexhibits better accuracy spectively. It is seen that the model

Fig. 4. Second-order tapped-line filter: initial fine model response (solid line) and coarse model R response (dashed line) at x .

than the model with respect to matching the fine model response. Note also that the response at its optimal solution does not satisfy the design specifications. Table I shows the optimization results. The optimized fine model responses are shown in Fig. 5. As we can see, the final specification error is almost independent of which a coarse model is used in the space-mapping algorithm. Different responses reflect different optima found by the algorithm in both for and for . cases: However, because model is more accurate than , the optimization result is obtained with a smaller number of fine is much model evaluations. On the other hand, because , the relative computational cost of cheaper to evaluate than solving the parameter extraction and surrogate model optimization sub-problems is much higher for the algorithm using than for the algorithm using (59% versus 10%). Hence, the than for . total optimization time is larger for It should also be mentioned that model does not need to achieve mesh convergence because it is a coarse model. However, as an effect of the lack of mesh convergence, the mesh topology and number of mesh elements vary due to the variation of geometrical design parameters during optimization. Conseis more difficult to optimize than , which quently, model

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TABLE I SPACE-MAPPING OPTIMIZATION RESULTS FOR THE SECOND-ORDER TAPPED-LINE MICROSTRIP FILTER

Includes fine model evaluation at the starting point

Fig. 6. Grid example for the 2-D case.

Fig. 7. Example of the base set for fuzzy-system interpolation n = 2.

be the function interpolating the data pairs denotes the value of the function at point . , we define as For each . In other words, is the result of “rounding” to one of the grid points. We define an interpolated coarse model as follows: Let

.

Fig. 5. Second-order tapped-line filter: final fine model response at the solution obtained with space mapping using the R model (solid line) and the R model (dashed line).

is reflected in a larger number of evaluations while performing parameter extraction and surrogate optimization. is accurate, but expensive As mentioned earlier, model (and not easy to optimize), while model is cheap, but not accurate. It can be inferred from the data in Table I that the cost of space-mapping optimization can be substantially reduced if we can provide a coarse model that is accurate, cheap, and easy to optimize. Section III introduces the concept that satisfies these conditions. III. INTERPOLATED COARSE MODELS A. Notation and Concept , be an original coarse Let model, which will typically be the model evaluated by the same simulator as the fine model, but using a coarse be a grid mesh (as in the example of Section II). Let , where is a user-defined grid size and denotes the set of integers; is the number of design variables. into hypercubes with points being Grid divides , we define corners of these hypercubes. For each as the center of the corresponding hypercube, and denote by the hypercube itself. Fig. 6 shows an example of the grid and hypercubes for . , we associate a base set , which With each . We is a set of points located in the hypercube with center a set of responses of the original coarse will denote by model evaluated at points from .

(1) In the remaining discussion here, we consider the realization of this concept, as well as implementation details. We employ fuzzy systems, techniques successfully used in the computeraided design of microwave structures by other authors (e.g., [34] and [35]). B. Realization It is desirable that the model is a continuous function, as this will facilitate further optimization of the space-mapping surrogate. This can be achieved using a fuzzy-system interpolation based on the points located at the corners of the hypercubes defined by the grid . In particular, we have , . An example of the base set for is shown in . Fig. 7. Note that the number of base points is In this study, we use a fuzzy system with triangle membership functions and centroid defuzzification [36]. The fuzzy , where and system uses data pairs , . In our realization, each , contains only one fuzzy interval region (i.e., the whole interval). Membership functions for the th variable are defined as shown in Fig. 8. Having defined membership functions, we need to generate fuzzy rules from given data pairs. We use if–then rules of the is in THEN , where is the output of form IF the rule. At the level of vector components, this means is in

AND AND

is in

is in

AND THEN

(2)

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3. Generate the base set 4. For each

calculate

; create

5. Prepare fuzzy rules; prepare interpolating function 6. Update (save and corresponding ) 7. Return

Fig. 8. Input interval tions.

[y ; y + ] and the corresponding membership func-

where , are components of vector . In our case, all rules are conflicting because they have the same IF part, but a different THEN part. However, each rule has a different set of associated membership functions. In particular, if , then the membership function associated with component of the th rule is , otherwise it is . Each rule has a degree that is assigned in the following way. is in AND is in For the rule “IF AND AND is in THEN ”, the degree of this rule for any , , is defined as denoted by (3) , are coeffiwhere cients in the following expansion of , . The output of our fuzzy system is determined using centroid defuzzification (4) (1) and which is a realization of an interpolated coarse model since can be written as is a function of both and . As mentioned before, in this paper, we only use triangular is a continuous membership functions. This assures that function over the whole domain of the coarse model (regard). Other less of the continuity of the original coarse model choices, e.g., z-shaped membership functions, would permit keeping first-order differentiability of the interpolated model, which may be important for some problems. C. Implementation Details In order to reduce the number of evaluations of the original coarse model , the interpolated model is implemented as a database of interpolating functions (4), which is updated if the coarse model needs to be evaluated. The coarse model evaluation process can be described by the following algorithm ( denotes the model database). 1. Get (the point to evaluate the coarse model at) , retrieve the interpolation function and 2. If return

In other words, if belongs to a hypercube for which the interpolating function is already set, the response of is obtained as the value of the interpolating function corresponding to this hypercube. Otherwise, must first be created (which requires setting up the base set, acquiring the original coarse model data, and calculating the necessary coefficients), then evaluated, and finally, stored in the database. As mentioned before, the number of base points for each hypercube is , i.e., the number grows exponentially with the number of design variables. However, in practice, many hypercubes considered during the optimization process are adjacent to each other. This means that many corner points are shared between hypercubes. Due to this, the actual average number of original coarse model evaluations per hypercube is smaller than . We observed that, depending on the problem, the figure is , where is typically from 2 to 4. It should also be noted that there is a tradeoff between the accuracy of the interpolated coarse model and its computational cost. On one hand, we want to take advantage of the accuracy of the original coarse model, as this would allow us to maintain the number of fine model evaluation as low during the space-mapping optimization process. On the other hand, we need to keep the interpolated coarse model fast; otherwise the benefits of using space-mapping optimization are lost due to the computational overhead related to parameter extraction and surrogate model optimization. Both model accuracy and speed depend on the user-defined grid , and the grid size should be adjusted so that both the accuracy and computational cost of the interpolated coarse model are sufficient. This may be easily achieved if the number of design variables is small, such as two or three. For larger values of , due to the exponential growth of the number of base points for each hypercube, the number of actual evaluations of the original coarse model may be too large and all the benefits of using our interpolation scheme may be lost. In pracunless the model tice, our method should not be used for is not highly nonlinear. Another method working regardless of the number of design variables will be described elsewhere. IV. EXAMPLES As a first example, consider again the second-order tapped-line microstrip filter described in Section II. We optimized this filter again, using the interpolated model (4) based on with grid size mm. the original coarse model Table II shows the optimization results (the optimized design ). It is seen that space-mapping optimization is with the interpolated model gives the same final specification and (cf. Section II), error as optimization with models but with substantially smaller computational cost. The reduction of the total optimization time in comparison to optimization with is 69% (56%). Most of the savings arise from using the interpolated coarse model. This resulted in

KOZIEL AND BANDLER: INTERPOLATED COARSE MODELS FOR MICROWAVE DESIGN OPTIMIZATION WITH SPACE MAPPING

TABLE II SPACE-MAPPING OPTIMIZATION RESULTS FOR THE SECOND-ORDER TAPPED-LINE MICROSTRIP FILTER (WITH INTERPOLATED COARSE MODEL)

Includes fine model evaluation at the starting point

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TABLE III SPACE-MAPPING OPTIMIZATION RESULTS FOR THE PATCH ANTENNA

Includes fine model evaluation at the starting point

Fig. 9. Geometry of the patch antenna.

a reduction of the total number of evaluations of to 8 (versus in space-mapping optimization). 115 when directly using Consider the patch antenna shown in Fig. 9. This antenna is printed on a substrate with relative dielectric constant and height mm. The design parameters are the patch . The objective is to obtain length and width, i.e., is simulated 50- input impedance at 2 GHz. The fine model in FEKO [32]. The number of meshes for the fine model is 1024, which ensures mesh convergence for the structure. Simulation time for the fine model is 34 s. is the structure in We consider two coarse models. Model Fig. 9 also simulated in FEKO, however, the number of meshes is 0.45 s. Model is only 100. Simulation time for model is an interpolated model (4) based on the original coarse model with grid size mm. The number of meshes for and are measured at the optimal solution of , which mm. The fine model response at is is 38.8 . For this problem, we used the same space-mapping surrogate model as described in Section II. We perform space-mapping and then using model . optimization twice: using model . In both cases, we use the same starting point Table III shows the optimization results. As we see, the final value of the input impedance is similar in both cases (the for and corresponding final designs are for ), although the accuracy is better for than for (a specification error of 0.05 for versus 0.09 for ). The computational cost of space-mapping than optimization is also more than two times smaller for for , which is because the total number of evaluations of the original coarse model has been reduced from more than 500 in space-mapping optimization) to 29 (when directly using (when using ). Consider the microstrip notch filter with mitered bends [37] shown in Fig. 10. The design parameters are mil. Other parameters are mil,

Fig. 10. Microstrip notch filter with mitered bends [37].

mil, and (loss tangent 0.0009). The model is simulated in Sonnet em [38] with a fine grid of fine 0.5 mil 0.5 mil. The simulation time for is 1 h and 34 min (12 points per frequency sweep). The design specifications are for for for

GHz GHz GHz

GHz GHz GHz

We consider the original coarse model , which is also simulated in Sonnet em, however, with a coarse grid of 5 mil 1 mil. The simulation time for is 65 s. Obviously, cannot be directly used in the optimization process because it is avail, which able only on a coarse grid. Instead, we use model with grid size is an interpolated model (4) based on mil. The optimal solution of this model is mil. Fig. 11 shows the fine and coarse model re. sponses at , which is the We also use another coarse model, i.e., circuit model implemented in Agilent ADS [33] and shown in is approximately 1.5 s. Fig. 12. The evaluation time for has its substrate permittivity tuned to , Model which allows us to shift the center frequency of its response to 13.2 GHz at . Without tuning, the center frequency of is approximately 11.12 GHz. This causes severe misalignment between the fine model and and makes it unsuitable for space-mapping optimization. Fig. 13 shows the responses of at before and after the tuning of . To solve our problem, we used the same space-mapping surrogate model described in Section II. We perform with space-mapping optimization twice: using model

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TABLE IV SPACE-MAPPING OPTIMIZATION RESULTS FOR THE MICROSTRIP NOTCH FILTER

Includes fine model evaluation at the starting point

Fig. 11. Microstrip notch filter: initial fine model response (solid line) and coarse model R response (dashed line) at x .

Fig. 14. Microstrip notch filter: final fine model response at the solution obtained with space mapping using R model (solid line) and R model (dashed line).

Fig. 12. Coarse model R

of the notch filter (Agilent ADS).

substantially smaller for than for . This is because is more accurate than . Note that although the evaluation time for the original coarse is 65 s, the total time required for parameter extraction model and surrogate optimization is only 28 min. This is because our interpolated model only required 26 evaluations of the original coarse model. V. CONCLUSION

Fig. 13. Microstrip notch filter: response of the coarse model R at x without tuning of " (dashed line) and with " tuned to 1.46 (solid line).

starting point

and then using model with starting point (the optimal solution of the (tuned) ). Table IV shows the optimization results. As we can see, the final solutions (the responses shown in Fig. 14) satisfy the design specification in both cases (the corresponding final designs for and for ). The are computational cost of space-mapping optimization, however, is

An interpolation technique for creating coarse models suitable for space-mapping optimization has been presented. Our technique allow us to build models that are tradeoffs between accuracy and computational cost. As a result, we are able to reduce the computational cost of space-mapping optimization by decreasing the number of fine model evaluations necessary to obtain satisfactory solutions (because of good coarse model accuracy), as well as by reducing the total cost of solving the parameter extraction and surrogate optimization sub-problems (because the interpolated coarse model is faster than the original coarse model). Examples demonstrate the robustness of our approach. ACKNOWLEDGMENT The authors thank Sonnet Software Inc., Syracuse, NY, for em, and Agilent Technologies, Santa Rosa, CA, for making ADS available. The authors acknowledge discussions with

KOZIEL AND BANDLER: INTERPOLATED COARSE MODELS FOR MICROWAVE DESIGN OPTIMIZATION WITH SPACE MAPPING

J. E. Rayas-Sánchez, Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, Mexico, and his help on the notch filter example. REFERENCES [1] J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R. H. Hemmers, “Space mapping technique for electromagnetic optimization,” IEEE Trans. Microw. Theory Tech., vol. 4, no. 12, pp. 536–544, Dec. 1994. [2] J. W. Bandler, Q. S. Cheng, N. K. Nikolova, and M. A. Ismail, “Implicit space mapping optimization exploiting preassigned parameters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 378–385, Jan. 2004. [3] J. W. Bandler, Q. S. Cheng, D. H. Gebre-Mariam, K. Madsen, F. Pedersen, and J. Søndergaard, “EM-based surrogate modeling and design exploiting implicit, frequency and output space mappings,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1003–1006. [4] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 337–361, Jan. 2004. [5] D. Echeverria and P. W. Hemker, “Space mapping and defect correction,” Int. Math. J. Comput. Methods Appl. Math., vol. 5, no. 2, pp. 107–136, 2005. [6] N. M. Alexandrov and R. M. Lewis, “An overview of first-order model management for engineering optimization,” Optimization Eng., vol. 2, no. 4, pp. 413–430, Dec. 2001. [7] A. J. Booker, J. E. Dennis, Jr., P. D. Frank, D. B. Serafini, V. Torczon, and M. W. Trosset, “A rigorous framework for optimization of expensive functions by surrogates,” Structural Optimization, vol. 17, no. 1, pp. 1–13, Feb. 1999. [8] J. E. Dennis and V. Torczon, “Managing approximation models in optimization,” in Multidisciplinary Design Optimization, N. M. Alexandrov and M. Y. Hussaini, Eds. Philadelphia, PA: SIAM, 1997, pp. 330–374. [9] S. J. Leary, A. Bhaskar, and A. J. Keane, “A knowledge-based approach to response surface modeling in multifidelity optimization,” Global Optimization, vol. 26, no. 3, pp. 297–319, Jul. 2003. [10] S. E. Gano, J. E. Renaud, and B. Sanders, “Variable fidelity optimization using a Kriging based scaling function,” presented at the 10th AIAA/ISSMO Multidisciplinary Anal. Optimization Conf., Albany, NY, 2004, Paper AIAA-2004-4460. [11] T. W. Simpson, J. Peplinski, P. N. Koch, and J. K. Allen, “Metamodels for computer-based engineering design: Survey and recommendations,” Eng. Comput., vol. 17, no. 2, pp. 129–150, Jul. 2001. [12] N. V. Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidynathan, and P. K. Tucker, “Surrogate-based analysis and optimization,” Progress Aerosp. Sci., vol. 41, no. 1, pp. 1–28, Jan. 2005. [13] M. A. Ismail, D. Smith, A. Panariello, Y. Wang, and M. Yu, “EMbased design of large-scale dielectric-resonator filters and multiplexers by space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 386–392, Jan. 2004. [14] K.-L. Wu, Y.-J. Zhao, J. Wang, and M. K. K. Cheng, “An effective dynamic coarse model for optimization design of LTCC RF circuits with aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 393–402, Jan. 2004. [15] S. Amari, C. LeDrew, and W. Menzel, “Space-mapping optimization of planar coupled-resonator microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2153–2159, May 2006. [16] M. Dorica and D. D. Giannacopoulos, “Response surface space mapping for electromagnetic optimization,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1123–1126, Apr. 2006. [17] J. Zhu, J. W. Bandler, N. K. Nikolova, and S. Koziel, “Antenna optimization through space mapping,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 651–658, Mar. 2007. [18] S. J. Leary, A. Bhaskar, and A. J. Keane, “A constraint mapping approach to the structural optimization of an expensive model using surrogates,” Optimization Eng., vol. 2, no. 4, pp. 385–398, Dec. 2001. [19] M. Redhe and L. Nilsson, “Using space mapping and surrogate models to optimize vehicle crashworthiness design,” presented at the 9th AIAA/ISSMO Multidisciplinary Anal. Optimization Symp., Atlanta, GA, Sep. 2002, Paper AIAA-2002-5536.

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[20] H.-S. Choi, D. H. Kim, I. H. Park, and S. Y. Hahn, “A new design technique of magnetic systems using space mapping algorithm,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3627–3630, Sep. 2001. [21] S. Koziel, J. W. Bandler, and K. Madsen, “A space mapping framework for engineering optimization: Theory and implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3721–3730, Oct. 2006. [22] S. Koziel and J. W. Bandler, “Space-mapping optimization with adaptive surrogate model,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 3, pp. 541–547, Mar. 2007. [23] S. Koziel, J. W. Bandler, A. S. Mohamed, and K. Madsen, “Enhanced surrogate models for statistical design exploiting space mapping technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 1609–1612. [24] S. Koziel, J. W. Bandler, and K. Madsen, “Theoretical justification of space-mapping-based modeling utilizing a data base and on-demand parameter extraction,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4316–4322, Dec. 2006. [25] V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, and Q.-J. Zhang, “Advanced microwave modeling framework exploiting automatic model generation, knowledge neural networks, and space mapping,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1822–1833, Jul. 2003. [26] J. E. Rayas-Sánchez, “EM-based optimization of microwave circuits using artificial neural networks: The state-of-the-art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 420–435, Jan. 2004. [27] J. E. Rayas-Sánchez, F. Lara-Rojo, and E. Martinez-Guerrero, “A linear inverse space-mapping (LISM) algorithm to design linear and nonlinear RF and microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 960–968, Mar. 2005. [28] L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2752–2767, Sep. 2005. [29] S. Koziel, J. W. Bandler, and K. Madsen, “Towards a rigorous formulation of the space mapping technique for engineering design,” in Proc. Int Circuits Syst. Symp., Kobe, Japan, May 2005, pp. 5605–5608. [30] K. Madsen and J. Søndergaard, “Convergence of hybrid space mapping algorithms,” Optimization Eng., vol. 5, no. 2, pp. 145–156, Jun. 2004. [31] A. Manchec, C. Quendo, J.-F. Favennec, E. Rius, and C. Person, “Synthesis of capacitive-coupled dual-behavior resonator (CCDBR) filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2346–2355, Jun. 2006. [32] “FEKO User’s Manual, Suite 4.2,” EM Softw. Syst. S.A. (Pty) Ltd., Stellenbosch, South Africa, 2004. [Online]. Available: http://www.feko.info [33] Agilent ADS. ver. 2003C, Agilent Technol., Santa Rosa, CA, 2003. [34] V. Miraftab and R. R. Mansour, “Computer-aided tuning of microwave filters using fuzzy logic,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2781–2788, Dec. 2002. [35] V. Miraftab and R. R. Mansour, “A robust fuzzy-logic technique for computer-aided diagnosis of microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 450–456, Jan. 2004. [36] L.-X. Wang and J. M. Mendel, “Generating fuzzy rules by learning from examples,” IEEE Trans. Syst., Man, Cybern., vol. 22, no. 6, pp. 1414–1427, Nov./Dec. 1992. [37] J. E. Rayas-Sánchez and V. Gutiérrez-Ayala, “EM-based Monte Carlo analysis and yield prediction of microwave circuits using linear-input neural-output space mapping,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4528–4537, Dec. 2006. [38] em. ver. 10.53, Sonnet Softw., North Syracuse, NY, 2006.

Slawomir Koziel (M’03–SM’07) received the M.Sc. and Ph.D. degrees in electronic engineering from Gdansk University of Technology, Gdansk, Poland, in 1995 and 2000, respectively, and the M.Sc. degrees in theoretical physics and in mathematics and Ph.D. degree in mathematics, from the University of Gdansk, Gdansk, Poland, in 2000, 2002, and 2003, respectively. He is currently a Research Associate with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada. He has authored or coauthored over 100 papers. His research interests include space mapping, circuit theory, analog signal processing, evolutionary computation, and numerical analysis.

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John W. Bandler (S’66–M’66–SM’74–F’78– LF’06) studied at Imperial College. He received the B.Sc. (Eng.), Ph.D., and D.Sc. (Eng.) degrees from the University of London, London, U.K., in 1963, 1967, and 1976, respectively. In 1969, he joined McMaster University, Hamilton, ON, Canada, where he is currently Professor Emeritus. He was President of Optimization Systems Associates Inc., which he founded in 1983, until November 20, 1997, the date of acquisition by the Hewlett–Packard Company. He is President of Bandler Corporation, Dundas, ON, Canada, which he founded in 1997. Dr. Bandler is a Fellow of several societies including the Royal Society of Canada. He was the recipient of the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Microwave Application Award.

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Compact Planar Quasi-Elliptic Function Filter With Inline Stepped-Impedance Resonators Jen-Tsai Kuo, Senior Member, IEEE, Ching-Luh Hsu, and Eric Shih

Abstract—Compact microstrip bandpass filters of orders = 4 6 and 8 with quasi-elliptic function responses are synthesized with inline stepped-impedance resonators. The filters have enhanced transition responses since two transmission zeros are generated on both sides of the passband. Existence of these two zeros is investigated by formulating the -matrix of the equivalent circuit for the filter and taking adjacent and nonadjacent coupling into account. Some matrix elements +2 representing coupling between two nonadjacent resonators are shown to have the opposite signs in both the lower and upper sides of the center frequency. This property leads to creation of the two transmission zeros on both sides of the passband. It is demonstrated that one more zero can be created in the rejection band by the tapped-line input/output scheme. Experimental circuits on substrates with = 2 2 and 10 2 are measured to validate the theory and design. Index Terms—Bandpass filter, elliptic function response, inline, stepped-impedance resonator, transmission zero.

I. INTRODUCTION

I

N THE RF front-ends of recent wireless communication systems, bandpass filters are required to have several essential properties, such as high selectivity, wide upper stopband, and compact circuit size. Based on the traditional parallel-coupled configuration [1], [2], however, the circuit may become very long since an th-order filter consists of a cascade of sections. To tackle this problem, folded stepped-impedance resonator filters are proposed to reduce the circuit size [3], [4]. Generally speaking, the stepped-impedance resonators are suitable for designing filters with a good transition response and an extended rejection band since its first higher order resonance can be easily tuned to much higher than twice the fundamental frequency. To enhance filter performance in the transition and rejection bands, however, one of the most effective ways is to insert transmission zeros. The insertion can be realized by the cross coupling [5], [6] or the source–load coupling [7], [8]. In this paper, we explore a simple filter structure using stepped-impedance resonators as building blocks. Fig. 1 plots two fourth-order circuits with both symmetric and Manuscript received February 4, 2007; revised April 25, 2007. This work was supported in part by the Ministry of Education under the ATU Program and by the National Science Council, Taiwan, R.O.C., under Grant NSC 95-2221-E009-037 and Grant NSC 95-2752-E-009-003-PAE. J.-T. Kuo and C.-L. Hsu are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, 300 Taiwan, R.O.C. (e-mail: [email protected]). E. Shih is with Foxconn Electronics Inc., Hsinchu, 300 Taiwan, R.O.C. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.901604

Fig. 1. Two inline fourth-order filters with two tapped input/output schemes: symmetric (A–B ) and skew–symmetric (A–B ) feeds.

skew-symmetric feeds [4]. The circuit exhibits several attractive properties, such as compact size, wide upper stopband, elliptic function passband response, and plural transmission zeros. In Fig. 1, all resonators form an inline array so that the circuit occupies a compact area. When order is increased, the circuit size grows only in the direction of the width, which is usually much smaller than the length of the resonator. Besides, in the array, adjacent and nonadjacent coupling simultaneously exists among the resonators. Although the analysis becomes more complicated, creation of certain transmission zeros in the filter response indeed relies on nonadjacent coupling. In the combline design [9]–[11], grounded line sections of different lengths are loaded with various capacitors. It can be easily identified in Fig. 1 that at the fundamental resonance, the low- and high- segments of the resonator behave as capacitors and inductors, respectively. Although the circuit layout looks quite similar to that of a combline structure, it has at least several distinct features. First, all the resonators have identical geometry and exhibit an electric length of 180 at the design frequency. Second, multiple coupling exists among the lowsection array, as well as the high- array. Third, the circuit needs neither a lumped element, nor grounding via so that filter fabrication is easier and more reliable. The price paid for the full-length resonators is that the circuit area is slightly larger than twice the size of a quarter-wave combline counterpart [10]. The use of full-length resonators, however, brings one more degree of freedom to circuit designers in choosing symmetric or skew-symmetric feed [4]. It will be shown later that existence and location of certain zeros are subject to the symmetry used in the tapped input/output arrangement. Here, we limit ourselves to exploring inline steppedimpedance resonator filters only of orders four, six, and eight. Some results for circuits of lower orders can be referred to [12], [13]. In [12], second-order filters are developed to achieve a wide upper stopband. The circuit in [13] also possesses a quasi-elliptic function response. Its analysis by the theory of multiple coupled microstrips, however, lacks for a design concept for filter synthesis. This paper is organized as follows. Section II briefly describes the passband synthesis procedure

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Fig. 2. Generic coupling structure of the inline bandpass filter.

and discusses the coupling properties among the inline resonators. Section III explores the existence of the zeros in terms of a -parameter matrix by taking the adjacent and nonadjacent coupling into account. Section IV addresses the creation of zero by the tapped input/output structure, Section V presents measured results for three experimental circuits. Section VI draws the conclusion. II. PASSBAND SYNTHESIS Fig. 2 shows the generic basic coupling structure for the filters in Fig. 1. Each resonator has one high- and two low- secand tions. The former has physical (electric) lengths each of the later has , and their respective widths are and with corresponding characteristic impedances and . In addition to , the gap size can be tuned to establish necessary coupling for synthesizing the passband. Choice of the geometrical dimensions for the resonator has been extensively and studied in [1] and [2]. The impedance ratio are the key parameters to deterlength ratio mine its resonant spectrum. If and are properly chosen, the first spurious resonance can be pushed far beyond twice the fun[2]. For example, if the first higher damental frequency or is desired, and order resonance occurring at can be used. When GHz, geometry parameters can be mm, mm, and mm for and thickness mm. a substrate with The next step is to determine spacing between each pair of adjacent resonators. The coupling coefficient between the th th resonators, i.e., , is given by [14] and (1) where is the th element value of the low-pass filter prototype and is the fractional bandwidth. To realize this coefficient for coupled resonators in Fig. 2, the test method in [6] can be invoked. Through weak gap feeds to the coupled resonators, the simulated transmission response will present two peaks. If the peaks are at and , the coefficient can be calculated as (2) For the fourth-order circuits in Fig. 1, the coupling matrix is symmetric about its two main diagonals, i.e., and . Thus, only and need specifying are zero and is negligible since all diagonal entries owing to the relatively large space between resonators 1 and 4.

Fig. 3. Coupling coefficients of two stepped-impedance resonators against D for various D . L = L = 7:6; W = 0:4; W = 2:0 (all in millimeters). Substrate: " = 2:2; thickness = 0:508 mm.

will not significantly It can be anticipated that change of alter the first resonance of the resonator, but will change magnitude, and even polarity of the coupling coefficient of two coupled resonators. This property is useful for adjusting the resonator geometry when more than one coupling coefficients have to be simultaneously considered in filter synthesis. An example will be given in Section V for such a demonstration. For mm to , Fig. 3 plots coupling coefficients of two resonators against . Except for , each curve runs is increased up to 4 from positive to negative values when mm. Generally speaking, the structure consists of both electric and magnetic coupling, called mixed coupling in [6]. When is small, magnitude of magnetic coupling due to current on the thin sections is larger than that of electric coupling between is increased to be the low- sections at both ends. When , on the other hand, electric coupling large enough for small becomes dominant. The coefficient calculated by (2) is the net or magnetic . coupling, which can be electric The use of curves in Fig. 3 can be demonstrated as follows. Suppose we are designing a fourth-order Chebyshev filter with %. From (1), the three interstage a 0.1-dB ripple and and coupling coefficients are . If mm is chosen, we have , and the zero-crossing point is at mm. It is obvious that both electric and magnetic coupling can be used value. Thus, there are at least two possible to realize each and , and the other designs: one uses and . The former and latter are, uses respectively, referred to as the - and -type filters herein. For the input and output coupling, the tap positions, i.e., in Fig. 2, should be determined by matching the singly loaded of the tapped resonator with the passband specification [15]. The singly loaded is defined as dB

(3)

is the impedance seen by the resonator looking towhere is the operation frequency, and is the ward the source, input susceptance of the resonator seen at the tap point. The

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Fig. 5. (a) Two coupled resonators with coupling. (b) Equivalent circuit.

Fig. 4. Simulation responses of the two fourth-order filters. (a) M -type: D = 0:28; D = 1:0; L = 3:2. (b) E -type: D = D = 0:82; D D 0:37; L = 3:2 (all in millimeters).

= =

derivation of (3) for a stepped-impedance resonator can be referred to [2]. Both - and -type circuits can be designed with or the skew-symmetric feeds the symmetric values and, hence, identical passband re[4] with identical sponses. In the rejection bands, nevertheless, they exhibit quite redifferent characteristics. Fig. 4 shows the simulated sponses of the fourth-order - and -type filters. The four dB, show very good agreepassbands, say, before ment. In Fig. 4(a), both the -type filters have a transmission zero in the upper stopband. The circuit with the symmetric feed, however, has one more zero in the lower stopband. In Fig. 4(b), the two -type filters exhibit sharp transition bands, like those of an elliptic function response, since and , are created on both sides of the two zeros, i.e., passband. In addition, there is an extra transmission zero in the lower stopband and in the upper stopband for the and feeds, respectively. Obviously, the -type filters possess better frequency selectivity in the stopband than the -type ones. Thus, the -type filters are investigated in detail as follows. III. TRANSMISSION ZEROS OF THE

-TYPE FILTERS

The resonators of a cross-coupled four-pole filter [4]–[6], [16] are arranged in a 2 2 configuration to achieve a quasi-elliptic

function response. The occurrence of the zeros relies on the elec, which causes two split signals to be out-oftric coupling phase at the output port. In the -type filters, however, the elliptic function-like response is clearly resulted from a different in Fig. 1 can be negligible, while the nonscheme since the and should be taken into adjacent coupling coefficients account. In addition, for predicting the zeros of the particular filter configuration, based on the -parameter of the equivalent circuit of the filter, an analysis method is developed as follows. The equivalent lumped-circuit model of two coupled resonators in Fig. 5(a) is shown in Fig. 5(b). Each resonator is modeled with a parallel LC network, and there are magnetic and electric coupling between the inductors and capacitors, respectively. From the circuit theory, the two-port -parameters can be derived as follows: (4) (5) where (6) (7) (8) (9) (10) (11) in (9) and in (10), respectively, repreThe coefficients sent magnetic and electric coupling between the two resonators. They are assumed constants over a certain frequency range centered at the design frequency. The natural frequencies and in (2) of the coupled system can be determined by two conditions: , which are obtained by enforcing the

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-matrix to zero. It can be validated that

(12)

The approximation is valid since . The result in (12) . means that the net coupling calculated by (2) should be Note that all -parameters in (4) and (5) are purely imaginary since the circuit is assumed lossless. In (5), has a . Thus, its sign over zero-crossing point at is opposite to that over . This property is unusual since, in a conventional coupling matrix, nondiagonal elements are usually assumed independent of frequency [6]. For investigating the possible occurrence of transmission zeros, deand as fine the relative phase between

(13)

when It can be deduced from this equation that or , and when if (i.e., net coupling is magnetic) or if (i.e., net coupling is electric). The identification rule for determining the type of coupling proposed in [17] can also be justified by (13) as well. Fig. 6 plots the simulated responses for the three basic coupled structures of the -type filter in Fig. 4(b). The -parameters are obtained by Zeland Software Inc.’s software package due to the IE3D [18]. Each response shows a jump at phase change of the denominator of (13). Based on Fig. 6(a), one can assure that the coupling between resonators 1 and 2 is magfor 2 GHz GHz. netic dominant and . Similarly, the response in Fig. 6(b) guarantees In Fig. 6(c), there are extra phase jumps at 2.08 and 2.75 GHz. Two important properties of this coupled structure should be coupling is identified by the latter jump. First, the type of GHz magnetic, as indicated in Fig. 6(a). The jump at changes sign, by (13), since when indicates that . Also, by (13), the and values can be extracted since and is known by (2) from simulation data. The jump at 2.08 GHz, however, cannot be explained by (13). It could be due to the fact that the equivalent circuit in Fig. 5(b) has a lower frequency limit for modeling the distributed coupled resonators with a relative large distance in Fig. 6(c). is further Identifying the type of coupling and value of mm and mm by Fig. 7. From investigated for is electric coupling and . SimFig. 7(a), is of the magnetic type and ilarly, Fig. 6(b) indicates that . It is important to identify and from the responses of coupled resonators in Figs. 6(c) and 7 since analysis of the transmission zeros relies on it. By analyzing the

0

Fig. 6. Responses of Y Y for investigating occurrence of the transmission zeros of the E -type filter in Fig. 4(b). (a) Resonators 1 and 2, D = 0:82 mm. (b) Resonators 2 and 3, D = 0:37 mm. (c) Resonators 1 and 3, D = 3:19 mm.

0

Fig. 7. Y Y responses for identification type of coupling between resonators 1 and 3. (a) D = 0:6 mm. (b) D = 1:5 mm.

phase relation of two split signals in the main and cross-coupled GHz paths, a zero in the upper stopband can occur at [17]. This zero can also be validated by the -matrix method % and ripple given below. Let the filter bandwidth dB, and then the external . The -matrix for the can be expressed circuit, normalized with respect to

KUO et al.: COMPACT PLANAR QUASI-ELLIPTIC FUNCTION FILTER

Fig. 8.

jS

j

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Fig. 9. Responses of higher order inline filters with m =

responses based on coupling matrices in (14).

as

(14)

where and . These values are derived from (1). The entries in the first off-diagonal use the following approximation. For example, since and the last term in (7) is neglected. Values of and can be obtained by prescribed zeros at . is required. If the two zeros are symmetric about is determined by Note that the sign of the elements . When and , and can be obtained from the responses based on (14), respecresponses from the matrices with and tively. Fig. 8 shows . Note that when without the nonadjacent coupling , the response will have no transmission zero. It can be seen from this example that values of and can be varied for controlling these two transmission zeros. One possible way to adjust and is to slide or deform the high- section of one of the coupled resonators, as shown in Fig. 1.

e

= 0:011.

The -matrix in (14) can be easily extended to circuits of and with the quasi-elliptic response. The -maorder trix can be established and the frequencies of the zeros can is be predicted. For example, the coupling matrix for shown in (15) at the bottom of this page, where and , and . Fig. 9 plots the responses for filters of order and with GHz, %, and a 0.1-dB ripple. For all nonadjacent elements is used. IV. TRANSMISSION ZEROS DUE TO TAPPED INPUT/OUTPUT responses of the -type filFig. 10 plots simulation and mm. ters with skew-symmetric feed for Impedance transformers are added to keep the value of each tapped resonator unchanged for the three tap positions. It can be and , as well as the passband do seen that frequencies of not vary significantly with the changes of . However, the zero moves to higher frequency when tap point is moved away from the center to the edge of the resonator. It reflects the fact can be dominated by . In the parthat determination of allel-coupled stepped-impedance resonator filters in [2], a zero can be created at a frequency where the electric length of the arm between the open end of the tapped resonator and the tap point is one quarter-wavelength long. The arm used for coupling with an adjacent resonator, however, does not create a zero. For

(15)

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Fig. 10. Moves of the tunable transmission zeros due to the slide of tap point for the E -type filters with skew-symmetric feed.

(c) Fig. 12. X responses for the test circuit and the E -type circuit. (a) Test circuit. (b) Skew-symmetric feed. (c) Symmetric feed.

Fig. 11(a) are set to zero. The transfer impedance , can be written as as

Fig. 11. Analysis of f and f . (a) Four-port network. (b) Responses for X of the E -type filters. For (X ) , only the important part is shown.

0

the structure in Fig. 1, both open ends of the input and output resonators are coupled with their adjacent resonators. Thus, crein Fig. 10 needs further investigation. ation of the zero The four-port network in Fig. 11(a) is employed for the prediction. Two more ports are added to the circuit in Fig. 1 since, in analysis, the whole circuit can be reduced by half due to the symmetry (dashed line). Let be the total current flowing into or . It can be derived that port (16) represents symmetric feed and and where subscript denote that the dashed line is a magnetic and electric and in wall, respectively. For the skew-symmetric feed,

, denoted

(17) For both feeds, zeros of the responses can be obtained by enforcing (16) and (17) to zero. The conditions are (18) where the plus and minus signs apply to the skew-symmetric and symmetric feeds, respectively. Obviously, complete formulas of and will be tedious and complicated since four-microstrip structures are involved [13]. The transfer impedances are purely reactive for lossless struc. tures. Let the reactance be denoted by and Simulated and responses are shown in Fig. 11(b), where each intersection point indicates a zero in response. Note that the zeros and are in the the lower and upper rejection bands for the symmetric and skew-symmetric feeds, respectively. To further investigate the property of the zeros, the behavior of the circuit in Fig. 12(a) is tested. It is the -type filter in Fig. 4(b) that is altered by

KUO et al.: COMPACT PLANAR QUASI-ELLIPTIC FUNCTION FILTER

Fig. 13. Group delay and S -parameter responses of the symmetric feed. All circuit parameters are in Fig. 4(b).

E

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-type filters with

Fig. 15. Layout and performances of the sixth-order E -type filters with skewsymmetric feed. (a) Circuit layout. Dimensions in millimeters: L = L =

6:32; L = 3:2; W = 0:2; W = 2:5; D = D = 0:14; D 0:82; D = 0:23. (b) Group delay, jS j and jS j responses.

=D

=

V. SIMULATION AND MEASUREMENT

Fig. 14. Layout and performances of the E -type filters with skew-symmetric feed. (a) Circuit layout. Dimensions in millimeters: L = 3:48; L =

3:78; L = 2:2; L = 1:48; L = 4:1; L = 1:5; L = 2:13; W = 0:2; W = 2:5; D = 0:6; D = 0:5. (b) Group delay, jS j and jS j

responses.

moving the high- sections of resonators 1 and 4 outward. responses of the test circuit Fig. 12(b) and (c) plots the (solid lines) and the -type filter (dashed lines). It can be seen that the -type filter shows three transmission zeros. For the and disappear, although test circuit, however, both and exist, respectively, for symmetric and skew-symmetric feeds. Based on the results in Fig. 12(b) and (c), must must be capacitive in the transition be negative or and in the design of the -type bands for creation of filters.

Fig. 13 plots the simulation and measured responses of the -type filters with a symmetric feed. All geometric parameters are referred to Fig. 4(b). The tap points are chosen to match to value for 50- reference impedance. The measured rethe . The jection levels are better than 40 dB up to 5 GHz or extra zero is at 1.9 GHz. It is found that rejection levels of better than 60 dB can be achieved within the bands covering from 1.82 to 2.18 and 2.9 to 3.01 GHz. It can be observed that has a response with sharp transition bands and good symmetry about the center frequency. The measured and simulated are also given. group delays The second design demonstrates control of the transmission and . The center frequency GHz zeros and fractional bandwidth % with a 0.1-dB ripple level. GHz and GHz, the coupling coLet and efficients . The geometric dimensions of the end resonators and are chosen to locate the zero at with the tap position no transformer. For reducing the circuit size, a substrate with and thickness mm is used. Fig. 14(a) shows the circuit layout with deformed resonators 2 and 3 to simultaneously fulfill required magnitudes of all the coupling coefficients , the high- sections including and . To have of two middle resonators are bent to a U shape. The distance can be readily determined by the results shown in Fig. 3. At the same time, the low- sections are moved inwardly by and a distance to simultaneously realize the specified values. It could be due to the right-angled bends in the high-

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sections (linewidth mm) that and are trimmed by increasing 0.105 and 0.425 mm, respectively, for recovering the resonant frequency shift by the resonator deformation. In the previous example, there is no such problem. The simulated and measured filter responses in Fig. 14(b) show good agreement. In measurements, the passband insertion loss is approximately notch at 4.9 GHz is approximately 65 dB. 2.2 dB and the The total circuit size is approximately 1.5 1.5 cm . The third example is a sixth-order filter built on a substrate and thickness mm. The center frequency with GHz, ripple dB, and %. The resonator geometry is chosen to push the first spurious passband to . In the filter, the resonators are configured with alternating electric and magnetic types of coupling for establishing all coupling coefficients with proper magnitudes and phases. Note that the and are of an electric type. coupling coefficients Simulation and measured results with a skew-symmetric feed are plotted in Fig. 15. The insertion loss is 2.5 dB at , in-band return loss is better than 15 dB, and the stopband with a rejecand 30 dB to tion level of 50 dB is extended to 7.5 GHz . 12.5 GHz An eighth-order filter is synthesized with a skew-symmetric feed. The circuit simulation exhibits a similar response to that of a sixth-order -type filter in Fig. 15, but has better rejection rates in transition bands. VI. CONCLUSION Stepped-impedance resonators have been arranged in an inline configuration to make the entire circuit a compact size. The use of the resonators has assured a wide upper stopband and the inline resonator array facilitates new coupling schemes for producing a quasi-elliptic function passband response. Creation of transmission zeros has been investigated by -matrix parameters of the equivalent circuit of the filter. It has been shown that are key factors for creating proper nonadjacent elements the transmission zeros on both sides of the passband for fourth-, sixth-, and eighth-order filters. An enhanced attenuation rate in transition bands can then be obtained. Formulation of the conditions of the extra zero in rejection bands has also been given. It has been demonstrated for the particular inline structure that the extra zero can be placed in the lower and upper stopbands by symmetric and skew-symmetric feeds, respectively. For demonstrative purposes, measured results for three experimental filters have been compared with simulation data. REFERENCES [1] M. Makimoto and S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 12, pp. 1413–1417, Dec. 1980. [2] J.-T. Kuo and E. Shih, “Microstrip stepped-impedance resonator bandpass filter with an extended optimal rejection bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1554–1559, May 2003. [3] M. Sagawa, K. Takahashi, and M. Makimoto, “Miniaturized hairpin resonator filters and their application to receiver front-end MIC’s,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1991–1997, Dec. 1989. [4] C.-M. Tsai, S.-Y. Lee, and C.-C. Tsai, “Performance of a planar filter using a 0 feed structure,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2362–2367, Oct. 2002.

[5] R. M. Kurzrok, “General four-resonator filters at microwave frequencies,” IEEE Trans. Microw. Theory Tech., vol. MTT-14, no. 6, pp. 295–296, Jun. 1966. [6] J.-S. Hong and M. J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 2099–2109, Nov. 1996. [7] S. Amari, “Direct synthesis of folded symmetric resonator filters with source–load coupling,” IEEE Microw. Wireless Compon. Lett, vol. 11, no. 6, pp. 264–266, Jun. 2001. [8] C.-K. Liao and C.-Y. Chang, “Design of microstrip quadruplet filters with source–load coupling,,” IEEE Trans. Microwave Theory and Tech., vol. 53, no. 7, pp. 2302–2308, Jul. 2005. [9] W.-T. Lo and C.-K. C. Tzuang, “ -band quasi-planar tapped combline filter and diplexer,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 2, pp. 215–223, Feb. 1993. [10] T. Kitamura, Y. Horii, M. Geshiro, and S. Sawa, “A dual-plane combline filter having plural attenuation poles,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1216–1219, Apr. 2002. [11] W. Menzel and M. Berry, “Quasi-lumped suspended stripline filters with adjustable transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 1601–1604. [12] C.-L. Hsu and J.-T. Kuo, “A two-stage SIR bandpass filter with an ultrawide upper rejection band,” IEEE Microw. Wireless Compon. Lett, vol. 17, no. 1, pp. 34–36, Jan. 2007. [13] E. Shih and J.-T. Kuo, “A new compact microstrip stacked-SIR bandpass filters with transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1077–1080. [14] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Network, and Coupling Structures. Norwood, MA: Artech House, 1980, ch. 8, p. 432. [15] J. S. Wong, “Microstrip tapped-line filter design,” IEEE Trans. Microw. Theory and Tech., vol. MTT-27, no. 1, pp. 44–50, Jan. 1979. [16] C.-L. Hsu and J.-T. Kuo, “Design of cross-coupled quarter-wave SIR filters with plural transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig, San Francisco, CA, Jun. 2006, pp. 1205–1208. [17] C.-C. Chen, Y.-R. Chen, and C.-Y. Chang, “Miniaturized microstrip cross-coupled filters using quarter-wave or quasi-quarter-wave resonators,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 120–131, Jan. 2003. [18] IE3D Simulator. Zeland Softw. Inc., Freemont, CA, Jan. 2002.

K

Jen-Tsai Kuo (S’88–M’92–SM’04) received the Ph.D. degree from the Institute of Electronics, National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1992. Since 1984, he has been with the Department of Communication Engineering, NCTU, where he is currently a Professor. From 1995 to 1996, he was a Visiting Scholar with the Electrical Engineering Department, University of California at Los Angeles (UCLA). His research interests include analysis and design of microwave integrated circuits (MICs) and numerical techniques in electromagnetics. Dr. Kuo was the recipient of the 2006 Thomson Scientific Citation Laureate Award.

Ching-Luh Hsu received the B.S. degree in communication engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1989, the M.S. degree in electrical engineering from National Sun Yet-Sen University (NSYSU), Kaohsiung, Taiwan, R.O.C., in 1994, respectively, and is currently working toward Ph.D. degree at NCTU. From 1994 to 1999, he was a RF Engineer with Microelectronic Technology Incorporation (MTI), where he developed transceivers for point-to-point digital microwaves. In 1999, he joined the faculty of the Department of Electronic Engineering, Ta Hwa Institute of Technology, Hsinchu, Taiwan, R.O.C., where he is currently a Lecturer. His research interests include the design of planar circuits for microwave and millimeter-wave applications.

KUO et al.: COMPACT PLANAR QUASI-ELLIPTIC FUNCTION FILTER

Eric Shih was born in Taoyuan, Taiwan, R.O.C., on April 12, 1976. He received the B.S. degree in engineering and system science from National Tsing Hua University (NTHU), Hsinchu, Taiwan, R.O.C., in 1998, and the Ph.D. degree in communication engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 2003. He is currently with Foxconn Electronics Inc., Hsinchu, Taiwan, R.O.C. His research interests include the design of microwave planar filters, RF modules, and antennas for microwave and millimeter-wave applications.

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Balanced Coupled-Resonator Bandpass Filters Using Multisection Resonators for Common-Mode Suppression and Stopband Extension Chung-Hwa Wu, Student Member, IEEE, Chi-Hsueh Wang, Member, IEEE, and Chun Hsiung Chen, Fellow, IEEE

Abstract—Novel fourth-order balanced coupled-resonator bandpass filters are proposed using suitably designed half-wavelength ( 2) multisection resonators for common-mode suppression. By properly designing the input/output (I/O) resonators associated with the filter composed of four bi-section resonators, a balanced filter with good common-mode suppression is realized, but its rejection bandwidth is rather limited. To widen the rejection bandwidth, the I/O bi-section resonators are replaced by the tri-section ones so that a balanced filter with good common-mode suppression and wide rejection bandwidth may be realized by suitably arranging the composed bi-/tri- section resonators. Specifically, a stopband-extended balanced filter with good common-mode suppression ( 50 dB) within the differential-mode passband is implemented and its stopbands are also extended up to 5 0 with a rejection level of 30 dB, where 0 is the center frequency in differential-mode operation. Index Terms—Balanced filter, common-mode suppression, coupled-resonator bandpass filter, half-wavelength ( 2) resonator, stopband extension.

I. INTRODUCTION ALANCED circuits are essential in building a modern communication system. Recently, under the trend of system-on-chip, it requires the integration of RF and analog circuits onto the digital baseband processor, thus the problem of interference and crosstalk from substrate coupling between components is getting more and more serious. A fully balanced transceiver architecture [1] with differential operation such as the one illustrated in Fig. 1 shows higher immunity to the environmental noise when compared with the unbalanced topology with single-ended signaling. Among various basic components, low-noise amplifiers, power amplifiers, mixers, and voltage-controlled oscillators have rapidly been developed as balanced circuits over the past few years. To establish a balanced system, the development of balanced filters is also necessary. A well-designed differential-to-differential balanced bandpass filter should exhibit the desired differential-mode frequency response and should also be capable of reducing the

B

Manuscript received January 24, 2007. This work was supported in part by the National Science Council of Taiwan under Grant NSC 95-2752-E-002-001-PAE and in part by National Taiwan University under Excellent Research Project NTU-ERP-95R0062-AE00-00. The authors are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, 106 Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.901609

Fig. 1. Simplified architecture of the balanced transceiver.

common-mode signal, which is essential in increasing the signal-to-noise-ratio in the receiver and improving the efficiency of the dipole antenna in the transmitter. Furthermore, a well-designed balanced filter should also possess excellent out-of-band rejection and high selectivity. In particular, the wide-stopband bandpass filters are usually needed in conjunction with the nonlinear components (e.g., mixers or power amplifiers) so as to eliminate the undesired interference or noise in the stopband. However, pervious studies on the balanced filters with good common-mode suppression, high selectivity, and wide rejection bandwidth are rather limited [2]–[5]. Many single-ended filters with specified filtering characteristics have been developed in the past. Among them, parallel-coupled-line bandpass filters have the advantages of wide realizable bandwidth and simple synthesis procedures [6], [7]. Other popular ones are the coupled-resonator filters [8]–[10], particularly the narrowband bandpass filters that play a significant role in many applications. The coupled-resonator filters are implemented using the design technique based on the coupling coefficients and external quality factors, which may be applied to any type of resonator with different physical structures. Recently, a fourth-order balanced coupled-line bandpass filter using parallel-coupled-line structures was proposed in [4], which presents high selectivity and good common-mode suppression within the passband. However, this filter has several drawbacks such as requiring a via-hole process and having . To overcome the a common-mode passband around shortcomings in [4], another balanced filter was proposed in [5]. With the adoption of coupled stepped-impedance resonators,

0018-9480/$25.00 © 2007 IEEE

WU et al.: BALANCED COUPLED-RESONATOR BANDPASS FILTERS USING MULTISECTION RESONATORS

it is possible to extend the differential- and common-mode and also to avoid the undesired via-hole stopbands up to process. However, only the differential-mode quality factor is properly designed so that an acceptable common-mode rejection level of 34.46 dB is achieved around . In this study, two novel fourth-order balanced bandpass filmultisection resonators are demonstrated to imters using prove the common-mode suppression in [5]. In the first design, by properly designing the impedance ratio, length ratio, and tap position of the input/output (I/O) bi-section resonators, one may present the desired external quality factor in differentialmode operation and also give the complete reflection condition , ) in common-mode operation. Thus, a ( very low common-mode signal level may be achieved without any degradation of the insertion loss in differential-mode operation; however, its rejection bandwidth is still limited. Note that by using the bi-section resonators solely in the filter design, one may only achieve a balanced filter with either an extended stopband [5] or a good common-mode suppression, as demonstrated in this study, but not both. To realize a balanced filter with good common-mode suppression and stopband extension, both bi- and tri-section resonators are needed in the filter design. In the second design, to further extend the differential- and common-mode stopbands, the tri-section resonators are adopted for the I/O resonators. Specifically, by properly adjusting the I/O tri-section resonators, one may suitably misalign the corresponding differential- and common-mode higher order spurious resonance frequencies [11] so that a balanced filter with good common-mode suppression, as well as wider differential- and common-mode stopbands, may be realized.

Fig. 2. Physical layout bi-section resonators (W W = 0:8 mm, W L = 11:9 mm, L G = 1:4 mm, G =

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of the proposed fourth-order balanced filter using = 1:9 mm, W = 11:5 mm, W = 0:4 mm, = 6:9 mm, L = 15:5 mm, L = 42 mm, = 11:1 mm, L = 2:2 mm, G = 0:5 mm, 1:4 mm, D = 0:8 mm).

Fig. 3. Basic structure of =2 bi-section resonator in Fig. 2.

II. BALANCED FILTER USING BI-SECTION RESONATORS A. Filter Structure The proposed fourth-order balanced filter shown in Fig. 2 is bi-section resonators. With the composed of four symmetric adoption of the symmetric structure, the proposed filter presents a perfect electric conductor wall under differential-mode excitation and a perfect magnetic conductor wall under commonmode excitation along the line of symmetry of the structure. Thus, it is possible to reduce the level of common-mode noise in addition to possessing the desired bandpass frequency response in differential-mode operation. bi-section resonator is adjusted for In this study, the common-mode suppression, while in [5], it is used for stopband bi-section resonator in Fig. 2, as illusextension. Each trated by Fig. 3, is symmetric and has different characteristic , and lengths , , where the subscript impedances denotes the resonators and , respectively. The associated and the length ratio parameters such as the impedance ratio are defined by (1) (2) Under differential-mode operation, a virtual short (perfect electric wall) would appear along the line of symmetry, therefore,

Fig. 4. Tapped I/O resonator (resonator a). (a) Common-mode equivalent half-circuit. (b) Differential-mode equivalent half-circuit. Here, = k , = k , and k = f =f .

each resonator, resonating at , may be treated as a shorted resonator [see Fig. 4(b)]. Alternaquarter-wavelength tively, under common-mode operation, a virtual-open (perfect magnetic wall) would be present along the line of symmetry, , may thus each resonator, now resonating at resonator with both ends opened [see be regarded as a Fig. 4(a)]. The proposed fourth-order balanced coupled-resonator bandpass filter in Fig. 2 is implemented in the microstrip structure

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on the FR4 substrate (substrate thickness mm, dielectric , loss tangent , and metal thickness constant mm). In the filter design, the required coupling coefficients and external quality factors are extracted from the corresponding differential- and common-mode excitation circuits, and , rewhich have the termination impedances spectively, as in [5]. In this study, the extract process is accomplished by the full-wave simulator ADS Momentum. As to the measurement, the balanced structure, as a four-port device, is first measured by the Agilent E5071B network analyzer to give . The two-port differthe standard four-port -parameters ential- and common-mode -parameters, i.e., and , may , as given then be deduced from the four-port -parameters in [4] and [12].

Fig. 5. Parameters to give complete reflection condition in common-mode operation.

B. Design Procedure The fourth-order balanced bandpass filter is designed to possess a quasi-elliptic response in the differential-mode operaand the fraction with the center frequency at tional bandwidth of 10%. The corresponding element values , , of the low-pass prototype are , and . To determine the physical dimensions of the balanced filter, the differential-mode coupling and I/O external quality factors need to coefficients be calculated [13] as follows:

Fig. 6. Simulated Q impedance ratio R .

versus the length ratio

under different values of

Additionally, for the differential-mode equivalent half-circuit of the I/O resonators [see Fig. 4(b)] to possess the differenGHz , the resotial-mode fundamental frequency at nance condition given by (5)

(3) To improve the common-mode suppression in addition to possessing the desired differential-mode bandpass response, the I/O resonators (resonator a) are both designed to possess the dein differential-mode sired external quality factors , operation and also to present a complete reflection ( ) in common-mode operation. For this purpose, the impedance ratio, length ratio, and tap position of the I/O resonators should properly be arranged. Fig. 4(a) and (b) shows the corresponding common- and differential-mode equivalent half-circuits for the tapped I/O and are resonators (resonator a), in which the corresponding differential- and common-mode electrical lengths, and is the parameter associated with the tap position. By letting the common-mode input impedance be equal to in Fig. 4(a), one may obtain the condition of zero , complete reflection in common-mode operation which may be expressed as (4)

should also be satisfied. Thus, by combining (4) and (5), one may obtain the desired relation between the tap position under different values of parameter and the length ratio , as depicted in Fig. 5, which is useful in impedance ratio designing the I/O bi-section resonator (resonator a) for good common-mode suppression. Alternatively, the differential-mode external quality factor may be characterized by the ratio of and [13], where is the center frequency in the phase response of with reference to the differential-mode equivalent half-circuit should be determined from shown in Fig. 4(b), and has a phase shift of the frequencies at which the parameter 90 with respect to the absolute phase at . Hence, the coris responding differential-mode external quality factor and may related to the differential-mode input impedance be adjusted according to the given specification by properly and choosing the values of impedance and length ratios ( ), as shown in Fig. 6. With the goal of giving , , impedance ratio , and tap the length ratio are chosen as the initial values position parameter in determining the physical dimensions of the I/O resonators , , , and ) and the tap position . ( For easy comparison with [5], the physical dimensions of the , , , and ) are deterinter-coupled resonators ( mined with the parameters and . Finally,

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Fig. 7. Simulated differential-mode coupling coefficients versus the gaps between adjacent resonators.

the gaps ( , , , and ) between adjacent resonators are obtained according to the corresponding differential-mode in (3) . The differencoupling coefficients may be evaluated from the tial-mode coupling coefficients two split resonance frequencies associated with the coupled-resto the onator structure [13]. The design curves relating gaps between adjacent resonators are illustrated in Fig. 7. In the initial design phase, it is rather difficult to fully consider the discontinuity and coupling effects associated with the bi-section resonators. Thus, a fine-tune process based on full-wave simulation is needed to give the final physical dimensions, which are deviated approximately 10% from the initial values. C. Differential-Mode Response Fig. 8(a) and (b) shows the measured and simulated differential-mode frequency responses of the proposed balanced filter in Fig. 2. The measured differential-mode center frequency is at 1.025 GHz, the measured 3-dB bandwidth is 10.5%, and the minimum insertion loss is 4 dB. This higher loss is mainly due to the use of a low-cost FR4 substrate, which has a high loss tangent of 0.02. The fabricated balanced filter is compact and has (31 mm 53.26 mm), where is a size of the guided wavelength of the microstrip structure at the differen. Fig. 8(b) shows the tial-mode center frequency wideband differential-mode frequency responses ranging from 0.5 to 8.5 GHz. Note that this proposed balanced filter has only pushed the differential-mode stopband up to 3.1 GHz with a rejection level of 30 dB. The implement filter has created two transmission zeros at 0.9 and 1.17 GHz, as expected. Moreover, an additional transmission zero is observed at 0.68 GHz, which is produced by the in-phase cancellation between two dif, ) and is not conferential ports ( trollable. Fig. 9(a) shows the distribution of differential-mode spurious resonance frequencies of each resonator up to 8.5 GHz. Note that the first and second spurious resonances of the I/O resonator (resonator a), which have been moved toward the center freGHz , are now located at 2.315 and 3.16 GHz. quency Therefore, the differential-mode response may have a repeated . passband around 3 GHz D. Common-Mode Response Fig. 8(a) and (c) shows the measured and simulated commonmode frequency responses of the proposed balanced filter in Fig. 2. The measured common-mode response is suppressed

Fig. 8. Measured and simulated responses of the proposed balanced filter shown in Fig. 2. (a) Narrowband responses. (b) Wideband differential-mode responses. (c) Wideband common-mode responses.

Fig. 9. Fundamental and harmonic frequencies of each resonator for the proposed balanced filter in Fig. 2. (a) Differential mode. (b) Common mode.

below 54 dB within the passband. The proposed balanced filter shows a good common-mode signal suppression around (1 GHz); however, it only pushes the common-mode stopband up to 2.9 GHz with a rejection level of 30 dB. From

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the distribution of common-mode spurious resonance frequencies of each resonator, as shown in Fig. 9(b), one can find that the common-mode fundamental resonance frequencies of the resonators are located at 3 and 3.43 GHz, respectively. Therefore, the common-mode response has a passband around 3 GHz . As suggested by [4], the common-mode rejection ratio (CMRR) defined by dB

(6)

is adopted as a figure-of-merit for qualitative characterizing the level of common-mode suppression around the passband of balanced filters. Specifically, the proposed balanced filter in Fig. 2 has a maximum CMRR of 58 dB at with all values of CMRR above 54 dB from 0.97 to 1.08 GHz. The measured results shown in Fig. 8 demonstrate that the design of I/O resonators to give proper external quality factors and ) is feasible to improve the common( mode suppression without degrading the differential-mode response. However, by the adoption of the bi-section resonators , the higher order spuwith impedance ratio rious resonance frequencies are moved toward the fundamental resonance frequency [14], [15]. Thus, the common-mode stopband of proposed balanced filter has only been pushed up to , which is not satisfactory for modern wireless 2.9 GHz communication system.

Fig. 10. Physical layout of the proposed stopband-extended fourth-order = 1:9 mm, balanced filter using bi-/tri-section resonators (W = 0:3 mm, W = 9:3 mm, W = 13 mm, W = 0:8 mm, W = 6:6 mm, L = 4:9 mm, L = 8:7 mm, L = 10:8 mm, W = 11:9 mm, L = 11:3 mm, L = 1:3 mm, G = 0:4 mm, L = 1:4 mm, G = 1:6 mm, D = 0:5 mm). G

Fig. 11. Basic structure of the tri-section resonator (resonator a).

III. STOPBAND-EXTENDED BALANCED FILTER USING BI-/TRI-SECTION RESONATORS design. To simplify the idea, a proper approximation is introduced in the design phase.

A. Filter Structure To further extend the stopbands of the proposed balanced filter in Fig. 2, the corresponding differential- and commonmode higher order resonance frequencies of the I/O resonators should properly be arranged [11]. Here, by incorporating the tri-section resonators into the I/O resonators, one may suitably separate the corresponding differential- and common-mode higher order resonance frequencies so that the stopbands may be further extended. Shown in Fig. 10 is the proposed stopband-extended fourthbi-section order balanced filter, which is composed of two tri-section resonators (resresonators (resonator b) and two onator a). Different from the one proposed in Fig. 2, the I/O restri-section resonators. The onators are replaced by the tri-section resonator (resonator a) is symmetric and has different , , and and lengths , characteristic impedances , and , as illustrated in Fig. 11. Here, the lengths and are initially set identical in order to simplify the preliminary design. The associated parameters such as the and are defined by impedance ratios (7) and the length ratio is again defined by (2). Note that the adoption of tri-section resonators is critical for stopbands extension; however, it also increases the complexity in the filter

B. Design Procedure and of the tri-section resFor simplicity, the lengths in the initial onators are assumed to be identical design phase, meanwhile, the width is set to be 0.3 mm , which is the minimum width for fabrication. , , , and ) need Therefore, only four variables ( to be determined. Fig. 12 shows the tapped structure for the I/O resonators (resonator a). By letting the common-mode input , ) in Fig. 12(a), one impedance equal to zero ( may obtain the complete reflection condition in common-mode operation (8) From (8), the impedance ratio versus under different values of can be plotted as shown in Fig. 13, which is useful in designing the I/O resonators for good common-mode suppression. Note that if the tap position parameter is chosen, then is required to obtain . or is chosen, becomes a nonWhen linear function of ; however, it would nearly approach to a is rather small (such as less than 40 ). As constant when a result, one may make the approximation that when is selected, or when is selected. Under such an approximation, one may provide a

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Fig. 12. Tapped I/O resonator (resonator a). (a) Common-mode equivalent half-circuit. (b) Differential-mode equivalent half-circuit. Fig. 14. Simulated impedance ratios R

Q

versus the length ratio under different values of and R . (a) = 1=2. (b) = 1=3.

Fig. 13. Parameters to give complete reflection condition in common-mode operation.

large reflection in common-mode operation so that a very large may be achieved to suppress the common-mode signal. Note that the above assumption and approximation would largely simplify the initial design procedure, especially in designing the narrowband filter, as discussed in this study, which . Once the variable and the impedance requires are assigned, is almost fixed, and beratio comes a function of , , and , which can be adjusted according to the given specification. The simulated design curves against the length ratio under different values of for and are plotted in Fig. 14(a) and Fig. 14(b) . From Fig. 14(a) and (b), one may find that for the case is too small to meet the specification , thus, the case is more suitable for realizing the desired . Based on the design curves in Fig. 14(b), the , , , , physical dimensions of I/O resonators ( ) and the tap position may initially be determined. and , length ratio , Here, the tap position parameter and are chosen and impedance ratios as the initial values. , , , and ) of the The physical dimensions ( inter-coupled resonators (resonator ) are then determined based and as in the on the chosen parameters design of Fig. 2. Finally, the gaps ( , , , and )

Fig. 15. Simulated differential-mode coupling coefficients versus the gaps between adjacent resonators.

between adjacent resonators are obtained according to the dein (3). The design curves for relating to the sired gaps between adjacent resonators are illustrated in Fig. 15. Finally, a fine-tune process based on full-wave simulation is again required to relieve the assumption and approximation made in the simplified initial design and also to compensate for the discontinuity and coupling effects of the multisection resonators. Actually, this fine-tune process makes the final physical lengths not identical and causes a variation of almost 10% from the initial values. C. Differential-Mode Response Fig. 16(a) and (b) shows the measured and simulated differential-mode frequency responses of the proposed balanced filter in Fig. 10. The measured differential-mode center frequency is at 1.025 GHz, the measured 3-dB bandwidth is 11.5%, and the minimum insertion loss is 3.88 dB (again due to the high loss tangent of the FR4 substrate). The fabricated balanced filter has (38.4 mm 56 mm). Fig. 16(b) a size of shows the wideband differential-mode frequency responses ranging from 0.5 to 8.5 GHz. From the distribution of differential-mode spurious resonance frequencies of each resonator,

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Fig. 17. Fundamental and harmonic frequencies of each resonator for the stopband-extended balanced filter in Fig. 10. (a) Differential mode. (b) Common mode.

TABLE I PERFORMANCE COMPARISON WITH PREVIOUS STUDIES

Fig. 16. Measured and simulated responses of the stopband-extended balanced filter shown in Fig. 10. (a) Narrowband responses. (b) Wideband differentialmode responses. (c) Wideband common-mode responses.

shown in Fig. 17(a), the differential-mode higher order spurious resonances have been separated from each other before 5.5 GHz. Therefore, the proposed stopband-extended balanced filter has a stopband up to which is almost twice wider than the one in Fig. 2. In addition to the preselected transmission zeros at 0.88 and 1.18 GHz, here an additional transmission zero is again produced at 0.68 GHz, a consequence of in-phase cancellation between two differential ports. D. Common-Mode Response Fig. 16(a) and (c) shows the measured and simulated common-mode frequency responses of the proposed balanced filter in Fig. 10. The measured common-mode response is suppressed below 52.7 dB within the passband. Specifically, this proposed balanced filter shows good common-mode signal suppression around (1 GHz) and its common-mode stopband is pushed up to 5 GHz with a rejection level of 30 dB. From the distribution of common-mode spurious resonance frequencies of each resonator, shown in Fig. 17(b), one can

find that the fundamental resonance frequencies are located at 3.5 GHz (resonator b) and 5.2 GHz (resonator a). However, the resonance at 5.2 GHz would be the main factor in deciding the common-mode stopband response due to the stronger coupling between resonators a [5]. The balanced filter in Fig. 10 has a maximum CMRR of 49.3 dB at 1.025 GHz with all values of CMRR above 46.4 dB from 0.966 to 1.084 GHz. Note that the adoption of the I/O tri-section resonators not only provides a good common-mode suppression, but also possesses a higher flexibility in misaligning the differential- and common-mode spurious resonance frequencies so that the corresponding stopbands may also be extended. Two balanced filters realized in this study are summarized and compared with the pervious study in Table I. Note that the proposed stopband-extended balanced filter depicted in Fig. 10 shows good CMRR of 46.4 dB within the passband and has a with a rejection level of 30 dB. Until wider stopband up to now, the proposed stopband-extended balanced filter (Fig. 10) may be the one to give a better performance of good CMRR, high selectivity, and wide rejection bandwidth. IV. CONCLUSION In this paper, two novel balanced filters using suitably designed I/O multisection resonators for common-mode

WU et al.: BALANCED COUPLED-RESONATOR BANDPASS FILTERS USING MULTISECTION RESONATORS

suppression have been presented. Meanwhile, the design procedures and the corresponding differential- and common-mode frequency responses have also been carefully examined. By properly designing the I/O bi-section resonators associated with the filter composed of four bi-section resonators, it is possible to implement a balanced filter with good common-mode suppression; however, its rejection bandwidth is rather narrow (up ). To realize a balanced filter with good common-mode to suppression and wider rejection bandwidth, a novel filter structure composed of two I/O tri-section resonators and two inter-coupled bi-section resonators has been proposed. By properly designing the bi-/tri-section resonators, a stopband-exdB within tended balanced filter with good CMRR the passband has been implemented, and its stopband has been with a rejection level of 30 dB. extended up to

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[13] J. S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Application. New York: Wiley, 2001. [14] M. Makimoto and S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 12, pp. 1413–1417, Dec. 1980. [15] M. Sagawa, M. Makimoto, and S. Yamashita, “Geometrical structures and fundamental characteristics of microwave stepped-impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1078–1085, Jul. 1997.

Chung-Hwa Wu (S’06) was born in Tainan, Taiwan, R.O.C., in 1982. He received the B.S. degree in electrical engineering from National Chung Hsing University, Taichung, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design and analysis of microwave filters and passive circuits.

REFERENCES [1] C.-H. Wang, Y. H. Cho, C. S. Lin, H. Wang, C. H. Chen, D. C. Niu, J. Yeh, C. Y. Lee, and J. Chern, “A 60 GHz transmitter with integrated antenna in 0.18 m SiGe BiCMOS technology,” in IEEE Int. SolidState Circuit Conf. Tech. Dig., Feb. 2006, pp. 186–187. [2] A. Ziroff, M. Nalezinski, and W. Menzel, “A 40 GHz LTCC receiver module using a novel submerged balancing filter structure,” in Proc. Radio Wireless Conf., 2003, pp. 151–154. [3] Y.-S. Lin and C. H. Chen, “Novel balanced microstrip coupled-line bandpass filters,” in URSI Int. Electromagn. Theory Symp., 2004, pp. 567–569. [4] C.-H. Wu, C.-H. Wang, and C. H. Chen, “Novel balanced coupled-line bandpass filters with common-mode noise suppression,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 2, pp. 287–295, Feb. 2007. [5] C.-H. Wu, C.-H. Wang, and C. H. Chen, “Stopband-extended balanced bandpass filter using coupled stepped-impedance resonators,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 7, Jul. 2007. [6] S. B. Cohn, “Parallel-coupled transmission-line-resonator filters,” IRE Trans. Microw. Theory Tech., vol. MTT-6, no. 7, pp. 223–231, Apr. 1958. [7] C.-Y. Chang and T. Itoh, “A modified parallel-coupled filter structure that improves the upper stopband rejection and response symmetry,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 2, pp. 310–314, Feb. 1991. [8] J. S. Hong and M. J. Lancaster, “Coupling of microstrip square open-loop resonator for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2099–2109, Dec. 1996. [9] S. Y. Lee and C. M. Tsai, “New cross-coupled filter design using improved hairpin resonators,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2482–2490, Dec. 2000. [10] C. C. Chen, Y. R. Chen, and C. Y. Chang, “Miniaturized microstrip cross-coupled filters using quarter-wave or quasi-quarter-wave resonators,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 120–131, Jan. 2003. [11] S.-C. Lin, P.-H. Deng, Y.-S. Lin, C.-H. Wang, and C. H. Chen, “Wide-stopband microstrip bandpass filters using dissimilar quarter-wavelength stepped-impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1011–1018, Mar. 2006. [12] D. E. Bockelman and W. R. Eisenstant, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1530–1539, Jul. 1995.

Chi-Hsueh Wang (S’02–M’05) was born in Kaohsiung, Taiwan, R.O.C., in 1976. He received the B.S. degrees in electrical engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1997, and the Ph.D. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003. He is currently a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University. His research interests include the design and analysis of microwave and millimeter-wave circuits and computational electromagnetics.

Chun Hsiung Chen (SM’88–F’96) was born in Taipei, Taiwan, R.O.C., on March 7, 1937. He received the B.S.E.E. and Ph.D. degrees in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1960 and 1972, respectively, and the M.S.E.E. degree from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1962. In 1963, he joined the Faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. From August 1982 to July 1985, he was Chairman of the Department of Electrical Engineering, National Taiwan University. From August 1992 to July 1996, he was the Director of the University Computer Center, National Taiwan University. In 1974, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. From August 1986 to July 1987, he was a Visiting Professor with the Department of Electrical Engineering, University of Houston, Houston, TX. In 1989, 1990, and 1994, he visited the Microwave Department, Technical University of Munich, Munich, Germany, the Laboratoire d’Optique Electromagnetique, Faculte des Sciences et Techniques de Saint-Jerome, Universite d’Aix-Marseille III, Marseille, France, and the Department of Electrical Engineering, Michigan State University, East Lansing, respectively. His areas of interest include microwave circuits and computational electromagnetics.

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Dual-Mode Microstrip Open-Loop Resonators and Filters Jia-Sheng Hong, Senior Member, IEEE, Hussein Shaman, Student Member, IEEE, and Young-Hoon Chun, Member, IEEE

Abstract—A miniature dual-mode microstrip open-loop resonator is proposed. Distinct characteristics of this new type of dual-mode resonator are investigated using full-wave electromagnetic simulations. It is shown that the two operating modes, i.e., the even and odd modes, within a single dual-mode resonator of this type do not couple. It is also found that there is a finite-frequency transmission zero inherently associated with the even mode. Two two-pole filters using this type of dual-mode resonator are demonstrated with opposite asymmetric responses, which result from different locations of the transmission zero. Higher order filters of this type are also investigated. Both simulated and measured results are presented. Index Terms—Dual-mode resonators and filters, microstrip filters, microstrip resonators.

I. INTRODUCTION

M

ICROSTRIP filters have found wide applications in many RF/microwave circuits and systems [1]. In general, microstrip bandpass filters may be designed using singleor dual-mode resonators. Dual-mode microstrip resonators are attractive because each dual-mode resonator can be used as a doubly tuned resonant circuit and, therefore, the number of resonators required for a given degree of filter is reduced by half, resulting in a compact filter configuration. Several types of dual-mode microstrip resonators have been investigated, including the circular ring [2], square loop [3], circular disk, and square patch [4], [5]. More recently, the dual-mode microstrip triangular patch or loop resonators and filters have been reported [6]–[8]. In this paper, we present an investigation of a new type of miniature microstrip dual-mode resonator for filter applications. The proposed new dual-mode resonator is developed from a single-mode (operated) open-loop resonator [9]. The open-loop resonator is well known for its flexibility to design cross-coupled resonator filters, as well as its compact size, by , where is the guided wavewhich amounts to length at the fundamental resonant frequency. It will be shown that the proposed dual-mode open-loop resonator has a size that is the same as the single-mode open-loop resonator, which, Manuscript received February 2, 2007; revised May 1, 2007. This work was supported by the U.K. Engineering and Physical Science Research Council under Grant EP/C520289/1. The authors are with the Department of Electrical, Electronic and Computer Engineering, Heriot-Watt University, Edinburgh EH14 4AS, U.K. (e-mail: [email protected]) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.901592

Fig. 1. Layout of a dual-mode microstrip open-loop resonator (dimensions in millimeters).

however, is much smaller than the conventional dual-mode loop resonator [3]. The size of the dual-mode open-loop resonator is only approximately one-quarter of the dual-mode loop resonator, which is a significant size reduction. In Section II, we will show that the dual-mode open-loop resonator has some distinct characteristics, which are different from that of the conventional dual-mode loop resonator. For the applications of this new type of dual-mode resonator, some filter examples are described in Section III. Finally, conclusions are given in Section IV. II. DUAL-MODE OPEN-LOOP RESONATORS For our investigation, Fig. 1 shows the layout of a dual-mode microstrip open-loop resonator on a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm. All dimensions are in millimeters. A loading element with a variable is tapped from inside onto the open loop. When parameter varies, the modal resonant characteristic is changed. For the demonstration, we use a commercially available full-wave electromagnetic (EM) simulator [10] to simulate the resonant frequency response of the proposed dual-mode open-loop resonator. To excite the resonator, two ports are weakly coupled to the resonator, as shown. The simulated results are plotted in Fig. 2 for three different values of . As can be seen, for mm, two modes exhibit the same resonant frequency. When is decreased or increased from this value, the two mm and modes split. This is shown in Fig. 2 for

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Fig. 2. Modal resonant characteristic of the proposed dual-mode microstrip open-loop resonator for g = 0:9 mm and d = 1:1 mm.

Fig. 4. Current distribution. (a) Odd mode. (b) Even mode.

Fig. 3. Charge distribution. (a) Odd mode. (b) Even mode.

mm. The smaller results in a mode being shifted to shifts a mode to a lower a higher frequency, while the larger frequency. In all the cases, it is found that only one modal resonant frequency is affected, while the other one is hardly changed. It can be shown that the mode whose resonant frequency is being affected is an even mode, whereas the unaffected mode is an odd mode. To this end, Figs. 3 and 4 illustrate the charge and current distributions of the two modes, respectively. It is evident that the odd mode has a field distribution similar to that of the single-mode open-loop resonator. Hence, the tapping point of the loading element is actually a virtual ground for the odd

mode. As a consequence, the loading element does not affect the odd-mode characteristic, including its resonant frequency. On the contrary, we see a significant field distribution within the loading element for the even mode. This is the reason why the makes the resonant frequency of the even mode change of shifted. It is also interesting to notice from Fig. 2 that there is a finite-frequency transmission zero when the two modes split. The transmission zero is allocated on the high side of the two modes when the even-mode frequency is higher than the odd-mode frequency. On the other hand, the transmission zero is on the low side of the two modes when the even-mode frequency is lower than the odd-mode frequency. Thus, the transmission zero appears to be closely associated with the even mode, and this unique property of the proposed dual-mode resonator can be explored for designing filters with an asymmetrical frequency response, which will be demonstrated in Section III. Another distinct characteristic of the dual-mode open-loop resonator is that the two modes are not coupled to each other even after the modes are split. This is entirely different from the dual-mode square-loop resonator [3], for which once the modes are split, there will be coupling between the two modes. No coupling between the two modes of the dual-mode openloop resonator can be examined based on a theory of asynchronously tuned coupled resonators. The theory states that, if the two split-mode frequencies are equal to the two self-resonant

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Fig. 5. g -dependence of modal resonant characteristic of the proposed dualmode microstrip open-loop resonator for W = 8:1 mm and d = 1:1 mm.

Fig. 6. d-dependence of modal resonant characteristic of the proposed dualmode microstrip open-loop resonator for g = 0:9 mm and W = 8:1 mm.

frequencies, respectively, there is no coupling between the two resonators [1]. The equality between the self-resonant frequencies and split-mode frequencies for the dual-mode open-loop resonator has been investigated using full-wave EM simulations. The self-resonant frequencies of the even and odd modes are obtained in the simulation by placing a magnetic or electric wall along the symmetric axis of the resonator, respectively. Evidently, the EM-simulated results have shown that the two split-mode frequencies are equal to the self-resonant frequencies, which confirms that there is no coupling between the two modes. This characteristic is very important for developing a dual-mode filter using this kind of resonator, which is presented below. According to a general definition of the EM coupling of coupled resonators [1], no coupling between resonant modes may be explained by either an orthogonal nature of their fields or a cancellation of their electric and magnetic couplings.

mode resonant frequency, which can be seen in Fig. 2. In addition, the even-mode resonant frequency can be easily controlled by varying the dimension of , which is demonstrated in Fig. 6. As can be seen, when is changed from 0.3 to 1.3 mm, the even-mode resonant frequency is effectively shifted from 0.983 to 1.152 GHz, whereas the odd-mode resonant frequency is hardly changed. In fact, the smaller results in a larger inductive loading and, hence, a lower resonant frequency for the even mode. The easy control of modal resonant frequencies allows one to control filter bandwidth because the separation of the modal resonant frequencies is propositional to the bandwidth. The input/output (I/O) couplings to the even and odd modes can be characterized in terms of the even- and odd-mode external quality factors, which may be obtained in the simulation by placing a magnetic or electric wall along the symmetric axis of the excited resonator, respectively. Details of the implementation will be described in the following filter examples. Filter Example A

III. FILTER APPLICATIONS Based on the above discussion, a simple two-pole bandpass filter can be implemented using a single dual-mode open-loop resonator, while an -pole filter can be constructed using dual-mode open-loop resonators. Basic coupling structures of this type filter can be found in [6] where no coupling between the even and odd modes has been implicated. In general, to design this type of filter, one needs to allocate the modal resonant frequencies within the desired passband. For a given center frequency, the perimeter of the openloop is approximately a half-wavelength, which sets the resonant frequency of the odd mode. Once the loop size is found, the odd-mode resonant frequency can be tuned to the desired one by varying the gap of the dual-mode open-loop of Fig. 1. In Fig. 5, the simulated resonant frequency response is plotted for different values of . When is changed from 0.3 to 1.3 mm, the odd-mode resonant frequency is effectively shifted for 37 MHz, while the even-mode resonant frequency is much less affected. The even-mode resonant frequency is mainly determined by the size of the loading element and can easily be tuned by sevand of Fig. 1. The dimension of eral parameters such as controls the capacitive loading so as to change the even-

The first microstrip filter example to be described exhibits a finite-frequency zero at the upper stopband. Fig. 7 shows the filter layout on a 1.27-mm-thick dielectric substrate with a relative dielectric constant of 10.8. The filter uses a single dual-mode open-loop resonator with a size of 15.1 mm 15 mm. The I/O is implemented based on a coupled-line structure with a coupling gap of 0.2 mm. The filter is designed in the light of EM simulation with a center frequency of 1.07 GHz. The designed filter is fabricated and the measurement is taken with a microwave network analyzer. Both measured and simulated results are plotted in Fig. 8. Note that, owing to a tolerance in the substrate and fabricais measured to be 0.9975 GHz, tion, the center frequency which is slightly lower than the simulated one. For a better comparison of frequency response, a normalized frequency axis is used in this figure. Nevertheless, good agreement between the measured and simulated filtering responses is seen. The filter shows a nice asymmetric frequency response with a transmission zero on the high side of the passband, resulting in a higher selectivity on this side. The wideband response of the filter can be seen more clearly in Fig. 8(b) with the first spurious occurring at approximately 2.3 times the center frequency. This is a wider

HONG et al.: DUAL-MODE MICROSTRIP OPEN-LOOP RESONATORS AND FILTERS

Fig. 7. Layout of filter A on a 1.27-mm-thick dielectric substrate with a relative dielectric constant of 10.8 (all dimensions are in millimeters).

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Fig. 9. Layout of filter B on a 1.27-mm-thick dielectric substrate with a relative dielectric constant of 10.8 (all dimensions are in millimeters).

Filter Example B Filter B demonstrates a filtering characteristic with a finitefrequency zero on the low side of the passband. Fig. 9 illustrates the layout of the microstrip filter on a 1.27-mm-thick dielectric substrate with a relative dielectric constant of 10.8. The filter also uses a single dual-mode open-loop resonator, which has the same size as filter A, but with a different loading element, as shown. This is because the even-mode frequency for filter B has to be lower than the odd-mode frequency, as discussed in Section II, in order to place a desired transmission zero on the low side of the passband. The filter design is carried out by EM simulation. Fig. 10 shows the measured and simulated performance of the filter. Again, owing to the tolerance, the measured center frequency of 0.95 GHz is slightly lower than the simulated 1.01 GHz and, hence, a normalized frequency axis is used for comparison. It can be seen that the measured frequency response is very similar to the simulated one. The filter exhibits a higher selectivity on the low side of the passband because of the finite-frequency transmission zero. The first spurious does not occur at 2 , but at a higher frequency. Filter Examples C and D

Fig. 8. Measured and simulated frequency response of filter A in Fig. 7. (a) Passband. (b) Wideband (the inset shows a photograph of the fabricated filter).

upper stopband as compared with a filter using the single-mode open-loop or dual-mode square-loop resonator. The different wideband response of the dual-mode microstrip open-loop resonator filter is a direct consequence of reallocation of harmonics due to the loading element implemented.

Filter C has attempted to demonstrate a higher order filter of this type based on coupled dual-mode open-loop resonators. To this end, it is very important to prove that when two dual-mode open-loop resonators are coupled, there are only couplings between the same modes of the resonators, i.e., the odd mode of resonator 1 to the odd mode of resonator 2, and the even mode of resonator 1 to the even mode of resonator 2, respectively. For our purposes, an inter-resonator coupling structure is proposed in Fig. 11(a), where the two identical dual-mode open-loop resonators are approximately placed in opposite orientations with a spacing to facilitate the coupling. The two I/O ports with extended probes are arranged to weakly excite

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Fig. 10. Measured and simulated frequency response of filter B in Fig. 9. (a) Passband. (b) Wideband (the inset shows a photograph of the fabricated filter).

Fig. 11. (a) Coupled dual-mode open-loop resonators. (b) Coupled singlemode (odd-mode) open-loop resonators.

the coupled resonators for EM simulations. Fig. 12 shows typical simulated resonant frequency responses. The magnitude response in Fig. 12(a) clearly shows four resonant peaks (full line) for the coupled resonator structure of Fig. 11(a). The first pair of the peaks at lower frequencies results from a mode split of the two odd modes because of a coupling between them. The

Fig. 12. Resonant responses of the coupled resonator structures in Fig. 11. (a) Magnitude. (b) Phase.

second pair of the resonant peaks at higher frequencies is a result of the mode split of the two even modes when they couple each other. These two pairs of coupled resonant modes exhibit a phase response, as plotted in Fig. 12(b) via the full line, where the two 180 phase change regions correspond to the two pairs of mode splits in the magnitude response. Thus far, the question still remains about whether there is a coupling between the two different modes of the coupled dualmode open-loop resonators. Following the conclusion drawn from the discussion in Section II that the loading element of dual-mode open-loop resonator does not affect the odd mode, we can investigate the coupling between the odd modes alone by simply removing the loading elements from the coupled structure of Fig. 11(a), and simulating the resultant coupled structure of Fig. 11(b). The simulated results are also plotted in Fig. 12 (broken line). As can be seen, the resonant responses (both magnitude and phase) of the coupled structure of Fig. 11(b) match to that of the pair of odd modes of the coupled structure of Fig. 11(a). This implies that there is no coupling between the odd and even modes of the two coupled dual-mode open-loop resonators. Otherwise, we would have seen different degrees of mode splitting. Thus, the coupling structure proposed in [6] for a multipole dual-mode resonator filter is also applicable for this new type of filter composed of dual-mode resonator, and a four-pole two identical dual-mode open-loop resonators is developed for the demonstration. The layout of this filter (filter C) is illustrated in Fig. 13. The filter has a very compact size and uses a coupled

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Fig. 13. Layout of the four-pole dual-mode filter (filter C) on a 1.27-mm-thick dielectric substrate with a relative dielectric constant of 10.8 (all dimensions are in millimeters).

Fig. 14. Measured and simulated frequency response of filter C in Fig. 13 (the inset shows a photograph of the fabricated filter).

resonator structure of Fig. 11(a) with a small offset of the two coupled resonators for a fine adjustment of the couplings. The simulated and measured performances of the filter are shown in GHz while Fig. 14. The simulated performance is GHz. The measured the measured performance is midband insertion loss is 1.0 dB. The bandwidth measured at 15-dB return loss is from 0.965 to 1.035 GHz. The filter shows a finite-frequency zero on the high side of the passband, which is because the dual-mode open-loop resonators used have a higher even-mode frequency. The above four-pole filter exhibits a second-order finite-frequency transmission zero on one side of the passband for applications such as duplexers where asymmetric frequency selectivity is desired. As a matter of fact, a four-pole filter can also be designed to produce two transmission zeros, each on the one side of the passband, leading to a symmetric frequency response if required. This is demonstrated by filter D of Fig. 15(a) with its coupling structure shown in Fig. 15(b). In this case, the two dual-mode resonators without any direct couplings between them are cascaded thorough the two nonresonating nodes, as indicated. The two dual-mode resonators are also not the same as shown; one will produce a zero on the high side of the passband, while the other will produce a zero on the low side of the passband. The full-wave EM simulation has been carried

Fig. 15. (a) Layout of the four-pole dual-mode filter (filter D) on a 1.27-mmthick dielectric substrate with a relative dielectric constant of 10.8 (all dimensions are in millimeters). (b) Its coupling structure. (c) EM-simulated performance.

out to demonstrate its performance, and the result is plotted in Fig. 15(c), showing a symmetric passband with the two desired transmission zeros. IV. CONCLUSION A miniature dual-mode open-loop resonator has been proposed for filter applications. The characteristics of this new type of microstrip dual-mode resonator have been investigated. It has been shown that the loading element inside the open loop does not affect the odd-mode characteristic and the two modes of this type of dual-mode resonator do not couple, even with a mode split. It has also been shown that a finite-frequency transmission zero is inherently associated with the even mode. This makes it much easier to design bandpass filters with an asymmetric response, as no cross coupling is required to generate the transmission zero. Two two-pole filters of this type with opposite asymmetric responses have been demonstrated

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with both simulated and experimental results. For developing higher order filters using the proposed dual-mode open-loop resonators, the EM simulations have been carried out to investigate the inter-resonator coupling characteristics. A very important and useful characteristic has been found, which is that the two different modes of coupled dual-mode open-loop resonators also do not couple with each other. To this end, a compact four-pole filter using two dual-mode open-loop resonators has been demonstrated with a promising performance. In addition, by implementing nonresonating nodes, it has been shown that the four-pole filter can actually produce two separated transmission zeros, each on one side of the passband. It is envisaged that the research presented in this paper will stimulate further developments of these types of compact filters with different topologies and responses. REFERENCES [1] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [2] I. Wolff, “Microstrip bandpass filter using degenerate modes of a microstrip ring resonator,” Electron. Lett., vol. 8, no. 12, pp. 302–303, Jun. 1972. [3] J.-S. Hong and M. J. Lancaster, “Bandpass characteristics of new dualmode microstrip square loop resonators,” Electron. Lett., vol. 31, no. 11, pp. 91–892, Nov. 1995. [4] J. A. Curitis and S. J. Fiedziuszko, “Miniature dual mode microstrip filters,” in IEEE MTT-S Int. Microw. Symp. Dig., 1991, pp. 43–446. [5] R. R. Mansour, “Design of superconductive multiplexers using singlemode and dual-mode filters,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 7, pp. 411–1418, Jul. 1994. [6] J. S. Hong and S. Li, “Theory and experiment of dual-mode microstrip triangular patch resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1237–1243, Apr. 2004. [7] C. Lugo and J. Papapolymerou, “Bandpass filter design using a microstrip triangular loop resonator with dual-mode operation,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 7, pp. 75–477, Jul. 2005. [8] R. Wu and S. Amari, “New triangular microstrip loop resonators for bandpass dual-mode filter applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 41–944. [9] J.-S. Hong and M. J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 11, pp. 2099–2109, Nov. 1996. [10] “EM User’s Manual,” ver. 10, Sonnet Softw. Inc., Syracuse, NY, 2005.

Jia-Sheng Hong (M’94–SM’05) received the D.Phil. degree in engineering science from the University of Oxford, Oxford, U.K., in 1994. His doctoral dissertation concerned EM theory and applications. In 1994, he joined the University of Birmingham, where he was involved with microwave applications of high-temperature superconductors, EM modeling, and circuit optimization. In 2001, he became a faculty member with the Department of Electrical, Electronic and Computer Engineering, Heriot-Watt University, Edinburgh, U.K., where he leads a team for research into advanced RF/microwave device technologies. He has authored or coauthored over 130 journal and conference papers. He also authored Microstrip Filters for RF/Microwave Applications (Wiley, 2001) and RF and Microwave Coupled-Line Circuits, Second Edition (Artech House, 2007). His current interests involve RF/microwave devices, such as antennas and filters, for wireless communications and radar systems, as well as novel material and device technologies including RF microelectromechanical systems (MEMS) and ferroelectric and high-temperature superconducting devices.

Hussein Shaman (S’05) was born in Najran, Saudi Arabia, in 1973. He received the B.Eng. degree in electrical and electronic engineering from Heriot-Watt University, Edinburgh, U.K., in 2005, and is currently working toward the Ph.D. degree in electrical engineering at Heriot-Watt University. His research interest is ultra-wideband (UWB) microwave filters for radar and wireless communications.

Young-Hoon Chun (M’00) received the M.S. and Ph.D. degrees in electronic engineering from Sogang University, Seoul, Korea, in 1995 and 2000, respectively. From 2000 to 2005, he was with the research staff of the Millimeter-Wave Innovation Technology (MINT) Research Center, Dongguk University, Seoul, Korea. During 2004, he was a Visiting Scholar with Heriot-Watt University, Edinburgh, U.K. Since July 2005, he has been with the Department of Electronic Engineering, Heriot-Watt University, as a Research Associate. His research area includes microwave active filters, RF MEMS, passive and active millimeter-wave devices, and multifunctional integrated devices for RF front-ends.

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Design of Vertically Stacked Waveguide Filters in LTCC Tze-Min Shen, Chi-Feng Chen, Ting-Yi Huang, and Ruey-Beei Wu, Senior Member, IEEE

Abstract—This paper proposes four-pole quasi-elliptic function bandpass waveguide filters using multilayer low-temperature co-fired ceramic technology. The vertical metal walls of the waveguide resonators are realized by closely spaced metallic vias. Adjacent cavities are coupled by a narrow slot at the edge of the common broad wall or an inductive window on the sidewall. Two types of vertical coupling structures are utilized to achieve the cross coupling between nonadjacent resonators at different layers. With multilayer capability, there is more flexibility to arrange the cavities of coupled resonator filters in 3-D space. It is demonstrated by both the simulation and experiment that the proposed filter structures occupy a compact circuit area and have good selectivity. The filter with electric field cross coupling occupies a half area of a planar four-pole waveguide filter, while the filter with stacked vias cross coupling has 65% size reduction in comparison with a planar waveguide filter. Index Terms—Bandpass filter, cavity, coupling coefficient, lowtemperature co-fired ceramic (LTCC), stacked vias, quasi-elliptic function.

I. INTRODUCTION

M

ODERN microwave communication systems require high-performance bandpass filters with high selectivity, low insertion loss, and compact size. Filters with a waveguide structure can offer low loss and a high quality ( ) factor, but usually at the price of large size, heavy weight, and high cost. The manufacturing of the waveguide also needs sufficient accuracy in order to operate at the millimeter-wave frequency. Recently, the concept of synthesized rectangular waveguide structures [1] has attracted much interest. The waveguide is dielectric filled and embedded into a substrate. The sidewall of the rectangular waveguide can be realized by arrays of metallic via or metallic grooves. This kind of waveguide not factor, but also suits the realization only maintains a good of high-performance bandpass filters at the millimeter-wave frequency regime. Several direct-coupled cavity filters have been realized by the synthesized waveguide structures on flip-chip modules [2], printed circuit board [3], and thick-film technology [4]. These filters usually occupy a large circuit area because of the planar arrangement of the resonators. With the mature multilayer technology, synthesized waveguide filters are also fabricated on a Manuscript received February 6, 2007; revised May 10, 2007. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 93-2752-E-002-003-PAE and Grant NSC 94-2219-E-002-001 and by the Industrial Technology Research Institute. The authors are with the Department of Electrical Engineering and Graduate Institute Communication Engineering, National Taiwan University, Taipei, 10617 Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.902080

low-temperature co-fired ceramic (LTCC) technology [5] and micromachined process [6]. Multilayer filter technology provides significant benefits in terms of design flexibility and density. This makes vertical coupling between resonators possible and cavities can be piled up in 3-D space, which will largely reduce the circuit area. Frequency selectivity is also an essential feature of a highperformance filter. Quasi-elliptic or elliptic filters will have transmission zeros at finite frequencies and give more improved stopband rejection than conventional direct-coupled filters [7]–[9]. Such filter responses can be realized with cross coupling between nonadjacent resonators [10]. The zeros are then obtained by means of destructive interference between the different signal path connecting the input and output ports [11]. Recently, a conventional parallel coupled microstrip filter with a transmission line inserted inverter for realization of different advanced filtering characteristics was presented in [12]. With the additional cross-coupled transmission line, there is greater flexibility in the arrangement of the cross coupling path to achieve the desired frequency response. In this paper, quasi-elliptic bandpass filters with a cross-coupling architecture are developed in the multilayered LTCC technology, as shown in Figs. 1 and 2. An open-ended microstrip line is used to excite the filters by a narrow slot etched on the first/last cavity. The LTCC resonators can be stacked three-dimensionally to provide various coupling mechanisms required in the design of quasi-elliptic bandpass filters, while achieving compact sizes and good selectivity. The cross coupling between nonadjacent resonators is achieved by a square aperture at the center of the common wall in Fig. 1 or by additional stacked vias and short-circuited coplanar waveguides (CPWs) in Fig. 2. This paper is organized as follow. Section II describes the key design parameters required to realize the quasi-elliptic filter. Section III introduces several coupling structures and the relation of coupling coefficients versus physical dimensions. Section IV provides two design examples. The experiment data are presented and compared with simulation results. Finally, some brief conclusions are drawn in Section V.

II. FILTER DESIGN A general coupling structure of a quasi-elliptic filter is depicted in Fig. 3 [10], where each node represents a resonator, and the solid and dashed lines indicate the main and cross-coupling paths, respectively. It is essential that the signs of the coupling and are opposite in order to coefficients realize a pair of attenuation poles at the finite frequencies. This and means that the coupling routes of

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Fig. 3. General coupling structure of a quasi-elliptic filter.

The design parameters of bandpass filters, i.e., the coupling factor in Fig. 3, can be detercoefficients and the external mined in terms of the circuit elements of a low-pass prototype filter [13]. After determining the required coupling coefficients and external factor, the relationship between coupling coefficients and physical structures of coupled resonators should be established in order to determine the physical dimensions of the filter against the design parameters. The coupling coefficients of coupled resonators can be specified by two split resonate frequencies resulting from electromagnetic coupling [14], i.e., (1)

Fig. 1. Structure of a four-pole quasi-elliptic waveguide filter in a multilayer configuration. (a) 3–D overview. (b) Side view.

In (1), and are defined to be the lower and higher resonance frequencies, respectively. The sign of the coupling coefficient is dependent on the physical structure of the coupled resonators. For filter design, the meaning of positive or negative coupling is rather relative. The positive and negative coupling will have an opposite phase response, which can be found by the -parameter of the coupling structure. The external factor can be characterized by [13] (2) represent the resonance frequency and the where and 3-dB bandwidth of the input or output resonator. By (1) and (2), design curves of the coupling coefficients and external factor versus physical dimensions of coupled resonators can be established. The sizes of coupling structures are also obtained according to the design parameters.

(a)

III. REALIZATION OF COUPLING COEFFICIENTS A. LTCC Cavity Resonator The cavity resonator is formed by several stacked dielectric substrate with metal surfaces at the outer layers and via arrays as vertical sidewalls, which is shown in Fig. 4. The resonant frequencies of the cavity with a perfectly conducting wall can be obtained by [15] (b) Fig. 2. Structure of a vertically stacked quasi-elliptic waveguide filter. (a) 3-D overview. (b) Side view.

need to be out of phase. However, it does not matter which one is positive or negative, as long as their signs are opposite.

(3) where is the relative dielectric constant, is the speed of light, and are the width, height, and length of the cavity, respecand are the indices of the resonant mode. By tively, and (3), the initial dimensions of the synthesized waveguide cavity can be determined, and the final values are optimized by the

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Fig. 4. Cavity resonator with metallic plates and via arrays.

Fig. 6. Coupling coefficients of the slot coupling structure.

the common wall, a narrow slot at the edge of the common wall, or an additional thru via. B. Magnetic Coupling by Broad-Wall Slots

Fig. 5. Field patterns of TE mode of a single rectangular cavity. (a) Electric field. (b) Magnetic field. (c) Surface current on the metal plane.

eigenmode solution solver of a full-wave simulator, e.g., the High Frequency Structure Simulator (HFSS). Besides, the factor of a cavity resonator increases with the cavity height. To get a higher cavity, more substrate layers will be used to form a synthesized cavity according to the fabrication limitation. The cavity resonators in Figs. 1 and 2 operate in their fundaat a common center frequency. Fig. 5 shows mental mode the field patterns. The electric field is mainly concentrated at the center of the cavity and in the direction normal to the metal plane. The magnetic field is tangential to the metallic walls and rotates in the cavity. The magnetic field increases its strength gradually when approaching the sidewalls. The surface current will flow into the center of the metal plate in a radial shape. Next, several structures of coupled resonators in LTCC technology are introduced. The cavities in the same layer are coupled by an inductive window, while the cavities between different layers are coupled by a square aperture in the center of

To efficiently couple two adjacent cavities in different layers, a narrow slot in the common intermediate wall is placed near the sidewalls of the cavity and in the direction perpendicular to the surface current. It will significantly interrupt the surface current flow and introduce strong coupling, analogous to the design principle of waveguide slot antenna [16]. Hence, the coupling between adjacent cavities can be achieved by means of magnetic fields through the narrow slot in the common wall. The coupling coefficient is affected by the length and position of the narrow slot. To get strong direct coupling, the narrow slot should be located as close to the sidewall of the cavity as possible. The coupling strength is then controlled by the slot length. Fig. 6 shows the relation between the slot length and coupling coefficients of the stacked cavities. As mentioned in Section II, the coupling coefficients are calculated by two split resonate frequencies, which can be obtained by an eigenmode solution solver of HFSS. Each cavity resonates at 31 GHz, with mm and the the cavity size . According to fabrication relative dielectric constant limitation, the slot is located 0.2 mm from the cavity sidewall. Due to the presence of the slot, the length of each cavity should be adjusted to compensate for the shifted resonant frewill be the difference between the original cavity quency [2]. for the frequency length and the modified cavity length compensation. The relation between the coupling coefficients and the variation of the cavity length is also plotted in Fig. 6. Both the adjustments in slot length and cavity length are normalized to the cavity width in Fig. 6. C. Magnetic Coupling by Narrow-Wall Window The pair of vias composing the inductive window are used to control the coupling of the cavities at the same layer. It is a common coupling structure in the planar waveguide filter [3]. The coupling strength is controlled by the separation of the via pair. The wider the separation, the stronger the coupling can be.

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Fig. 7. Coupling coefficient of the inductive window coupling structure.

Fig. 9. Via coupling structure. (a) Overview. (b) Coupling coefficient.

Fig. 8. Coupling coefficients of the square aperture coupling structure.

Fig. 7 shows the relation of coupling coefficients versus the via . pitch (VP) and the cavity length variation D. Electric Coupling by Broad-Wall Aperture If a square aperture is opened at the center of two stacked cavities, where the electric field is a maximum, the coupling can be achieved in terms of an electric field normal to the aperture. Fig. 8 shows the aperture length (AL) and cavity length devi, which are both normalized to cavity width , versus ation the coupling coefficients. E. Cross Coupling by Vias Connecting Nonadjacent Cavities As shown in Fig. 2, the first and last cavities are coupled by an additional through via, which provides the cross-coupling path to achieving a quasi-elliptic frequency response [17]. The crosscoupling structure is mainly formed by short-circuited CPW feed lines with a main thru-hole via and two shorter buried vias beside the main through via. The CPW feed lines are connected to the first and last resonators and the main thru-hole via is connected to the CPW feed lines. In the LTCC process, the electric field of a grounded CPW is mainly concentrated under the signal line because of the small

substrate height and wide gap. The electric field distribution of of a cavity a grounded CPW will be similar to that of resonator and, therefore, energy can be gathered from the cavity easily with a CPW feed line. The energy will pass down along the main thru via and couple to the other cavity connected to the CPW feed line. Two shorter buried vias provide current return paths when energy is delivering. The coupling coefficient between the first and last cavities can be extracted by a very weak excitation with the same method described in [2]. Two split resonant frequencies can be seen clearly from the -parameter of the coupled resonators structure. The strength of the cross coupling can be controlled by the length of the CPW stretched into a cavity. When the short-circuited end of the CPW is closer to the center of the cavity where the electric field is strongest, more energy can be gathered from the resonator. The relation between the coupling coefficient and the CPW length is shown in Fig. 9. The coupling coefficient basically increases with the CPW length. Fig. 10 plots the -parameter of coupled resonators with a slot coupling structure and a via coupling structure. By comparing the phase responses in Fig. 10(a) and (b), it is clear that they are out-of-phase. That is to say, two extracted coupling coefficients have opposite signs [18]. Therefore, slot coupling and via coupling structures can be used to realize a quasi-elliptic filter.

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Fig. 11. External

Q factor of the microstrip line feeding structure.

also contribute to fabrication simplicity. The external factor of the feeding structure is controlled by the external slot length and position. Fig. 11 shows the relation of the external factor and the cavity length variation . versus the slot length B. Basic Stacked LTCC Filters Design

Fig. 10. Phase response of coupled resonators. (a) Slot coupling structure. (b) Via coupling structure.

IV. DESIGN EXAMPLES AND EXPERIMENTAL VERIFICATION Next, two kinds of cross coupling structures are utilized to realize the four-pole quasi-elliptic filters. When designing the filters, perfect conductor sidewalls are assumed for calculation efficiency. Based on the above procedure, the initial dimensions of the coupling slots, aperture, and inductive window are decided. The cavities sizes with small variation are also known. The entire filter structure is then optimized by HFSS to meet the design specification. After the initial design of the filter is accomplished, the metallic via arrays take the place of the perfect conductor sidewalls to complete the filter design. A. Feeding Structure The filter is excited by open-ended microstrip lines, as shown in Fig. 1. The slot discontinuity at the return path of the microstrip line causes strong coupling for the same reason that waveguide slot antenna radiates. To maximize the magnetic coupling, a virtual short is placed at the center of each slot by using a quarter-wavelength open stub beyond the slots center [19]. This kind of feeding structure can not only avoid dc power loss, but

A canonical waveguide filter with coupling between nonadjacent cavities can be utilized to achieve an elliptic-function filter response [20]. The cross couplings are achieved by a circle at the center of the common wall or by a narrow slot at the edge of the cavity. In the same concept, a quasi-elliptic filter realized by LTCC technology is presented here. The configuration of the basic stacked LTCC filter is shown in Fig. 1. The coupling produced by means of electric and magnetic fields have opposite signs [21]; therefore, the filter architecture of Fig. 1 will conform to the general coupling structure in Fig. 3, which results in a quasi-elliptic frequency response. The four-pole quasi-elliptic waveguide filter is designed and fabricated in LTCC. The specification of the filter is 10% fractional bandwidth centered at 31 GHz with 20-dB passband return loss. The element values of the low-pass prototype filter are found to be and . By [13], the coupling coefficients and I/O external factor are

(4) The relative dielectric constant of the substrate is 7.8 and its loss tangent is 0.0078 at 30 GHz. The thickness of each metal layer is 13 m and the dielectric layer thickness between two metal layers is 50 m. The cavity height is 250 m, while the microstrip substrate height is 150 m. The via diameter is 100 m. To allow on-wafer measurement by coplanar probes, the input and output probe pad should be on the same layer. Therefore, a vertical transition composed of thru-hole vias is utilized to connect the bottom microstrip line to the top layer. Eight grounded

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TABLE I GEOMETRIC PARAMETERS OF THE QUASI-ELLIPTIC FILTER

Fig. 14. Simulation and measurement results of the four-pole quasi-elliptic LTCC bandpass filter. Fig. 12. (a) Hole structure of a four-pole quasi-elliptic waveguide filter with a vertical transition. (b) Fabricated filter.

Fig. 13. Geometric parameters of the quasi-elliptic filter.

vias are located around the thru-hole via to mimic a coaxial transmission line effect. Fig. 12(a) shows the whole filter configuration and Fig. 12(b) is a photograph of the fabricated filter. Geometric parameters of the filter are illustrated in Fig. 13 and summarized in Table I. The overall size of the four-pole quasi-elliptic LTCC waveguide filter without the vertical transition is 4.46 2.72 0.8 mm , , where is the i.e., approximately guided wavelength on the substrate at the center frequency. A full-wave simulator HFSS is used to calculate the factor of a cavity. The factor is found to be approximately 103, which is similar to the measured data of approximately 99. The frequency response of the filter is shown in Fig. 14, where the solid and dashed lines denote the measured and simulated results, respectively. The dashed–dotted lines represent the ideal

circuit response. The simulation result is not fully identical with the theoretical response. It can be contributed to the replacement of vertical sidewalls of the cavities by via arrays in LTCC and the vertical transitions for on-wafer measurement. When the perfect sidewalls are substituted by the via arrays, the major difference is the in-band return loss. This may be contributed to the variations in the coupling coefficients and external factors, which make the frequency response deviated from that by the theoretical one. When the vertical transition is taken into consideration, The major discrepancy is the deterioration at the higher frequency side of the passband. The measured center frequency of the filter is 30.9 GHz and the 3-dB bandwidth is 3.85 GHz. The passband insertion loss is approximately 2.55 dB and the passband return loss is greater than 12 dB. Two attenuation poles near the cutoff frequencies of the passband can be clearly identified. The two attenuation poles are located at 28.2 and 34 GHz. The measured results are in good agreement with the full-wave simulation results by HFSS. C. Fully Stacked LTCC Filters Design The filter in Fig. 2 introduces a novel structure composed of vertically stacked cavities to realize a quasi-elliptic function filter. The configuration is re-plotted in Fig. 15, composed of four vertically stacked synthesized rectangular cavities. Adjacent cavities are coupled to each other by a narrow slot near the edge of the common wall. The cross-coupling path is realized by short-circuited CPW feed lines connected to the first and last resonators with a main thru-hole via connection. A vertically stacked four-pole quasi-elliptic waveguide filter is designed and fabricated by the same LTCC process in the previous design example. The dimensions of other coupling slots

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Fig. 15. Layer sketch of a fully stacked quasi-elliptic waveguide filter without via arrays.

Fig. 17. Geometric parameters of the vertically stacked quasi-elliptic filter. (a) Top view of the first and fourth cavity. (b) Top view of the second and third cavity.

TABLE II GEOMETRIC PARAMETERS OF THE VERTICALLY STACKED QUASI-ELLIPTIC FILTER

Fig. 16. (a) Entire structure of a fully stacked four-pole quasi-elliptic waveguide filter with a vertical transition. (b) Fabricated filter.

and cavities can be determined under the same design guide. Here, the specification of the filter is 10% fractional bandwidth centered at 30.2 GHz with 20-dB passband return loss. Same element values of the low-pass prototype filter in the previous example are used. The coupling coefficients and external factor are equal to (4) because the same fractional bandwidth is chosen. The whole filter configuration and the photograph of the fabricated filter are shown in Fig. 16. The cavity height is 150 m. The microstrip line substrate height is 100 m. To simplify the measurement, the input microstrip line at the top layer will feed the filter from the opposite direction. A vertical transition connecting microstrip lines at the top and bottom layers is used for on-wafer measurement, as mentioned in the previous example. Geometric parameters of the filter are illustrated in

Fig. 17 and summarized in Table II. The size of the vertically stacked four-pole quasi-elliptic LTCC waveguide filter without the vertical transition is 3.67 2.4 0.8 mm , i.e., approxi. mately The frequency response of the filter is shown in Fig. 18, where the solid and dashed lines denote measured and simulated results, respectively. The dashed–dotted lines represent the ideal circuit response. The measured center frequency of the filter is 29.5 GHz and the 3-dB bandwidth is 3.93 GHz. The passband insertion loss is approximately 2.8 dB and the passband return loss is greater than 12 dB. The two attenuation poles are located at 26.85 and 33.05 GHz. The measured center frequency has down shifted approximately 2% as compared to the simulation

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[2] M. Ito, K. Maruhashi, K. Ikuina, T. Hashiguchi, S. Iwanaga, and K. Ohata, “A 60-GHz-band planar dielectric waveguide filter for flip-chip modules,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2431–2436, Dec. 2001. [3] D. Deslands and K. Wu, “Single-substrate integration technique of planar circuits and waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 593–596, Feb. 2003. [4] D. Stephens, P. R. Young, and I. D. Robertson, “Millimeter-wave substrate integrated waveguides and filters in photoimageable thick-film technology,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3832–3838, Dec. 2005. [5] J.-H. Lee, S. Pinel, J. Papapolymerou, J. Laskar, and M. M. Tentzeris, “Low-loss LTCC cavity filters using system-on-package technology at 60 GHz,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3817–3824, Dec. 2005. [6] J. Papapolymerou, J.-C. Cheng, J. East, and L. P. B. Katehi, “A micromachined high-band resonator,” IEEE Microw. Guided Wave Lett., vol. 7, no. 6, pp. 168–170, Jun. 1997. [7] Z. C. Hao, W. Hong, X. P. Chen, J. X. Chen, K. Wu, and T. J. Cui, “Multilayered substrate integrated waveguide (MSIW) elliptic filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 95–97, Feb. 2005. [8] J. A. Ruiz-Cruz, M. A. E. Sabbagh, K. A. Zaki, J. M. Rebollar, and Y. Zhang, “Canonical ridge waveguide filters in LTCC or metallic resonators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 174–182, Jan. 2005. [9] M. M. Fahmi, J. A. Ruiz-Cruz, K. A. Zaki, and A. J. Piloto, “LTCC wideband canonical ridge waveguide filters,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 249–252. [10] R. Levy and S. B. Chon, “A history of microwave filter research, design, and development,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 9, pp. 1055–1067, Sep. 1984. [11] J. A. R. Cruz, K. A. Zaki, J. R. M. Garai, and J. M. Rebollar, “Rectangular waveguide elliptic filters with capacitive and inductive irises and integrated coaxial excitation,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 269–272. [12] J. S. Hong and M. J. Lancaster, “Transmission line filters with advanced filtering characteristics,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 319–322. [13] J. S. Hong and M. J. Lancaster, “Design of highly selective microstrip bandpass filters with a single pair of attenuation poles at finite frequencies,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1098–1107, Jul. 2000. [14] J. S. Hong and M. J. Lancaster, “Couplings of microstrip square openloop resonator for cross-couple planar microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2099–2109, Dec. 1996. [15] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, ch. 6. [16] R. S. Elliott, Antenna Theory and Design. New York: Wiley, 2003, ch. 3. [17] T.-M. Shen, T.-Y. Huang, C.-F. Chen, and R.-B. Wu, “Design of a vertically stacked waveguide filter with novel cross coupling structures in LTCC,” in Asia–Pacific Microw. Conf. Dig., Dec. 2006, pp. 1161–1164. [18] J. S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Application. New York: Wiley, 2001, ch. 8.5. [19] M. J. Hill, J. Papapolymerou, and R. W. Ziolkowski, “High- micromachined resonant cavities in a -band diplexer configuration,” in Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., Oct. 2001, vol. 148, no. 5, pp. 307–312. [20] A. E. Atia and A. E. Williams, “Nonminimum-phase optimum-amplitude bandpass waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 4, pp. 425–431, Apr. 1976. [21] T. Shen, H.-T. Shu, K. A. Zaki, and A. E. Atia, “Full-wave design of canonical waveguide filter by optimization,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 504–511, Feb. 2003.

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Fig. 18. Simulation and measurement results of the vertically stacked four-pole quasi-elliptic LTCC bandpass filter.

result. This may be contributed to the smaller LTCC shrinkage due to more metal plates used in this filter configuration. Therefore, the cavities are bigger than expected and will result in the down-shifted center frequency. V. CONCLUSION New structures have been proposed to realize the various coupling mechanisms required for quasi-elliptic bandpass filters design using stacked LTCC cavities. The idea has been validated by presenting two four-pole quasi-elliptic function bandpass filters in LTCC. Several coupling mechanisms between adjacent and nonadjacent resonators have been described in details. By this multilayer technology, the vertical coupling between cavities at different layers can be achieved and the filters will have compact size as compared to the conventional planar filters. The filter with the electric field cross-coupling structure approximately occupies the size of two cavities, while the footprint of the filter with fully stacked cavities and cross-coupling via structure can achieve nearly 65% size reduction as compared to the conventional planar four-pole waveguide filters. The cross coupling between nonadjacent resonators is introduced to exhibit a single pair of transmission zeros near the passband at finite frequencies and, thus, much better selectivity. As a result, the proposed structures of the filters occupy a compact circuit area and have a good stopband response. ACKNOWLEDGMENT The authors would like to thank Dr. H.-H. Lin, C.-L. Wang, and C.-C. Chuang, all with the Computer and Communication Laboratory, Institute of Technology Industrial Research, Hsinchu, Taiwan, R.O.C., for their help in the fabrication and measurement of the LTCC filters. REFERENCES [1] H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a laminated waveguide’,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2437–2443, Dec. 1998.

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Tze-Min Shen was born in Chiayi, Taiwan, R.O.C., on August 5, 1981. He received the B.S. degree in electrical engineering and M.S. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004 and 2006, respectively, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University. His research interests is the design of microwave filters.

SHEN et al.: DESIGN OF VERTICALLY STACKED WAVEGUIDE FILTERS IN LTCC

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Chi-Feng Chen was born in PingTung, Taiwan, R.O.C., on September 3, 1979. He received the B.S. degree in physics from Chung Yuan Christian University, Taoyuan, Taiwan, R.O.C., in 2001, the M.S. degree in electrophysics from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 2003, and the Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2006. His research interests include the design of microwave filters and associated RF modules for microwave and millimeter-wave applications.

Ruey-Beei Wu (M’91–SM’97) was born in Tainan, Taiwan, R.O.C., on October 27, 1957. He received the B.S.E.E. and Ph.D. degrees from National Taiwan University, Taipei, Taiwan, R.O.C., in 1979 and 1985, respectively. In 1982, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. He is also with the Graduate Institute of Communications Engineering, which was established in 1997. From March 1986 to February 1987, he was a Visiting Scholar with IBM, East Fishkill, NY. From August 1994 to July 1995, he was with the Electrical Engineering Department, University of California at Los Angeles. He was appointed the Director of the National Center for High-Performance Computing from May 1998 to April 2000 and the Directorate General of Planning and Evaluation Division from November 2002 to July 2004, both under the National Science Council. Since August 2005, he has been Chairperson of the Department of Electrical Engineering, National Taiwan University. He has authored or coauthored over 150 papers in international journals or conferences. He served as an Associate Editor of the Journal of Chinese Institute of Electrical Engineering in 1996. His research interests include computational electromagnetics, transmission line and waveguide discontinuities, microwave and millimeter-wave planar circuits, and interconnection modeling for computer packaging. Dr. Wu is a member Phi Tau Phi and the Chinese Institute of Electrical Engineers. He has been an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES since 2005. He is an elected Executive Committee member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Taipei Chapter. He is an elected Executive Committee member of the Institute of United Radio Science (URSI) Taipei Section. He was the recipient of the Distinguished Research Award presented by the National Science Council (1990, 1993, 1995, and 1997) and the Outstanding Electrical Engineering Professor Award presented by the Chinese Institute of Electrical Engineers (1999).

Ting-Yi Huang was born in Hualien, Taiwan, R.O.C., on November 12, 1977. He received the B.S. degree in electrical engineering and M.S. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University. His research interests include computational electromagnetics, the design of microwave filters, transitions, and associated RF modules for microwave and millimeter-wave applications.

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Wideband Bandstop Filter With Cross-Coupling Hussein Shaman, Student Member, IEEE, and Jia-Sheng Hong, Senior Member, IEEE

Abstract—A general circuit configuration for cross-coupled wideband bandstop filters is proposed. The distinct filtering characteristics of this new type of transmission line filter are investigated theoretically and experimentally. It is shown that a ripple stopband can be created, leading to a quasi-elliptic function response that enhances the rejection bandwidth. A demonstrator with approximately 80% fractional bandwidth at a mid-stopband frequency of 4 GHz is developed and presented. The proposed filter is successfully realized in theory and verified by full-wave electromagnetic simulation and the experiment. Theoretical, simulated, and measured results are in excellent agreement. Index Terms—Microstrip filters, transmission line filters, wideband bandstop filters.

I. INTRODUCTION ANDSTOP filters are in demand for use in many RF/ microwave applications to block undesired signals and to suppress spurious harmonics. Recently, many bandstop filters have been developed to realize a wideband stopband via different methods and structures [1]–[7]. In [1], a uniplanar bandstop filter using two bent open-end stubs is demonstrated where air-bridges are used to suppress the undesired coupled-slot line mode. The filter lacks sharpness at both sides of the stopband and shows narrow bandwidth at 30 dB. To enhance the stopband rejection level of a planar microwave filter, a slow-wave effect is realized by using loading capacitance [2] and a spiral compact microstrip resonator cell [3]. The effects of the spiral compact microstrip resonator cell and loading capacitance shift the spurious resonant frequency from integer multiples of the fundamental frequency. Hence, the stopbands are extended. Alternatively, a spurline is embedded between adjacent shunt stubs to obtain deeper rejection [4]. In [5], a low-loss bandstop filter is constructed by using modulated strips and slotted ground-plane structures. The filter exhibits poor performance at the upper skirt of the stopband. However, an improvement in frequency selectivity at the lower edge is obtained by increasing the slotted cells, which increases the size of the filter. Another wideband bandstop filter with a fractional bandwidth of approximately 120% at a center frequency of 6 GHz has been recently demonstrated in [6]. The filter is a simple design based on a single opencircuited stub; it requires a tight input/output coupling structure to obtain a wide bandwidth, which may limit the attenuation across the stopband. Furthermore, the high-frequency

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Manuscript received March 15, 2007; revised May 10, 2007. The work of H. Shaman was supported by the Saudi Interior Ministry. This work was supported in part by the U.K. Engineering and Physical Science Research Council. The authors are with the Department of Electrical, Electronic and Computer Engineering, Heriot-Watt University, Edinburgh EH14 4AS, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.901600

(upper passband) performance of the filter is not demonstrated. Standard conventional [8] or optimum [9] bandstop filters with shunt open-circuited stubs appear attractive for realization of wideband bandstop filters. However, the impedance of the connecting lines becomes unreasonably high if the bandwidth is very large. This problem may be overcome by using artificial transmission lines [7]. Bandstop filters can also be designed to exhibit ellipticor pseudoelliptic-function responses. This type of filter has a higher selectivity as compared to the conventional Chebyshev filter. Several authors have presented design procedures for such filters when applied to TEM distributed parameter networks. For example, the use of digital lines in a design based on a lumped-element prototype [10], [11] has found wide application but can only be constructed in machined form. In [12], a method is proposed in which TEM Foster sections are exchanged for pairs of noncommensurate stubs. However, high-impedance ( 150 ) line sections are often encountered in the implementation. Another elliptic-function bandstop filter using TEM transmission line resonators is reported in [13], but is inherently narrowband. More recently, bandstop filters using stepped-impedance resonators have been demonstrated [14], which are originally aimed to have extended passbands. A disadvantage of this type of filter is the rather poor upper stopband skirt. Nevertheless, improved characteristics may be obtained by allowing the loss poles to spread out at different frequencies across the stopband, resulting in a pseudoelliptic filter. Recently, we have investigated a cross-coupled ultra-wideband bandpass filter [15]. In this paper, we propose a general configuration for cross-coupled wideband bandstop filters based on an -stub optimum bandstop filter with cross-coupling between the input and output (I/O) feed lines. The characteristics of this type of filter are investigated theoretically in Section II. It is shown that a ripple stopband can be created, leading to an enhanced rejection bandwidth. In addition, the ripple level can easily be controlled by the cross-coupling for a desired rejection level across the stopband. A filter of this type is experimentally demonstrated in Section III, and is realized on a low-cost microstrip substrate having a relative dielectric constant of 3.05 and a thickness of 0.508 mm. The theoretical, simulated, and measured results obtained are presented. Finally, a conclusion is given in Section IV. II. CONFIGURATION OF PROPOSED FILTER Since both the standard conventional [8] and optimum [9] wideband bandstop stub filters have all attenuation poles centered at each mid-stopband frequency, they require more stubs in order to obtain a higher selectivity filtering characteristic. On the other hand, it is known that a filter with elliptic or quasi-elliptic function response exhibits ripples in both the

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TABLE I CIRCUIT PARAMETERS FOR A STANDARD BANDSTOP FILTER WITH ONE, THREE, AND FIVE STUBS FOR A WIDEBAND STOPBAND WITH A FRACTIONAL BANDWIDTH OF APPROXIMATELY 80%

shown in Fig. 2, are demonstrated in Table I for , and . Since the filter configuration of Fig. 1 is symmetrical, its is defined by [16] transmission coefficient (1) Fig. 1. Proposed configuration for the cross-coupled wideband bandstop filter.

with (2) (3)

Fig. 2. General circuit model for a standard bandstop filter with cuited stubs.

n open-cir-

stopband and passband, which improves the filter selectivity effectively. In order to create a ripple in the stopband of a standard conventional or optimum wideband bandstop filter to improve the performance, we propose a general cross-coupled wideband bandstop filter configuration, as shown in Fig. 1. In this configuration, the dashed box indicates a generic circuit model of standard bandstop filters, either conventional or optimum bandstop filters, with open-circuited stubs. A cross-coupling between the I/O feed lines is implemented with a parallel-coupled line section having an electrical length and a pair of even- and odd-mode impedances denoted by and . Assume that the resultant cross-coupled wideband bandstop filter is symmetrical with respect to the I/O ports, which and is the case for most practical filters of this type. Let represent the input impedances of the odd- and even-mode excitations of the proposed filter model, as indicated in Fig. 1. In order to understand the behavior of the proposed structure, a standard optimum bandstop filter whose connecting lines are nonredundant [9] is chosen for implementation. The chosen standard bandstop filter is designed to have a cascade of opennonredundant connecting circuited stubs separated by lines, as illustrated in Fig. 2 [16]. The characteristic impedances of the open-circuited stubs are , while the characteristic impedances defined by . for the connecting lines are defined by The terminal impedance is defined as . The open-circuited stubs and the connecting lines are a quarter-wavelength long at . The element values for this the mid-stopband frequency type of filter are available and tabulated in [16] for fractional bandwidths between 0.3–1.5. To obtain a fractional bandwidth of approximately 80%, the parameters of the circuit model,

where and denote the even- and odd-mode input impedances of the -stub bandstop filter presented by the dashed box in Fig. 1. Referring to the filter structure of Fig. 2, it may be and by and , easier to find where and are the even- and odd-mode input admit, we tances of the -stub bandstop filter. For example, for have (4)

(5) Assume that all transmission line elements have an electrical length at the midband frequency . The frequencyand , which are expressed dependent characteristics of in (2) and (3), can be calculated for . . Fig. 3 depicts To realize a cross-coupling, and . It can be seen that the typical responses of and are open circuits at , which results in an attenuation pole at . In addition, and are equal at two different frequencies, between the upper/lower cutoff frequencies and , as displayed in Fig. 4(a) and (b). Therefore, three transmission zeros at different frequencies can be generated according to (1) with I/O cross-coupling, whereas all attenuation poles are at the same frequency in the midband when there is no I/O cross-coupling. As a result of split attenuation poles, a ripple stopband is created. The level of ripple stopband can be controlled by controlling a cross-coupling coefficient defined by (6) To demonstrate this characteristic, Fig. 5 displays the frequency responses and group delay of the proposed circuit model with varying values of and . shown in Fig. 1 for

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Fig. 3. Z and Z of the proposed cross-coupled filter with Z = 51:5

and Z = 48:5 using the standard bandstop filter with three stubs (Z = Z = 73:54 ; Z = 42:48 ; and Z = Z = 73:88 ).

Fig. 5. (a) Magnitude responses and (b) group delay of the proposed circuit model with = 90 at 4 GHz and varying values of Z and Z using the standard bandstop filter with three stubs (Z = Z = 73:54 ; Z = 42:48 ; and Z = Z = 73:88 ).

Fig. 4. Z and Z of the proposed cross-coupled filter with Z = 51:5

and Z = 48:5 using the standard bandstop filter with three stubs (Z = Z = 73:54 ; Z = 42:48 ; and Z = Z = 73:88 ). (a) At the low frequency. (b) At the high frequency.

It should be noted that when , there is no cross-coupling between the I/O feed lines and the filter performs as a normal three-stub standard bandstop filter that has

only a single attenuation pole in the stopband. However, when the coupling coefficient between the parallel-coupled lines increases, the two additional transmission zeros move towards the edges of the stopband, resulting in sharper skirts and a wider ripple bandwidth. In this case, as can be seen from Fig. 5(a), the 30-, 40-, and 50-dB rejection bandwidth are increased by 32%, 42%, and 48%, respectively. These enhanced ripple bandwidths are more useful for practical applications. The proposed filter configuration of Fig. 1 is also advantageous in that different levels of ripple stopband can easily be obtained by adjusting and of the parallel-coupled lines without changing the parameters of the original standard bandstop filter. and of the proposed cirTable II demonstrates the cuit model using standard bandstop filters with and . These parameters associated with those demonstrated in Table I are chosen to obtain a wideband stopband of approximately 80% fractional bandwidth and 30-dB attenuation inside the stopband. Fig. 6 depicts the magnitude responses of the proposed circuit model using the parameters, which are shown in Tables I and at 4 GHz. It can be seen that the coupling coII for is inversely proportional to the number of stubs efficient , e.g., increasing the degree of the standard bandstop filter

SHAMAN AND HONG: WIDEBAND BANDSTOP FILTER WITH CROSS-COUPLING

TABLE II PARAMETERS OF THE PROPOSED CIRCUIT MODEL FOR WIDEBAND STOPBAND OF 30-dB ATTENUATION LEVEL USING STANDARD BANDSTOP FILTERS WITH ONE, THREE, AND FIVE STUBS

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TABLE III CIRCUIT PARAMETERS FOR DIFFERENT ATTENUATION LEVELS

Fig. 7. Magnitude responses of the proposed circuit model with = 90 at 4 GHz using standard bandstop filter with three stubs for the three cases shown in Table III.

III. IMPLEMENTATION AND EXPERIMENTAL PERFORMANCE

Fig. 6. Magnitude responses of the new circuit model for n = 1; 3; and 5 with = =4 at 4 GHz. (a) Insertion loss. (b) Return loss.

reduces the required coupling for a determined attenuation inside the stopband. Another advantage of this structure is that an equal ripple for at the stopband and for at the adjacent passbands can be obtained by slightly tuning and of the standard bandstop filter. Table III demonstrates the circuit parameters for various attenuation levels, i.e., 30, 40, and 50 dB. The equal ripple frequency responses of the proposed circuit model using the parameters in Table III are demonstrated in Fig. 7. For given circuit parameters, these frequency responses are readily computed using a circuit simulator such as AWR’s Microwave Office [17].

A standard bandstop filter with three open-circuited stubs is chosen for experiment. The filter is designed to have 80% fractional bandwidth at 4 GHz. Based on the element table and design equations in [16], the parameters for the standard bandstop filter are calculated as and . In order to maintain two new transmission zeros and more than a 30-dB attenuation level inside and are chosen to be 51.5 and 48.5 , the stopband, respectively. Unlike synthesis of narrowband filters based on the so-called general coupling matrix [18], direct synthesis of the proposed wideband transmission line filter may be difficult. An alternative approach using synthesis by optimization can be used to design this type of filter. Fig. 8 displays the magnitude responses for the proposed cross-coupled filter and the standard filter without I/O cross-coupling. It should be noted that the proposed filter achieves an improvement in the filter selectivity at both sides of the stopband and can extend the stopband by approximately 32% at 30 dB over the standard bandstop filter without cross-coupling between I/O feed lines [16]. This filter stub resonators and nonredundant example for unit elements (UEs) is actually a filter of degree because each stub and UE contributes a loss pole. The three stub loss poles are in the stopband, but the two UE loss poles are reflection in the complex plane. That is why there are zeroes shown in the upper passband in Fig. 8. Before the introduction of the cross-coupling coupled-line structure, the three stub loss poles are at the same frequency, showing a single notch in the stopband. When the cross coupling is introduced,

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Fig. 8. Comparison between the proposed cross-coupled filter with Z = 51:5 and Z = 48:5 and the standard filter without I/O cross-coupling.

Fig. 9. Final layout of the proposed wideband bandstop filter (unit: millimeters).

Fig. 11. Magnitude responses of the theory, full-wave EM simulation, and experiment.

of 3.05 and a thickness of 0.508 mm. The final layout of the completed design, including the physical dimensions, is shown in Fig. 9. The filter is fabricated using print circuit board (PCB) technology and Fig. 10 depicts the photographed fabricated filter with attached subminiature A (SMA) connectors. The complete filter, including the feed lines, occupies a compact size 0.45 , of 33.6 mm 12.5 mm, which amounts to 1.2 is the guided wavelength of a 50- line at the where mid-stopband frequency on the substrate. Fig. 11 demonstrates the frequency responses of the theory, full-wave EM simulation, and the experiment where excellent agreement is obtained. The EM simulation is done using a commercially available tool [19]. The filter exhibits a highly selective wideband bandstop performance with a fractional bandwidth of approximately 80% at a mid-stopband frequency of 4 GHz. The measured insertion loss is found to be less than 0.75 dB and less than 2 dB over the lower and upper passbands, respectively. Moreover, the measured insertion loss shows new attenuation poles inside the stopbands, which improve the frequency selectivity and widen the 30-dB stopband without any particular change in the 3-dB bandwidth. IV. CONCLUSION

Fig. 10. Fabricated wideband bandstop filter with attached SMA connectors.

two of those poles split off, resulting in three distinct attenuation poles. This is the difference between the standard bandstop filter without cross-coupling and the proposed new filter with cross-coupling. The proposed wideband bandstop filter is realized on a low-cost microstrip substrate with a relative dielectric constant

A general circuit configuration for cross-coupled wideband bandstop filters has been proposed. The characteristics of this type of filter based on an -stub standard bandstop filter with cross-coupling between the I/O feed lines have been investigated. It is shown that a ripple stopband can be created, leading to an enhanced rejection bandwidth. In addition, the ripple level can easily be controlled by the cross-coupling for a desired rejection level across the stopband. An experimental demonstrator of this type of filter with approximately 80% fractional bandwidth at a mid-stopband frequency of 4 GHz has been presented. The proposed filter is successfully realized in theory and verified by full-wave EM simulation and the experiment where excellent agreement was obtained. It has been demonstrated that the use of I/O cross-coupling not only improves the filter selectivity, but also increases the 30-dB bandwidth of the stopband by more than 32% over the standard design without cross-coupling.

SHAMAN AND HONG: WIDEBAND BANDSTOP FILTER WITH CROSS-COUPLING

REFERENCES [1] A. Gorur and C. Karpuz, “Uniplanar compact wideband bandstop filter,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 3, pp. 114–116, Mar. 2003. [2] J. Chen and Q. Xue, “Compact microstrip lowpass filter using slowwave resonator,” in Proc. Eur. Microw. Conf., 2005, pp. 929–930. [3] J. Gu and X. Sun, “Compact lowpass filter using spiral compact microstrip resonant cells,” Electron. Lett., vol. 41, no. 19, pp. 1065–1066, Sep. 2005. [4] W. Tu and K. Chang, “Compact microstrip bandstop filter using open stub and spurline,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 268–270, Apr. 2005. [5] J. Kim and H. Lee, “Wideband and compact bandstop filter structure using double-plane superposition,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 7, pp. 279–280, Jul. 2003. [6] M. Hsieh and S. Wang, “Compact and wideband microstrip bandstop filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 7, pp. 472–474, Jul. 2005. [7] J. Park, G. Kim, and P. Kim, “Miniaturization of lowpass filter by using artificial transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 2227–2230. [8] B. Schiffman and G. Matthaei, “Exact design of band-stop microwave filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-12, no. 1, pp. 6–15, Jan. 1964. [9] M. Horton and R. Menzel, “General theory and design of optimum quarter wave TEM filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 5, pp. 316–327, May 1965. [10] M. C. Horton and R. J. Wenzel, “The digital elliptic filter—A compact sharp cutoff design for wide bandstop or bandpass requirements,” IEEE Trans. Microw. Theory Tech., vol. MTT-15, no. 5, pp. 307–314, May 1967. [11] R. J. Wenzel, “Small elliptic-function low-pass filters and other applications of microwave C sections,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 12, pp. 1150–1158, Dec. 1970. [12] J. G. Malherbe, “TEM pseudoelliptic-function bandstop filters using non-commensurate Lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, no. 5, pp. 242–248, May 1976. [13] H. C. Bell, “L-resonator bandstop filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2669–2672, Dec. 1976. [14] R. Levy, R. V. Snyder, and S. Shin, “Bandstop filters with extended upper passbands,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 2503–2515, Dec. 2006. [15] H. Shaman and J.-S. Hong, “A novel ultra-wideband (UWB) bandpass filter (BPF) with pairs of transmission zeros,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 121–123, Feb. 2007.

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[16] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [17] Microwave Office. Appl. Wave Res., El Segundo, CA, 2005. [18] R. J. Cameron, M. Yu, and Y. Wang, “Direct-coupled microwave filter with single and dual stopband,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3288–3297, Nov. 2005. [19] “EM User’s Manual, Version 10,” Sonnet Softw. Inc., North Syracuse, NY, 2005.

Hussein Shaman (S’05) was born in Najran, Saudi Arabia, in 1973. He received the B.Eng. degree in electrical and electronic engineering from Heriot-Watt University, Edinburgh, U.K., in 2005, and is currently working toward the Ph.D. degree in electrical engineering at Heriot Watt University. His research interest is ultra-wideband (UWB) microwave filters for radar and wireless communications.

Jia-Sheng Hong (M’94–SM’05) received the D.Phil. degree in engineering science from the University of Oxford, Oxford, U.K., in 1994. His doctoral dissertation concerned electromagnetic (EM) theory and applications. In 1994, he joined the University of Birmingham, Edgbaston, Birmingham, U.K., where he was involved with microwave applications of high-temperature superconductors, EM modeling, and circuit optimization. In 2001, he joined the Department of Electrical, Electronic and Computer Engineering, Heriot-Watt University, Edinburgh, U.K., as a faculty member leading a research team into advanced RF/microwave device technologies. He has authored or coauthored over 130 journal and conference papers. He also authored Microstrip Filters for RF/Microwave Applications (Wiley, 2001) and RF and Microwave Coupled-Line Circuits, Second Edition (Artech House, 2007). His current interests involve RF/microwave devices such as antennas and filters for wireless communications and radar systems, as well as novel material and device technologies including RF microelectromechanical systems (MEMS) and ferroelectric and high-temperature superconducting devices.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 8, AUGUST 2007

60-GHz System-on-Package Transmitter Integrating Sub-Harmonic Frequency Amplitude Shift-Keying Modulator Dong Yun Jung, Student Member, IEEE, Won-il Chang, Ki Chan Eun, Student Member, IEEE, and Chul Soon Park, Senior Member, IEEE

Abstract—This paper proposes a simple low-temperature co-fired ceramic (LTCC) integrated transmitter using sub-harmonic amplitude shift-keying modulation for 60-GHz wireless communications applications. The transmitter system-on-package (SoP) has been monolithically implemented with a six-layer LTCC block embedding a resonator, modulator, and antenna and two active circuits, including a negative resistance generator and frequency doubler on the block. The transmitter SoP integrating whole millimeter-wave circuitry is as small as 26 18 0.6 mm3 , which needs external interfaces only for supplying dc power and digital input signal. The fabricated transmitter SoP reveals a bit error rate of 10 11 and good eye pattern through a 2.5-m transmission of 800-Mb/s data. Index Terms—Amplitude shift-keying (ASK) modulator, antenna, low-temperature co-fired ceramic (LTCC), resonator, system-on-package (SoP), transmitter.

I. INTRODUCTION

T

HE 60-GHz band has advantages in terms of broadband, frequency reuse, circuit size, power, unlicensed band, etc. [1]–[9]. Recently, the practical utilization of the 60-GHz band has been a focus of research, and several applications have been proposed for wireless personal area network (WPAN) [3], wireless local area network (WLAN) [4], millimeter-wave video transmission [5], indoor wireless systems [6], and wireless gigabit Ethernet links [7]. System-on-package (SoP) integration using multilayer low-temperature co-fired ceramic (LTCC) technology presents a challenge for those applications in that a small size system integration is required [5]–[10]. The use of multilayer LTCC makes it possible to both make interconnections from or to activate monolithic microwave integrated circuits (MMICs) and embed passive components in themselves, thus dramatically Manuscript received August 19, 2006; revised January 5, 2007. This work was supported by the Ministry of Science and Technology/Korea Science and Engineering Foundation as the Intelligent Radio Engineering Center of Korea under the Science Research Center/Engineering Research Center Program. D. Y. Jung, K. C. Eun, and C. S. Park are with the Intelligent Radio Engineering Center, School of Engineering, Information and Communications University, Daejeon 305-732, Korea (e-mail: [email protected]). W. Chang was with the Intelligent Radio Engineering Center, School of Engineering, Information and Communications University, Daejeon 305-732, Korea. He is now with the Brodern Company, Anyang-City, GyeongGi-Do 430-817, Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.901596

reducing package size and manufacturing cost. In addition, the low-loss characteristics of the LTCC dielectric and conductor allow low-loss transmission lines and high quality factor passive devices for millimeter-wave frequency circuits. A compact LTCC integration of a 60-GHz transmitter and receiver was previously published for a millimeter-wave video transmission system [5]. In that study, frequency multiplication by 16 from a 1.8-GHz phase-locked-loop oscillator and several stages of filtering and amplification were used repeatedly, making the RF system architecture complex and power dissipation serious. A 1-Gb/s indoor wireless system for high-definition television (HDTV) transmission was successfully demonstrated adopting a simple amplitude shift-keying (ASK) modulation with 1.5-GHz bandwidth, eliminating the phase-locked-loop circuits [6]. The RF size was shrunk significantly using direct ASK modulation. However, the passive circuits, i.e., the resonator, filter, and antenna, were not implemented monolithically within the LTCC circuits. Flip-chip bonding was made to integrate the off-chip components on to the LTCC block so that the system had disadvantages in terms of signal loss and manufacturing cost. We have studied 60-GHz SoP transmitters using LTCC multilayer technologies [8]–[10]. A single block transmitter with sub-harmonic mixing from an IF of 2.4 GHz had been reported, which revealed an output power of 13 dBm at 62 GHz and an overall gain of 11 dB, and the RF–local oscillator (LO) isolation was 21.4 dBc [8]. By embedding the bandpass filter, we improved the RF–LO isolation as much as 33.4 dBc in [9]. Although these two studies were the first achievement for a 60-GHz SoP embedding a bandpass filter, as well as an antenna, they required an external supply of a 15-GHz oscillator. In [10], we proposed the system architecture for a 60-GHz direct conversion ASK transmitter, and reported a simulation on ASK modulation. In this paper, a fully integrated 60-GHz transmitter SoP including an oscillator is presented with a sub-harmonic frequency ASK modulator. All passive circuits, i.e., the resonator, ASK modulator, and antenna, are embedded within the LTCC multilayer circuits, where the antenna is placed on the reverse surface from the RF circuits in order to eliminate a possible interference. This paper is organized as follows. Section II discusses the oscillator integrating the negative resistance generator circuit and embedded LTCC resonator. Section III demonstrates the LTCC ASK modulator including a modified rat-race coupler and two Schottky diodes. Section IV describes the development of

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Fig. 1. Integrated oscillator circuit composed of an LTCC resonator and negative resistance generator.

the 60-GHz LTCC embedded patch antenna. Finally, Section V demonstrates the 60-GHz sub-harmonic frequency ASK transmitter SoP on LTCC multilayer circuits. II. LTCC CIRCULAR RESONATOR The sub-harmonic oscillator has been designed with a pseudomorphic HEMT (pHEMT) MMIC generating negative resistance and an LTCC embedded resonator. The MMIC generates an input reflection coefficient larger than unity and makes oscillation combined with the LTCC resonator, as shown in Fig. 1. The oscillation of the integrated oscillator is expressed in (1) [11]. Equation (1) implies (2) and (3) as follows, which signify that the product of the magnitude and summation of the phase of reflection coefficients of the two elements are unity and zero, respectively: (1) (2) (3) Equation (2) is changed to (4) as follows because several decibels of startup gain are needed to start oscillation: (4) Once the oscillation starts and the active part goes to saturation state, (2) is met. We used a commercial 0.25- m GaAs pHEMT (-)R MMIC, which generates the negative resistance from 25 to 35 GHz. The -parameters of the (-)R MMIC have been measured using a vector network analyzer for the different bias conditions, and which was used for the frequency control of the integrated oscillator with the LTCC circular resonator. The measured MMIC V and V has negative resisat bias points tance at the whole frequency band and an especially strong negative resistance corresponding to the input reflection coefficient larger than 5 at a frequency between 30–31 GHz in Fig. 2. By designing a resonator with an LTCC structure having a proper reflection at negative resistance frequency, we can devise a high quality factor monolithic oscillator. A circular resonator for 30-GHz resonance is designed using three layers of LTCC. It can be readily implemented in LTCC by replacing the vertical metallic walls by closely spaced via posts, as shown in Fig. 3. The coplanar waveguide (CPW) for the coupling-fed circular resonator is located on the top layer, and coupling occurs through the center strip of the CPW. (where and are the height and raWhen dius of the circular resonator, respectively), the first fundamental

Fig. 2. Measured magnitude and phase of the input reflection coefficient of the active negative resistance generator MMIC integrated on LTCC @ V = 4 V and V = 0:5 V).

0

Fig. 3. Circuit diagram of an LTCC circular resonator.

transverse magnetic onance frequency of in (5) as follows, where respectively:

is in dominant mode [12]. The resfor the cylinder cavity is expressed is permeability and is permittivity,

(5) The LTCC resonator is designed with a relative dielectric constant of 7.4 and a loss tangent of 0.002. For 30-GHz resonance, is calculated as 1410 m from (5). The signal linewidth of the CPW is 269 m; , the space between the signal line and ground plane of the CPW, is 146 m; and is 300 m for 50 . The CPW line length is 3703 m. We used CST Microwave Studio for a 3-D electromagnetic simulation. The simulation result of the resonator is compared with the measurement result in Fig. 4. Compared to the design, frequency is shifted to a lower frequency by approximately 700 MHz due to experimental deviation such as dielectric shrinkage and fabrication accuracy, and return loss is 4.73 dB at 30.3 GHz. The oscillation frequency of the integrated oscillator can be adaptively tuned through control of the -parameter of the (-)R MMIC satisfying (2) and (3) by changing the bias to the MMIC. The 4.73-dB loss of the resonator still supplies negative resistance for oscillation because the total loop gain, which is defined as a product of the reflection coefficients of

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Fig. 6. Measured phase results of the circular resonator. Fig. 4. Measurement and simulation results of the LTCC circular resonator.

Fig. 5. Product of the measured gains of the resonator and negative resistance generator.

Fig. 7. Measurement output power of the oscillator integrated the (-)R MMIC and the LTCC resonator.

the active MMIC and resonator, is over 0 dB at 30.5 GHz, as shown in Fig. 5. Fig. 6 shows the measured phase results of the LTCC circular resonator. From the measured results, the phase sum is approx57.4 170.4 at 30.3 GHz. To null the phase imately 113 sum of the resonator and negative resistance generator circuit, the length of the CPW feeding line of the resonator and the input CPW transmission line of the MMIC are controlled when we design the monolithic transmitter SoP. Fig. 7 shows the output power of around 11 dBm including 10-dB loss of the attenuator at 29 GHz. The oscillation frequency was shifted down from the resonant frequency of the resonator due to inductance by bond wires and parasitic capacitance of the (-)R MMIC. III. LTCC ASK MODULATOR OF THE MODIFIED RAT-RACE COUPLER TYPE A simple gigabit/second rate modulator has been designed with a modified rat-race coupler and two Schottky diodes connected in different polarity to the ground, as shown in Fig. 8. The IN and OUT ports mean the input port connected to the direct modulating oscillator and the modulated output port, re-

Fig. 8. Configuration of ASK modulator (D: diode, MN: matching network, RFC: RF choke).

spectively. The digital signal supplied through the bottom port switches the diodes.

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Fig. 9. Layout of 30-GHz LTCC ASK modulator (D: diode, MN: matching network, RFC: RF choke).

Fig. 10. Measured insertion and output return loss of ON/OFF states.

When a 0.8-V digital signal is supplied (high state), diode 1 is forced to be off state, while diode 2 is on state. The reflected signals from diode 1 and diode 2 are exactly in-phase because there is an 180 difference in the transmission line length and polarity reversion of the reflection coefficients between the open and short diode ends. The in-phase signals from the two diodes are combined at the sum port of the rat-race coupler [13], which is an output port in this circuit. When the digital signal is low state (zero voltage), the impedances of the two diodes are the same. The reflected signals from diode 1 and diode 2 are out-of-phase with each other. The reflected power is combined at the difference port, which is an input port in this circuit, and there is no power at the output port. Four matching networks in the modulator consist of distributed elements for ON/OFF isolation with over 30 dB and input/output 50- matching. We design and measure the ASK modulator on the two LTCC layers. Fig. 9 shows the layout of the designed 30-GHz ASK modulator for evaluation. The digital signal is supplied to the modulator through an RF choke using two radial stubs. To match the impedance for both on and off states simultaneously, the matching networks are designed using distributed elements on the LTCC. The size of the fabricated modulator is 25 11 0.2 mm . Fig. 10 shows the measurement result of the on and off states, revealing excellent isolation as much as 32.7 dB and output return loss of less than 15 dB. Although the ASK modulator is matched at 28.5 GHz, as shown in Fig. 10, the final dimension of the modulator has been modified for 30.5-GHz operation in the integrated transmitter.

Fig. 11. (a) LTCC modulator on test fixture. (b) Digital signal supplied from signal generator. (c) Dynamic modulation output signal.

Fig. 11(a) shows the test fixture of the fabricated modulator circuit for the dynamic modulation test. The signal linewidth

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2

Fig. 12. Structure of 60-GHz LTCC 4 4 patch array antenna (e-MSL: embedded microstrip line, GND: ground plane).

of the IN/OUT port of the modulator implemented on the LTCC is designed with 200 m; the -parameters are measured using ground–signal–ground (GSG) probes and the network analyzer. The transmission line with signal linewidth over 700 m is needed to connect the external K-connector for the dynamic modulation test. An additional 50- transmission line with 750 m is implemented on an RO4003 substrate with a thickness of 8 mil. The signal power of 5 dBm at 28.5 GHz is supplied from a signal generator into the input port of the modulator. In the digital signal port, the signal of the duty cycle is 20% with 0.8-V amplitude at a clock frequency of 3 MHz entered from a function generator. Fig. 11(b) and (c) shows the digital input signal and the modulated signal in the frequency domain. IV. LTCC EMBEDDED 60-GHZ PATCH ANTENNA A high gain antenna was designed with a 4 4 microstrip patch array structure and schematic, as shown in Fig. 12. Three layers of 100- m LTCC are used. The radiating patches and feeding network are placed on the third and second layers, respectively. The ground plane is on the back side of the first layer. The feeding network consists of embedded microstrip line T-combiners. The dividers are located on the second layer as an embedded microstrip line because of the dispersion effect at 60 GHz. Proximity-coupled feeding is used to feed the patches on the third layer. is determined by The initial width of the rectangular patch (6) [14] as follows:

2

Fig. 13. (a) Fabricated 60-GHz LTCC 4 4 patch array antenna. (b) Simulated and measured gain of the -plane beam pattern. (c) Measured return loss and the antenna on test board (e-MSL: embedded microstrip line).

H

(6) where is the velocity of light, is the resonance frequency, is the relative dielectric constant of the substrate. The and is an eflength of the patch is determined from (7), where fective dielectric constant defined by (8), is the height of the rectangular patch, and is defined by (9) [14] as follows: (7)

(8) (9) The patch and feeding dimension is finally optimized 640 m through rigorous electromagnetic as 1250 m simulation.

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Fig. 14. Structure for sub-harmonic ASK transmitter SoP using six LTCC layers (Res: resonator, GND: ground, e-MSL: embedded microstrip line).

Fig. 16. (a) Top and (b) bottom of the LTCC transmitter SoP on the test board.

Fig. 15. (a) Top and (b) bottom views of the fabricated sub-harmonic frequency transmitter SoP using six LTCC layers (the active MMICs are not attached).

The size of the LTCC transmitter antenna is 20 20 0.3 mm . Fig. 13 shows a photograph of the fabricated antenna, the simulated and measured results of the -plane beam patterns, and the measured return loss. For the measurement, we designed and fabricated a test board with an embedded microstrip line-to-WR 15 transition, as shown in the inset of 13(c). The gain and 3-dB beam width are 14 dBi and 19 at 60 GHz, respectively. The return loss of the antenna on the test board is below 15 dB from 59.75 to 60.25 GHz. V. GIGAHERTZ SUB-HARMONIC FREQUENCY ASK TRANSMITTER ON LTCC Up to Section IV, we introduced and verified the design and characteristics of each sub-block for 60-GHz applications. Section V describes a 60-GHz ASK transmitter integrating the sub-blocks previously described. Slight modifications of the design parameters of each component are made to adjust the transmission frequency. The vertical structure of the multilayer 60-GHz sub-harmonic ASK transmitter is shown in Fig. 14. The oscillator along with

the LTCC circular resonator, modulator, and frequency doubler are located on the top sixth layer. The antenna is located on the bottom layers through connection of via transitions. The main ground plane is located on the third layer. The circular resonator consumes three layers over the main ground plane. The oscillator MMIC and attenuator are placed on the top surface layer with an elevated ground plane, which is connected to the main ground plane through three stacked ground vias. The modulator is placed on the top two layers with one layer being an elevated ground plane. An additional coupler for monitoring the 30-GHz signal is inserted next to the modulator. The pHEMT frequency-doubler MMIC is placed on the fifth layer to reduce bonding wire inductance at 60-GHz frequency. The output of the frequency doubler is connected to the e-MSL of the antenna on the first layer through via transition. The 4 4 rectangular patch array antenna is located on the bottom surface. Fig. 15 shows top and bottom views of the fabricated transmitter SoP LTCC block before attaching the MMICs. The total size of the transmitter is 26 18 0.6 mm . The MMICs and required components for supplying dc bias are die attached on the LTCC using silver epoxy and bond wire, and the LTCC substrate is mounted on the FR4 test board where required bias circuits are placed. The LTCC transmitter is placed on the metal test board for measurement, as shown in Fig. 16. The transmitter SoP integrates all microwave and millimeter-wave frequency circuits on the LTCC block and the external interface to

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Fig. 17. Measurement setups of the proposed transmitter SoP (Tx: transmitter, Rx: receiver, SoP: system-on-package).

Fig. 19. (a) Eye diagram fed from signal source generator with an 800-Mb/s data rate. (b) Demodulated signal at the commercial receiver.

Fig. 18. Measured 60-GHz output spectrum from the transmitter antenna with a digital signal of 800 Mb/s.

the FR4 test board is comprised of circuits for dc supply and digital signal only. The measurement setups of the proposed transmitter SoP are shown in Fig. 17. To measure the 60-GHz output spectrum, a standard horn antenna is used to receive the transmitted 60-GHz signal, as shown in Fig. 17 (setup a), and the corresponding 60-GHz output spectrum with 800-Mb/s digital signal is shown in Fig. 18. The total power dissipation of the transmitter is 1.1 W and the output power in front of the transmitting antenna is 14 dBm. It is evaluated at the output pads of the MMIC doubler prior to the antenna transition. The received power through the horn antenna with a gain of 20 dBi located at a distance of 2.5 m away from the transmitter is 30.83 dBm at 60.939 GHz. To measure the eye pattern, an 800-Mb/s digital signal is fed into the digital signal port of the transmitter from the pulse pattern generator under synchronization with an oscilloscope. The transmitted signal from the designed transmitter is received by a standard horn antenna and down converter at around 2.5 m away from the transmitter through the air interface, as shown in Fig. 17 (setup b). The eye pattern of the digital input signal of 800 Mb/s fed from the pulse pattern generator and the de-

Fig. 20. Measured bit error rate at 800 Mb/s.

modulated signal through the transmitter and down converter are shown in Fig. 19(a) and (b), respectively. When the digital signal of 800 Mb/s with a time jitter of around 100 ps is supplied to the digital port of the transmitter SoP, as shown in Fig. 19(a), the eye diagram shows a maximum time jitter of 600 ps, including distortion by the receiver components, as shown in Fig. 19(b). Finally, the bit error rate is measured as shown in Fig. 17 (setup c), at 800 Mb/s. and Fig. 20 describes the bit error rate of 10 These results reveal that they are suitable for 60-GHz radio with a high data rate of the transmitter implemented on the LTCC SoP using the sub-harmonic frequency ASK modulator. VI. CONCLUSION A simple cost-effective 60-GHz transmitter using a sub-harmonic ASK modulator is proposed. Each sub-block is designed

JUNG et al.: 60-GHz SOP TRANSMITTER INTEGRATING SUB-HARMONIC FREQUENCY ASK MODULATOR

and verified. For 60-GHz wireless transmission, the integrated transmitter requires only two MMICs: one for the negative resistance generator and the other for the frequency doubler. The monolithic transmitter SoP using six LTCC layers measuring 26 18 0.6 mm integrates entire microwave and millimeterwave functions including the oscillator with an LTCC circular resonator, an LTCC ASK modulator, a frequency doubler, and a 4 4 LTCC rectangular patch array antenna. The external interface consists of circuits to supply a dc and digital signal only so that the 60-GHz SoP can be mounted on a plain circuit board. From the 60-GHz output spectrum with 800-Mb/s data rate, the harmonic of the ON/OFF signal is verified to have little harmonic after frequency doubling. The fabricated transmitter SoP reveals and a good eye pattern through 2.5-m a bit error rate of 10 transmission of an 800-Mb/s data rate. REFERENCES [1] H. Ogawa, “Millimeter-wave personal area network systems,” in IEEE Radio Freq. Integrated Circuits Symp., Jun. 2006, pp. 379–382. [2] S. Sarkar, D. A. Yeh, S. Pinel, and J. Laskar, “60-GHz direct-demodulator on liquid-crystal polymer,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1245–1252, Mar. 2006. [3] B. Razavi, “A 60-GHz CMOS receiver front-end,” IEEE J. Solid-State Circuits, vol. 41, no. 1, pp. 17–22, Jan. 2006. [4] N. Deparis, A. Bendjabballah, A. Boe, M. Fryziel, C. Loyez, L. Clavier, N. Rolland, and P. A. Rolland, “Transposition of a baseband UWB signal at 60 GHz for high data rate indoor WLAN,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 609–611, Oct. 2005. [5] A. Yamada, E. Suematsu, K. Sato, M. Yamammoto, and H. Sato, “60 GHz ultra compact transmitter/receiver with a low phase noise PLLoscillator,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 3, pp. 2035–2038. [6] K. Maruhashi, S. Kishimoto, M. Ito, K. Ohata, Y. Hamade, T. Morimoto, and H. Shimawaki, “Wireless uncompressed-HDTV-signal transmission system utilizing compact 60-GHz-band transmitter and receiver,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, 4 pp. [7] K. Ohata, K. Maruhashi, M. Ito, S. Kishimoto, K. Ikuina, T. Hashiguchi, K. Ikeda, and N. Takahashi, “1.25 Gbps wireless gigabit Ethernet link at 60 GHz band,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 1, pp. 373–376. [8] Y. C. Lee, W.-I. Chang, Y. H. Cho, and C. S. Park, “A very compact 60 GHz transmitter integrating GaAs MMICs on LTCC passive circuits for wireless terminals applications,” in IEEE Compound Semicond. Integrated Circuit Symp., Oct. 2004, pp. 313–316. [9] Y. C. Lee, W.-I. Chang, and C. S. Park, “Monolithic LTCC SiP transmitter for 60 GHz wireless communication terminals,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, Paper WE4F-3. [10] D. Y. Jung, W. I. Chang, and C. S. Park, “A system-on-package integration of 60 GHz ASK transmitter,” in IEEE Radio Wireless Symp., Jan. 2006, pp. 151–154. [11] G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design. Englewood Cliffs, NJ: Prentice-Hall, 1997. [12] A. M. El-Tager and L. Roy, “Study of cylindrical multilayered ceramic resonators with rectangular air cavity for low-phase noise K=Ka-band oscillators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 2211–2219, Jun. 2005. [13] D. M. Pozar, Microwave Engineering. New York: Wiley, 2005. [14] C. A. Balanis, Antenna Theory Analysis and Design. New York: Wiley, 2005.

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Dong Yun Jung (S’05) received the B.S. degree in electronics and materials engineering from Kwangwoon University, Seoul, Korea, in 2001, the M.S. degree in electronics engineering from the Information and Communications University (ICU), Daejeon, Korea, in 2003, and is currently working toward the Ph.D. degree in electronic engineering at ICU. In 2003, he joined the Basic Research Laboratory, Electronics and Telecommunications Research Institute (ETRI), as a Member of the Engineering Staff. His research interests include MMICs and 3-D module design using LTCC technologies for millimeter-wave applications.

Won-il Chang received the B.S. degree in electronics engineering from Sogang University, Seoul, Korea, in 1997, and the M.S. degree in electronics engineering from the Information and Communications University (ICU), Daejeon, Korea, in 2006. In 1996, he joined Hyundai Electronics Industries as a Development Engineer. In 1997, he joined KMW 1997, as a Development Engineer. From 1998 to 2004, he was with Millitron Inc., as a Senior Engineer. Since 2005, he has been with the Brodern Company, Anyang-City, GyeongGi-Do, Korea, as a Senior Engineer. His research interests include MMICs and 3-D module design using LTCC technologies for millimeter-wave applications.

Ki Chan Eun (S’07) received the B.S. degree in electronics engineering from Chonbuk National University, Jeonju, Korea, in 2001, the M.S. degree in electronics engineering from the Information and Communications University (ICU), Daejeon, Korea, in 2003, and is currently working toward the Ph.D. degree in electronic engineering at ICU. In 2003, he joined the Radio Technology Group, Digital Broadcasting Research Division, Electronics and Telecommunications Research Institute (ETRI), as a Member of the Research Staff. His research interests include MMIC design and their 3-D integration using LTCC-based system-in-package (SIP) technology for millimeter-wave applications.

Chul Soon Park (M’97–SM’07) received the B.S. degree in metallurgical engineering from Seoul National University, Seoul, Korea, in 1980, and the M.S. and Ph.D. degrees in materials science from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1982 and 1985, respectively. From 1985 to 1999, he was with the Electronics and Telecommunication Research Institute (ETRI), where he contributed to the development of semiconductor devices and circuits. From 1987 to 1989, he studied the very initial growth of group IV semiconductors during a visit to AT&T Bell Laboratories, Murray Hill, NJ. Since 1999, he has been with the Information and Communications University (ICU), Daejeon, Korea, where he is a Professor with the Engineering School and Director of the Intelligent Radio Engineering Center. His research interests include power amplifiers and reconfigurable RFICs and their system-on-chip (SoC)/SoP integration.

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Left-Handed Metamaterial Coplanar Waveguide Components and Circuits in GaAs MMIC Technology Wei Tong, Student Member, IEEE, Zhirun Hu, Member, IEEE, Hong. S. Chua, Member, IEEE, Philip D. Curtis, Andrew. A. P. Gibson, Senior Member, IEEE, and Mohamed Missous, Senior Member, IEEE Abstract—A set of novel left-handed LH) basic block monolithic-microwave integrated-circuit (MMIC) components (transmission line, open and short stubs) are presented with applications to the RF/microwave filter and power divider. These new open and short LH MMIC resonators have compact sizes of 0.11 and 0.22 mm2 , respectively. The prototypes of an LH bandpass filter and LH power divider constructed from these basic components have been designed, fabricated, and characterized. The LH filter consisting of an LH transmission line and four LH open resonators exhibits transmission zeros at both sides of the passband from 1.02 to 1.42 GHz with 33.3% fractional bandwidth. The LH power divider including two LH branches shows equal power split from 2.8 to 3.72 GHz. The filter and power divider benefit from the miniaturized LH MMIC components and have compact sizes of 2.85 and 2.3 mm2 , respectively. Loss issues of these LH structures and, more importantly, the effect of skin depth at low frequency are investigated. Full-wave simulations for conductor thicknesses of 0.2, 1.4, and 5 m are presented for the proposed LH MMIC components and circuits. There is a good agreement between the full-wave simulations, equivalent-circuit model, and measurement results with 1.4- m metal thickness demonstrating the effectiveness of the models. Index Terms—Bandpass filter, left-handed (LH) metamaterial, power divider, resonators.

I. INTRODUCTION

A

RTIFICIAL metamaterials exhibiting simultaneously negative electric permittivity and permeability (doublenegative index), now usually termed left-handed (LH) metamaterials, have generated considerable interest. Starting with Veselago’s seminal theoretical analysis in 1968 [1], Pendry et al. introduced metallic structures with a negative permittivity [2] and developed a periodic nonmagnetic structure of a split-ring resonator with a negative permeability [3] in the late 1990s. Smith et al. [4] and Shelby et al. [5] experimentally realized an LH medium, which was composed of an array of metallic wires to attain negative and an array of split-ring resonators to achieve negative . Subsequently, a number of novel LH devices and applications based on the split-ring resonator or

Manuscript received November 16, 2006; revised May 2, 2007. This work was supported by the U.K. Electro Magnetic Remote Sensing Defence Technology Centre under Grant EMRS/DTC/3/79 and by the Engineering and Physical Sciences Research Council under Grant EP/C015339/1. The authors are with the School of Electrical and Electronic Engineering, The University of Manchester, Manchester M60 1QD, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.902579

complementary split-ring resonator technology were developed, which include bandstop, bandpass filters, and spurious suppression [6]–[10]. The first papers based on a transmission line approach of metamaterials were published by Caloz and Itoh [11], Iyer and Eleftheriades [12], and Oliner [13]. Since then, many discrete LH transmission line circuits, such as stubs, couplers, phase shifters, and filters, have been developed [14]–[24]. To fully utilize the LH transmission line properties in size reduction and performance enhancement, it is a natural step forward to incorporate LH transmission line metamaterials with integration technologies such as monolithic microwave integrated circuit (MMIC) and low-temperature co-fired ceramic (LTCC) [25]–[29]. In this paper, a set of novel LH basic MMIC components, such as LH transmission line and open and short stubs, are reported. Series metal–insulator–metal (MIM) capacitors and spiral shunt inductors are realized in a GaAs MMIC process, offering not only compact dimensions, but also significant reduction in right-handed (RH) parasitic effects. As a demonstration of the applications of these basic LH MMIC components, both the LH MMIC bandpass filter and power divider have been designed and fabricated. Since in standard MMIC technology the metal thickness is limited, usually up to 5 m for the top metal layer and 1 m for the bottom layer, metal loss is an inevitable issue for the realization of these components, especially skin effect at lower operating frequencies. This is unlike printed circuit board (PCB) structures, where metal thickness is usually more than 30 m. To understand how metal thickness may affect LH MMIC circuits, tolerance analysis has been carried out by means of full-wave simulation, equivalent-circuit modeling, and experimental measurements. This paper is organized as follows. Section II describes the proposed LH transmission lines for MMIC applications with six and eight unit cells. This is followed by a brief description of the fabrication process of the MMIC coplanar waveguide (CPW) structure. Based on the proposed unit cell, two basic LH components, i.e., open and short stubs, are presented in Section III. To illustrate the case of utilizing the proposed LH MMIC basic components, a super compact LH bandpass filter and LH power divider based on this technology are presented in Sections IV and V, respectively. Finally, conclusions are discussed in Section VI. II. INTEGRATED LH TRANSMISSION LINE A. LH Transmission Line An LH transmission line is an artificial transmission line structure created by the repetition of series capacitance and and of Fig. 1(a) are due to the shunt inductance . The

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Fig. 2. Cross-sectional view of a 3-D MMIC incorporating passive CPW integrated technology. (M1 and M2: bottom metal layer and top metal layer, P: polyimide layer, GND: CPW ground).

The center bandgap is determined by the series and shunt res, shunt onant frequencies. The series resonant frequency resonant frequency , and two cutoff frequencies and are given by [21] (1) (2) (3) (4) The dispersion of the LH transmission line can be split into additive positive linear RH and negative hyperbolic LH terms [21], and is given as (5)

Fig. 1. (a) Equivalent-circuit model of an integrated LH transmission line unit cell: LH: series capacitance C , LH: shunt inductance L , RH-series inductance L , RH: shunt capacitance C and C (including fringing capacitance and series resistance of the spiral inductor C , and R). (b) Unit cell structure and (c) fabricated prototype of LH MMIC transmission line with six unit cells.

RH parasitic effects. As a result, an LH band at lower frequencies and an RH band at higher frequencies are present. Fig. 1(a) illustrates the corresponding equivalent circuit of the unit cell. In GaAs MMIC technology, the capacitance and inductance are typically implemented in the form of two metal layer capacitors and spiral inductors, as shown in Fig. 1(b). The corresponding photograph of a fabricated six-unit cell LH MMIC transmission line is depicted in Fig. 1(c). A full equivalent-circuit model of the spiral inductor [30] was adopted in the analysis. Theoretically, an LH transmission line has two operating reand the LH gions, corresponding to the RH mode mode , respectively. Notice that these two regions are bounded by a bandgap and two cutoff frequencies determined by the RH circuit elements within the unit cell (low-pass filter) and LH circuit elements within the unit cell (high-pass filter).

where is the length of the unit cell. When , the propagation constant is negative, meaning that the transmission line operates in the LH range, otherwise the transmission line operates in the RH ranges. The input feeding line presented here was implemented by a CPW transmission line with a width of 100 m and gaps of 70 m, having a characteristic impedance of 50 . The LH MMIC components were fabricated using two metal layers sandwiched between a polyimide dielectric, which was placed on a GaAs substrate. A cross-sectional view of the 3-D MMIC incorporating passive CPW integrated technology is shown in Fig. 2. In realizing these MMIC structures, several processing aspects have to be taken into account, such as polyimide spin, curing, etching, and metal contact formation. In these structures, different layers need to be interconnected properly through the etched widows of the polyimide insulating layers. Ti/Au layer (M1 and M2) metallization was formed by first evaporating a 40-nm thin Ti film layer onto the substrate, and then a 170-nm Au metal layer is deposited onto the Ti layer. The polyimide layer (PI 2610) is spun on the Ti/Au metal layer at a speed of 8000 r/min and cured in a vacuum oven at 200 C for 2 h. The polyimide interconnection layer was formed using oxygen plasma reactive ion etching (RIE) through a photoresist protecting layer patterned using a lithography process. The

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Fig. 3. Experimental results for the six-unit cell LH MMIC transmission line compared with the equivalent-circuit model and full-wave HFSS simulation results. The values of parameters extracted by the technology presented in [21] for the equivalent-circuit mode are C = 1:1 pF, L = 1 nH, C = 0:06 pF, L = 0:075 nH, and R = 28 .

Fig. 5. Phase shift for six-cell ( respectively.

) and eight-cell ( ) LH transmission lines,

TABLE I RESISTANCES OF THE PROPOSED 1-nH SPIRAL INDUCTOR FOR 0.2-, 1.4-, AND 5-m METAL THICKNESSES AT 1 GHz

Fig. 4. Dispersion diagram of the six-unit cell LH MMIC transmission line, where d is the total length of the line.

2.3 to 17.5 GHz. To further illustrate the left-handedness of the transmission line, the phase advance of the eight-cell transmission line with respect to six cells is shown in Fig. 5. As can be seen, the longer the transmission line (more unit cells), the more the phase advance will be, demonstrating the existence of a negative phase velocity. B. Conductor Loss

thickness of the polyimide layers is 1 m and its dielectric constant is 3.8. The process is terminated by evaporating a final Ti/Au metal layer. The thickness of the GaAs substrate is 625 m. Fig. 3 shows the simulated and measured transmission coefficients of the six-unit cell LH MMIC transmission line. The full-wave simulation was conducted using Ansoft’s High Frequency Structure Simulator (HFSS). In this case, the capacitor of the unit cell occupies 0.031 mm , providing a capacitance of approximately 1.1 pF. The spiral inductor, with 10- m strip width and 10- m gap, has an area of 0.022 mm and an inductance of 1 nH. The metal thickness is 0.2 m for both top and bottom metal layers. As can be seen in this figure, a very good agreement between the measured and simulated results, from both equivalent-circuit model and full wave simulation, is achieved. Fig. 4 illustrates the dispersion characteristics of the LH MMIC transmission line, obtained from the unwrap phase of by [21]. The negative sign of the slope for an operating range from 2.3 to 17.5 GHz demonstrates the existence of a negative phase velocity in this frequency range. Thus, the structure exhibits ultra-broad LH frequency bandwidth from

In a typical MMIC technology, the thicknesses of the metal conductor are usually up to 5 m for M2 (the top metal layer) and 1 m for M1 (the bottom layer). Thinner metal layers will cause significant conductor loss, especially at lower microwave frequency due to the skin effect. This is particularly true for the MMIC spiral inductors. in Fig. 1, which takes into account the conductor loss, can be derived as [31] (6) where is a normalized frequency, and are the width and thickness of the conductor strip, and and are the permeability and conductivity of the conductor, respectively. A good agreement between the results of the equivalent-circuit model and measurement data can be seen in Fig. 3. Due to the thin metallization of the LH MMIC transmission line, slowly the conductor loss is relatively large, resulting in rising near the cutoff frequency. Table I shows the values of resistances for metal thicknesses of 0.2, 1.4, and 5 m for M2 and 0.2 m for M1 at 1 GHz. It can be seen that, as the metal thickness increases, the conductor loss significantly reduces.

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Fig. 6. Fabricated prototype of the one-unit cell LH MMIC open stub. Fig. 8. Fabricated prototype of LH MMIC short stub.

Fig. 7. Measured and simulated results of the LH MMIC open stub with 0.2-, 1.4-, and 5-m metal thicknesses. Fig. 9. Measured and simulated results of the LH MMIC short stub with 0.2-, 1.4-, and 5-m metal thicknesses.

III. INTEGRATED LH STUBS A. LH MMIC Open Stub As a basic microwave component, the resonant frequency of open stub depends on the length of the the conventional transmission line. In contrast, the resonant frequency of an LH transmission line open stub is only determined by the LC values. Thus, an open stub can achieve a rather compact dimension with appropriate values of L and C. To minimize the size and loss, the proposed LH open stub consists of only one LH unit cell with a series capacitor of 0.76 pF and a shunt inductor of 1.82 nH. The top metal thickness is 1.4 m and the bottom is 0.2 m. The fabricated prototype is shown in Fig. 6. The LH MMIC open stub occupies a very small area equal to 0.11 mm . The simulated and measured results of are shown in Fig. 7. reaches a minimum with a value close to zero, at 4.2 GHz, as expected. This means the LH MMIC open stub can perform as well as a transmission line open stub. Meanwhile, as conventional the metal thickness increases, the performance of the LH open stub gets better, as shown in Fig. 7. B. LH MMIC Short Stub Besides the open stub, the short stub is another common component for microwave devices. An LH MMIC short stub consists of two unit cells, with the series capacitor and shunt inductors having the same values as those of the open stub, as depicted

in Fig. 8. In this case, such an LH MMIC short stub is equivalent to a short-circuited stub with large input impedance at resonance. The short stub has a size of 0.22 mm and resonates at of our fabricated stub reaches a maximum value of 3 GHz. 142 , which is lower than expected. This is due to our fabrication limitations. In our fabrication process, the thickness of metallization we can achieve is 1.4 m for M2 and 0.2 m for M1 with our current technology. Fig. 9 shows the measured and simulated results of the short stub. It can be seen that there is a good agreement between the full-wave simulation and measured results for the stub with a metal thickness of 1.4 m (M2) and 0.2 m (M1). The importance of metal thickness should not be underestimated, as it can be seen in Fig. 9, a short stub with 0.2- m metal thickness for both M1 and M2 cannot perform as a resonator, as the losses are too high. On the other hand, however, if the metal conductors can be fabricated to be as thick as 5 m increases significantly at resonant fre(M2) and 1 m (M1), quency. In other words, high- inductors are highly desirable for LH MMIC applications, which can be achieved with thicker metal. IV. LH MMIC BANDPASS FILTER As a demonstration of utilizing the basic LH MMIC components, a novel LH MMIC bandpass filter has been designed, fab-

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Fig. 10. Fabricated LH MMIC bandpass filter with center frequency of 1.2 GHz and 33% fractional bandwidth. Open stubs 1 and 2 have resonant frequencies at 0.76 and 1.75 GHz, respectively.

ricated, and measured. The bandpass filter topology adopts the LH open stubs connected to an LH transmission line to achieve the transmission zeros at both sides of the passband. To enhance the rejection at transmission zeros, each zero pole is led by two identical LH open stubs. The design steps are thereby as follows. Step 1) Designing the LH open stubs with resonant frequencies at 0.76 and 1.75 GHz, which means that the proposed LH stubs provide 90 phase shift at these frequencies. The parameters of these stubs can be obtained from [21] (7) (8)

Fig. 11. Measured and simulated results of the LH MMIC bandpass filter with 1.4- and 5-m metal thicknesses.

In contrast to a conventional microwave transmission line bandpass filter, the bandwidth and center frequency of the LH bandpass filter do not depend on the length of the transmission lines. Transmission poles and zeros can be created and tuned by adjusting the LC values of the open stubs. Fig. 11 illustrates the measured and simulated frequency responses of the LH MMIC bandpass filter. It can be seen that a good agreement between simulation and measurement has been achieved. The relatively high insertion loss of the filter is due to the thin metallization. The thickness of our evaporated Ti/Au is 1.4 m (M2), which is approximately five times less than the normal thickness associated with three skin depth penetration on gold at 1 GHz. A full-wave simulation result for the filter with 5 m (M2) and 1 m (M1) is also shown in Fig. 11, indicating less than 2-dB insertion loss. V. LH MMIC POWER DIVIDER

where is the number of unit cells. Step 2) Designing an LH transmission line with cutoff frequency lower than that of the first resonant frequency, 0.76 GHz. The parameters of this LH transmission line can be obtained from (7) and (9) Step 3) Connecting the above open stubs to transmission line. A little tuning would be expected to get a better matching performance. pF, Finally, the designed bandpass filter has pF, pF, pF, nH, nH, nH, and nH. These values have been realized by MMIC technology, as described in Section II-A. Therefore, our LH MMIC bandpass filter consists of four LH open stubs and one LH transmission line, as shown in Fig. 10. The filter has a size of 2.85 mm , with an operating center frequency of 1.2 GHz and 33.3% fractional bandwidth. Two zeros occur at 0.76 and 1.75 GHz, respectively, as designed, giving a 3-dB passband from 1.02 to 1.42 GHz.

A 3-dB LH MMIC power divider, consisting of LH MMIC transmission lines, has been designed and fabricated to provide an equal power split between the two output branches at branches are replaced by 3.1 GHz. The two conventional two LH transmission lines, each of which has two unit cells with a characteristic impedance of 70.7 , as shown in Fig. 12. In order to achieve 90 phase shift at 3.1 GHz and the required characteristic impedance, each unit cell of the LH transmission line consists of a 0.95-pF capacitor with a size of 0.028 mm and a 4.75-nH spiral inductor (8- m strip width and 6- m gap) with an area of 0.053 mm . The values of LC parameters can be calculated from (7) and (8) as well. The fabricated LH MMIC power divider has a very compact size of 2.3 mm with a center frequency at 3.1 GHz, as shown in Fig. 13. It can also be seen in this figure that equal power split between the two output ports and a good return loss has been achieved. The measured return loss is below 10 dB from 2.75 to 3.55 GHz, indicating that the GHz with device is well matched, especially around dB. It is evident that the meaa minimum value of sured and simulated results agree well. Again, the loss is relatively high due to the thin metallization. To achieve 3-dB power

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ACKNOWLEDGMENT The authors wish to thank Dr. J. Sly, School of Electrical and Electronic Engineering, The University of Manchester, Manchester, U.K. REFERENCES

Fig. 12. Fabricated LH MMIC 3-dB power divider with center frequency at 3.1 GHz.

Fig. 13. Measured and simulated frequency responses for the thrus (S S ), and the return loss (S ) of the LH MMIC power divider.

and

split, much thicker metal layers are needed. This is indicated by the simulation results with a metal thickness of 5 m (M2) and 1 m (M1).

VI. CONCLUSION In this paper, a set of novel LH basic components, transmission line, and open- and short-circuited stubs have been realized in an in-house MMIC technology. We have demonstrated the applications of these LH basic components by designing, fabricating, and testing a miniaturized LH MMIC bandpass filter and power divider. The measured results show that the LH MMIC bandpass filter has a passband from 1.02 to 1.42 GHz, while covering an area equal to 2.85 mm . The LH MMIC power divider achieves an equal power split from 2.8 to 3.72 GHz with a total area of 2.85 mm . In addition, the effect of conductor loss was addressed. Full-wave simulations were confirmed via measurement results. Investigations concluded that the loss issue can be resolved by increasing the metal thicknesses.

[1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative value of " and ,” (in Russian) Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] J. B. Pendry, A. J. Holden, D. J. Robbinst, and W. J. Stewart, “Low frequency plasmons in thin-wire structure,” J. Phys, Condens. Matter, pp. 4785–4809, 1998. [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, May 2000. [5] R. A. Shelby, D. R. Smith, and S. Schulta, “Experimental verification of a negative index refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [6] J. D. Baena, J. Bonache, F. Martin, R. M. Silleo, F. Falcone, T. Lopeteg, M. A. G. Laso, J. Garcia-Garcia, I. Gill, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1451–1461, Apr. 2005. [7] F. Falcone, T. Lopetegi, J. D. Baena, R. Marques, F. Martin, and M. Sorolla, “Effective negative- " stopband microstrip lines based on complementary split ring resonators,” Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 280–282, Jun. 2004. [8] J. Bonache, I. Gil, J. Garcia-Garcia, and F. Martin, “Novel microstrip bandpass filters based on complimentary split-ring resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 265–271, Jan. 2006. [9] J. Garcia-Garcia, F. Martin, F. Falcone, J. Bonache, I. Gil, T. Lopetegi, M. A. G. Laso, M. Sorolla, and R. Marques, “Spurious passband suppression in microstrip coupled line bandpass filters by means of split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 416–418, Sep. 2004. [10] J. Garcia-Garcia, F. Martin, F. Falcone, J. Bonache, J. D. Baena, I. Gil, E. Amat, T. Lopetegi, M. A. G. Laso, J. A. M. Iturmendi, M. Sorolla, and R. Marques, “Microwave filters with improved stopband based on sub-wavelength resonators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1997–2006, Jun. 2005. [11] C. Caloz and T. Itoh, “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 2002, pp. 412–415. [12] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index media supporting 2-D waves,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, pp. 1067–1070. [13] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 2002, pp. 41–44. [14] M. A. Antoniades and G. V. Eleftheriades, “A broadband Wilkinson balun using microstrip metamaterial lines,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 209–212, 2005. [15] M. A. Antoniades and G. V. Eleftheriades, “Compact linear lead/lag metamaterial phase shifter for broadband applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 103–106, 2003. [16] M. A. Antoniades and G. V. Eleftheriades, “A broadband series power divider using zero-degree metamaterial phase-shifting lines,” IEEE Antennas Wireless Compon. Lett., vol. 15, no. 11, pp. 808–810, Nov. 2005. [17] R. Islam and G. V. Eleftheriades, “Printed high-directivity metamaterial MS/NRI coupled-line coupler for signal monitoring applications,” IEEE Antennas Wireless Compon. Lett., vol. 16, no. 4, pp. 164–166, Apr. 2006. [18] H. Lin, M. De Vincentis, C. Caloz, and T. Itoh, “Arbitrary dual band components using composite right/left-handed transmission lines,” IEEE Trans. Microw. Theory. Tech., vol. 52, no. 4, pp. 1142–1149, Apr. 2004. [19] H. Okabe, C. Caloz, and T. Itoh, “A compact enhanced-bandwidth hybrid using a left-handed transmission line,” IEEE Trans. Microw. Theory. Tech., vol. 52, no. 3, pp. 798–804, Mar. 2004.

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[20] C. Caloz, A. Sananda, and T. Itoh, “A novel composite right-/left-handedcoupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [21] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. New York: Wiley, 2005. [22] W. Tong and Z. Hu, “Left-handed L band bandstop filter with significantly reduced-size,” Proc. Inst. Elect. Eng.—Antennas Propag., to be published. [23] S. G. Mao and M. S. Wu, “Equivalent circuit modeling of symmetric composite right/left-handed coplanar waveguides,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1953–1956. [24] S. G. Mao, M. S. Wu, Y. Z. Chueh, and C. H. Chen, “Modeling of symmetric composite right/left-handed coplanar waveguides with applications to compact bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3460–3466, Nov. 2005. [25] W. Tong and Z. Hu, “Left-handed multilayer super compact coupledline directional coupler,” in Proc. Eur. Microw. Assoc., Mar. 2006, vol. 2, pp. 60–65. [26] W. Tong and Z. R. Hu, “A super compact multilayer broadband lefthanded metamaterial for RF/MMIC applications,” in Proc. AP-S Int. Symp., Washington, DC, Jul. 2005, pp. 656–659. [27] P. D. Curtis, Z. Hu, W. Tong, H. S. Chua, A. P. Gibson, and M. Missous, “Integrated left-handed metamaterials for RF/MMIC miniaturization and performance enhancement,” in 36th Eur. Microw. Conf., Manchester, U.K., Sep. 10–15, 2006, pp. 940–942. [28] Y. Horii, C. Caloz, and T. Itoh, “Super-compact multilayered left-handed transmission line and diplexer application,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1527–1534, Apr. 2005. [29] J. Perruisseau-Carrier and A. K. Skrivervik, “Composite right/lefthanded transmission line metamaterial phase shifters (MPS) in MMIC technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1582–1589, Apr. 2006. [30] R. P. Ribas, J. Lescot, J.-L. Leclercq, J. M. Karam, and F. Ndagijimana, “Micromachined microwave: Planar spiral inductors and transformers,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 8, pp. 1326–1335, Aug. 2000. [31] E. Pettenpaul, H. Kaputsa, A. Weisgerber, H. Mampe, J. Luginsland, and I. Wolff, “CAD models of lumped elements on GaAs up to 18 GHz,” IEEE Trans Microw. Theory Tech., vol. 36, pp. 294–304, Feb. 1988. Wei Tong (S’04) was born in Chengdu, Sichuan, China, in 1982. He received the B.S. degree in electronic engineering from the University of Electronic and Science Technology of China, Chengdu, Sichuan, China, in 2004, and is currently working toward the Ph.D. degree at The University of Manchester, Manchester, U.K. His research interests include microwave and millimeter-wave circuit designs.

Zhirun Hu (M’99) received the B.Eng. degree in telecommunication engineering from Nanjing University of Posts and Telecommunications, Nanjing, China, in 1982, and the Master in Business Administration degree and Ph.D. degree in electrical and electronic engineering from the Queen’s University of Belfast, Belfast, U.K., in 1988 and 1991, respectively. In 1991, he joined the Department of Electrical and Electronic Engineering, University College of Swansea, as a Senior Research Assistant in computational semiconductor device simulation. In 1994, he rejoined the Department of Electrical and Electronic Engineering the Queen’s University of Belfast, as a Research Fellow involved with silicon MMIC design, realization, and characterization. In 1996, he joined GEC Marconi, as a Microwave Technologist involved with microwave/millimeter-wave circuit design and characterization.

From 1998 to 2003, he was a Lecturer with the Department of Electronic Engineering, King’s College London. He is currently a Senior Lecturer with the School of Electrical and Electronic Engineering, The University of Manchester, Manchester, U.K.

Hong. S. Chua (S’03–M’06) received the B.Eng. (Hons.) degree in electrical and electronic engineering (with first-class honors) and Ph.D. degree from The University of Manchester Institute of Science and Technology (UMIST), Manchester, U.K., in 2002 and 2006, respectively. From 2002 to 2006, he was involved in a program of research investigating different methods such as microwave, MRI, and terahertz to characterize wheat grain. His current research involves integrated LH metamaterials for RF/MMIC applications.

Philip. D. Curtis received the B.Eng. degree in electronic engineering and Ph.D. degree in electronic and electrical engineering from The University of Leeds, Leeds, U.K., in 1999 and 2003, respectively. Since 2003, he has been a Research Associate with the School of Electronics and Electrical Engineering, The University of Manchester, Manchester, U.K.

Andrew. A. P. Gibson (M’04–SM’05) was born in Dunfermline, U.K., in 1962. He received the M.Eng. degree in electrical and electronic engineering and Ph.D. degree from Heriot-Watt University, Edinburgh, U.K., in 1985 and 1988, respectively. He is currently a Professor with The University of Manchester, Manchester, U.K. His research has included finite-element analysis of gyromagnetic waveguides. Prof. Gibson was the recipient of the 1985 Institution of Electrical Engineers (IEE), U.K., Institute Prize.

Mohamed Missous (M’95–SM’05) received the Ph.D. degree from The University of Manchester, Manchester, U.K., in 1985. His doctoral research concerned the growth and characterization of epitaxial Al grown by molecular beam epitaxy (MBE) onto GaAs. In 1991, he became a Lecturer. In 1996, he became a Reader 1996. In 2001, he became a Professor of semiconductor materials and devices. All of his recent research has been closely associated with the MBE effort at The University of Manchester. He is actively involved as a consultant to both the microwave industry in the U.K. and the U.S. and to the optoelectronics industry in Japan and the U.S. on the growth of lasers and detectors in the InGaAs/InAlAs systems. He has authored or couathored over 170 papers on MBE materials, devices, and circuits related topics. He has recently concentrated with considerable success on establishing practical approaches and techniques required to meet stringent doping and thickness control for a variety of advanced quantum devices. This has included research on GaAs, AlGaAs, InGaAs, InAlAs, InGaP, and InAlP. His principal research interests are interfaces, both metal–semiconductor and semiconductor–semiconductor, and MBE growth mechanisms, especially under conditions of exact stoichiometry at low temperatures.

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Virtual Auxiliary Termination for Multiport Scattering Matrix Measurement Using Two-Port Network Analyzer Chih-Jung Chen and Tah-Hsiung Chu, Member, IEEE

Abstract—Almost all methods for measuring the scattering matrix of an -port device with the use of a two-port vector network 2 ports analyzer (VNA) require one to terminate the other in the fully characterized auxiliary terminations and prefer auxiliary terminations with small or moderate reflection coefficients. In this paper, a technique is presented to measure the auxiliary terminations indirectly. It not only eases the measurement of auxiliary terminations, but also makes the concept of virtual auxiliary termination realizable. In practice, examples of virtual auxiliary terminations can be the inherent connectors or bonding pads of the test device. Also studied is the remedy for the numerical pitfall possibly coexisting with the use of strongly reflecting auxiliary terminations such as virtual auxiliary terminations. The scattering matrix of a four-port circulator is then acquired accordingly from measurements using a two-port VNA and virtual auxiliary terminations. Index Terms—Auxiliary termination, multiport network, scattering matrix measurement.

I. INTRODUCTION

In (1),

.. .

.. .

..

.

..

.

..

.

(2)

The diagonal elements in (2) are the reflection coefficients of the auxiliary terminations, which are connected in sequence to the and are partitioned materminated ports. trices of the true -port -matrix and are defined as (3), shown at the bottom of the following page. as measured Taking a three-port network with terminated in as an exports and the other port ample, we can write the measured two-port -matrix according to (1)–(3) as

W

HEN IT comes to -port scattering matrix ( -matrix) measurement with the use of a two-port vector network analyzer (VNA), one can follow the mathematical definition [1] combinations of two-port measurements with and perform unused ports terminated in perfectly matched the other loads, where . Nonetheless, a matched load is not always accessible to applications such as millimeter wave or broadband. By taking advantage of the formulas derived in [2], the following two paragraphs are given to explain how an imperfectly matched load degrades the accuracy of the measured -parameters, and to introduce the nomenclature in this paper. For an -port device-under-test (DUT), the true -port -matrix and its element in the th row and th column are denoted and , respectively. For each two-port measurement, the is divided into two disjoint set of number of ports for the two measured ports and subsets, which are for the other terminated ports. Therefore, the possible combinations of and are both . The measured two-port -matrix is given by [2] (1) Manuscript received December 11, 2006; revised April 3, 2007. This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC 95-2221-E-002-086-MY3 and Grant NSC 95-2752-E-002-004-PAE. The authors are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C. (e-mail: [email protected] tw). Digital Object Identifier 10.1109/TMTT.2007.901603

(4) It is observable that four elements of the true three-port -matrix can be obtained as the first matrix on the right side of (4) . Otherwise, the imperfectly matched load causes if mismatch-induced errors and makes the measured scattering parameters ( -parameters) deviate from the true ones. Thanks to the efforts of pioneering researchers, many mathematical formulas for correcting the mismatch-induced errors have been developed to resolve the predicament of measuring multiport devices with the use of a two-port VNA [2]–[11]. Among them, each method asks one to deploy auxiliary terminations according to its own strategy. However, the need for all [2]–[9] or partial [10], [11] auxiliary terminations to be fully characterized in terms of their complex reflection coefficients is in common. In addition, we observe that most of the methods prefer auxiliary terminations with small or moderate reflection coefficients. It is believed that they can alleviate possible numerical difficulties by those auxiliary terminations. Thus, there are two issues worthy of our attention. Firstly, depending on the types of the ports of the DUT, measuring the reflection coefficient of auxiliary termination may be easy or tedious. For instance, it could be beneficial for not directly probing auxiliary terminations like surface mount devices. Secondly, in contrast to

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the processes of soldering surface mount resistors for a printed circuit board (PCB) and fabricating poly resistors for a monolithic microwave integrated circuit (MMIC), which suffer from large variation, the process of implementing strongly reflecting termination like a short-circuited or an open-circuited transmission line is repeatable and reliable for both applications. The applicability of existing methods is then limited, as they do not function normally with the use of auxiliary terminations having large reflection coefficients. Only recently have these two issues been tackled in [11]. It proposes a method to acquire the -matrix of a multistage line coupler using a two-port VNA and , in short-circuited reflective terminations with which all but one are unknown. Inspired by an attempt to deal with the first issue mentioned above, we propose an indirect measurement technique to circumvent those direct measurements of auxiliary terminations. It is later learned that the synergy of this technique and the method in [2] allows one to dispense with the use of auxiliary terminations. In other words, the DUT can have its unused ports left unconnected even though it may give large reflection coefficients. Since the unused ports have no terminations to be specified, they are denoted as virtual auxiliary terminations in this paper. The inherent subminiature A (SMA) connector of the DUT is an example of virtual auxiliary termination. It should be noted that the reflection coefficient of a virtual auxiliary termination usually cannot be measured directly. Although the concept of virtual auxiliary termination is verified by reconstructing the three-port -matrix of a Mini-Circuits D17I 17 dB directional coupler in [12], numerical experiments to be conducted in Section II reveal that the utilization of virtual auxiliary terminations could entrap the method in [2] into numerical pitfalls for low-loss DUTs at certain frequency points. A remedy for the numerical difficulties is then studied. The practical applicability of virtual auxiliary termination is demonstrated by reconstructing the four-port -matrix of a nonreciprocal component, i.e., Narda COF-2040 circulator, in Section III. A successful fulfillment of using only virtual auxiliary terminations can simplify the measurement arrangement and procedures, as well as broaden the applicability of the method in [2]. II. FORMULATION Here, a technique for indirectly measuring an auxiliary termination is described. The concept of virtual auxiliary terminations and how their utilization impacts the accuracy of the method in [2] are elucidated in detail. A corresponding remedy

based on Newton’s method and simple moving average is then formulated. A. Indirect Measurement of Auxiliary Termination When the third to the th ports of an -port network are termi, one can measure ports 1 and 2 and obtain nated in . One then terminates the second the two-port -matrix and measures port 1 to get the one-port -parameter port in . By performing signal flow graph analysis, can be caland as [12] culated from the elements of (5) Therefore, through proper permutations, all without directly measuring them.

can be acquired

B. Virtual Auxiliary Termination According to [2], the -matrix of an -port network can be reconstructed using combinations of two-port measurements unused ports terminated in fully characterwith the other ized auxiliary terminations. It is given by (6) In (6),

.. .

.. .

..

.

..

.

..

.

(7)

diagonal matrix, whose diagonal elements are the reis an flection coefficients of the auxiliary terminations in which ports are invariably terminated while not being connected to the VNA. The elements of to be expressed in Section II-C are related to the reflection coefficients of the auxiliary terminations and the measured two-port -matrices so that the reconstructed -matrix is mathematically identical to the true -port -matrix . This method imposes no restriction on the deployment of auxiliary terminations, except each port should be terminated in the same auxiliary termination while not being connected to the VNA. Accordingly, a logical and handy choice of auxiliary terminations is to use the inherent connectors for components with coaxial-type connectors or probe pads for the MMIC. In other words, the DUT leaves its unused ports with no connection to

(3) .. .

.. .

.. .

.. .

..

.

.. .

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the specified terminations. Instead of knowing the reflection coefficients of these auxiliary terminations in advance, one can acquire them by following the procedures of developing (5). By this approach, one can leave those ports not connected to the VNA unconnected during each two-port measurement. These terminations are then dubbed as the virtual auxiliary terminations. Note that the virtual auxiliary terminations usually have reflection coefficients close to one. Studies of the possible computation difficulties are presented in the following. C. Accuracy Analysis Since both calculations of and (6) involve matrix inversion, it could be numerically difficult under certain circumstances [3], [4], [13]–[15]. Judging by [15], one can tell the accuracy of the method in [2] from the two-norm condition number of . A matrix with a large condition number is known as ill conditioned, one whose solution to an inverse matrix is overly sensitive to perturbations in its elements. In our case, numerical perturbations mostly originate from the measurement errors and the finite precision of the computer. Before the discussion of the condition number of , the calculation of itself is given in the following. For a two-port measurement with the measured ports and the termi, in (7) can be partitioned nated ports accordingly as

.. .

.. .

.. .

..

.

..

.

..

.

.. .

(8)

By defining [2], (9) and substituting (1) into (9), one can show that [2]

(10) can be completely known from While (9) signifies that measurements, (10) relates it to the true -port -matrix. In the meantime, one can also define [2]

(11)

Fig. 1. Port enumeration of the four-port circulator.

The formula for matrix inversion in [16] is then applied to express (11) in the form of

(12) and are of no significance for the following disAs cussion, their explicit expressions are not shown. By making a comparison between (10) and (12), the identity between and concludes that is a partitioned matrix of . Concombinations of sequently, one can piece together from , which can be evaluated from the measured two-port -matrices and the reflection coefficients of auxiliary terminations by using (9). Through trying to expand (10), one finds that the elements of are nonlinear complex functions of -parameters of the -port network and reflection coefficients of the auxiliary terminations. It is difficult, if not impossible, to figure out what tends to make ill conditioned by interpreting and deciphering the equations. In contrast, it is practicable to perform quantitative analyses by conducting computer simulation of measurement. Designating a Narda COF-2040 four-port circulator, which operates between 2–4 GHz as the experimental sample, we illustrate its port enumeration in Fig. 1 and prepare its -matrices at different frequencies for the ensuing simulations by following the mathematical definition of the -parameter [1] and performing six combinations of two-port measurements with unused ports terminated in the loads of an Agilent 85052D calibration kit. The first computer simulation is to study how different reflection coefficients of auxiliary terminations affect the condition numbers of . Although auxiliary terminations are permitted to be all identical, partially distinct, or all distinct, the case of being all identical is of interest and is simulated. Defining the average power dissipation of an -port network as (13)

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Fig. 2. Average power dissipation of the circulator.

and with Fig. 2 depicting that of the circulator, we group the -matrices of the circulator into three types. The corresponding features of the condition number of are individually studied from their sampled -matrices. The type 1 -matrix located between 1–1.5 GHz is high loss. The -matrix sampled at 1.2 GHz is shown in (14) at the bottom of this page. The type 2 -matrix residing between 2–4 GHz is in-band and has characteristics of low loss, good matching, low backward leakage, high isolation, and high forward transmission. Consequently, the measurement of type 2 -matrix requires careful arrangement due to the high dynamic range in the -parameters. The -matrix sampled at 2.95 GHz is shown in (15) at

the bottom of this page. The type 3 -matrix lying between 4.5–5 GHz is low loss. The -matrix sampled at 4.75 GHz is shown in (16) at the bottom of this page. The two-port measurements are then simulated by partitioning the sampled -matrices properly and substituting these partitioned matrices and the specified reflection coefficients of auxiliary terminations into (1). After six combinations of are evaluated from (9), is acquired and its two-norm condition numbers for various reflection coefficients of auxiliary terminations over the Smith chart are depicted for the sampled -matrices in Figs. 3–5. The condition number distribution pattern of the type 1 sample shows only one peak, whereas those of types 2 and 3 have four relatively higher peaks. After running more simulations at other frequencies for each type of -matrix, we observe that the resulting patterns are similar to those given in Figs. 3–5. Moreover, even though peaks of the patterns change positions for different cases, they never move inward on the Smith chart. Three implications can then be drawn from the results of the first experimental simulation. Firstly, the method in [2] prefers auxiliary terminations with small reflection coefficients. Secondly, when auxiliary terminations with large reflection coefficients are taken, it may encounter numerical difficulty whose destructiveness depends inversely on the power dissipation of the DUT. Thirdly, since the reflection coefficient of virtual auxiliary termination is close to that of an open circuit, which is at the furthest right point in the Smith chart in Fig. 4(b), the second sampled -matrix is a specimen of the numerical difficulty for using virtual auxiliary terminations. The second computer simulation explores the compatibility between virtual auxiliary terminations and the method in [2]. Assuming the reflection coefficients of virtual auxiliary terminations to be unity, we evaluate the corresponding condition numbers of with the given -matrices of the circulator from

(14)

(15)

(16)

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Fig. 3. Condition number distribution patterns in: (a) 3-D plot and (b) contour plot of the circulator at 1.2 GHz.

1 to 5 GHz. In Fig. 6, one observes that peaks with a condition number greater than 30 appear almost periodically within the operation bandwidth of the circulator. This simulation study manifests the necessity of a remedy for those ill-conditioned points. D. Remedy In (1), a measured two-port -matrix is related to the true -port -matrix of the DUT and the reflection coeffiof auxiliary terminations. With combinations cients of two-port measurements, there are equations available, which can be written concisely in the form of a system of nonlinear equations as

Fig. 4. Condition number distribution patterns in: (a) 3-D plot and (b) contour plot of the circulator at 2.95 GHz.

from the first port of a four-port DUT. The repetitions of reflecredundant equations in (17). tion terms sum up to By keeping only one of the reflection terms of each port and dropping the others, the number of equations is then identical to that of the unknowns, which herein are the elements of . The reduced version of (17) is arranged as

.. . (18)

.. . (17) Among these measured two-port -parameters, the times with difreflection term of each port is measured ferent ports terminated in auxiliary terminations. For example, and are the three reflection terms measured

These equations are rational functions concerning each element of the true -matrix, therefore they are analytic over the complex plane, except for those points making the denominator equal to 0 [17]. The existence of the partial derivatives hints that Newton’s method is a candidate for solving (18). By using the vector notation and defining (19) (20)

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the solution is denoted by [18] (22) from . which is an iteration operation to calculate The attribute of starting with an initial guess and iterating (22) to approach the true solution makes the success of Newton’s method greatly dependent on the initial guess and the convergence criterion. If the threshold of the condition number is set appropriately, the -matrix reconstructed using (6) before the onset of numerical difficulty is reliable and is a good initial guess due to the continuity property of -parameters and the smooth variation of the condition number, as shown in Fig. 6. Since the magnitudes of the measured -parameters could differ by as much as many tens of decibels, the relative error

(23)

Fig. 5. Condition number distribution patterns in: (a) 3-D plot and (b) contour plot of the circulator at 4.75 GHz.

is adopted to establish the convergence criterion as . After taking into account the measurement errors possibly involved in the measured two-port -parameters and the matrix inversion involved in (22), it is proposed to perform a simple moving average to remove the glitches at certain frequencies. A -point simple moving average is the unweighted mean of neighboring ones, and is the datum of interest and its the simplest form of a low-pass filter from the perspective of discrete-time signal processing [19]. It is given as

(24) where and are the indices, and is an odd number. In analogy to the design of a low-pass filter, is chosen to be reasonably large to make the curve of the -parameter smooth and not to distort those segments occupying the valleys of condition numbers in Fig. 6, which are reliably reconstructed using (6). III. EXPERIMENT

Fig. 6. Computer-simulated condition numbers of the circulator with the reflection coefficients of auxiliary terminations to be unity.

and

.. .

.. .

..

.

.. .

(21)

The following experiment is conducted to demonstrate the applicability of virtual auxiliary termination. The DUT is a Narda COF-2040 four-port circulator, which is the experimental sample in Section II. It has four SMA female connectors. In the experiment, its four-port -matrix is going to be reconstructed from measuring two-port -matrices while leaving the other two ports unconnected as the virtual auxiliary terminations. The reason for using a four-port circulator in this experiment is that it is a nonreciprocal device with -parameters having a large dynamic range to verify the applicability of our approach. One needs to measure six combinations of two-port -matrices, namely, and , to get . Additionally, the one-port -parameters for characterizing the unknown reflection coefficients of virtual auxiliary terminations are acquired by connecting one port to the VNA while

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Fig. 7. Typical reflection coefficient of the virtual auxiliary termination.

Fig. 8. Condition number of the circulator terminated in virtual auxiliary terminations.

Fig. 9. Reflection coefficient S

leaving the other three ports unconnected. The unknown reflection coefficients of the virtual auxiliary terminations at the other three ports are then derived based on the procedures of developing (5). For example, if the one-port -parameter of the first , the reflection coefficients port is measured and denoted as of the virtual auxiliary terminations at the second, third, and fourth ports are given by

cient of the virtual auxiliary terminations. However, measuring the one-port -parameters four times from four ports individually and evaluating the 12 possible values of the reflection coefficient help get a refined one and improve the accuracy of the calculation followed. With and in hand, (6) is then carried out to reconstruct the four-port -matrix of the DUT, which is called the result of the first step in the process of reconstruction. Once arriving at this stage, one reconstructs the -matrices for those frequency points with condition numbers low enough. The remedy described in Section II-D is then applied for those ill-conditioned points by executing Newton’s method given in (22) and the simple moving average given in (24) with an appropriate value to acquire the final results. In this experiment, each two- or one-port measurement produces a set of raw data for further processing. As a result, measurement precision plays a crucial role in lowering the numerical error. Special attention should be paid in the arrangements of calibration and measurement [20]–[22].

(25) (26) and (27) Since the four connectors of the DUT are the same type of female SMA, the one-port -parameter measured from any port is algebraically capable of characterizing the reflection coeffi-

in: (a) magnitude and (b) phase.

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Fig. 10. Forward transmission coefficient S

in: (a) magnitude and (b) phase.

IV. RESULTS As expected, the reflection coefficient of the virtual auxiliary termination is identified as being close to unity after comprehensively surveying the 12 possible values. Typical results of the virtual auxiliary termination are shown in Fig. 7. Fig. 8 shows the condition numbers of the DUT when terminating in virtual auxiliary terminations. The similarity between the pattern of Fig. 8 and that of Fig. 6 validates the conclusion from the computer simulation in Section II. The port enumeration of the DUT illustrated in Fig. 1 suggests that elements of each column of the -matrix can be categorized into four operation coefficients as reflection, forward transmission, isolation, and backward leakage. Although the operation coefficients of one port are not identical to those of another port, they do behave in a similar way. For the sake of brevity, only the operation coefficients of the first port are presented in Figs. 9–12. Fig. 9–12 shows the benchmark, first step result based on (6), and final result. The benchmark for verifying the results is

Fig. 11. Isolation coefficient S

in: (a) magnitude and (b) phase.

the four-port -matrix prepared for the computer simulation in Section II. They are obtained from two-port measurements with other two ports terminated with loads of a calibration kit. The aim of showing results of the first step is twofold. Although it shows that the utilization of virtual auxiliary termination possibly suffers from numerical difficulty, it serves to specify the threshold of the condition number by clarifying the interrelation between the magnitude of the condition number and the extent of -parameters runaway. From observing the results of the first step in Figs. 9–12 and the condition number in Fig. 8, the coincidence between the soar of the condition number and the runaway of the reconstructed -parameters is apparent. For those ill-conditioned -parameters, we set the threshold of the condition number to be 5 and apply Newton’s iteration given in (22) maximally seven times to meet the convergence . The simple moving average criterion of defined in (24) is then performed to give the final results shown in Fig. 9–12. Each solid line in these figures consists of 801 data points, and that in Fig. 10 and those in other figures are and , respectively. In smoothed using (24) with

CHEN AND CHU: VIRTUAL AUXILIARY TERMINATION FOR MULTIPORT SCATTERING MATRIX MEASUREMENT

Fig. 12. Backward leakage coefficient S

in: (a) magnitude and (b) phase.

most situations, the final results show good agreement with the benchmark. V. CONCLUSION A technique has been described to measure the auxiliary terminations indirectly. In practice, it could be beneficial for not directly probing unknown auxiliary terminations. Based on this technique and the method in [2], the concept of virtual auxiliary termination is introduced and applied in the multiport -matrix measurement with the use of a two-port VNA and leaving the rest of the ports of the DUT unconnected. The two-norm condition number of is studied as an indicator for one to tell the accuracy of the result evaluated from (6). If the calculation of (6) suffers from numerical difficulty at certain frequencies, it is complemented by a remedy that uses both Newton’s method given in (22) to solve those points iteratively and the simple moving average given in (24) to remove the remaining glitches in the curve of the -parameter. The formulation and approach presented in this paper can acquire the -matrix of an -port network with the use of a

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two-port VNA and virtual auxiliary terminations. By reviewing the DUTs on which we have experimented, we then give the following observation on a three-level difficulty in implementing virtual auxiliary terminations. The three-port -matrix of a Mini-Circuits D17I 17-dB directional coupler can almost be reconstructed by (6) due to having peak values of the condition numbers lower than 40 over the operational bandwidth [12]. Using (6) and (22) enables reconstructing of the three-port -matrix of a DiTom three-port circulator D3C2040, though not shown here. The peak values of the condition numbers of this device are below 90 from 1 to 5 GHz. For the four-port circulator Narda COF-2040 given in Section III, the moving average operation defined in (24) is further applied to remove the remaining glitches after performing (6) and (22). In other words, moving average is a complementary approach to (6) and (22) for better reconstructing of the -matrix of a DUT with the highest level of difficulty, which is observed as a four-port device with the peak values of the condition numbers greater than 70. Bearing in mind that one may distort the -parameters of a DUT by mistaking rapidly changing -parameters for the remaining glitches and misusing moving average, two possible cases are discussed concerning what condition numbers the sharpest segment of the -parameters is with. For the case with the condition number lower than 5, the sharpest segment is reliably reconstructed by (6) and surely not the remaining glitches. The in (24) is then chosen to be reasonably large to remove the suspected glitches occurring near the peaks of the condition numbers and not to distort the sharpest segment. As the sharpest segment is with the condition numbers larger than 5, it is possibly masked by the suspected glitches and turns out to be distorted. To prevent this scenario from happening, a priori knowledge about the characteristics of the DUT is helpful to distinguish the rapidly changing magnitudes or phases from the remaining glitches. In this paper, the practical applicability of virtual auxiliary termination is demonstrated by an experiment for measuring a four-port circulator. The final results show the agreement with those of the benchmark, and should be a creditable -matrix of the DUT, by which one can acquire the characteristics of the DUT and use it in system design. It then leads to the ease of multiport network characterization by leaving the unused ports unconnected as the virtual auxiliary terminations. REFERENCES [1] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2003. [2] S. Sercu and L. Martens, “Characterizing n-port packages and interconnections with a 2-port network analyzer,” in 1997 IEEE 6th Elect. Performance Electron. Packaging Top. Meeting, Oct. 1997, pp. 163–166. [3] J. C. Tippet and R. A. Speciale, “A rigorous technique for measuring the scattering matrix of a multiport device with a 2-port network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 5, pp. 661–666, May 1982. [4] J. C. Rautio, “Techniques for correcting scattering parameter data of an imperfectly terminated multiport when measured with a two-port network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 5, pp. 407–412, May 1983. [5] M. Davidovitz, “Reconstruction of the S -matrix for a 3-port using measurements at only two ports,” IEEE Trans. Microw. Guided Wave Lett., vol. 5, no. 10, pp. 349–350, Oct. 1995.

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[6] H. C. Lu and T. H. Chu, “Port reduction methods for scattering matrix measurement of an n-port network,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 6, pp. 959–968, Jun. 2000. [7] D. Woods, “Multi-port-network analysis by matrix renormalization employing voltage-wave S -parameters with complex normalization,” Proc. IEEE, vol. 124, no. 3, pp. 198–204, Mar. 1977. [8] D. F. Williams and D. K. Walker, “In-line multiport calibration algorithm,” in 51st ARFTG Conf. Dig., Jun. 12, 1998, pp. 88–90. [9] W. Lin and C. Ruan, “Measurement and calibration of a universal sixport network analyzer,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 4, pp. 734–742, Apr. 1989. [10] H. C. Lu and T. H. Chu, “Multiport scattering matrix measurement using a reduced-port network analyzer,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1525–1533, May 2003. [11] I. Rolfes and B. Schiek, “Multiport method for the measurement of the scattering parameters of n-ports,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1990–1996, Jun. 2005. [12] C. J. Chen and T. H. Chu, “Multiport scattering matrix measurement with a two-port network analyzer using only virtual auxiliary termination,” in Asia–Pacific Microw. Conf., Dec. 2006, pp. 1583–1586. [13] H. Dropkin, “Comments on ‘A rigorous technique for measuring the scattering matrix of a multiport device with a 2-port network analyzer’,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 1, pp. 79–81, Jan. 1983. [14] E. Vanlil, “Comments on ‘A rigorous technique for measuring the scattering matrix of a multiport device with a 2-port network analyzer’,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 3, pp. 286–287, Mar. 1985. [15] J. R. Rice, Matrix Computations & Mathematical Software. New York: McGraw-Hill, 1985. [16] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas With Applications to Linear Systems Theory. Princeton, NJ: Princeton Univ. Press, 2005. [17] D. G. Zill and M. R. Cullen, Advanced Engineering Mathematics, 2nd ed. Boston, MA: Jones & Bartlett, 2000. [18] S. C. Chapra and R. P. Chnale, Numerical Methods for Engineers, 4th ed. New York: McGraw-Hill, 1989.

[19] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. [20] “Specifying calibration standards and kits,” Agilent Technol., Santa Clara, CA, Applicat. Note 1287-11, 2006. [21] “10 hints for making better VNA measurements,” Agilent Technol., Santa Clara, CA, Applicat. Note 1291-1B, 2001. [22] “Understanding and improving dynamic range,” Agilent Technol., Santa Clara, CA, Applicat. Note 1363-1, 2000. Chih-Jung Chen was born in Taichung, Taiwan, R.O.C., in 1973. He received the B.S. degree in electronic engineering from National Yunlin University of Science and Technology, Yunlin, Taiwan, R.O.C., in 1996, the M.S.E.E. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 2000, and is currently working toward the Ph.D. degree at National Taiwan University. His research interests include microwave circuits and subsystems, and microwave measurements.

Tah-Hsiung Chu (M’87) received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1976, and the M.S. and Ph.D. degrees from the University of Pennsylvania, Philadelphia, in 1980 and 1983, respectively, all in electrical engineering. From 1983 to 1986, he was a Member of Technical Staff with the Microwave Technology Center, RCA David Sarnoff Research Center, Princeton, NJ. Since 1986, he has been on the faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor of electrical engineering. His research interests include microwave-imaging systems and techniques, microwave circuits and subsystems, microwave measurements, and calibration techniques.

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Letters Corrections on “Low-Loss Patterned Ground Shield Interconnect Transmission Lines in Advanced IC Processes” Luuk F. Tiemeijer, Ralf M. T. Pijper, Ramon J. Havens, and Olivier Hubert In the above paper [1, Sec. III], there is an error in the basic equation used to capture current crowding effects. Thus, [1, eq. (2)] should have read [2]

Z s = Rs Z g = Rg where

!s

Rs and Rg

1 + !|!s + |! Ls 0 L2sc

1 + !|!g + |! Lg 0 L2gc

(1)

denote the signal and ground line resistances, and

= Rs =Lsc and !g = Rg =Lgc are the corresponding transition

corner frequencies.

REFERENCES Manuscript received April 19, 2007. L. F. Tiemeijer and R. M. T. Pijper are with the Research Department, NXP Semiconductors, 5656 AE Eindhoven, The Netherlands (e-mail: [email protected]). R. J. Havens is with the Innovation Center RF, NXP Semiconductors, 6534 AE Nijmegen, The Netherlands. O. Hubert is with the Process and Library Technology Department, NXP Semiconductors Caen, 14079 Caen, France. Digital Object Identifier 10.1109/TMTT.2007.901593

[1] L. F. Tiemeijer, R. M. T. Pijper, R. J. Havens, and O. Hubert, “Low-loss patterned ground shield interconnect transmission lines in advanced IC processes,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 3, pp. 561–570, Mar. 2007. [2] L. F. Tiemeijer, R. J. Havens, R. de Kort, Y. Bouttement, P. Deixler, and M. Ryczek, “Predictive spiral inductor compact model for frequency and time domain,” in Proc. Int. Electron Device Meeting, 2003, pp. 875–878.

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Digital Object Identifier 10.1109/TMTT.2007.905569

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H. Kurebayashi K. Kuroda N. Kuster M. Kuzuhara Y. Kwon G. Kyriacou M. K. Kärkkäinen F. Ladouceur K. Lakin P. Lampariello M. Lancaster U. Langmann G. Lapin J. Larson L. Larson J. Laskar C. L. Lau A. Lauer D. Lautru P. Lavrador G. Lazzi C. H. Lee J. F. Lee R. Lee S. Lee S. Y. Lee T. Lee T. C. Lee Y. Lee Y. H. Lee D. Leenaerts Z. Lei G. Leizerovich Y. C. Leong S. Leppaevuori G. Leuzzi Y. Leviatan B. Levitas R. Levy G. I. Lewis H. B. Li H. J. Li L. W. Li X. Li Y. Li H. X. Lian C. K. Liao S. S. Liao D. Y. Lie L. Ligthart E. Limiti C. Lin F. Lin H. H. Lin J. Lin K. Y. Lin T. H. Lin Y. S. Lin E. Lind L. Lind D. Linkhart P. Linnér A. Lipparini D. Lippens A. S. Liu J. Liu L. Liu P. K. Liu Q. H. Liu S. I. Liu T. Liu T. P. Liu I. Lo J. LoVetri S. Long N. Lopez M. Lourdiane G. Lovat D. Lovelace Z. N. Low H. C. Lu K. Lu L. H. Lu S. S. Lu V. Lubecke S. Lucyszyn N. Luhmann A. Lukanen M. Lukic A. D. Lustrac J. F. Luy G. Lyons J. G. Ma Z. Ma S. Maas G. Macchiarella J. Machac M. Madihian K. Maezawa G. Magerl S. Mahmoud F. Maiwald A. H. Majedi M. Makimoto J. Malherbe V. Manasson T. Maniwa R. Mansour D. Manstretta M. H. Mao S. G. Mao A. Margomenos R. Marques G. Martin E. Martinez K. Maruhashi J. E. Marzo D. Masotti G. D. Massa D. Masse A. Materka B. Matinpour A. Matsushima S. Matsuzawa G. Matthaei J. Mayock J. Mazierska

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