IEEE MTT-V055-I06A (2007-06A) [55, 06A ed.]

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Citation preview

JUNE 2007

VOLUME 55

NUMBER 6

IETMAB

(ISSN 0018-9480)

PART I OF TWO PARTS

PAPERS

Active Circuits, Semiconductor Devices, and Integrated Circuits Power Harvester Design for Passive UHF RFID Tag Using a Voltage Boosting Technique ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ... A. Shameli, A. Safarian, A. Rofougaran, M. Rofougaran, and F. De Flaviis A Compact Quadrature Hybrid MMIC Using CMOS Active Inductors .. ...... ... ... H.-H. Hsieh, Y.-T. Liao, and L.-H. Lu Design of a High-Efficiency and High-Power Inverted Doherty Amplifier ...... ......... ......... ........ ......... ......... .. .. . G. Ahn, M. Kim, H. Park, S. Jung, J. Van, H. Cho, S. Kwon, J. Jeong, K. Lim, J. Y. Kim, S. C. Song, C. Park, and Y. Yang Field Analysis and Guided Waves Inverted Slot-Mode Slow-Wave Structures for Traveling-Wave Tubes ... . V. L. Christie, L. Kumar, and N. Balakrishnan A Periodically Loaded Transmission Line Excited by an Aperiodic Source—A Green’s Function Approach .. ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... I. A. Eshrah and A. A. Kishk Magnetic-Type Dyadic Green’s Functions for a Corrugated Rectangular Metaguide Based on Asymptotic Boundary Conditions ...... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... I. A. Eshrah and A. A. Kishk CAD Algorithms and Numerical Techniques Efficient Cartesian-Grid-Based Modeling of Rotationally Symmetric Bodies ... ......... ......... ........ ... D. M. Shyroki Filters and Multiplexers Physical Interpretation and Implications of Similarity Transformations in Coupled Resonator Filter Design ... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ S. Amari and M. Bekheit A Reconfigurable Micromachined Switching Filter Using Periodic Structures .. ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... M. F. Karim, A. Q. Liu, A. Alphones, and A. Yu A Synthesis Method for Dual-Passband Microwave Filters ...... ......... ........ ......... ......... J. Lee and K. Sarabandi

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Theoretical and Experimental Studies of Flip-Chip Assembled High- Suspended MEMS Inductors . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... J. Zeng, C. Wang, and A. J. Sangster Mutual Synthesis of Combined Microwave Circuits Applied to the Design of a Filter-Antenna Subsystem .... ......... .. .. ........ ......... ......... .. M. Troubat, S. Bila, M. Thévenot, D. Baillargeat, T. Monédière, S. Verdeyme, and B. Jecko Analysis of Multiconductor Coupled-Line Marchand Baluns for Miniature MMIC Design ..... ........ ......... ......... .. .. ........ ......... ......... ........ ......... . C.-S. Lin, P.-S. Wu, M.-C. Yeh, J.-S. Fu, H.-Y. Chang, K.-Y. Lin, and H. Wang Instrumentation and Measurement Techniques Submillimeter-Wave Phasor Beam-Pattern Measurement Based on Two-Stage Heterodyne Mixing With Unitary Harmonic Difference ... ........ ......... ......... ... Y.-J. Hwang, R. R. Rao, R. Christensen, M.-T. Chen, and T.-H. Chu Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators ..... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... M. Oxborrow Microwave Photonics Multilevel Modulated Signal Transmission Over Serial Single-Mode and Multimode Fiber Links Using Vertical-Cavity Surface-Emitting Lasers for Millimeter-Wave Wireless Communications .... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . A. Nkansah, A. Das, N. J. Gomes, and P. Shen MEMS and Acoustic Wave Components Thermally Actuated Multiport RF MEMS Switches and Their Performance in a Vacuumed Environment ...... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ........ M. Daneshmand, W. D. Yan, and R. R. Mansour High-Performance CMOS-Compatible Solenoidal Transformers With a Concave-Suspended Configuration ... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. L. Gu and X. Li Biological, Imaging, and Medical Applications Fringe Management for a T-Shaped Millimeter-Wave Imaging System .. ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... . Y. Li, J. W. Archer, G. Rosolen, S. G. Hay, G. P. Timms, and Y. J. Guo

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LETTERS

Corrections to “CMOS Low-Noise Amplifier Design Optimization Techniques” ........ ......... . ....... ......... . N.-J. Oh Corrections to “A Low-Power CMOS Direct Conversion Receiver With 3-dB NF and 30-kHz Flicker-Noise Corner for 915-MHz Band IEEE 802.15.4 ZigBee Standard” .... ......... ......... ........ ......... ......... ........ ......... . N.-J. Oh

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

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Digital Object Identifier 10.1109/TMTT.2007.901534

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

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Power Harvester Design for Passive UHF RFID Tag Using a Voltage Boosting Technique Amin Shameli, Aminghasem Safarian, Ahmadreza Rofougaran, Senior Member, IEEE, Maryam Rofougaran, and Franco De Flaviis, Senior Member, IEEE

Abstract—This paper presents a guideline to design and optimize a power harvester circuit for an RF identification transponder. A power harvester has been designed and fabricated in a CMOS 0.18- m process that operates at the UHF band of 920 MHz. The circuit employs an impedance transformation circuit to boost the input RF signal that leads to the improvement of the circuit performance. The power harvester has been optimized to achieve maximum sensitivity by characterizing both the impedance transformation network and the rectifier circuit and choosing the optimum values for the circuit parameters. The measurement results show sensitivity of 14.1 dBm for dc output voltage of 1 V and the output current of 2 A that corresponds to the output power of 2 W. Index Terms—Charge pump, impedance transformation, power harvester, RF identification (RFID) system, transponder, UHF RFID.

I. INTRODUCTION

HE RF identification (RFID) system is an automatic identification method using RF waves to transfer data between reader units and moveable objects called a transponder or tag. The RFID tag can be attached to almost anything such as pallets or cases of product, documents, electronic devices, luggage, people, or pets in order to identify, track, or categorize them. In comparison to other identification methods, RFID is faster, more reliable, more secure, and does not require physical sight or contact between the reader unit and the tagged object. There are different frequencies proposed for RFID operation ranging from 125 kHz to 5.8 GHz [1]. The high frequency band RFID operating at 13.56 MHz has been widely used for industrial applications. On the other hand, the UHF band RFID operating at 900 MHz provides a higher data rate, as well as smaller antenna size compared to 13.56-MHz RFID systems. There are also other frequencies suggested for RFID operation such as 2.4 and 5.8 GHz that result in smaller antenna size, as well as utilizing less populated frequency ranges. These frequencies, how-

T

Manuscript received September 20, 2006; revised February 6, 2007. A. Shameli and A. Safarian were with the Department of Electrical Engineering and Computer Science, University of California at Irvine, Irvine, CA 92697 USA. They are now with the Broadcom Corporation, Irvine, CA 92617 USA (e-mail: [email protected]; [email protected]). A. Rofougaran and M. Rofougaran are with the Broadcom Corporation, Irvine, CA 92617 USA (e-mail: [email protected]; maryam@broadcom. com). F. De Flaviis is with the Department of Electrical and Computer Science, University of California at Irvine, Irvine, CA 92697 USA (e-mail: franco@uci. edu). Digital Object Identifier 10.1109/TMTT.2007.896819

ever, are not of great interest primarily due to high electromagnetic loss and multipath fading in air associated with them. In general, RFID tags can be categorized as active and passive [2], [3]. The active tags get their energy completely or partially from an integrated power supply, i.e., battery, while the passive tags do not have any power supply and rely only on the power extracted from the RF signal transmitted by the reader. However, both active and passive tags are required to operate at extremely low power levels in order to increase the battery life time and the sensitivity for active and passive tags, respectively. In addition, the high volume use of RFID requires the tag to be very small and low cost in order to make it possible to use it on every single object. Due to its low cost and high integration capability, CMOS technology is one of the best candidates for implementation of the RFID tag. Additionally, extensive prior research on CMOS ultra-low power analog/digital integrated circuit (IC) design has laid the groundwork to design micropower baseband and radio units in CMOS process [4], [5]. However, the design of the power harvester circuit, which extracts the power from the receiving RF signal and generates the supply voltage, is an issue in the CMOS process. This is mainly due to the nonidealities of the diode connected MOSFET transistors, which leads to the efficiency degradation of the circuit. In this paper, we mainly focus on addressing the design issues of the power harvester circuit in CMOS technology. It is shown that the circuit performance will be improved using an impedance transformation circuit to provide voltage gain from the antenna to the input of the rectifying circuit. Additionally, both the rectifier and impedance transformation circuits have been characterized in order to choose the optimum value for the circuit parameters, which leads to maximum circuit sensitivity. The remainder of this paper is organized as follows. Section II presents a summary of the RFID system, as well as of prior research and the design challenges of the RFID transponder in CMOS technology. Section III presents the design of the power harvester circuit for passive RFID tags operating at a UHF band of 920 MHz. Section IV shows the measurement results of the designed circuit. Section V then provides the summary and conclusions. II. RFID OVERVIEW, PRIOR STUDY, AND DESIGN ISSUES RFID systems generally employ either near- or far-field coupling techniques for power and data transmission [3]. In nearfield applications, the communication is taking place using inductive coupling between the reader and the tag coils, while in far-field RFID systems, the communication between the reader

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Fig. 1. Far-field RFID system.

and tag is performed through transmission, propagation, and reception of electromagnetic waves [3]. The near-field RFID systems mainly operate at relatively low frequencies, i.e., LF and HF bands, with the reading distance well within the radian [6]. The far-field RFID, on the other sphere defined by hand, operates at higher frequencies with a higher read range compare to the near-field systems. The main theme of this paper, however, is focused on the design of the power generating circuit for far-field RFID applications operating at the UHF band. Shown in Fig. 1 is the block diagram of a far-field RFID system. The block of a passive transponder is also shown in Fig. 1, including a power harvester unit, radio unit, and processor unit. The role of the power harvester unit is to rectify the receiving signal and to generate the supply voltage. The radio unit includes an envelop detector and a comparator in order to receive the data transmitted from the reader. The receiving data is processed in the processor unit and the tag responds to the reader by changing its input impedance, which can be sensed in the reader side. One of the main issues in the implementation of the passive RFID tag in the CMOS process is the efficiency of the power harvester circuit. The power harvester circuit consists of multiple rectifier cells in a stacked configuration that accumulates the dc voltage of each stage to build the supply voltage. A very popular circuit to implement the power harvester is the charge-pump structure shown in Fig. 2 [7]–[10]. In this cirand ) and cuit, each cell consists of a dc-level shifter ( and ). In the negative half period of the a peak detector ( will be charged up to input signal, the sampling capacitor through the diode connected transistor , where is the amplitude of the input signal, is the voltage , and is the dc reference voltage at the drain drop over node of that is provided from the previous stage. As a result, the dc level of the signal applied to the peak detector is and, therefore, the dc voltage over shifted to in the positive half period of the input signal is the capacitor equal to . As shown in Fig. 2, the dc voltage generated at each stage of the charge-pump configuration is applied as the dc reference to its following stage. Therefore, the dc output voltage of an -stage charge-pump circuit is expressed as follows: (1)

Fig. 2. Charge-pump circuit.

Equation (1) implies that the performance of the charge-pump circuit suffers from the voltage drop over the diode connected MOSFET transistor. This voltage drop is mainly due to the threshold voltage of the transistor and the resistance of its channel. It has been suggested in previous studies [7], [11] to use highly efficient diodes, i.e., Schottky diode, to implement the power harvester circuit. Although this method effectively improves the efficiency of the power harvester, it is not available in the standard CMOS process and requires a special fabrication process, which increases the cost of the circuit. In addition, the threshold voltage of a MOSFET transistor can be reduced by injecting some charges on its gate oxide [2]. This can be done by applying a high voltage to the gate, which causes the electrons to be trapped in the oxide due to tunneling effect. This method reduces the threshold voltage and improves the efficiency of the circuit. However, it is not cost effective since the adjustment is required for each single transistor used in the diode connected configuration. The trapped charges will also be released gradually over temperature and time, which causes the threshold voltage to be increased to its initial value. Therefore, this method is not very reliable for long-term operation. In Section III, we will discuss the design of the power harvester in CMOS technology. III. CIRCUIT DESIGN The threshold voltage of the transistors can be reduced in some processes providing native devices [9]. Such devices provide lower threshold voltage compared to regular MOSFET transistors and are available in many CMOS processes, i.e., TSMC 0.18 m. The I–V curves of diode-connected MOSFET transistors in the TSMC 0.18- m process, for both native and regular devices, are shown in Fig. 3. However, the efficiency of the charge-pump circuit using native devices is still much lower than the circuits using Schottky diodes due to the high channel resistance of the MOSFET transistors for a weak input signal. To improve the efficiency of the circuit, an effective way is to use a passive network to boost the voltage amplitude at the input of the charge-pump circuit [12]. By increasing the amplitude of

SHAMELI et al.: POWER HARVESTER DESIGN FOR PASSIVE UHF RFID TAG USING VOLTAGE BOOSTING TECHNIQUE

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Fig. 3. I–V curves of the diode connected MOSFET transistors in TSMC 0.18-m process for both regular and native devices with L = 0:5 m.

the input signal, the diode connected transistors would rectify a greater portion of the input signal and, therefore, the efficiency of the circuit would be increased. As the first step to design a power harvester circuit, one needs to characterize both the impedance transformation and the charge-pump circuits for a given process. The results can be used for the optimization of the overall circuit that leads to the maximum sensitivity of the power harvester. Next, the behavior of the impedance transformation circuit is studied by taking into account the effects of the parasitic elements on the gain of the circuit. In addition, the input impedance of the charge-pump circuit is modeled and the effect of the circuit parameters on the performance of the charge-pump circuit is characterized by assuming the circuit to employ native MOSFET transistors in the TSMC 0.18- m process. Finally, the overall power harvester circuit is optimized to achieve the maximum sensitivity for dc output voltage of 1 V with the output dc current of 2 A. A. Impedance Transformation Circuit Shown in Fig. 4(a) is an ideal impedance transformation to the input circuit that transforms the load impedance . In this circuit, the passive components are impedance assumed to be lossless and, therefore, the power delivered to the input is equal to the output power of the circuit. This implies that a proper ratio between the impedance value at the input and output ports of the circuit results in the magnification of the voltage amplitude at the output. This technique can be used in the design of the power harvester circuit since the efficiency is a function of the input amplitude. The input and output power of the impedance transformation circuit can be expressed as (2) and (3), respectively, as follows: (2) (3) is the input admittance, is the In the above equations, and repimpedance seen from the input source, and resent the source amplitude and output amplitude of the circuit, respectively. To achieve the maximum amplitude at the should be matched to the source output, the input impedance , which results in the maximum power at the impedance

Fig. 4. Impedance transformation circuit.

output node of the circuit. Using (2) and (3), the voltage gain of the matching network can be derived as follows: (4) On the other hand, the input admittance can be expressed as (5), assuming the input impedance to be matched to the source , which is a real value impedance (5) In the above equation, and are the real and imaginary is the parallel admittance of parts of the load admittance and the matching circuit, as shown in Fig. 4(a). Substituting (5) into (4) leads to (6) for the gain of the matching circuit (6) Equation (6) can be expressed as (7) where is the quality factor at the output of the circuit [12]. Equation (7) implies that the maximum gain of the matching circuit is achieved for the maximum quality factor at the output. In reality, the loss of the passive elements and parasitic components should be taken into account in (7). Shown in Fig. 4(b) is an impedance transformation circuit with a resistive load equal . In this circuit, the loss of the inductor is modeled as a to and the parasitic capacitance introshunt parasitic resistor . The parduced by the charge-pump circuit is modeled as asitic components effectively reduce the quality factor at the output port, which leads to the degradation of the matching circan cuit’s gain. The effective quality factor of the circuit at be expressed as follows: (8)

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Fig. 5. Effect of parallel load capacitance on the voltage gain of the impedance transformation circuit for different values of the inductor quality factor.

Fig. 6. Effect of the parallel load resistance on the voltage gain of the impedance transformation circuit for different load capacitance.

The value of in (7) can be replaced by in (8) for the gain adjustment due to the parasitic components. The effects of the parasitic capacitor and resistor on the gain of the matching circuit are shown in Figs. 5 and 6, respectively. As shown in these figures, the gain of the matching circuit will be increased and maxiby minimizing the parasitic shunt capacitance and . These values are remizing the shunt resistances lated to the parameters of the charge-pump and the impedance transformation circuits. On the other hand, the efficiency of the charge-pump circuit is also a function of the circuit parameters. Therefore, it is required to characterize the charge-pump circuit in order to optimize the sensitivity of the overall power harvester circuit [13]. B. Charge-Pump Characterization A key parameter of the charge-pump circuit is the voltage sensitivity. This parameter is defined as the minimum voltage amplitude required at the input of the circuit to achieve a specific output dc voltage for a given load current. The voltage sensitivity is a function of the circuit parameters such as the number of stages and device sizes. It also depends on the fabrication

Fig. 7. Minimum input voltage amplitude required to achieve V = 1 V and I = 2 A for charge-pump circuit with W = 3 m and different values of N , using native devices in CMOS 0.18-m process with L = 0:5 m.

Fig. 8. Minimum input voltage amplitude required to achieve V = 1 V and I = 2 A for a four stage charge-pump circuit with different values of W , using native devices in CMOS 0.18-m process with L = 0:5 m.

process and is lower for the processes providing diodes with lower threshold voltage. In this paper, the voltage sensitivity of the charge-pump circuit is calculated for output dc voltage of 1 V with load current of 2 A. It is assumed that the circuit employs native devices in the TSMC CMOS 0.18- m process with the channel length of 0.5 m. The voltage sensitivity of the charge-pump circuit is shown in Fig. 7 for different number of stages as a function of the output current. In this figure, the transistors widths are equal to 3 m. In addition, the effect of the device size on the sensitivity of a four stage charge-pump circuit is shown in Fig. 8 as a function of the output current. As can be seen in this figure, the sensitivity of the circuit increases by increasing the size of the diode connected transistors. However, the sensitivity improvement almost saturates for device sizes greater than 6 m, as shown in Fig. 8. The sensitivity of the power harvester circuit is defined as the minimum input power required to achieve a certain dc output voltage and current [13]. This parameter can be calculated by dividing the voltage sensitivity of the charge pump circuit by the voltage gain of the matching circuit. It results in the minimum

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Fig. 9. Charge-pump circuit model.

voltage amplitude required at the input port of the power harvester circuit. This voltage can be translated to power, which defines the sensitivity of the power harvester circuit. On the other hand, the gain of the matching circuit is a function of the input impedance of the charge pump. Therefore, it is required to calculate the input impedance of the charge pump circuit as a function of its input voltage amplitude and output dc current. In order to find the input impedance of an -stage charge pump, one should consider the circuit as a parallel combination of single-stage rectifier cells [2]. In this circuit, the output dc currents of all the stages are equal, and the voltage amplitudes at the input of all rectifier cells are also the same. Therefore, it is only required to characterize the input impedance of a single stage rectifier cell as a function of the output dc current and the input voltage amplitude. A rectifier cell, however, is a nonlinear circuit that its bias point changes with time. In addition, the amplitude of its input signal is large and none of the small-signal models can be used to model the circuit. Therefore, the input impedance needs to be derived using the large-signal analysis [13], [14]. The circuit model of a one stage rectifier is shown in Fig. 9. and in Fig. 9(a) are large and act as short The capacitors circuit at the frequency of operation. However, at each half period of the input signal, one of the diode connected transistors is in the forward-bias region and the other one is in the reverse-bias circuit, as region. The circuit can be modeled as a shunt is the effective resistance shown in Fig. 9(b). In this model, of the two diode connected transistors that is a function of the output dc current and the input signal amplitude. The capacitor is the total parasitic capacitance at the input node, which includes the effective parasitic capacitance of the MOSFET transistors and the parasitic capacitance from each terminal of the in Fig. 9(a). The values of and sampling capacitor are shown in Figs. 10 and 11, respectively, for different output dc current and device sizes. It is assumed that the circuit employs native MOSFET devices with the channel length of 0.5 m and the amplitude of the input signal is equal to the voltage sensitivity of the charge pump circuit for output dc voltage of 1 V with a 2- A load current. As can be seen in Fig. 10, the value of the capacitor is mainly determined by the device size and is almost constant by the variation of the output dc current. Howwith respect to the dc current ever, the slight variation of

Fig. 10. Input shunt capacitance of a single-stage charge-pump circuit using native devices in CMOS 0.18-m process with L = 0:5 m.

Fig. 11. Input shunt resistance of a single stage charge-pump circuit using native devices in CMOS 0.18-m process with L = 0:5 m.

Fig. 12. Circuit model of an N -stage power harvester circuit.

is mainly due to the variation of the device capacitance in different regions of operation [15]. On the other hand, the value of the shunt resistance is changing with the size of the device and the output current. In general, a diode connected transistor can be modeled as an ideal diode in series with a resistor that can be expressed as follows [15]: (9) where width,

is the transistor’s channel length, is the transistor is the dc current, is the carrier mobility, and

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TABLE I MINIMUM REQUIRED POWER TO ACHIEVE 1-V dc OUTPUT WITH 2-A CURRENT FOR DIFFERENT PARAMETERS OF THE CHARGE-PUMP CIRCUIT

is the oxide capacitance per unit area. The effective resistance of a diode connected transistor in response to a large sinusoidal voltage can be expressed as follows:

TABLE II SUMMARY OF THE DESIGNED CIRCUIT PARAMETERS

(10) , is the resistance due to the nonlinear The first part, i.e., is a function of the behavior of the diode. The resistance input voltage amplitude and the threshold voltage of the tran, is sistor. The second part of the effective resistance, i.e., the resistance of the diode connected transistor, which is a function of the bias current and the transistor’s size, as shown in (9). In large-signal analysis, however, the bias point is varying with time and, therefore, it is required to use the effective dc current in (9) to calculate . The analytical expression for the effective large-signal resistance seen from the input of the charge-pump circuit is complicated and is beyond the extent of this paper. However, we expect the series resistance of the diodes to be dominant for the low output dc current. Therefore, the effective input resistance of the chare pump drops rapidly by increasing the output dc current and device width. On the other hand, the effective resistance due to the nonlinear behavior of the circuit is dominant at high values of the output dc current. As a result, the parallel effective resistance of the circuit is almost constant at high dc output current levels. Fig. 11 confirms this analysis. Recall from Figs. 5 and 6 that the gain of the impedance transformation circuit is increased by increasing the value of the shunt resistance and decreasing the value of the shunt capacitance at the output port of the circuit. Therefore, according to Figs. 10 and 11, the dc output current, the width of the transistors, and also the number of rectifier stages employed in the charge-pump circuit should be reduced in order to increase the gain of the matching circuit. On the other hand, reducing the device width leads to the increase of the voltage drop over the diode connected transistors and, therefore, higher amplitude is required to achieve a certain dc voltage at the output. In addition, reducing the number of stages employed in the charge-pump structure leads to the reduction of the gain of the charge pump and, therefore, reduces the sensitivity of the circuit. In conclusion, there is an optimum value for the circuit parameters, i.e., transistor’s width and the number of stages that results in the maximum sensitivity of the circuit. C. Circuit Optimization To complete the design of the power harvester circuit, we need to use the results of the matching circuit and charge-pump

characterization to optimize the overall circuit for a given output dc voltage and current. In addition, it is required to design an inductor for the impedance transformation circuit with the highest possible quality factor to achieve the maximum voltage gain from the antenna to the input of the charge pump. Shown in Fig. 12 is the model of the power harvester circuit with all the parasitic components. In real circuits, there is a tradeoff between the performance of the charge pump and the gain of the matching circuit. For example, according to Fig. 7, increasing the number of stages lowers the minimum swing required at the input of the charge pump to achieve a certain output dc voltage and current. On the other hand, this increases the parasitic capacitance and decreases the shunt resistance seen from the input of the charge-pump circuit and, therefore, reduces the gain of the matching circuit. The same situation is also true according to Figs. 5, 6, 11, and 12. for transistors width Taking into account all these parameters, we calculated the sensitivity of the power harvester circuit to achieve the output dc voltage of 1 V and the dc current of 2 A. The results are shown in Table I for different circuit parameters. In the calculations, it is assumed that the inductor presents the quality factor of 7 with self-resonance frequency of 2 GHz. The inductor is designed such that the input impedance of the circuit matches with the impedance of the antenna. As shown in Table I, the power harvester circuit achieves the maximum sensitivity for and m. needs to be large enough so The sampling capacitor that the voltage division between the sampling capacitor and impedance introduced by two diode connected transistors is negligible. On the other hand, the sampling capacitor introduces a parasitic capacitor from each of its terminals to ground that is proportional to the value of the capacitor itself. According to the simulation results, the value of the sampling capacitor for m should be bigger than 200 fF. In this design, the is defined to be equal to 250 fF. The summary of value of the circuit parameters is shown in Table II. The value of the inductor is determined from the input impedance of the charge-pump circuit by using (5) to match the real part of the input admittance of the circuit to the admittance

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Fig. 13. Inductor designed for the impedance transformation circuit using top tick metal with trace width of 10 m and area of 455 m 455 m.

2

Fig. 16. Measured sensitivity of the fabricated power harvester circuit for different values of output load.

Fig. 14. Equivalent circuit of the designed inductor.

Fig. 17. Measured output dc voltage for different input frequency with input power level of 14.1 dBm and output load of 500 k .

0

Fig. 15. Die photograph of the fabricated power harvester circuit.

of the antenna. As shown in Fig. 13, the inductor is designed using a top thick metal to increase the self-resonance frequency and to improve the quality factor. To precisely predict the value and quality factor of the inductor, it is required to use full-wave analysis. The Zeland IE3D tool has been used, which provides very accurate results by taking in to account metal thicknesses, as well as conductor losses. The designed inductor occupies 455 m and has a value of 23.3 nH an area of 455 m with a quality factor of 6.8 at a frequency of 920 MHz. At this frequency, the measured inductance and quality factor of the inductor were within 1.3% and 4% of the simulated data, respectively. The equivalent circuit of the designed inductor is shown in Fig. 14. The power harvester circuit is fabricated using a TSMC CMOS 0.18- m process with native device option. The fabricated circuit occupies an area of 800 m 1000 m. A photograph of the fabricated die is shown in Fig. 15. IV. MEASUREMENT RESULTS The measurement results and simulated data of the fabricated power harvester circuit are shown in Figs. 16–19. The circuit ex-

hibits the sensitivity of 14.1 dBm for the output dc voltage of 1 V with output current of 2 A at a frequency of 920 MHz, which corresponds to an output power of 2 W. The sensitivity measurement of the power harvester circuit is performed by measuring the generated dc voltage at the output for different levels of the input signal power. The results are shown in Fig. 16 for different load resistance values. In addition, the output dc voltage of the circuit is measured at different input frequencies for the input power level of 14.1 dBm and the output load of 500 k . The results are shown in Fig. 17. As can be seen in Fig. 17, the circuit exhibits the maximum dc voltage at the input frequency of 925 MHz. In general, the rectifying block of the power harvester circuit is not very sensitive to the frequency of operation. However, the impedance transformation circuit is tuned to resonate at the frequency of 920 MHz in order to provide the input impedance matched to the impedance of the antenna. The input return loss of the circuit is shown in Fig. 18 for different load resistance and input power levels. The impedance of the diode connected transistors is a function of the input power and the output dc current. As a result, the input return loss changes by the variation of the input

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V. CONCLUSIONS A power harvester circuit for RFID application has been designed and fabricated in the CMOS process. It has been shown that the use of an impedance transformation circuit can effectively improve circuit efficiency and that its use makes it possible to design an RFID transponder in the CMOS process. The designed circuit operates at a UHF band of 900 MHz with sensitivity of 14.1 dBm for a 2- W output power level.

ACKNOWLEDGMENT

Fig. 18. Measured input return loss of the fabricated power harvester circuit for different combinations of input power level and output load.

The authors would like to thank Prof. N. G. Alexopoulos, University of California at Irvine, for his valuable comments on this work. The authors also appreciate the valuable help from Dr. J. Castaneda, Broadcom Corporation, Irvine, CA, and E. Roth, Broadcom Corporation, for the measurement of the circuit. In addition, the authors would like to acknowledge the support from the Zeland Corporation, Fremont, CA, for providing the IE3D tool for the simulation of passive structures.

REFERENCES

Fig. 19. Measured input impedance for various values of load impedance and input power level of 14.1 dBm.

0

TABLE III SUMMARY OF THE FABRICATED POWER HARVESTER CIRCUIT

power and the output load of the circuit. According to Fig. 15, the circuit exhibits the minimum return loss at the frequency of 925 MHz. The circuit also has maximum sensitivity at this frequency. The bandwidth of the circuit that is defined as the frequency range where the input return loss is less than 10 dB is around 100 MHz. The input impedance of the circuit is also shown in Fig. 19 for different values of load resistance and an input power level of 14.1 dBm. The summary of the fabricated chip is shown in Table III.

[1] K. Finkenzeller, RFID Handbook: Fundamentals and Applications in Contactless Smart Cards and Identification, 2nd ed. New York: Wiley, 2003. [2] S. Mandal, Far Field RF Power Extraction Circuits and Systems. Cambridge, MA: MIT Press, 2004. [3] T. A. Scharfeld, An Analysis of the Fundamental Constrains on Low Cost Passive Radio-Frequency Identification System Design. Cambridge, MA: MIT Press, 2001. [4] D. Friedman, H. Heinrich, and D.-W. Duan, “A low-power CMOS integrated circuits for field-powered radio identification tags,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 6–8, 1997, pp. 294–295. [5] A. Shameli and P. Heydari, “Ultra-low power RFIC design using moderately inverted MOSFETs: An analytical/experimental study,” in Radio Freq. Integrated Circuits Symp., Jun. 2006, pp. 470–473. [6] S. C. Q. Chen and V. Thomas, “Optimization of inductive RFID technology,” in IEEE Int. Electron. and Environ. Symp., May 2001, pp. 82–87. [7] U. Karthaus and M. Fischer, “Fully integrated passive UHF transponder IC with 16.7-W minimum RF input power,” IEEE J. Solid-State Circuits, vol. 38, no. 10, pp. 1602–1608, Oct. 2003. [8] J.-P. Curty, N. Joehl, C. Dehollain, and M. J. Declercq, “Remotely powered addressable UHF RFID integrated system,” IEEE J. Solid-State Circuits, vol. 40, no. 11, pp. 2193–2202, Nov. 2005. [9] F. Kocer and M. P. Flynn, “A new transpounder architecture with on-chip ADC for long-range telemetry applications,” IEEE J. Solid-State Circuits, vol. 41, no. 5, pp. 1142–1148, May 2006. [10] T. Umeda, H. Yoshida, S. Sekine, Y. Fujita, T. Suzuki, and S. Otaka, “A 950-MHz rectifier circuit for sensor network tags with 10-m distance,” IEEE J. Solid-State Circuits, vol. 41, no. 1, pp. 35–41, Jan. 2006. [11] M. Ghovanloo and K. Najafi, “Fully integrated wideband high-current rectifiers for inductively powered devices,” IEEE J. Solid-State Circuits, vol. 39, no. 11, pp. 1976–1984, Nov. 2004. [12] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 2004. [13] G. De Vita and G. Iannaccone, “Design criteria for the RF section of UHF and microwave passive RFID transponders,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2978–2990, Sep. 2005. [14] K. K. Clark and D. T. Hess, Communication Circuits: Analysis and Design. Reading, MA: Addison-Wesley, 1971. [15] Y. P. Tsividis, Operation and Modeling of the MOS Transistor, 2nd ed. New York: McGraw-Hill, 1999.

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Amin Shameli received the B.S. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 2001, the M.S. degree in electrical engineering from the Isfahan University of Technology, Isfahan, Iran, in 2003, and the Ph.D. degree in electrical engineering from the University of California at Irvine, in 2007. From 2004 to 2006, he was with the Broadcom Corporation, Irvine, CA, as an Intern involved with an RFID research and development project. He is currently a Staff Scientist with the Broadcom Corporation, where he is involved in the design of transmitters for cellular wireless applications. His research interests include polar transmitters, RFID systems, and ultra-low power RF circuits. Dr. Shameli was the recipient of the Gold Medal in the 1997 National Computer Olympiad in Iran.

Aminghasem Safarian received the B.S. and M.S. degrees in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 2000 and 2002, respectively, and the Ph.D. degree in electrical engineering from the University of California at Irvine, in 2006. He is currently a Staff Scientist with the Broadcom Corporation, Irvine, CA. His research interest is RF ICs for wireless communication systems.

Ahmadreza (Reza) Rofougaran (S’93–M’95– SM’05) received the B.S.E.E., M.S.E.E., and the Ph.D. degree from the University of California at Los Angeles (UCLA), in 1986, 1988, and 1998, respectively. In July 2000, he joined the Broadcom Corporation, Irvine, CA (through Innovent System’s acquisition), where he is currently a Fellow/Chief Technologist. Since 2000, he has been in charge of all RF CMOS radios for Bluetooth, wireless local area networks (WLANs), and cellular. In addition to product development, he also leads the research and development of next-generation software defined RF cellular radios, RFID products, and smart/software define antennas. He has authored or coauthored over 45 technical papers. He holds over 75 issued or pending U.S. patents. Dr. Rofougaran was the recipient of several premium international IEEE awards. His technical contributions in RF CMOS have also been recognized worldwide by both industry and academia.

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Maryam Rofougaran received the B.S. and M.S. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1992 and 1995, respectively. While with UCLA, she played a significant role in the pioneering design of a single-chip 900-MHz RF CMOS transceiver. She cofounded Innovent Systems, which was acquired by the Broadcom Corporation, Irvine, CA, in 2000. She is currently a Senior Director of engineering with the Broadcom Corporation, where she is involved with wireless products. She has authored or coauthored numerous papers. She holds several patents in the area of wireless systems and IC design. Ms. Rofougaran was the recipient of the 1995 European International Solid-State Circuits Conference (ISSC) Best Paper Award, the 1996 International Solid State Circuits Conference ISSCC Jack Kilby Award for Outstanding Paper, the 1997 ISSCC Jack Raper Award for Outstanding Technology Direction, and the 1998 Design Automation Conference Best Paper Award.

Franco De Flaviis (M’99–SM’07) was born in Teramo, Italy, in 1963. He received the Laurea degree in electronics engineering from the University of Ancona, Ancona, Italy, in 1990, and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA) in 1994 and 1997, respectively. In 1991, he was an Engineer with Alcatel, where he specialized in the area of microwave mixer design. In 1992, he was Visiting Researcher with UCLA, where he was involved with low intermodulation mixers. He is currently an Associate Professor with the Department of Electrical Engineering and Computer Science, University of California at Irvine. He has authored or coauthored over 100 papers in reference journals and conference proceedings. He has authored three book chapters. He has filed several international patents. His research interest include the development of microelectromechanical systems (MEMS) for RF applications fabricated on unconventional substrates such as printed circuit board and microwave laminates with particular emphasis on reconfigurable antenna systems. He is also active in the research field of highly integrated packaging for RF and wireless applications. Dr. De Flaviis is a member of the URSI Commission B.

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A Compact Quadrature Hybrid MMIC Using CMOS Active Inductors Hsieh-Hung Hsieh, Student Member, IEEE, Yu-Te Liao, Student Member, IEEE, and Liang-Hung Lu, Member, IEEE Abstract—A miniaturization technique for the quadrature hybrid is proposed for monolithic-microwave integrated-circuit applications. By using active inductors for the circuit implementation, a significant area reduction can be achieved due to the absence of distributed components and spiral inductors. Using a 0.18- m bulk CMOS process, a 4.2-GHz quadrature hybrid is implemented for demonstration. The chip size measures 0.72 0.64 mm2 where the hybrid circuit only occupies active area of 0.4 0.2 mm2 . With the enhanced factor provided by the active inductors, the fabricated circuit has an insertion loss less than 0.4 dB for 3-dB coupling while maintaining excellent return losses and port isolation in the vicinity of the center frequency. Owing to the reconfigurable capability provided by the CMOS active inductors, the center frequency of the quadrature hybrid can be varied from 3.6 to 4.7 GHz by adjusting the bias currents, exhibiting a tuning range of 26% at -band. Index Terms—CMOS active inductors, insertion loss, lumped components, miniaturization, monolithic microwave integrated circuit (MMIC), quadrature hybrid.

I. INTRODUCTION

T

HE quadrature hybrid is a four-port network, which behaves as a directional coupler. Within the operating bandwidth, equal power dividing with a 90 phase difference is provided at the outputs of the through and the coupled port, while zero signal coupling is exhibited at the isolation port. Since the quadrature hybrid is suitable for balanced circuit operations, it has been widely used in microwave systems for decades. In a planar process technology, the circuit is typically realized by the branch-line architecture. Owing to the use of four quarter-wavelength transmission line components in conventional circuit implementations, the size of the quadrature hybrid is prohibitively large, making it impractical for monolithic-microwave integrated-circuit (MMIC) applications, especially at frequencies below 10 GHz. In order to reduce the chip area, techniques have been proposed for the quadrature hybrids by using multilayer configurations [1], [2], 3-D technologies [3], and lumped [4]–[7] or lumped-distributed equivalence [8]. However, due to the lack of high- on-chip passive components, most of the miniaturized circuits suffer from high insertion losses, large phase error, Manuscript received December 1, 2006; revised February 8, 2007. This work was supported in part by the National Science Council under Grant 94-2220E-002-026 and Grant 94-2220-E-002-009. H.-H. Hsieh and L.-H. Lu are with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: [email protected]). Y.-T. Liao was with the Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, Taiwan, 10617, R.O.C. He is now with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA. Digital Object Identifier 10.1109/TMTT.2007.896815

and unequal coupling at the thru and coupled port. On the other hand, the existing techniques still require a considerably large chip area, leading to a high implementation cost for system integration. To overcome the design limitations imposed by the lumped passive components, the concept of quadrature hybrids with active inductors has been adopted, and a controllable attenuator has been realized in a pseudomorphic HEMT (pHEMT) foundry process [9]. In this paper, a miniaturized technique is presented for the design of fully integrated CMOS hybrids. By utilizing active inductors in the lumped equivalent circuit, a significant area reduction can be achieved while maintaining enhanced circuit performance at multigigahertz frequencies. In addition, the tunable inductance of the active inductors enables the control of the center frequency by the bias currents, which can be utilized for the realization of reconfigurable RF frontends in multistand wireless systems. Based on the proposed technique, a -band hybrid circuit was designed and fabricated in a 0.18- m bulk CMOS process for demonstration. This paper is organized as follows. Section II presents the realization of the CMOS active inductors, including the circuit topology, theoretical analysis, design considerations, and chip implementation. The design of the fully integrated quadrature hybrid using CMOS active inductors is described in Section III, followed by the experimental results in Section IV. Finally, conclusions are briefly outlined in Section V. II. CMOS ACTIVE INDUCTORS To alleviate the limitations imposed on the chip area and the quality ( ) factors of the spiral inductors, active designs [10]–[15] were proposed to implement the required on-chip inductance. For RF applications, the regulated cascode topology is commonly used in the design of CMOS active inductors [16]. Fig. 1(a) shows the schematic of the active inductor. As the input voltage applies to the gate terminal of the common-source tran, the transconductance converts the voltage to a sistor of transistor . drain current charging the capacitance is then converted to the The voltage established across , emulating the curinput current by the transconductance of rent–voltage characteristics of a shunt inductance. Note that the is used as the gain-boosting stage to enhance the transistor factor of the active inductor, while the required bias currents and are provided by the current mirrors . for From the small-signal analysis, the nodal voltages and can be expressed as

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(1) (2)

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Fig. 2. Die photograph of the fabricated active inductor.

TABLE I CIRCUIT PARAMETERS OF THE FABRICATED ACTIVE INDUCTOR

Fig. 1. (a) Schematic and (b) equivalent circuit of the regulated cascode active inductor.

and the input admittance of the active inductor is given by

(3) Assuming that the operating frequency of the active inductor , which can be is much lower than the cutoff frequency of expressed as , the input admittance of the active inductor can be approximated by the equivalent circuit, as , , and shown in Fig. 1(b), and the expressions of the , are provided as

To optimize the inductor performance, the transistor parameters compensates for the loss can be properly chosen such that from at the frequency of interest

(9) (4) (5) (6)

As a result, a peak

factor can be achieved at

(10)

(7) For an active inductor, the value of the inductance is deand , termined by the small-signal circuit parameters of while the factor is strongly influenced by the values of and . In the equivalent circuit, represents the shunt conductance, accounting for the loss of the active inductor. On the is a negative resistance with frequency-depenother hand, dent characteristics. Based on the simplified circuit model, the resonant frequency of the active inductor is given by

(8) Note that and are the cutoff frequencies of and , respectively, which impose a fundamental limitation on the operating frequencies of the active inductors. Typically, the active inductors are operated at frequencies much lower than the resonant frequency to ensure the desirable circuit characteristics.

In order to evaluate the broadband characteristics of the active inductors, the quality factor is defined as the ratio of the imaginary part to the real part of the input impedance, which can be approximated by

(11) Other than the reduced chip area and the enhanced factor, another advantage of using active inductors is the tuning capability of the inductance values. From (4), the value of is governed and . Hence, the equivalent by the transconductances and , reinductance can be adjusted by the bias currents sulting in another degree of freedom for the circuit operation. In (4), the value of the inductance is determined by the and , which are considered constant transconductance under small-signal approximation. However, it is not very accurate for the active inductors operating in the large-signal

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Fig. 3. Simulated and measured small-signal characteristics of the active inductor with I = 1:5 mA and I = 1:6 mA.

Fig. 5. Measured equivalent inductance at various bias currents (I , I ).

Fig. 6. Port assignment of the quadrature hybrid.

Fig. 7. Equivalent circuit model of a lumped hybrid.

Fig. 4. Measured: (a) equivalent inductance and (b) Q factor of the regulated cascode active design with various incident power levels.

mode. As the amplitude of the signal power increases, the excess voltage swing leads to a decrease in the transconductance thus deviates from its of the transistors. The inductance small-signal value due to the nonlinear characteristics, resulting in undesirable signal distortion. Therefore, the impact of the linearity issues on the circuit performance should be carefully examined when the active inductors are used to replace the spiral inductors in MMIC designs. For a CMOS active inductor using the regulated cascode topology, the deviation in the transconductance due to large-signal operations can be effectively minimized by increasing the overdrive voltage and

by reducing the transistor size at the expense of an elevated supply voltage. In order to characterize the small- and large-signal performance of the active inductors for RF applications, a standard 0.18- m CMOS process was employed for the circuit implementation. The microphotograph of the fabricated circuit is shown in Fig. 2, while the design parameters are tabulated in mA and mA, the small-signal Table I. With characteristics of the active inductor are illustrated in Fig. 3. is defined here as Note that the equivalent inductance (12) Based on the experimental results, the fabricated circuit exhibits an equivalent inductance of 0.88 nH with a maximum factor of 59 at 2.8 GHz. In such a specified bias condition, the large-signal behavior of the active inductor is also characterized by measuring the input reflection coefficient with various incident signal power. The impacts of the incident power level on the equivalent inductance and factor are shown in

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Fig. 8. Complete circuit schematic of the fully integrated quadrature hybrid.

Fig. 4(a) and (b), respectively. It is noted that the equivalent inductance increases with the signal power level due to the decreasing transconductances of the MOS devices. Since the transistors may operate in the triode regions at large-signal operations, the factor tends to degrade at higher input power level. Another important feature of the active inductor is the tunability. By varying the bias currents of and , the measured equivalent inductance as a function of frequency is illustrated in Fig. 5, indicating a wide inductance tuning range at multigigahertz frequencies.

TABLE II CIRCUIT PARAMETERS OF THE QUADRATURE HYBRID

III. DESIGN OF MMIC QUADRATURE HYBRID The port assignment and functionality of a quadrature hybrid are shown in Fig. 6. For a planar process technology, the circuit is conventionally implemented by branch-line couplers due to its simplicity [17], [18]. In consideration of the prohibitively large chip area, lumped-element topologies have been proposed to realize the hybrid circuits for applications below 10 GHz [4]–[7]. With reactive elements including shunt inductors and series capacitors, the equivalent circuit model of a lumped hyis shown in Fig. 7, where the brid at its center frequency values of the components are given as (13) (14) To demonstrate the potential of monolithic system integration, a 1P6M 0.18- m CMOS process provided by a commercial foundry is used to fabricate the proposed quadrature hybrid. With a device layout optimized for the RF performance, the MOSFETs exhibit a maximum oscillation frequency up to 60 GHz. As for the backend process technology, a top AlCu metallization layer of 2- m thickness is available for the interconnection, while a metal–insulator–metal (MIM) structure with a capacitance density of 1 fF m is also provided. With a center frequency of 4 GHz, the required inductances and capacitances in this design can be calculated from (13) and (14).

The obtained and are 2.0 and 1.4 nH, respectively, while is 0.8 pF and is 1.1 pF. For the circuit implementation, the shunt inductances in the lumped circuit model are realized by the regulated cascode active inductors, and MIM capacitors provided by the standard CMOS process are used for the capacitive elements. The complete circuit schematic of the fully integrated quadrature hybrid is depicted in Fig. 8, where the design parameters are tabulated in Table II. Due to the use of active inductors, no distributed elements or spiral inductors are required. A significant reduction in chip area can be achieved. In addition, the resulting high- factors of the active inductors also enhance the hybrid performance compared with the lumped-element topology. Another important feature of the proposed technique is the reconfigurability in the operating frequency bands. As indicated in (4), the inductance of an active inductor is strongly influenced by the bias currents of the MOSFETs. Therefore, the center frequency of the quadrature hybrid can be effectively tuned by the bias currents to compensate for the process variation or to realize a multiband RF functionality.

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(a)

Fig. 9. Die photograph of the fabricated quadrature hybrid.

IV. EXPERIMENTAL RESULTS Fig. 9 shows a microphotograph of the fabricated circuit. In this design, the total chip area including the pads is 0.72 0.64 mm , where the active circuit area occupies only 0.4 0.2 mm . Compared with conventional branch-line hybrids at the same frequency band, the chip area is reduced by several orders of magnitude. To characterize the -parameters of the quadrature hybrid, on-wafer probing was performed by using a multiport vector network analyzer. No on-chip terminations are required and the impedance mismatch during the measurement can be minimized for better accuracy. When the circuit is biased for operation and at a center frequency of 4.2 GHz, the bias currents are both 1.5 mA. The hybrid circuit consumes a dc power of 20.4 mW from a 1.8-V supply voltage. The measured -parameters and output phase of the fabricated circuit are shown in Fig. 10(a) and (b), respectively. The insertion losses at the thu and the coupled port are 3.3 and 3.2 dB, respectively, with a phase difference of 91.2 at the center frequency, while the return loss and port isolation are 28 and 15 dB, respectively. The bandwidth, which is defined as the frequency range, where dB, is 9.5%. Within this bandwidth, the insertion loss maintains lower than 4.0 dB and the phase error is less than 3.5 . Due to the use of active devices in the circuit design, the nonlinear behavior of the quadrature hybrid is characterized by its between the input-referred 1-dB compression point input and output port. With ports 3 and 4 terminated, Fig. 11 at shows the measured output power and the associated the thu port versus the input power level. From the measureis 16 dBm, and tends to decrease ment results, as the input power level increases. For a conventional quadrature hybrid with passive components, the noise figure is simply evaluated by the insertion loss from the input to the output port. As for the active hybrid circuit, however, excess noise sources from the MOSFETs have to be taken into account. It is more complicated to define and to extract the noise figure experimentally for the four-port network. In this study, a simplified measurement setup has been adopted to evaluate the noise figure. By treating the quadrature hybrid as a two-port network with 50- terminations for the isolation and the coupled port, the measured noise figure from the input to the through port is 17 dB at the center

(b) Fig. 10. Measured: (a) quadrature hybrid.

S -parameters and (b) output phase of the fabricated

Fig. 11. Measured large-signal behavior of the quadrature hybrid.

frequency of 4 GHz. It is noted that the measurement accuracy may be influenced by the port reflections due to impedance mismatch and by the thermal noise from the resistive loads. Therefore, it tends to overestimate and can only be considered an approximated specification for design reference. One of the unique features for the proposed hybrid circuit is the reconfigurability provided by the tunable active inductors. By adjusting the equivalent inductance through the bias curand , the center frequency can be effectively tuned rents over a wide frequency range. For the operation at the lower and are set to 1.1 and frequency band, the bias current

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proposed circuit is suitable for reconfigurable RF applications with a center frequency from 3.6 to 4.7 GHz. Within the entire and of frequency range of operation, the measured the quadrature hybrid are higher than 4 dB with an output phase error less than 4 , while the return loss and isolation are generally better than 20 and 10 dB, respectively. V. CONCLUSION A tunable active inductor has been implemented in a standard 0.18- m CMOS process. With the analysis and characterization of the fabricated circuit, the small-signal performance and nonlinear behavior of the CMOS active inductors have been presented for MMIC designs at multigigahertz frequencies. Based on the development of the active inductor, a compact quadrature hybrid has been demonstrated at a center frequency of 4.2 GHz. Due to the absence of distributed components and spiral inductors, the fully integrated hybrid only occupies an active chip area of 0.4 0.2 mm . With the high- factors and tunable inductance provided by the active inductors, the quadrature hybrid exhibits enhanced performance in terms of insertion loss, impedance matching, and port isolation while maintaining a frequency tuning range of 1.1 GHz. ACKNOWLEDGMENT

Fig. 12. Measured S -parameters of the quadrature hybrid at the: (a) lower and (b) upper operating frequency bands.

TABLE III PERFORMANCE SUMMARY

The authors would like to thank the National Chip Implementation Center (CIC), Hsinchu, Taiwan, R.O.C., for chip fabrication and National Nano Device Laboratories (NDL), Hsinchu, Taiwan, R.O.C., for chip measurement. The authors are also grateful to Y.-C. Tarn, National Taiwan University, Taipei, Taiwan, R.O.C., for measurement support. REFERENCES

1.4 mA, respectively. The resulting inductances of and are approximately 2.4 and 1.8 nH, respectively, leading to a shift of the center frequency from its midband value of 4.2 of 1.9 mA and of to 3.6 GHz. On the other hand, with and are 1.4 and 3.0 mA, the equivalent inductances of 1.1 nH, respectively, effectively moving the center frequency of the hybrid to 4.7 GHz. The measured -parameters for the lower and the upper operating frequency bands are illustrated in Fig. 12(a) and (b), respectively. Table III summaries the performance of the fabricated quadrature hybrid at various operating frequency bands. According to the measurement results, the

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[1] T. Tokumitsu et al., “Multilayer MMIC using a 3 m 3-layer dielectric film structure,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1990, vol. 2, pp. 831–834. [2] I. Toyoda et al., “Multilayer MMIC branch-line coupler and broad-side coupler,” in IEEE Microw. Millimeter-Wave Monolithic Circuits Symp., Jun. 1992, pp. 79–82. [3] C. Y. Ng, M. Chongcheawchamnan, and I. D. Robertson, “Lump-distributed hybrids in 3D-MMIC technology,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 151, no. 4, pp. 370–374, Aug. 2004. [4] Y. C. Chiang and C. Y. Chen, “Design of lumped element quadrature hybrid,” Electron. Lett., vol. 34, no. 5, pp. 465–467, Mar. 1998. [5] L.-H. Lu et al., “Design and implementation of micromachined lumped quadrature (90 ) hybrids,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, vol. 2, pp. 1285–1288. [6] R. C. Frye, S. Kapur, and R. C. Melville, “A 2-GHz quadrature hybrid implemented in CMOS technology,” IEEE J. Solid-State Circuits, vol. 38, no. 3, pp. 550–555, Mar. 2003. [7] W.-S. Tung, H.-H. Wu, and Y.-C. Chiang, “Design of microwave wideband quadrature hybrid using planar transformer coupling method,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1852–1856, Jul. 2003. [8] T. Hirota, A. Minakawa, and M. Muraguchi, “Reduced-size branchline and rat-race hybrids for uniplanar MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 3, pp. 270–275, Mar. 1990. [9] G. Avitabile et al., “A 5.8 GHz ISM band active 90 hybrid and variable attenuator,” Microw. Opt. Technol. Lett., vol. 36, no. 5, pp. 325–327, Feb. 2003. [10] L.-H. Lu, H.-H. Hsieh, and Y.-T. Liao, “A wide tuning-range CMOS VCO with a differential tunable active inductor,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 9, pp. 3462–3468, Sep. 2006. [11] L.-H. Lu, Y.-T. Liao, and C.-R. Wu, “A miniaturized Wilkinson power divider with CMOS active inductors,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 775–777, Nov. 2005.

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[12] 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. [13] R. Mukhopadhyay et al., “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] Y. Wu et al., “RF bandpass filter design based on CMOS active inductors,” IEEE Circuits Syst. II, Analog Digit. Signal Process., vol. 50, no. 12, pp. 942–949, Dec. 2003. [15] C.-C. Hsiao et al., “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. [16] A. Thanachayanont and A. Payne, “VHF CMOS integrated active inductor,” Electron. Lett., vol. 32, no. 11, pp. 999–1000, May 1996. [17] C. Collado, A. Grau, and F. D. Flaviis, “Dual-band planar quadrature hybrid with enhanced bandwidth response,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 180–188, Jan. 2006. [18] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. Hsieh-Hung Hsieh (S’05) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree in electronic engineering at National Taiwan University. His research interests include the development of low-voltage and low-power RF integrated circuits, multiband wireless systems, RF testing, and monolithic microwave integrated circuit (MMIC) designs.

Yu-Te Liao (S’05) was born in Taichung, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 2003, the M.S. degree in electronics engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2005, and is currently working toward the Ph.D. degree at the University of Washington, Seattle. His research interests include RF integrated circuits, wideband frequency synthesizers, and wireless sensor network interface designs.

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 Graduate Institute of Electronics Engineering and the Department of Electrical 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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Design of a High-Efficiency and High-Power Inverted Doherty Amplifier Gunhyun Ahn, Min-su Kim, Hyun-chul Park, Student Member, IEEE, Sung-chan Jung, Ju-ho Van, Hanjin Cho, Sung-wook Kwon, Jong-hyuk Jeong, Kyung-hoon Lim, Jae Young Kim, Sung Chan Song, Cheon-seok Park, Member, IEEE, and Youngoo Yang, Member, IEEE

Abstract—In this paper, we present a design method for a compact inverted Doherty power amplifier (IDPA), which has high-efficiency and high-power characteristics. An optimum load matching network and an additional offset line, after the matching network of the carrier amplifier, dynamically modulate the load impedance according to the input power drive, while the conventional Doherty power amplifier uses a quarter-wave line to do it. The operational principles and design guide are provided. For experimental verification, a 50-W Doherty amplifier was designed 4 differential quadrature phase-shift keying application for the at the 860-MHz band. The measured performance of the IDPA was compared with that of the balanced class-AB amplifier with the same output matching network. At an output power of 50 W, the IDPA performs with 3.16 dB better adjacent channel power ratio ( 28 versus 24.84 dBc) and 6.15% higher power-added efficiency (59.02 versus 52.87%) than the class-AB amplifier does. Index Terms—Doherty power amplifier, inverted Doherty power amplifier (IDPA), load modulation, offset line, 4 differential quadrature phase-shift keying (DQPSK).

I. INTRODUCTION

T

HE MICROWAVE Doherty amplifier was first invented with a capability of efficiency improvement at the output power backoff condition for vacuum tube amplifiers by Doherty in 1936 [1]. Since then, many publications and subsequent inventions, related to the Doherty amplifier concept have been made [2]–[14]. A microwave Doherty amplifier having a better compromise of efficiency and linearity at a given output power level than the conventional class-AB counterpart was demonstrated [2]. They used an optimized output matching network and a specific length of offset line after it. This allowed to extract proper load impedance modulation, as well as full output power from the Doherty amplifier [2]–[6]. Linearity was also improved by optimizing the bias conditions of the carrier and peaking amplifiers, which makes a tradeoff with efficiency enhancement. Explanations for the

Manuscript received January 18, 2007; revised March 11, 2007. This work was supported by the Samsung Thales Company Ltd. under the Core-Tech Program. G. Ahn, M. Kim, H. Park, S. Jung, J. Van, H. Cho, S. Kwon, J. Jeong, K. Lim, C. Park, and Y. Yang are with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea (e-mail: [email protected]). J. Y. Kim and S. C. Song are with the Samsung Thales Company Ltd., Yongin 449-712, 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.2007.896807

linearity improvement mechanism were also provided in [3]. The microwave Doherty amplifier has been extensively employed for wideband code division multiple access (WCDMA) or Wibro/WiMax base-station power amplifiers by virtue of the new design techniques and newly discovered possibilities for obtaining an increased output power and an improved linearity. The inverted Doherty power amplifier (IDPA) has a compact inverted load network configuration in comparison with the normal Doherty power amplifier [4], [5]. Operational principles and circuit configuration were provided in [4] and good experimental results were demonstrated using the IDPA concept for handset applications in [5]. In this paper, we present a selection guide between the inverted and normal Doherty power amplifiers and a design guide for fully taking advantage of the inverted load network. We propose offset lines for the IDPA that have been already employed for the normal Doherty amplifier design [2], [3]. An IDPA design method also employs the output impedance or output reflection coefficient of the amplifier as a design reference in order to initially determine the required lengths of the offset lines for both carrier and peaking amplifiers. However, the IDPA requires the offset line to rotate the angle of output reflection coefficient . The output matching network and addito not 0 , but tional offset line dynamically modulate the load impedance of the carrier amplifier. A quarter-wave transformer does it for the conventional Doherty amplifiers. The operational principles and circuit configuration of the IDPA will be presented. Using an ideal class-B amplifier model, the performance of the IDPA was simulated according to var. Based ious angles of the output reflection coefficient , a rough length on the simulated results and the measured of offset line can be determined. For experimental validation, a 50-W IDPA was designed and fabricated at the 860-MHz band. The measured efficiency and linearity performance of the differential quadraDoherty amplifier, using two-tone and ture phase-shift keying (DQPSK) signals, will be compared with those of the conventional class-AB amplifier adopting the same devices and matching networks. II. IDPA A. Configuration and Operation Fig. 1 shows a simplified load network and its operation for the IDPA at a low power level. The peaking amplifier has an additional quarter-wave line after its offset line, while the carrier amplifier has it for the conventional Doherty power amplifier. In addition, the initial length of the offset line for the IDPA is set

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Fig. 2. Selection criterion between the inverted and normal output networks after the output matching network. using the position of 0

Fig. 1. Simplified load network configuration and its operation for the IDPA.

in order to make the , while the conventional to be 0 . Doherty amplifier requires If we assume that the transistors are unilateral, the internal parasitic and pre-matching components, inside a packaged transistor, can be simplified to an equivalent shunt component with , which becomes the output an arbitrary impedance of impedance of the transistor. The optimum power matching network and a proper offset line follow right after the transistor in order to extract the maximum output power from the device and to rotate the output impedance near to zero, respectively. At a high power level, if the phase difference between the carrier and peaking paths corresponds with the input circuits and the current sources of the carrier and peaking amplifiers drive the same currents, then the load impedance seen from each . The internal load amplifier after the offset line becomes becomes opimpedance seen from the current source timum with the optimum matching circuits. At a low power level, the peaking amplifier is assumed to be turned off. The impedance seen from the load into the peaking amplifier becomes very high, i.e., close to infinity thanks to the additional quarter-wave line, which minimizes the power leakage from the carrier amplifier. The combination of internal components, matching network, and offset line for the carrier ) to amplifier transforms the load impedance from half ( ). twice of the optimum internal load impedance ( We can select a better topology between the normal and for the fully inverted load networks after considering matched carrier amplifier, as shown in Fig. 2. If the measured or of the carrier amplifier locates from simulated to (clockwise), we have to choose an inverted load network that requires shorter offset lines to rotate the to . Otherwise, the normal Doherty network requires shorter offset lines. If the overruns from 180 to

Fig. 3. (a) Quarter-wave line with characteristic impedance of R versus (b) matching network and offset line consisting of transmission lines and open stub with a characteristic impedance of R .

180 (clockwise), we may lose some performance of the IDPA and obtain circuit simplicity in return, which will be discussed in Section II-B. A matching network with a proper offset line can work exactly like a quarter-wave line in the impedance transformation. In Fig. 3, a quarter-wave line, having characteristic impedance of , is compared with a matching network. This is followed . If by an offset line having characteristic impedance of is required to be transformed to , must be given as (1) From (1), if the load impedance goes down from to , quarter-wave line transforms the input impedances then the to [see Fig. 3(a)]. up from can be matched to In the same way, the load impedance of the input impedance of using transmission line matching

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Fig. 4. Simulated transmission coefficients for the quarter-wave transformer (T ), matching network (T ), and matching network with a proper offset line (T ).

networks such as a series line and an open-stub having an ar). The angle bitrary characteristic impedance (in this case, then does not generally of the transmission coefficient reach to 90 where the angle of the transmission coefficient for a quarter-wave line is located (see Fig. 4). To send the angle to 90 without disturbing the matching impedance, a characteristic impedance proper length of the offset line with should be inserted. It then has the same electrical characteristics as a quarter-wave line even if the load impedance is changed to [see Fig. 3(b)]. If the internal circuit components, which are assumed to in Fig. 1, do not significantly rotate the angle be of the transmission coefficient of the overall load network internal components matching network offset line , the electrical length of the offset line can be adjusted to have a net electrical length for the load network of 90 , which is equivalent to that of a quarter-wave line. B. Offset Lines for the Inverted Load Network As explained using transmission coefficients in Fig. 4, a simple L-section matching network generally has a shorter effective electrical length than that of a quarter-wave line. of the amplifier Without additional offset lines, the could have a serious effect on performances of the Doherty , a harmonic balamplifier. To examine the effect of ance simulation was conducted. The simulation setup for the IDPA was built for Agilent’s Advanced Design System (ADS) using an ideal class-B amplifier model for both the carrier and peaking amplifiers. A matching network was designed using for the carrier a series line and open-stub topology. amplifier was swept using an offset line after the matching network. The inverted load network then follows.

Fig. 5. Simulated characteristics of the IDPA using ideal class-B amplifiers ’s of the carrier amplifier. (a) Normalized RF according to the various 0 output currents for the carrier and peaking amplifiers. (b) Normalized RF output voltages for the carrier and peaking amplifiers. (c) Efficiency.

Fig. 5 shows the simulated performances of an ideal IDPA according to the various ’s from 120 to 180 . The normalized RF output currents of and for carrier and peaking amplifiers significantly vary against , as shown in Fig. 5(a). Fig. 5(b) plots the normalized RF output voltages of and

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Fig. 6. Schematic diagram of the IDPA including offset lines.

for the carrier and peaking amplifiers. Finally, the efficiency trend is presented in Fig. 5(c). As shown, the current, voltage, and efficiency characteristics are not affected at the peak output power level. On the contrary, the efficiency is significantly degraded at the backed-off power level as goes counterclockwise on the polar chart from 180 toward 120 . The same degradation trend for efficiency must be found in case goes clockwise from 180 . Here, we find a tradeoff between circuit complexity and performance for the Doherty amplifier. When locates slightly over 180 , an almost quarter-wave offset line is required to make it a normal Doherty combiner network. In this case, we can still use an inverted load network with no offset line only for the sake of the circuit’s simplicity in spite of a certain amount of efficiency degradation, which can be predicted using Fig. 5(c). For example, if the is 160 , a normal Doherty design requires an approximately 80 offset line without having any performance degradation or an inverted Doherty design could be directly applied with no offset line. In this case, an efficiency of approximately 6.6% at a 6-dB input backoff point is degraded for the ideal Doherty amplifier [see the data for a of 160 in Fig. 5(c)]. Using these simulation results, we can also predict a broadband response of the Doherty amplifier by measuring the profile over the frequency range of interest. These results can be applied to a normal Doherty amplifier design as well.

III. CIRCUIT DESIGN AND IMPLEMENTATION Fig. 6 shows a schematic diagram of the IDPA including offset lines. Identical class-AB amplifiers, which have optimized input/output matching networks, are used for the carrier and peaking amplifiers, respectively. Contrary to conventional Doherty power amplifiers, a quarter-wave line is located after the peaking amplifier to transform a very low output impedance after the offset line to a high impedance seen from the load junction. The phase difference between the two paths because of the additional quarter-wave line and offset lines can be compensated for using a quarter-wave line and additional line at the input of the carrier amplifier. The electrical length of

Fig. 7. Measured mented amplifier.

0

including a load matching network for the imple-

Fig. 8. Implemented 50-W IDPA for 860-MHz =4 DQPSK application.

the additional line becomes to match the phases between two paths. Class-AB amplifiers were designed and implemented using Freescale’s MRF9030 30-W PEP LDMOSFET for the 860-MHz band. A simple single-section low-pass matching topology, using a series transmission line and a shunt capacitor, was utilized to reduce the equivalent electrical length of the matching network. was obtained using a one-port -parameter measurement from the 820–900-MHz frequency span. The measured locates near to the short-circuited port, as shown in Fig. 7, which allows to choose an inverted load network. As a result, the circuit was significantly simpler for this case. The input splitter, delay line, and Doherty combiner circuit were also implemented. Bias conditions for both the carrier and peaking amplifiers were individually adjusted to achieve a maximum composite performance for efficiency and linearity for DQPSK excitation. The optimized quiescent currents were 283 and 0 mA for the carrier and peaking amplifiers, respectively. For comparison, the balanced class-AB amplifier was set to have a quiescent current of 566 mA (283 mA each). Fig. 8 is a photograph of the implemented IDPA.

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Fig. 9. Measured performances for the IDPA and class-AB amplifier in the case of two-tone excitation with a center frequency of 860 MHz and a tone spacing of 25 kHz. (a) IMD . (b) PAE.

IV. EXPERIMENTAL RESULTS A two-tone test was performed for both the IDPA and class-AB amplifier. Fig. 9 shows the measured and power-added efficiency (PAE) performances. In Fig. 9(a), we can see that the IDPA outperforms for when the output power goes over 43 dBm. For PAE in Fig. 9(b), the IDPA has a significantly improved performance through the overall output power range compared with the conventional class-AB configuration. At an output power of 45 dBm, the IDPA has 1.83 dB better ( 27 versus 25.17 dBc) and 8.21% better PAE (52.70 versus 44.49%). The measured performances for the IDPA and class-AB amplifier using DQPSK excitation are presented in Fig. 10. The IDPA gets better ACLR for higher output power levels from 44.5 to 47 dBm at 25-kHz offset [see Fig. 10(a)]. It also has higher PAE for an overall output power level, as shown in Fig. 10(b). Fig. 10(c) shows the power spectral density (PSD) with 1-kHz resolution bandwidth at an output power level of 47 dBm (50 W). Table I and II summarize the representative performances and the performance comparison with the previously published studies, respectively.

Fig. 10. Measured performances for the IDPA and class-AB amplifier in case of =4 DQPSK excitation. (a) ACLRs with an offset of 25 and 50 kHz. (b) PAE. (c) PSD with 1-kHz resolution bandwidth at an output power level of 50 W.

TABLE I PERFORMANCE SUMMARY: CLASS-AB AMPLIFIER VERSUS IDPA

y Two-tone: 25-kHz tone spacing, ACLR: 25-kHz offset.

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TABLE II PERFORMANCE COMPARISON

V. CONCLUSIONS An inverted load network including offset lines was presented for a Doherty amplifier design. The IDPA utilizes an output matching network and offset line to modulate the load impedance of the carrier amplifier, while the conventional Doherty amplifier uses an additional quarter-wave transmission line to modulate the impedance. The operational principles and circuit configuration were presented. Selection criterion between the inverted and conventional Doherty amplifiers was also explained. The harmonic-balance simulation setup was built using an ideal class-AB amplifier model. The simulation results verified the proper operation of the inverted load network with a proper offset line. Furthermore, a tradeoff between performance and circuit size was found. For experimental validation, a 50-W IDPA for the DQPSK application was designed and fabricated using two 30-W PEP LDMOSFETs at the 860-MHz band. The impleDQPSK mented IDPA was evaluated using two-tone and signals. It performed better in linearity and efficiency at the same time than the conventional class-AB amplifier. The IDPA including offset lines, presented in this paper, completes the Doherty amplifier design in conjunction with the conventional Doherty amplifier. Both have the same capabilities for load modulation and efficiency improvement. After checking , we can choose a better topology considthe position of ering size and performance for each case.

REFERENCES [1] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [2] Y. Yang, J. Yi, Y. Y. Woo, and B. Kim, “Optimum design for linearity and efficiency of microwave Doherty amplifier using a new load matching technique,” Microw. J., vol. 44, no. 12, pp. 20–36, Dec. 2001. [3] Y. Yang, J. Cha, B. Shin, and B. Kim, “A fully matched n-way Doherty amplifier with optimized linearity,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 986–993, Mar. 2003. [4] R. F. Stengel, “High efficiency power amplifier having reduced output matching networks for use in portable devices,” U.S. Patent 6 262 629 B1, Jul. 17, 2001. [5] L. F. Cygan, “A high efficiency linear power amplifier for portable communications applications,” in IEEE CSIC Dig., 2005, pp. 153–157. [6] Y. Yang, J. Cha, B. Shin, and B. Kim, “A microwave Doherty amplifier employing envelope tracking technique for high efficiency and linearity,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 9, pp. 370–372, Sep. 2003. [7] F. H. Raab, P. M. Asbeck, S. Cripps, P. B. Kenington, Z. B. Popovic´ , N. Pothecary, J. F. Sevic, and N. O. Sokal, “Power amplifiers and transmitters for RF and microwave,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 814–826, Mar. 2002. [8] D. M. Upton, “A new circuit topology to realize high efficiency, high linearity, and high power microwave amplifiers,” in Proc. RAWCON’98, 1998, pp. 317–320.

[9] S. C. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House Inc., 1999. [10] F. H. Raab, “Efficiency of Doherty RF power-amplifier systems,” IEEE Trans. Broadcasting, vol. BC-33, no. 3, pp. 77–83, Sep. 1987. [11] M. Iwamoto, A. Williams, P. Chen, A. G. Metzger, L. E. Larson, and P. M. Asbeck, “An extended Doherty amplifier with high efficiency over a wide power range,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. [12] Y. Zho, A. Metzger, P. J. Zampardi, M. Iwamoto, and P. M. Asbeck, “Linearity improvement of HBT-based Doherty power amplifier based on a simple analytical model,” in IEEE MTT-S Int. Microw. Symp. Dig., 2006, pp. 877–880. [13] J. Jung, U. Kim, J. Kim, K. Kang, and Y. Kwon, “A new ‘seriestype’ Doherty amplifier for miniaturization,” in IEEE RFIC Symp. Dig., 2005, pp. 259–262. [14] I. Kim, J. Cha, S. Hong, J. Kim, Y. Y. Woo, C. S. Park, and B. Kim, “Highly linear three-way Doherty amplifier with uneven power drive for repeater system,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 176–178, Apr. 2006.

Gunhyun Ahn was born in Seoul, Korea, in 1978. He received the B.S. degree in electronic engineering from Anyang University, Anyang, Korea, in 2005, and is currently working toward the M.S. degree in information and communication engineering at the Sungkyunkwan University, Suwon, Korea. His research interests include Doherty amplifier design and linearization techniques.

Min-su Kim was born in Seoul, Korea, in 1978. He received the B.S. degree in electronic engineering from Incheon University, Incheon, Korea, in 2005, and is currently working toward the M.S. degree from the Sungkyunkwan University, Suwon, Korea. His current research interests include transmitter and linearization techniques.

Hyun-chul Park (S’07) was born in Seoul, Korea, in 1980. He received the B.S. degree in information and communication engineering from the Sungkyunkwan University, Suwon, Korea, in 2006, and is currently working toward the M.S. degree in information and communication engineering at Sungkyunkwan University. His research interests include design of high linear and efficient handset power amplifiers with HBT or CMOS processes.

Sung-chan Jung was born in Seoul, Korea, in 1973. He received the B.S. degree in electronic engineering and M.S. and Ph.D. degrees in information and communication engineering from Sungkyunwan University, Suwon, Korea, in 1998, 2000, and 2006, respectively. He is currently a Post-Doctoral Researcher with the Microwave Circuits and Systems (MCS) Laboratory, Sungkyunkwan University. His current research interests include design of high power amplifiers, linearization techniques, and efficiency enhancement techniques for base stations and mobile terminals.

AHN et al.: DESIGN OF HIGH-EFFICIENCY AND HIGH-POWER INVERTED DOHERTY AMPLIFIER

Ju-ho Van was born in Iksan, Korea, in 1982. He is currently working toward the B.S. degree in information and communication engineering at Sungkyunkwan University, Suwon, Korea. His research interests include high-efficiency switching-mode power amplifiers.

Hanjin Cho was born in Seoul, Korea, in 1972. He received the B.S. degree in electronic engineering and M.S. degree in electronic engineering from Sungkyunkwan University, Suwon, Korea, in 1997 and 1999, respectively, and is currently working toward the Ph.D. degree in information and communication engineering at Sungkyunkwan University. He has developed code division multiple access (CDMA) and global system for mobile communications (GSM)/global positioning remote sensing (GPRS) RF modules for mobile phones with Samsung Electromechanics. His research interests include polar transmitter system and design of CMOS power amplifiers.

Sung-wook Kwon was born in Seoul, Korea, in 1981. He received the B.S. degree in information and communication engineering from Sungkyunkwan University, Suwon, Korea, in 2007, and is currently working toward the M.S. degree in information and communication engineering at Sungkyunkwan University. His research interests include Doherty amplifier design and GaN device modeling.

Jong-hyuk Jeong was born in Junju, Korea, in 1977. He received the B.S. degree in information and communication engineering from Sungkyunkwan University, Suwon, Korea, in 2007, and is currently working toward the M.S. degree in information and communication engineering at Sungkyunkwan University. His research interests include Doherty amplifier design and microcontrollers.

Kyung-hoon Lim was in Seoul, Korea, in 1979. He received the B.S. degree in electrical and electronic engineering from DanKook University, Seoul, Korea, in 2007, and is currently working toward the M.S. degree in information and communication engineering at Sungkyunkwan University, Suwon, Korea. His research interests include Doherty amplifier design and linearization techniques.

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Jae Young Kim was born in Ulsan, Korea, in 1971. He received the B.S. degree in electrical and electronic engineering from Keungbook National University, Taegu, Korea, in 1998. From 1998 to 2000, he was with Samsung Electronics, Giheung, Korea, Since 2000, he has been the Samsung Thales Company Ltd., Yongin, Korea, where he is currently a Senior Engineer with the RF Team of the research center. His research interests include design of linearization techniques and efficiency enhancement techniques for base stations.

Sung Chan Song was born in Hamyang, Korea, on February 2, 1976. He received the B.S. and M.S. degrees in electronic engineering from Hankuk Aviation University, Goyang, Korea, in 2001 and 2003, respectively. In 2001, he joined the Samsung Thales Company Ltd., Yongin, Korea, where he is currently a engineer with the RF Team of the research center. His research interests include computational electromagnetics and analysis and design of radar transmitters.

Cheon-seok Park (M’02) was born in Seoul, Korea, in 1960. He received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 1988, and the M.S. and Ph.D. degrees in electrical and electronic engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1990 and 1995, respectively. He is currently a Professor with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea. His research interests include design of RF power amplifiers, linearization techniques, and efficiency enhancement techniques.

Youngoo Yang (S’99–M’02) was born in Hamyang, Korea, in 1969. He received the Ph.D. degree in electrical and electronic engineering from the Pohang University of Science and Technology (Postech), Pohang, Korea, in 2002. From 2002 to 2005, he was with Skyworks Solutions Inc., Newbury Park, CA, where he designed power amplifiers for various cellular handsets. Since March 2005, he has been with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea, where he is currently an Assistant Professor. His research interests include power amplifier design, RF transmitters, RFIC design, and modeling of high power amplifiers or devices.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Inverted Slot-Mode Slow-Wave Structures for Traveling-Wave Tubes V. Latha Christie, Lalit Kumar, Member, IEEE, and N. Balakrishnan

Abstract—The dispersion and impedance characteristics of an inverted slot-mode (ISM) slow-wave structure computed by three different techniques, i.e., an analytical model based on a periodic quasi-TEM approach, an equivalent-circuit model, and 3-D electromagnetic simulation are obtained and compared. The comparison was carried out for three different slot-mode structures at -, -, and -bands. The approach was also validated with experimental measurements on a practical -band ISM traveling-wave tube. The design of ferruleless ISM slow-wave structures, both in circular and rectangular formats, has also been proposed and the predicted dispersion characteristics for these two geometries are compared with 3-D simulation and cold-test measurements. The impedance characteristics for all three designs are also compared. Index Terms—Dispersion, equivalent circuits, measurement, simulation, slow-wave structures, traveling-wave tubes (TWTs).

I. INTRODUCTION

E

MERGING applications in airborne radars and electronic countermeasures require lightweight, compact, and narrowband sources of high average power. The inverted slot-mode (ISM) traveling-wave tube (TWT) [1], [2] is an excellent candidate for such applications, as it offers the advantages of higher efficiency, reduced cross section, and lighter weight at a modest bandwidth. It belongs to the “fundamental backward” class of coupled-cavity (CC) TWT structures, which operate in the slot passband as opposed to cavity passband of conventional CC structures [3]–[5]. The special features of an ISM structure are: a smaller cavity size and vanishing interaction impedance at the upper cutoff, which provides freedom from band-edge oscillations, normally encountered in a CC TWT. However, the smaller interaction impedance at high frequencies leads to a steep fall in the gain with frequency. The smaller cavity radius allows the cavity partition walls to be made of copper and the inner diameter of the magnetic focusing system to coincide with the cavity outer diameter instead of the beam hole. This simplifies the thermal and focusing design of the TWT, as the magnetic period can be decoupled from the cavity period. In this paper, a basic design procedure, with approximate analytic expressions, to obtain the physical dimensions, and an equivalent-circuit approach to compute the dispersion and Manuscript received August 16, 2006; revised January 19, 2007. This work was supported by the Defence Research and Development Organization, Ministry of Defence, Government of India. V. L. Christie and L. Kumar are with the Microwave Tube Research and Development Centre, Defense Research Development Organization, Bangalore560013, India (e-mail: [email protected]; [email protected]). N. Balakrishnan is with the Aerospace Department, Indian Institute of Science, Bangalore-560012, India (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.897661

impedance for an ISM structure is presented. The analytic approach is based on the periodic quasi-TEM approach [6]–[8] and the equivalent-circuit approach is based on Curnow’s approach [9]. The initial design obtained by these methods can be refined using numerical models and finalized by experimentation. This method of designing the ISM structure is demonstrated on three different structures in the -, -, and -bands. The -band structure is that of a practical -band ISM TWT [2] and the computed dispersion and impedance were fed into a large-signal program [10], and the predicted power and gain matched quite closely to the experimental power and gain. Two new slot-mode structures have been proposed, namely: the circular and rectangular slot-mode structures without ferrules. They reduce the extremely tight tolerance requirements of fabrication, assembly, and brazing associated with slot-mode structures with ferrules. The rectangular slot-mode structures without ferrules gave the same impedance as the conventional slot-mode structure and the circular slot-mode structure without ferrules was found to give an improvement in the impedance with a tradeoff in the width of the dispersion characteristics. The circular slot-mode structure with and without ferrules and the rectangular slot-mode structure without ferrules were designed at the -band frequency and fabricated. The experimental results were compared with the results from the periodic quasi-TEM approach and 3-D MAFIA simulation and were found to be closely matching. II. DESIGN PROCEDURE The schematic of a circular ISM slow-wave structure with ferruled cavities is given in Fig. 1(a). The procedure for designing an ISM slow-wave structure [8] involves determination of cavity radius , finger width , wall thickness , cavity pitch , gap length , tunnel radius , and ferrule radius . The lowest cutoff frequency of the second passband is the cavity resonant frequency that determines the cavity radius. This is chosen as three to four times of the center operating frequency resulting in a smaller cavity radius. Due to the steep fall in the interaction impedance with the phase shift per cavity, the phase shift/cavity is chosen near to the lowest cutoff point of the lower passband in the determination of the cavity pitch. A. Periodic Quasi-TEM Approach A periodic quasi-TEM approach [6], [7] can be used to analyze this structure since, along the -direction of the cavity, a TEM wave can be assumed to propagate, as the components

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CHRISTIE et al.: ISM SLOW-WAVE STRUCTURES FOR TWTs

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is inductive at all frequencies The waveguide impedance below the cutoff frequency of the waveguide and is given as (5) The periodic quasi-TEM approach can also be used to design ISM structures with ferruleless cavities and with rectangular cavities [see Fig. 1(b)]. In the case of an ISM slow-wave structure with ferruleless cavities, the gap length in (3) is replaced by the cavity height . For ISM slow-wave structures with rectangular cavities, the cutoff wavelength is equal to . twice the slot length, which is given by B. Equivalent-Circuit Approach

Fig. 1. Schematic of: (a) circular ISM structure with ferruled cavities. (b) Rectangular ISM structure with ferruleless cavities.

and are negligible compared to the corresponding transverse components. The two basic assumptions made here are: 1) the structure can propagate an electromagnetic wave in the direction transverse to the array of parallel conductors forming it and 2) the field distribution is represented by a TEM wave or a sum of such waves propagating in the direction of conductors. Thus, the structure is assumed to consist of a periodic array of a TEM-wave line of length and a below-cutoff rectangular waveguide of length (Fig. 1). The delay ratio for a wave prop, and the phase shift per cavity agating along the structure is is given by [8]

The equivalent-structure procedure [11] used for the design of the ISM slow-wave structure [see Fig. 1(a)] is Curnow’s approach for the staggered CC structure with an overlap [9]. For this structure, the coupling coefficient is greater than 0.5 and the part of the circulating current not involved in coupling is zero. Hence, phase shift per cavity is given by [9]

(6) where

is the bandwidth parameter given as (7)

and the impedance

is defined by Curnow [9] as (8)

where is the power carried by the wave and is the corresponding voltage amplitude appearing across the cavities (9)

(1) is the characteristic impedance of the TEM is the impedance of the below-cutoff rectanwave line and gular waveguide with propagation constant given by (2)

The cavity resonant frequency , cavity admittance , slot resonant frequency , and slot admittance are calculated as , per [12]. The coupling coefficient , the slot capacitance needed to compute and are and the slot inductance calculated as follows: (10)

where the cutoff wavelength slot length given by [8]:

with as the effective . is given as follows

(11) (12)

(3)

The slot width to finger width ratio is multiplied by a correction factor of 1.05, which was empirically determined to take into account the fringing effects. C. Numerical Simulation

(4)

For the computation of phase velocity and impedance by numerical simulations, an eight-cavity structure with cavity walls of half the thickness at both ends is modeled using the 3-D

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TABLE I NORMALIZED DIMENSIONS AND OPERATING CONDITIONS OF ISM SLOW-WAVE STRUCTURE

Fig. 2. 3-D arrow plot of ISM structure at 1:5 phase shift.

software MAFIA. Fig. 2 gives the 3-D arrow plot of the ISM . When slow-wave structure for a phase shift per cavity of magnetic boundary conditions are applied at both ends of the structure, nine resonant frequencies corresponding to the phase at an interval of are obtained. shifts of to The phase shift per cavity corresponding to the resonant . Curnow’s frequencies is impedance is given as [12] (13) where is the group velocity and is the time-averaged stored electromagnetic energy per unit length of the structure. is the voltage appearing across the cavity walls of a single cavity. Let be the axial electric field, then is given as (14) D. Measurements The dispersion characteristic was experimentally measured on a section of the ISM structure consisting of eight full cavities with cavity walls of half the thickness at both ends. The structure was excited through loop coupling and this section gave seven to at an resonant frequencies corresponding to on an HP8757D scalar network analyzer. The interval of slot mode is characterized by a strong radial component of the electric fields in the coupling slots. The axial electric field for the slot mode has an extra phase reversal in each cavity in consistency with field orthogonality and is zero near the axis at a phase shift of . Thus, the resonances corresponding to the and phase shifts are not observed in the experiment as both ends are shorted. The measurement of interaction impedance follows the conventional technique of passing a dielectric rod along the axis of the structure and measuring the shift in resonant frequencies [13]. The impedance is computed using Vaughan’s procedure [14]. III. RESULTS AND DISCUSSION A. Validation of the Approaches To validate the approaches presented, the circular ferruled ISM slow-wave structure [see Fig. 1(a)] was designed in three different frequency bands: -, -, and -bands. The design was

Fig. 3. Comparison of dispersion and impedance characteristics by periodic quasi-TEM (dispersion only), equivalent circuit, and MAFIA for an S -band ISM slow-wave structure.

carried out to achieve a minimum bandwidth of 2% in all three cases. The specifications and dimensions of the -band ISM slow-wave structures are those of a practical ISM TWT [2]. In the - and -bands of frequencies, the beam voltage is chosen as that of practical CC TWTs. The dimensions of the structures given in Table I are normalized with respect to the wavelength , corresponding to the operating frequencies given in Table I. Figs. 3 and 4 give a comparison of the results of dispersion from the periodic quasi-TEM approach, equivalent-circuit approach, and MAFIA and impedance from the equivalent-circuit approach and MAFIA for - and -band ISM slow-wave structures, respectively. The computational time taken for the periodic quasi-TEM approach and for the equivalent-circuit approach in a 2.66-GHz Windows PC with a RAM of 632 MB is within 2 s. MAFIA simulation run for the -band ISM structure with ferrules took 8.67 h for a total mesh of 11, 57, 184 using

CHRISTIE et al.: ISM SLOW-WAVE STRUCTURES FOR TWTs

Fig. 4. Comparison of dispersion and impedance characteristics by periodic quasi-TEM (dispersion only), equivalent circuit, and MAFIA for C -band ISM slow-wave structure.

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Fig. 6. Comparison of dispersion characteristics of circular ( c) with rectangular ( r) ferruleless inverted slot-mode slow-wave structure obtained from periodic quasi-TEM, MAFIA, and measurements.

into the large-signal analysis of the device using this structure (Table I), agreed well with the experiment with respect to the saturated power and gain versus frequency characteristics [2]. B. Ferruleless ISM Structures

Fig. 5. Comparison of dispersion and impedance characteristics by periodic quasi-TEM (dispersion only), equivalent circuit, MAFIA, and measurements for X -band ISM slow-wave structure.

a Sun Ultra 60 Workstation for nine frequency points. The frequencies are normalized with respect to the operating frequencies, given in Table I. Fig. 5 gives a comparison of the results of dispersion from the periodic quasi-TEM approach, equivalent-circuit approach, MAFIA, and measurements and the comparison of impedance from the equivalent-circuit approach, MAFIA, and measurements for the -band ISM slow-wave structure. Interestingly, the theoretical dispersion and impedance versus phase-shift characteristics of the ISM structure (Fig. 5), when fed back

The ferrules are generally used in CC structures to concentrate the electric field. However, at higher frequencies or for slow-wave structures with smaller dimensions, such as the ISM slow-wave structure, they also reduce the tolerance requirements and increase fabrication problems. In the ISM slow-wave structure, as the operating frequency is quite close to the band-edge frequency, a slight shift in the frequency due to the errors in the cavity height and gap length shifts the total frequency band. This calls for extremely tight tolerances for fabrication, assembly, and brazing. Hence, the ferruleless ISM slow-wave structures both in the circular and rectangular format is of interest at higher frequencies. The schematic of a rectangular ISM slow-wave structure is given in Fig. 1(b). This can be fabricated using ladder technology [15] where the complete slow-wave structure is fabricated in two halves and then cover plates are added to close off the structure, thus simplifying the tolerance requirements. The rectangular ISM slow-wave structure is designed in the -band such that the interaction impedance is the same as that of the practical ISM slow-wave structure, and square cavities are used whose width is equal to the diameter of the circular cavity (Table I). Interestingly, a circular ferruleless ISM slow-wave structure with the finger width reduced as given in Table I for achieving the same operating band was found to give higher impedance over the operating band in all three frequency bands. This is due to an increase in the axial electric field and the reduction of group velocity by the reduction in the width of the dispersion curve. Unlike the cavity mode slow-wave structure, in an

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

in all the three cases. In the prediction of dispersion, the analytical approach based on a periodic quasi-TEM gives an error that is 4%, whereas the equivalent-circuit gives an error of 6% when compared with 3-D simulations in all the three cases. Thus, the first cut design of an ISM slow-wave structure can be obtained from analytical approaches and then the structure can be optimized with 3-D simulation without the necessity of costly cold-test experiments. The predicted value of dispersion and impedance from 3-D simulation, when fed into large-signal analysis, predicted the power within 0.3 dB and gain within 1 dB over most of the experimental data. It is also shown that the ISM structures without ferrules may have advantages over the ferruled structures in certain situations, particularly at high frequencies, and reduce the fabrication cost and time and improve the impedance. ACKNOWLEDGMENT

Fig. 7. Comparison of impedance characteristics obtained from MAFIA between the circular inverted slot-mode slow-wave structure ferruled (ferruled c), ferruleless (ferruleless c), and rectangular ISM slow-wave structure (ferruleless r).

The authors are thankful to Dr. S. Kamath, Dr. S. K. Datta, M. Sumathy, M. Raghavendra, and B. Sujatha, all with the Microwave Tube Research and Development Centre, Defense Research Development Organization (DRDO), Bangalore, India, for their help. REFERENCES

ISM slow-wave structure, the reduction in the width of the dispersion characteristics does not matter, as the impedance drops to only are very quickly and the frequencies from useful. The two ferruleless designs have been carried out using the periodic quasi-TEM approach, and finalized by MAFIA simulation and measurements. Fig. 6 gives a comparison of the results of dispersion characteristics from the periodic quasi-TEM approach, MAFIA, and measurements for the ferruleless structures in circular and rectangular formats. Fig. 7 gives a comparison of the impedance computed from MAFIA for the circular ISM slow-wave structure with and without ferrules and for the rectangular ISM slow-wave structure without ferrules at -band. The circular -band ISM slow-wave structure without ferrules gave an improvement of impedance by 30% that resulted in an output power of 8 kW over 3% bandwidth and the design of the rectangular ISM slow-wave structure gave the same power and gain as the practical ISM TWT when computed using large-signal analysis.

IV. CONCLUSION The analyses of an ISM slow-wave structure using the periodic quasi-TEM approach, equivalent-circuit approach, 3-D simulations, and measurements have been presented and their results compared. The approaches developed have been validated by application to three different structures operating in the -, -, and -bands of frequencies. The error in prediction of impedance from the equivalent-circuit approach is within 15%, as compared to 3-D simulations with MAFIA, and the 3-D simulation results match with experimental results within 2%

[1] J. R. Frey and L. Tammaru, “A coupled-cavity TWT operating in an inverted slot mode,” in Int. Electron. Device Meeting Tech. Dig., 1997, pp. 504–506. [2] S. Kamath, S. Karmakar, R. Hemamalini, R. Seshadri, M. Santra, M. Ramaswamy, V. L. Christie, B. M. Fazlunissa, C. Srinivasacharyulu, and L. Kumar, “Design and development of 6.5 kW -band inverted slot-mode coupled-cavity TWT,” in IEEE IVEC Dig., 2004, pp. 22–23. [3] M. Chodorow and R. A. Craig, “Some new circuits for high power traveling-wave tubes,” Proc. IRE, vol. 45, no. 8, pp. 1106–1118, Aug. 1957. [4] R. W. Gould, “Characteristics of TWT with periodic circuits,” IRE Trans. Electron Devices, vol. ED-6, no. 7, pp. 186–194, Jul. 1959. [5] A. Staprans, E. W. Mccune, and J. A. Ruetz, “High-power linear-beam tubes,” Proc. IEEE, vol. 61, no. 3, pp. 299–330, Mar. 1973. [6] R. C. Fletcher, “Broadband interdigital circuit for use in travellingwave amplifier,” Proc. IRE, vol. 40, no. 8, pp. 951–958, Aug. 1952. [7] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [8] J. Hirano, “Characteristics of interdigital circuits and their use for amplifiers,” Proc. Inst. Elect. Eng., vol. 105, pt. B, pp. 780–785, Dec. 1958., Suppl. 12. [9] H. J. Curnow, “A generalized equivalent circuits for coupled cavity slow-wave structures,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 5, pp. 671–675, May 1965. [10] S. K. Datta, “Microwave tube research and development centre,” unpublished. [11] V. L. Christie, L. Kumar, and N. Balakrishnan, “Design of an interdigital structure for TWTs,” in Proc. 27th IEEE Int. Infrared Millimeter Waves Conf., 2002, pp. 341–342. [12] V. L. Christie, L. Kumar, and N. Balakrishnan, “Improved equivalent circuit model of practical coupled-cavity slow-wave structures for TWTs,” Microw. Opt. Technol. Lett., vol. 35, pp. 322–326, Nov. 2002. [13] E. J. Nalos, “Measurement of circuit impedance of periodically loaded structures by frequency perturbations,” Proc. IRE, vol. 4, pp. 1508–1511, Oct. 1954. [14] J. R. M. Vaughan, “Calculation of coupled cavity TWT performance,” IEEE Trans. Electron Devices, vol. 22, no. 10, pp. 880–889, Oct. 1975. [15] B. G. James and P. Kolda, “A ladder circuit coupled cavity TWT at 80–100 GHz,” in Int. Electron. Device Meeting Tech. Dig., 1986, pp. 494–497.

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V. Latha Christie received the B.E. degree (Hons.) from Government College of Engineering (GCE), Tirunelveli, India, in 1986, the M.Sc. degree (Eng.) from the Indian Institute of Science (IISc), Banglaore, India, in 2000, and is currently working toward the Ph.D. degree in aerospace at IISc, Bangalore, India. Since 1987, she has been a Scientist with the Microwave Tubes Research and Development Centre, Defense Research Development. Organization (DRDO), Bangalore, India. Her current research interest includes discrete electromagnetic methods for modeling and simulation of vacuum electronic devices and computer-aided design (CAD) of microwave components and tubes, particularly slow-wave structures. Ms. Christie is a Fellow of the Vacuum Electronics Devices and Application Society (VEDAS). She is a member of the Institution of Electronics and Telecommunication Engineers (IETE). She was a recipient of the 2003 DRDO Agni Award for Excellence in Self-Reliance and the 2003 and 2005 Microwave Tube Research and Development Centre (MTRDC) Best Paper Award.

Lalit Kumar (M’79) received the M.Sc. degree from Meerut University, Meerut, India, and the Ph.D. degree in physics from the Birla Institute of Technology and Science (BITS), Pilani, India. He is currently the Director of the Microwave Tube Research and Development Centre (MTRDC), Defense Research Development. Organization (DRDO), Bangalore, India. From 1978 to 1997, he was a Scientist with the Central Electronics Engineering Research Institute (CEERI), Pilani, India. From 1983 to 1985, he was a Deutscher Akademischer Austausch Dienst (DAAD) Research Fellow in Germany, with the University of Tuebingen, Technical University Hamburg-Harburg and Valuo-Philips, Hamburg. From 1990 to 1991, and in 1994, he was Honorary Visiting Research Fellow at Lancaster University and worked on the European Space Agency project. His current interests include TWTs, microwave power modules, and ultrahigh-power microwave devices. Dr. Kumar is a Fellow of the Institution of Electronics and Telecommunication Engineers (IETE). He is a Member of the Indian Physics Association, the Indian Vacuum Society Indo-French Technical Association, and the Mag-

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netics Society of India. He is currently president of the Vacuum Electronics Devices and Application Society, Bangalore, India, which was founded in 2004. He was the recipient of the 1993 J. C. Bose Memorial Award of the IETE for Best Paper, the 1993 Best Project Award of the CEERI, the 2001 IETE–International Radar Symposium of India (IRSI) (83) Award, and the 2003 DRDO Agni Award for Excellence in Self-Reliance. He was also the recipient of the Indian National Academy of Engineering–All Indian Council of Technical Education (INAE–AICTE) Distinguished Visiting Professorship at the Institute of Technology (IT), Banaras Hindu University (BHU), Varanasi, India (2005–2007).

N. Balakrishnan received the B.E degree (Hons.) and Ph.D. degree from the Indian Institute of Science (IISc), Banglaore, India, in 1972 and 1979, respectively. He is currently the Professor Satish Dhawan Chair Professor and Associate Director of the IISc. He is also a Professor with the Department of Aerospace Engineering and with the Supercomputer Education and Research Centre. He is also a Visiting Professor with Carnegie–Mellon University, Pittsburgh, PA. He is an Honorary Professor with the Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR) and the National Institute of Advanced Studies. He has authored or coauthored over 200 publications in the international journals and international conferences and reports. He is the Editor of Electromagnetics, the International Journal of Computational Science and Engineering, and the International Journal on Distributed Sensor Networks. His areas of research include numerical electromagnetics, high-performance computing and networks, polarimetric radars and aerospace electronic systems, information security, and digital library and speech processing. Dr. Balakrishnan was the recipient of the 2002 Padmashree Award, the 2004 Homi J. Bhabha Award for Applied Sciences, the 2004 Hari Om Ashram Trust Awards, the 2001 IISc Alumni Award for Excellence in Research for Science and Engineering, the 2000 Millennium Medal presented by the Indian National Science Congress, the 2001 Indian National Science Academy (INSA) Jawaharlal Nehru Centenary Lecture Award, the 1987 and 2000 J. C. Bose Memorial Award for Best Paper, the 1998 Excellence in Education Award presented by the Aeronautical Society of India, and the 1986 UNESCO Regional Office of Science and Technology for South and Central Asia (ROSTSCA) Young Scientist Award.

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A Periodically Loaded Transmission Line Excited by an Aperiodic Source—A Green’s Function Approach Islam A. Eshrah, Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract—A simple and closed-form expression for the Green’s function of a periodically loaded infinite transmission line is derived. The spatial-domain formulation is based on distinguishing three coordinate systems, i.e., the observation, primary source, and secondary source coordinates. The Fourier transform is subsequently employed to transform the primary source spatial domain to a spectral domain, wherein Floquet’s theorem is applicable. While applying Floquet’s theorem, the observation coordinate is Fourier transformed to another spectral parameter that is a function of the primary source spectral parameter. The spatial-domain expression for the Green’s function is obtained upon identifying the poles in the spectral parameter complex plane. The derived expression is verified by comparing the values of the voltage along the line obtained analytically with those obtained using a circuit simulator. The effect of the various parameters on the voltage distribution along the line and the dispersion curves is investigated. Index Terms—Aperiodic source, Green’s function, periodic structure, transmission line.

I. INTRODUCTION

P

ERIODIC structures have a variety of applications in electromagnetics, ranging from microwave filters [1] and slow-wave structures [2] to frequency-selective surfaces [3], electromagnetic bandgap structures, and metamaterials [4]. Recently, periodically loaded transmission lines or guided-wave structures, in general, were introduced as structures that support left-hand propagation in addition to the conventional right-hand propagation regime [5]–[9]. The analysis of such transmission lines is typically performed by invoking the Bloch–Floquet theorem in order to determine the propagation constant of the supported wave [1]. For periodically loaded waveguide problems, Floquet’s theorem is used to obtain the distribution of the Floquet harmonics, as well as the dispersion relation [9]. The applicability of Floquet’s theorem is limited by the assumption that there is a fixed progressive phase shift from one unit cell to the other, which is the case in the source-free problem or when the impressed source exhibits a fixed phase shift all over the Manuscript received December 2, 2006; revised March 2, 2007. This work was supported in part by the National Science Foundation under Grant ECS0220218. I. A. Eshrah was with the Department of Electrical Engineering, The University of Mississippi, University, MS 38677 USA. He is now with the Department of Electronics and Communications Engineering, Cairo University, Giza 12261, Egypt (e-mail: [email protected]). A. A. Kishk is with the Department of Electrical Engineering, The University of Mississippi, University, MS 38677 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.2007.897667

structure. A well-known example for the latter situation is the frequency-selective surface under plane-wave excitation. In view of the limitations of Floquet’s theorem, the analysis of a periodic structure excited by an aperiodic or localized source is a challenging problem. The straightforward approach to analyze such problems is to use numerical techniques to solve the whole structure if it is finite, or truncate it with the appropriate boundaries if it is infinite. Either way is usually a time-consuming process that demands considerable computational resources, not to mention the lack of physical insight into the problem. In [10], the study of a 2-D periodic structure excited by a magnetic line source was presented, where a sampling technique was used to solve the problem of a grounded dielectric slab covered by a periodically slotted conducting plane. The radiation characteristics of an elementary antenna on a photonic bandgap structure were studied in [11], where infinite sources identical to the original single source (the antenna) were introduced with the same periodicity as the periodic structure. The introduced sources have a progressive phase shift relative to the reference source, which has zero phase. Since the sources are now periodic, Floquet’s theorem can be applied to compute the field in the periodic structure. An integration process then follows to cancel the contribution of the fictitious sources and leave behind only the contribution of the zero-phase source. The previous process requires careful and intensive numerical calculations, even for problems such as the one at hand. In this study, a periodically loaded transmission line excited by a localized source is analyzed. Though interesting per se, this 1-D problem is the natural first step in the analysis of 3-D periodic structures with the goal of obtaining a closed-form expression for the Green’s function (as in the current 1-D case) or developing a simple numerical approach in determining it for more complicated problems. In Section II, a Green’s function approach is used to solve the classical problem of a sourcefree periodically loaded transmission line. In Section III, periodic sources are introduced to the line and a brief description of the solution procedure is given. The basic idea in the analysis of a periodically loaded transmission line excited by a localized source, as presented in Section IV, is to distinguish the coordinate system of the observation point, primary source point, and secondary source point. The latter coordinate system, which does not exist in traditional Green’s function problems, is necessary to be introduced to access the equivalent sources that account for the scattering from the periodic loads or scatterers. Another key point is the fact that the equivalent sources are also a function of the location of the primary or impressed source. Upon establishing the mathematical formulation for this problem, it is readily noticed that a Fourier transform applied to

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In view of the compensation theorem (the analog of the equivalence principle in circuit theory), the periodic shunt load may be substituted by equivalent sources and, thus, the voltage along the line may be determined by using (2) upon writing the source as (4) Fig. 1. Geometry of a source-free periodically loaded transmission line.

the primary source coordinate transforms the problem to one where Floquet’s theorem is applicable to the spectral secondary sources. In applying Floquet’s theorem, the observation coordinate is Fourier transformed to another spectral parameter, which is a function of the primary source spectral parameter and the index of the Floquet harmonics. Upon identifying the poles in the complex plane and taking the inverse Fourier transform of the obtained expression, a simple closed-form expression is obtained for the periodic Green’s function. In Section V, the derived expressions are verified by comparing the voltage distribution on the line obtained using the current theory to that obtained from Agilent’s Advanced Design System (ADS) [12]. A summary and some concluding remarks are given in Section VI. II. SOURCE-FREE PERIODICALLY LOADED TRANSMISSION LINE Wave propagation in a periodically loaded transmission line (depicted in Fig. 1) is a classical problem that is well documented in numerous books, such as [1]. Conventionally, the Bloch–Floquet theorem is used to determine the dispersion charmatrices of acteristics of such a structure where the the blocks that constitute the unit cell are evaluated and multimatrix of the unit cell. Using plied to obtain the overall the propagation relation between the voltage (or current) at the nodes, the propagation constant of the periodic structure is then determined as a function of the propagation constant of the unloaded line, as well as the load parameters and period. In [1], a transmission line loaded with a normalized shunt susceptive load is analyzed and the dispersion relation was found to be

Using Parseval’s theorem [14], the integral in (2) may be expressed as the inverse Fourier transform of the product of two Fourier transforms viz. (5) where the tilde and the bar denote the Fourier transforms as follows:

(6) Following the procedure detailed in Appendix I, the integral in (5) will reduce to a series of the form (7) where is the Fourier transform of , which is the equivalent source within the reference period (centered about ) and . Upon determining the Fourier transforms, the expression in (7) reduces to (8) The previous expression gives the amplitude of the Floquet harmonics in terms of the voltage at the reference load. Evaluating (8) at yields

(1) (9) where and are the propagation constants in the loaded and unloaded transmission line, respectively, and is the period of the normalized load . It is interesting to see how a Green’s function approach to the same problem leads to the same dispersion relation, but with more physical insight along the way. Using the 1-D Green’s along a transmission line excited function [13], the voltage may be obtained by evaluating the integral by a source (2) where

The series in (9) can be summed as given in Appendix II to give the same dispersion relation in (1). To the best of the authors’ knowledge, a closed-form expression for the series in (8) with and is not available in any of the known books such as [15]. However, a simple expression for this series can be found as outlined in Section III and is given in Appendix II. Thus, the voltage distribution given in (8) may be conveniently rewritten in the form

is the characteristic impedance and (3)

(10)

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Fig. 2. Periodically loaded transmission line excited by a periodic source having the same periodicity as the load.

Fig. 3. Periodically loaded transmission line excited by a localized aperiodic source.

where is the remainder of the division of over . Notice that the dispersion relation (1) was used to simplify (10). The previous expression is that of a traveling wave having an amplitude that is an explicit function of and , and depends implicitly on , which determines the value of . It can be easily , i.e., when the loads are open circuited, verified that at is equal to , and the expression in (10) reduces to . It is important here to notice that whereas (8) expresses the voltage as a superposition of the individual Floquet harmonics, the expression in (10) suggests that the contribution of the highorder harmonics may be summed and appears in the amplitude of the fundamental Floquet harmonic.

the Green’s function of the periodic structure needs to be derived. The scattering superposition method [13] lends itself to this problem, but one point needs to be taken into consideration: since there are two types of sources in this problem, three coordinate systems will be employed, namely, the unprimed, primed, and double-primed, as follows:

III. PERIODICALLY LOADED AND EXCITED TRANSMISSION LINE If the transmission line is periodically excited by current sources (in phase or with progressive phase shift), in addition to the periodic loading, as shown in Fig. 2, the voltage along the line may be determined in a similar approach to that detailed in Section II. In this case, the expression of the voltage will have two terms corresponding to the contribution of the periodic sources and the periodic loads. If the sources have the same periodicity as the loads, and have a progressive phase shift , then the Floquet modes supported by the structure will have propagation constants . Thus, the value of the propagation constant is dictated by the source and not by a dispersion relation, as in the source-free case. However, if the operating wavenumber satisfies the dispersion relation in (1), the structure will resonate. This can be easily observed from the expression of the voltage distribution in this case, which will not be given here for brevity. Although this structure is not practical, the analysis thereof served in obtaining a simple closed-form expression for the series in Appendix II by comparing the resulting expression of the voltage to that obtained using the approach adopted in [1], which is based on the Bloch–Floquet theorem and the use of -parameters to relate the voltages at two successive the nodes. IV. PERIODICALLY LOADED TRANSMISSION LINE WITH APERIODIC SOURCE

(11) represents the secondary (or equivalent) sources, where which are functions of the primed coordinate, i.e., the location . The evaluation of the primary (or impressed) source of the integral in (11) is not straightforward since Floquet’s theorem does not apply to the secondary sources excited by the delta source. However, upon taking the Fourier transform of both sides of (11), one obtains

(12) is due to In the spectral domain, the secondary source , which exhibits a progresthe spectral impressed source sive phase shift between the successive periods. Thus, the expansion in (21) may be used to express the secondary source and the Fourier transform may be evaluated using the same approach given in Appendix I. A similar procedure to that applied in Section II may be employed to reduce the integral in (12) to (13) where (14) Upon evaluating the Fourier transforms, evaluating the Green’s function at the reference load (at ) to obtain an , and substituting in (13), one gets expression for

A. Spectral-Domain Solution The ultimate goal of examining the previous two problems is to facilitate the analysis of the more involved problem of a periodically loaded transmission line excited by an aperiodic source, as shown in Fig. 3. To be able to obtain the voltage in this case,

(15)

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B. Transformation to Spatial Domain By inspecting the expression in (15), two sets of poles can be identified: the poles due to the zeros of the characteristic function (16) which occur at and the poles that occur when . While evaluating the inverse Fourier transform, it can be shown that the residues due to the latter set of , where the residue cancels the first poles vanish, except at term, which represents the direct contribution of the primary contribute to source. Hence, only the former set of poles at the periodic Green’s function, which can be ultimately given by

Fig. 4. Circuit simulated using Agilent’s ADS2004. The circuit is terminated with the characteristic impedance computed using (20).

(17) where . Again, substituting for the series in (17) from Appendix II results in

(18)

where . The characteristic impedance the periodic structure may be evaluated using

of

(19) Halfway between two loads, the characteristic impedance assumes a real value in the passband, i.e., for real values of , and is given by (20) It is thus possible to simulate a finite structure by terminating both ends of a few cells cascaded together by the characteristic impedance . V. RESULTS To validate the derived Green’s function, the circuit shown in Fig. 4 was simulated in Agilent’s ADS for different values

Fig. 5. Voltage distribution on the periodically loaded transmission line for different source locations. (a) Magnitude. (b) Phase. Solid line: current theory. Markers: Agilent’s ADS circuit simulation ( : z = p=4 and : z = p=2).



2

of the load parameters and compared to the theoretical results. As an example, consider the case with and A. Fig. 5 shows the voltage distribution over two periods of the structure with the source which is, by reciprocity, equal to the located at when the source is swept in voltage observed at . Excellent agreement is observed the interval between both results.

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Fig. 6. Dispersion characteristics of the periodically loaded transmission line. The solid and dashed lines correspond to the real and imaginary parts, respectively. The thick lines (solid or dashed) highlight the values of p= .

For the same parameters, Fig. 6 plots the dispersion characteristics for the structure and highlights the relevant solution for . For small values of , the bandgap tends to disappear and the dispersion characteristics approach the light line. As increases, the bandgap grows wider until it eventually covers all frequencies as approaches , except for the resonance frequency points where . If the sign of is changed, i.e., if the loads are inductive, the new dispersion characteristics will have a stopband first, followed by a passband, which is the reverse order compared to the capacitive susceptance. The voltage maxima and minima occurrences will also be reversed. Practically, the load susceptance is frequency dependent. If the load susceptance increases (decreases) with frequency, i.e., if the load is capacitive (inductive), the bandgap grows wider (narrower) as the frequency increases. If the line is loaded with lossy loads, the amplitude of the voltage wave will be damped as the observer gets further from the source position, as shown in Fig. 7, where the source is loand the normalized susceptance is complex cated at with a negative imaginary part. VI. CONCLUSION The Green’s function for a periodically loaded transmission line was derived and verified through a comparison with a circuit simulator, where the circuit was terminated on both ends by the characteristic impedance obtained from the current theory. The results show that the voltage distribution along the line is that of a traveling wave with a spatially modulated amplitude. APPENDIX I FLOQUET HARMONIC EXPANSION OF THE NODE VOLTAGE

Fig. 7. Voltage distribution on the periodically loaded transmission line for lossy loads b = 1:2 j 1. The source is located at z = p=2.

0

is the voltage at the reference load, i.e., the apewhere riodic version of . It can be easily shown that the function is periodic with period and, thus, its Fourier transform may be written as

(22)

where version of

and is the aperiodic and has a Fourier transform given by (23)

From the relation between (23), the Fourier transform of

and , and using (22) and follows directly as

(24) where

. APPENDIX II USEFUL SERIES EXPANSIONS

The following identities in (25)–(28) were used in simplifying the expressions for voltage distribution and the Green’s function of the periodically loaded transmission line:

(25)

The voltage at the secondary sources, i.e., the loads, in the case of the source-free periodically loaded transmission line, can be written as (21)

(26)

ESHRAH AND KISHK: PERIODICALLY LOADED TRANSMISSION LINE EXCITED BY APERIODIC SOURCE—GREEN’S FUNCTION APPROACH

Substituting with on the right-hand side of (26)–(28) makes them valid for all .

(27)

(28) REFERENCES [1] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [2] S. Y. Liao, Microwave Devices and Circuits, 3rd ed. New York: Pearson Educ., 1990. [3] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000. [4] A. Lai, C. Caloz, and T. Itoh, “Composite right/left-handed transmission line metamaterials,” IEEE Micro., vol. 5, no. 3, pp. 34–50, Sep. 2004. [5] 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. [6] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett., pp. 183 901–183 904, Oct. 2002. [7] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 110–119, Jan. 2005. [8] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Rectangular waveguide with dielectric-filled corrugations supporting backward waves,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3298–3304, Nov. 2005. [9] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Spectral analysis of left-handed rectangular waveguides with dielectric-filled corrugations,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3673–3683, Nov. 2005.

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[10] R. A. Sigelmann and A. Ishimaru, “Radiation from periodic structures excited by an aperiodic source,” IEEE Trans. Antennas Propag., vol. AP-13, no. 3, pp. 354–3664, May 1965. [11] H.-Y. D. Yang, “Application of photonic bandgap materials to printed antennas,” in Proc. Int. Electromagn. Adv. Applicat. Conf., Turin, Italy, Sep. 1999, pp. 395–398. [12] “Advanced Design System 2004A User’s Guide,” Agilent Technol., Palo Alto, CA, 2004. [13] G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics: An Introduction. New York: Springer-Verlag, 2001. [14] W. R. LePage, Complex Variables and the Laplace Transform for Engineers. New York: McGraw-Hill, 1961. [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products.. San Diego, CA: Academic, 2000. Islam A. Eshrah (S’00–M’06) was born in Cairo, Egypt, in 1977. He received the B.Sc. and M.Sc. degrees in electronics and communications engineering from Cairo University, Cairo, Egypt, in 2000 and 2002, respectively, and the Ph.D. degree in electrical engineering from The University of Mississippi, University, in 2005. He is currently an Assistant Professor with the Department of Electronics and Communications Engineering, Cairo University, Giza, Egypt. Dr. Eshrah was the recipient of the Young Scientist Award presented at the 2004 URSI International Symposium on Electromagnetics, the Young Scientist Award presented at the 2005 URSI General Assembly, the Best Student Paper Award presented at the 2005 Antenna Applications Symposium, and the Raj Mittra Junior Researcher Award presented at the 2005 IEEE Antennas and Propagation Society Symposium. He was also a finalist in numerous student paper contests.

Ahmed A. Kishk (S’84–M’86–SM’90–F’98) received the B.Sc. degree in electronics and communications engineering from Cairo University, Cairo, Egypt, in 1977, the B.Sc. degree in applied mathematics from Ain Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1983 and 1986, respectively. In 1981, he joined the Department of Electrical Engineering, University of Manitoba. He is currently a Professor with The University of Mississippi, University. He has authored or coauthored over 170 refereed journal papers and book chapters. He coauthored Microwave Horns and Feeds (IEE Press, 1994; IEEE Press, 1994). He has also authored several book chapters. He was an Editor-in-Chief of the ACES Journal from 1998 to 2001. Dr. Kishk is an editor of the IEEE Antennas and Propagation Magazine. He was the recipient of the 1995 and 2006 Outstanding Paper Award of the Applied Computational Electromagnetic Society Journal and the 2004 Microwave Prize.

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Magnetic-Type Dyadic Green’s Functions for a Corrugated Rectangular Metaguide Based on Asymptotic Boundary Conditions Islam A. Eshrah, Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract—The Green’s functions for the magnetic scalar and electric vector potentials are derived for a rectangular waveguide with dielectric-filled corrugations supporting left-hand, as well as right-hand propagation. The derivation of the Green’s functions for this pair of auxiliary potentials is more involved than their electric-type counterparts. An investigation of the divergence of the electric vector potential is performed with emphasis on the waveguide transverse electric modes. As a result of the divergenceless nature of the vector potential for these modes, the scalar potential is shown to vanish within the corrugations, and the rest of the derivation proceeds by employing the spectral representation then transforming back to the spatial domain. The derived expressions are verified by considering some examples involving discontinuities that can be modeled by equivalent magnetic currents in view of the equivalence principle. Comparison with full-wave simulation commercial packages validates the current theory. Index Terms—Auxiliary potentials, corrugated waveguide, Green’s function, metamaterial transmission line.

I. INTRODUCTION HE GREEN’S function method is a well-known classical method used to analyze a variety of canonical structures for which the Green’s function is available either in closedform expressions, infinite series form, or integral form resulting from an inverse Fourier transform procedure. Invoking the surface equivalence principle to solve problems of radiation and/or scattering in regions with known Green’s functions reduces the problem to that of determining the unknown currents introduced on the surface of the object in the background medium. The method of moments (MoM) is subsequently used in the solution of an integral equation in a procedure that is usually much faster in terms of computational time and more manageable in terms of memory storage requirements compared to other full-wave techniques that require the discretization of the whole region. The Green’s function for rectangular waveguides was developed in the eigenfunction expansion form and has since been

T

Manuscript received December 3, 2006; revised January 25, 2007. This work was supported in part by the National Science Foundation under Grant ECS0220218. I. A. Eshrah was with the Department of Electrical Engineering, The University of Mississippi, University, MS 38677 USA. He is now with the Department of Electronics and Communications Engineering, Cairo University, Giza 12261, Egypt (e-mail: [email protected]). A. A. Kishk is with the Department of Electrical Engineering, The University of Mississippi, University, MS 38677 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.2007.897669

used to analyze scattering problems in the waveguide environment [1]. Recently, periodically loaded rectangular waveguides were shown to exhibit backward-wave propagation [2]–[13]. Rectangular waveguides with dielectric-filled corrugations were investigated to show the possibility of their supporting righthand, as well as left-hand propagation. This was done using an equivalent-circuit model approach [6], [7], a modal solution (as an infinite periodic structure) [8], and the full-wave simulation of a unit cell using Ansoft’s High Frequency Structure Simulator HFSS [9] or of a finite section using a commercial package (QuickWave-3D [10]) based on the finite-difference time-domain (FDTD) method. A simplified modal solution was sought in [11] based on the asymptotic corrugation boundary conditions [14]–[16]. In [12] and its extension in [13], the Green’s functions for the electric-type auxiliary potentials (the electric scalar and magnetic vector potentials) were derived with the goal of speeding up the analysis of coaxial-to-waveguide transitions for corrugated waveguides. In those papers, the spectral-domain representation of the Green’s function was adopted to reduce the problem to a 1-D problem in the transverse direction perpendicular to the corrugated surface. The inverse Fourier transform was then used to obtain the spatial-domain expressions upon determining the poles in the spectral parameter complex plane. A polynomial expansion of the dispersion relation was used to determine the poles without the knowledge of an initial guess. Compared to their electric-type counterparts, the magnetic scalar and electric vector potential Green’s functions have a more involved derivation details, as will be shown below. Some important observations on the auxiliary potential representation of waveguide propagation will be discussed in Section II. Based on these observations, the problem will be formulated in Section III, and a solution procedure similar to that described in [13] will be employed. The application of the derived Green’s functions to the analysis of some waveguide discontinuities will be demonstrated in Section IV. In Section V, a comparison between the results obtained from the current theory and those obtained using FDTD will be drawn. Some remarks will be made in Section VI in addition to some discussions about possible extensions to this study. II. ON THE DIVERGENCE OF THE VECTOR POTENTIALS IN WAVEGUIDE PROPAGATION PROBLEMS Before proceeding to the derivation of the metaguide Green’s function, here we will examine the representation of electro-

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magnetic waves in conventional conducting waveguides. The construction of the solution for waveguide propagation problems using the auxiliary potentials was given in [17] by expressing the electric and magnetic fields in terms of the longitudinal components of the magnetic vector potential (for TM modes) and the electric vector potential (for TE modes). This representation was based on a source-free assumption and, thus, one was at liberty to choose the longitudinal potential components. It is interesting to see that the choice of other components of the vector potentials to represent the fields will result in divergenceless potentials, i.e., the contribution of the scalar potential vanishes. In the presence of a source, which is the case for the Green’s function derivation, specific components of the vector potentials will be forcibly employed as dictated by the source polarization. The goal here is to show that the scalar potentials do not contribute to the propagation of the waveguide modes; the modes that occur in the sufficiently narrow corrugations. Without loss of generality, consider the waveguide mode, with being the direction of propagation and the subscripts corresponding to the periodicities in the - and -directions, respectively. The field components may be written as

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Fig. 1. Cross section in the rectangular waveguide with dielectric filled transverse corrugation. (a) Longitudinal view. (b) Transverse view. The cross indicates that the source is in region 1 (the waveguide).

written as follows:

(4) with

(5)

(1)

where and . If generated by an electric source (within or external to the waveguide), the previous fields may be expressed in terms of as the magnetic vector potential

(2) with (3) . It is interesting to see that for this mode, where the electric scalar potential has no contribution in this representation. This conclusion is valid also for TEM wave propagation in a parallel-plate waveguide, which is often mistakenly assumed to have fields derived only from the scalar potential analogous to the static fields. By validating that is divergenceless (solenoidal) in this case, the electric scalar potential should . Thus, vanish by the Lorentz gauge condition: in [13], the fact that the electric scalar potential vanishes within the corrugations is better explained by the solenoidal nature of rather than the narrow corrugation width. For magnetic excitation, the electric vector potential may be used to express the fields in (1) in terms of only, only, or a combination of both. For the latter case, the fields may be

. Again, it can be easily shown that where is divergenceless in this case, which results in the fact that the magnetic scalar potential vanishes for this representation. It is worth mentioning that the previous representations for the fields in terms of the vector potentials are possible because the modes can be also viewed as or modes. III. GREEN’S FUNCTIONS FOR AN INFINITE CORRUGATED WAVEGUIDE A. Problem Formulation As previously mentioned in Section I, the derivation of the Green’s function for the magnetic vector potential and electric scalar potential for the corrugated waveguide was detailed in [13]. For the sake of completeness, the procedure will be briefly described here while deriving the Green’s functions for the magnetic scalar and electric vector potentials. For the corrugated waveguide shown in Fig. 1, the corrugation width and period are assumed to be sufficiently small (typically . Under these assumptions, the asymptotic corrugation boundary [11], [14], viz. conditions apply at (6a) (6b) (6c) where regions 1 and 2 correspond to the waveguide and corrugations, respectively.

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Using the relation between the fields and the auxiliary potentials, the boundary conditions may be translated to the potentials. The boundary conditions may be further simplified based on the conclusions reached in Section II, namely, that within the since each corrugation can be corrugations, viewed as a short-circuited dielectric-filled waveguide section. , the boundary conditions Thus, on the corrugated surface are

which result in (11) Notice that the derivatives with respect to vanish inside the corrugation. Thus, the nonvanishing components of the Green’s function satisfy Helmholtz differential equation

(7a) (12)

(7b) (7c) (7d) where where only

which can be solved using the partial eigenfunction expansion method as

. Within the corrugations modes are supported, we have

(13) where the eigenfunctions may be obtained by imposing the peras fect electric conductor boundary conditions at (8)

The conditions in (7) and (8) will be mapped to the corresponding Green’s functions in addition to the perfect electric and to conductor boundary conditions at yield the Green’s functions expressions.

(14) and the 1-D characteristic function, subject to the perfect electric conductor boundary conditions at , is found to be (15)

B. Magnetic Scalar Potential Since the scalar potential vanishes within the corrugations, then as far as the potential in the waveguide region is concerned, the corrugated surface is the same as a perfect electric conducting surface. Thus, the homogeneous Neumann boundary conditions apply as given in the boundary condition (7d). Using the partial eigenfunction expansion method [1], it can be shown that the magnetic scalar potential Green’s function for the corrugated waveguide is given by

where . The artificial -dependence introis due to the fact duced in the unknown amplitude function that the field within any individual corrugation is function of its location with respect to the source in region 1. Taking the Fourier transform of (13), the spectral representation of the corrugation Green’s function may be found as

(16) with (17) (9) where and is the Green’s function of is the Kronecker region due to a source in region , and delta function.

2) Waveguide Region: For region 1, the spectral representation of the Green’s function may be used to write the following differential equations:

(18)

C. Electric Vector Potential 1) Corrugated Region: From (8), the dyadic Green’s function should satisfy the following inside the corrugation:

. Similar to the procedure used for region where 2, the partial eigenfunction expansion can be used to obtain expressions for the spectral Green’s functions as

(10)

(19)

ESHRAH AND KISHK: MAGNETIC-TYPE DYADIC GREEN’S FUNCTIONS FOR CORRUGATED RECTANGULAR METAGUIDE

Substituting in (18) using the expansion in (19), it can be satisfies shown that the 1-D characteristic function

where the characteristic function are defined by function

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and the “transformer”

(20)

where written in the form

(26)

. The solution of (20) may be

Substituting from (24) and (25) in (21) and (16), along with the use of (22) yields the final expression for the spectral representation of the Green’s functions. (21)

are obtained in terms of via the use where the constants of the perfect electric conductor boundary conditions at as (22)

D. Spatial-Domain Green’s Functions Upon determining the poles of the spectral-domain representation, the inverse Fourier transform is applied to obtain the spatial-domain expressions [13]. It is worth mentioning that the residue integrals involve removable singularities, but no branch cuts. This corresponds to propagation in an equivalently closed structure with no direct radiation. The different components of the waveguide Green’s dyadic are thus given by

3) Corrugation Boundary Conditions: Assuming that , the boundary conditions at given by (7) are mapped to the Green’s dyadics as (23a) (23b) (23c) (23d)

(27)

Transforming the boundary conditions in (23) to the spectral domain and applying them yields the following expressions for the constants after lengthy yet straightforward algebraic manipulations:

(28)

(24) and

(25)

(29)

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(35)

(30) (36)

(37) (31)

(38)

(32) where and characteristic function The lower half of

is the th zero of the for a fixed value of . is related to the upper half by . Similarly, the following expressions may be obtained for the corrugation Green’s function components:

where . The factor that appears in the metaguide mode terms can be regarded as the equivalent height of the corrugated waveguide. This height ranges from to and from to for real and imaginary values of , to respectively. It can be easily verified that by setting in the series corresponding to the metaguide modes in (27)–(32), the diagonal tensor, which characterizes the perfectly conducting waveguide, will be obtained. The corrugation Green’s function components will also vanish. The Green’s functions for a semi-infinite waveguide or a cavity can be obtained using similar modifications to those in [13]. IV. APPLICATION OF THE DERIVED GREEN’S FUNCTION A. Series T-Junction

(33)

(34)

To verify the derived Green’s functions, the simple example of a series or -plane T-junction is considered. In the geometry depicted in Fig. 2(a), the dominant mode of the narrow feed waveguide (denoted as region 3) is incident at port 1. The diand for the feed and branch wavegmensions are uides, respectively. Invoking the surface equivalence principle [17], the problem may be solved by considering the equivalent problem where the aperture is shorted out, and unknown magnetic currents are introduced on both sides of the surface . For simplicity, let the magnetic current introduced on the feed waveguide side be expressed as follows: (39)

ESHRAH AND KISHK: MAGNETIC-TYPE DYADIC GREEN’S FUNCTIONS FOR CORRUGATED RECTANGULAR METAGUIDE

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is the amplitude of the incident magnetic field and . Thus, the reflection coefficient at port 1 with the reference plane at may be evaluated using

where

(44)

B. Waveguide Transition Fig. 2. Geometry of the problems used to verify the derived Green’s functions. (a) T-junction. (b) Waveguide transition. The metaguide is infinite (matched) in the T-junction problem.

The previous 1-D one-term approximation gives accurate results for sufficiently small . The mixed-potential formulation may be used to determine the scattered magnetic field due to a surface magnetic current in a region having and as Green’s functions for the electric vector and magnetic scalar potentials, respectively, as (40) where

The transition from a conventional waveguide to the corrugated waveguide is another interesting example that can be solved using the derived Green’s functions. For the geometry shown in Fig. 2(b), both waveguides, i.e., the corrugated and noncorrugated, have cross-sectional dimensions . The corrugated section has a length of . The noncorrugated waveguides are filled with a dielectric having permittivity of . In view of the equivalence principle, the cavity and the short-circuit waveguide Green’s functions will be used to analyze the corrugated section and the noncorrugated waveguide ports, respectively. The modifications to the derived Green’s functions to obtain its cavity counterpart can be found in [13]. Let the equivalent magnetic currents introduced on the waveguide be approximated as a finite weighted sum sides of of the waveguide eigenfunctions with only the fundamental mode in the -direction, viz.

(41) Enforcing the continuity of the tangential magnetic field on the aperture yields the following equation: (42) where is the incident dominant mode magnetic field, is the dominant mode magnetic field reflected from the shorted is the scattered magnetic field in region 3 due to surface , , and is the scattered magnetic field in waveguide 1 due to (the continuity of the tangential electric field). In substituting for the magnetic fields using the relations in (40) and and will be the semi-infinite conventional (41) into (42), and will be waveguide Green’s functions, while and derived in Section III. It is worth mentioning that the equivalent source in (39) does not excite waveguide modes; only metaguide modes contribute to the propagation in region 1. This means that this source cannot see the corrugated surface as a perfect conducting surface. To determine the scattered fields, the boundary condition in to yield (42) is solved for

(43)

(45) The previous approximation is justified in view of the orthogonality of the waveguide eigenfunctions and the fact that the waveguides have the same width . Thus, the incident dominant mode is not expected to excite modes with higher order in the -direction. The continuity of the tangential magnetic field on results in two coupled integral equations in . A Galerkin procedure is subsequently used to determine the unknown magnetic current coefficients from which the scattering parameters may be computed using

(46)

V. RESULTS The results obtained from the current theory for the examples in Section IV are compared to those obtained using QuickWave-3D [10], where every corrugation was modeled separately and the mesh was made fine enough to capture the variations in the structure geometry. For the following results, the corrugated mm, waveguide has the following parameters: mm, mm, mm, and . The noncorrugated waveguides are filled with a to support propagating dielectric having permittivity waves all over the frequency band where the corrugated waveguide supports left- and right-hand propagation.

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Fig. 3. Reflection coefficient for the T-junction of Fig. 2(a).

Fig. 3 depicts the reflection coefficient for the T-junction mm. problem of Fig. 2(a) where the feed waveguide has The oscillations in the results of the FDTD simulator are due to the imperfect absorption of the absorbing boundaries at the ends of the corrugated waveguide. At the frequency around (the input matching condition since the which is equal to the magnitude of the tangential electric field), the real part and the overall magnitude of the reflection coefficient decrease. If the previous condition is satisfied at a low value of , which is the case here, then it can be simplified into

(47)

The previous relation is expected since the junction is a series discontinuity and matching occurs when the characteristic impedance of the feed waveguide is equal to double that of the branch (corrugated) waveguide. Notice that the waveguide characteristic impedance is proportional to the product of the waveguide width and intrinsic impedance in view of the powervoltage definition. The fraction on the right-hand side of (47) is the characteristic impedance of the corrugated waveguide dominant mode, as given in [8]. The scattering parameters for the waveguide transition of Fig. 2(b) are given in Fig. 4. The corrugated section has a length of 21 mm. The results were obtained using a one-term approximation for the magnetic currents, which suggests that the interaction with the high-order modes is not significant in this case. The absence of the absorbing boundary conditions in this case yields excellent agreement between both results. The points of total transmission may be also identified by requiring . By considering only the that mode of the corrugated waveguide in the expression of under the one-term current expansion, it can be easily shown

Fig. 4. Scattering parameters of the waveguide transition of Fig. 2(b). (a) jS (b) jS j.

j

.

that the total transmission occurs when the effective characteristic impedance of the corrugated section is equal to the characteristic impedance of the port waveguides. VI. CONCLUSION Beginning with some observations on the representation of waveguide modes using the auxiliary potentials, the magnetic-type Green’s functions were derived for a rectangular waveguide with dielectric-filled corrugations. Some examples for waveguide discontinuities were considered to verify the obtained expressions. With the availability of the magnetic-type Green’s functions, in addition to their electric-type counterparts, many interesting applications for this guided-wave structure may be explored.

ESHRAH AND KISHK: MAGNETIC-TYPE DYADIC GREEN’S FUNCTIONS FOR CORRUGATED RECTANGULAR METAGUIDE

Waveguide slot arrays, filters, cavity resonators, and novel junctions may be investigated. With the proper engineering of the dispersion characteristics of this waveguide, improved designs for conventional waveguide devices are investigated to achieve better performance or more compact size. REFERENCES [1] G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics: An Introduction. New York: Springer-Verlag, 2001. [2] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett., pp. 183 901–183 904, Oct. 2002. [3] S. Hrabar, J. Bartolic, and Z. Sipus, “Waveguide miniaturization using uniaxial negative permeability metamaterial,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 110–119, Jan. 2005. [4] T. Ueda and M. Tsutsumi, “Left-handed transmission characteristics of rectangular waveguides periodically loaded with ferrite,” IEEE Trans. Magn., vol. 41, no. 10, pp. 2532–2537, Oct. 2005. [5] J. Esteban, C. Camacho-Penalosa, J. E. Page, T. M. Martin-Guerrero, and E. Marquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes-theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1506–1514, Apr. 2005. [6] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Evanescent rectangular waveguide with corrugated walls: A composite right/ left-handed metaguide,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 12–17, 2005, pp. 1745–1748. [7] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Rectangular waveguide with dielectric-filled corrugations supporting backward waves,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3298–3304, Nov. 2005. [8] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Spectral analysis of left-handed rectangular waveguides with dielectric-filled corrugations,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3673–3683, Nov. 2005. [9] HFSS: High Frequency Structure Simulator Based on the Finite Element Method. ver. 9.2.1, Ansoft Corporation, Pittsburgh, PA, 2004. [10] QuickWave-3D: A General Purpose Electromagnetic Simulator Based on Conformal Finite-Difference Time-Domain Method. ver. 2.2, QWED Sp. Z o.o, Warsaw, Poland, 1998. [11] I. A. Eshrah and A. A. Kishk, “Analysis of left-handed rectangular waveguide with dielectric-filled corrugations using the asymptotic corrugation boundary condition,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 153, no. 3, pp. 221–225, Jun. 2006. [12] I. A. Eshrah and A. A. Kishk, “Dyadic Green’s function for a right/lefthanded rectangular waveguide,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 11–16, 2006, pp. 1659–1662. [13] I. A. Eshrah and A. A. Kishk, “Electric-type dyadic Green’s functions for a corrugated rectangular metaguide based on asymptotic boundary conditions,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 355–363, Feb. 2007.

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˘ us, “Asymptotic boundary condi[14] P.-S. Kildal, A. A. Kishk, and Z. Sip˘ tions for strip-loaded and corrugated surfaces,” Microw. Opt. Technol. Lett., vol. 14, pp. 99–101, 1997. [15] A. A. Kishk, P.-S. Kildal, A. Monorchio, and G. Manara, “Asymptotic boundary condition for corrugated surfaces and its application to scattering from circular cylinders with dielectric filled corrugations,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 145, no. 1, pp. 116–122, Feb. 1998. [16] A. A. Kishk, “Electromagnetic scattering from transversely corrugated cylindrical structures using the asymptotic boundary conditions,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 3104–3108, Nov. 2004. [17] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. Islam A. Eshrah (S’00–M’06) was born in Cairo, Egypt, in 1977. He received the B.Sc. and M.Sc. degrees in electronics and communications engineering from Cairo University, Cairo, Egypt, in 2000 and 2002, respectively, and the Ph.D. degree in electrical engineering from The University of Mississippi, University, in 2005. He is currently an Assistant Professor with the Department of Electronics and Communications Engineering, Cairo University, Giza, Egypt. Dr. Eshrah was the recipient of the Young Scientist Award presented at the 2004 URSI International Symposium on Electromagnetics, the Young Scientist Award presented at the 2005 URSI General Assembly, the Best Student Paper Award presented at the 2005 Antenna Applications Symposium, and the Raj Mittra Junior Researcher Award presented at the 2005 IEEE Antennas and Propagation Society Symposium. He was also a finalist in numerous student paper contests.

Ahmed A. Kishk (S’84–M’86–SM’90–F’98) received the B.Sc. degree in electronics and communications engineering from Cairo University, Cairo, Egypt, in 1977, the B.Sc. degree in applied mathematics from Ain Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1983 and 1986, respectively. In 1981, he joined the Department of Electrical Engineering, University of Manitoba. He is currently a Professor with The University of Mississippi, University. He has authored or coauthored over 170 refereed journal papers and book chapters. He coauthored Microwave Horns and Feeds (IEE Press, 1994; IEEE Press, 1994). He has also authored several book chapters. He was an Editor-in-Chief of the ACES Journal from 1998 to 2001. Dr. Kishk is an editor of the IEEE Antennas and Propagation Magazine. He was the recipient of the 1995 and 2006 Outstanding Paper Award of the Applied Computational Electromagnetic Society Journal and the 2004 Microwave Prize.

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Efficient Cartesian-Grid-Based Modeling of Rotationally Symmetric Bodies Dzmitry M. Shyroki

Abstract—Axially symmetric waveguides, resonators, and scatterers of arbitrary cross section and anisotropy in the cross section can be modeled rigorously with use of 2-D Cartesian-grid-based codes by means of mere redefinition of material permittivity and permeability profiles. The method is illustrated by the frequencydomain simulations of resonant modes in a circular-cylinder cavity with perfectly conducting walls, a shielded uniaxial anisotropic dielectric cylinder, and an open dielectric sphere for which, after proper implementation of the perfectly matched layer boundary conditions, the radiation quality factor is also determined. Index Terms—Body of revolution (BOR), coordinate transformation, finite-difference frequency-domain (FDFD) method, Maxwell equations, perfectly matched layer (PML).

I. INTRODUCTION IMENSIONALITY reduction is a natural and elegant way to model electromagnetic resonators, waveguides, and scatterers possessing rotational symmetry. It has been used for nearly half a century with integral-equation methods [1], , as well as in the later finite-difference time-domain and finite-element studies—from modeling guided waves in particle accelerators [2, Ch. 12] to ultrashort pulse focusing [3], scattering from axially symmetric targets [4], and cavity mode analysis [5]. The efficiency of these techniques, as compared to straightforward 3-D finite-difference modeling, is pronounced at no accuracy sacrificed. Furthermore, with a “2.5-D” computation workload, the cylindric-grid-based simulators are able to treat systems exhibiting perturbed or discrete rotational symmetry. From the practical view, however, there is one dissatisfactory issue with all such schemes. A conventional approach to reduce dimensionality for rotationally symmetric problems is by performing differentiation with respect to the azimuthal coordinate analytically after casting the Maxwell equations in anholonomic cylindric frame—locally Cartesian—with vector differential operators modified accordingly [6]. The dimensions of the electromagnetic field components remain unchanged under such transformation, but the resulting form of the equations, as opposed to their Cartesian version, is modified. This obviates the need for a separate program implementation of the algorithms based on the Maxwell equations in anholonomic cylindric coordinate system.

D

Manuscript received November 12, 2006; revised March 13, 2007. The author is with the Department of Communications, Optics, and Materials, Technical University of Denmark, 2800 Lyngby, Denmark (e-mail: ds@com. dtu.dk). Digital Object Identifier 10.1109/TMTT.2007.897841

In this paper, a simple way to retain Cartesian-like Maxwell equations and, hence, to use standard Cartesian-grid-based algorithms while actually doing the finite-difference body-of-revolution modeling in cylindrical, spherical, or other rotationally symmetric coordinates, is proposed. We use the generally covariant formulation of Maxwell equations as was propelled, in particular, by Schouten [7] and Post [8] years ago; hence, our scheme involves appropriate transformation of the dielectric permittivity and magnetic permeability together with, where necessary, the electric and magnetic field components instead of modifying the core differential, or finite-difference, equations. It is interesting to note that, in spite of its obvious appeal, this is still a marginal approach in computational electrodynamics, utilized in only a few publications thus far. Thus, Ward and Pendry [9] realized the power of covariant formulations permitting a single “code to handle any coordinate system by adjusting the and ” in 1996, and implemented this idea in their nonorthogonal logically rectangular finite-difference time-domain code [10], [11]. Covariant 3-D Maxwell equations were used in a recent study [12] on the finite-element modeling of twisted optical fibers in nonorthogonal helical coordinates. Much earlier, in [13] and a series of subsequent papers, covariant transformations were employed to analyze diffraction gratings of complicated profiles in a coordinate system adjusted to the dielectric discontinuity surface. Very recently [14], Maxwell covariance was used for mapping the infinite free space onto the finite computational domain in order to avoid the notorious problem of mesh truncation in the finite-difference computations. In the following, we derive, after briefly recalling the generally covariant formulation of Maxwell equations, the exact equivalent permittivity and permeability profiles for rotationally symmetric body of arbitrary cross section and anisotropy in the cross section, and for the anisotropic perfectly matched layer (PML) medium used customarily in the finite-difference simulations of radiation and scattering losses. Afterwards, three validating numeric examples are provided via the Yee-mesh-based full-vector 2-D finite-difference frequency-domain calculations of resonant modes. The first example is a canonic circular-cylinder cavity with perfect electric conducting (PEC) walls for which an analytic solution is well known and the finite-difference treatment is not burdened by, e.g., material discontinuities or complicated boundary conditions, thus allowing to focus on how adequate for numeric modeling the proposed equivalent profile method itself is. The second example is a PEC-shielded uniaxially anisotropic dielectric resonator analyzed in [15] with the mode-matching method, and later modeled in [16] with the

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SHYROKI: EFFICIENT CARTESIAN-GRID-BASED MODELING OF ROTATIONALLY SYMMETRIC BODIES

finite-difference tool; with this example, we demonstrate the efficiency of a polarization-sensitive dielectric index averaging at discontinuities on a rectangular finite-difference grid. The third example is an open dielectric sphere, upon which resonant frequencies and -factors can be calculated analytically [17], [18] and were modeled with the finite-element method in cylindric coordinates [5]; several finite-difference grids to represent the cross section of the sphere are compared and the performance of cylindrical and spherical PMLs to simulate radiation losses is considered.

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Fig. 1. Cartesian and rotationally symmetric coordinate systems for the given body of revolution. The x-axis coincides with rotational symmetry axis of the body; the  = 0 plane corresponds to z = 0, y 0 in Cartesian coordinates.



II. FORMULATION A. Covariant Maxwell Equations Here we give a covariant formulation of Maxwell equations in 3-D form most resembling the customary 3-D Cartesian representation. We start from a system [7], [8] (1) (2) Here, the square brackets denote alternation and , and are the covariant electric vector and magnetic bivector, respectively, coinciding and in a with the electric and magnetic three-vectors and are the contravariant Cartesian frame, magnetic bivector density and the electric vector density of weight 1 corresponding to and , and are the electric . With current and charge densities, and , , and such transformation characteristics assigned to the electromagnetic field quantities, (1) and (2) are known to be form invariant [7], [8], i.e., they do not change their form under arbitrary reversible coordinate transformations. To convert (1) and (2) to the form directly reminiscent of the Maxwell equations in a Cartesian frame, we use the dual equivalents and , where and are pseudopermutation (hence, tildes) fields equal to the Levi–Civita symbol in any coordinate system

where

is the Jacobian transformation matrix, and (6)

In this formulation, geometry enters the Maxwell equations (3) and (4) through material fields and exclusively, while the form of (3) and (4) is precisely as if they were written in Cartesian components. If, in some chosen coordinate system , and happen to be independent of one of the coordinates, say, , that coordinate can be separated in the and usual manner with viewed as the permittivity and permeability profiles of the equivalent dimensionality reduced system in a Cartesian frame. Of course, modeling of rotationally symmetric bodies is only a subset of the possible applications of this technique. B. Equivalent Body-of-Revolution Profiles Depending on the shape of the given body of revolution and its anisotropy, one of many rotationally symmetric coordinate systems can be found advantageous to use. The transformation from orthogonal Cartesian to general rotationally symmetric coordinates, as shown in Fig. 1, is given by [19] (7)

(3)

, and are the coordinate lines in where half-plane—either orthogonal or nonorthogthe onal, analytically defined, or automatically generated boundaryfitted ones. The corresponding transformation matrix for contravariant components reads

(4)

(8)

while the constitutive relations are implied being , . We leave optical activity, nonlocality, and nonlinearity out of scope here, though these phenomena can be treated on a similar ground insofar as the governing Maxwell field [see (1) and (2)] and the accompanying constitutive relations adhere to the principle of general covariance. The transformation be, , , and stipulates that the permittivity havior of and permeability are contravariant tensor densities of weight 1 so that they are transformed as (5)

Before substituting this in (5) and (6), we factor out the depen, where denotes transposition, dence is the matrix of rotation around , and

(9) . For a body of arbitrary cross section is the Jacobian at and anisotropy in the cross section, if only invariant under rota, where tions around the axis, i.e.,

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, and similarly for identities

, one finally gets after ap-

(10) (11)

the plus sign before the imaginary parts implies the time dependence of the fields in the frequency domain. The equivalent permittivity (12) further transformed with within the PML regions becomes (17)

Since such transformed permittivity and permeability do not depend on the coordinate, the standard separation of variables in factored out. At the same time, (3) and (4) is in order with it should be emphasized that the form of the Maxwell equations (3) and (4) remains exactly Cartesian like, whatever the and coordinates and the associated “physical” grid in the body cross section is preferred to be employed. In the specific case of cylindric coordinates, one has and , hence, in the above formulas. For the diagonal-anisotropic media, i.e., if , as measured in a locally Cartesian reference frame, one thus obtains (12) and, similarly, for . In the case of a spherical coordinate system (we use it alongside with the cylindric and one in Section III-C) with , one gets for the equivalent profiles of isotropic media (13) and similarly for . The equivalent permittivity profile for the is now diagonal-anisotropic media represented by a nondiagonal matrix. C. PMLs The primary way to simulate radiation losses with the finite-difference or finite-element tools is by introducing the PML boundaries [2, Ch. 7]. We construct the “uniaxial” PMLs below by appropriate complex coordinate stretching [20], [21]. For cylindrical PMLs, for example, we employ the scaling from to via (14)

with the complex parameter

often chosen as for

and the similar scaling

(15)

with for

(16) is the dimensionless frequency-dependent maxwhere and are the PML imum PML conductivity, and are the PML widths in the - and -diinterfaces, rections, is the conductivity polynomial profile order, and

and, similarly, for . For the spherical PMLs imposed on the equivalent material profiles (13), one gets (18) with and defined similarly to (14) and (15). Extensions to the more complicated cases are clear. III. VALIDATION For the three numeric examples here, we specify the equivalent profiles of the resonators according to (12) or (13) and discretize them on a 2-D projected Yee grid with appropriate boundary conditions. The discretized version of Maxwell equations (3) and (4), or rather the reduced eigenproblem in the or , is then frequency squared and in solved iterationally via ARPACK [22] based eigs function in MATLAB.1 Although the time-domain simulations are certainly possible as well, the frequency domain is naturally preferred in finding resonant modes here. The choice for the finite-difference, as opposed to the integral-equation methods, is dictated by the fact that even for the homogeneous bodies, the equivalent profiles are continuously inhomogeneous in the radial direction; and this choice is supported, on the physical part, by the high efficiency of the finite-different solvers in dealing with the often complicated microstructured profiles of the current and the prospective real-world devices. A. Empty Cylinder Cavity With PEC Walls This canonic example [23], unobscured by any complexities such as nontrivial boundary conditions or material profile discontinuities, permits to scrutinize the core of the method—the equivalent-profile formulation itself, as implemented for the finite-difference modeling. We chose a cylinder with the height-to-radius ratio equal to the golden ratio and simulated, on the domain, several of the lowest resonant modes with precalculated frequencies (for the mode), 4.2956 , 5.6740 , 2.4048 , 3.0908 , and 5.4904 , all in the units. Convergence curves for these modes are plotted in Fig. 2, where all the modes show perfect second-order behavior, while the grid resolution increases by more than an order of magnitude. The higher order modes are found with systematically lower accuracy at a given resolution because of their more complicated oscillatory electric and magnetic field profiles (see Fig. 3). Overall, such convergence behavior, together with low absolute values for the errors, is very similar to the results obtained for a straight rectangular resonator with the same finite-difference code after 1[Online].

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SHYROKI: EFFICIENT CARTESIAN-GRID-BASED MODELING OF ROTATIONALLY SYMMETRIC BODIES

Fig. 2. Convergence of the eigenfrequencies with resolution for some of the lowest TE and TM modes in the air-filled circular-cylinder cavity with perfectly conducting walls.

Fig. 3. Axial half cross sections of the electric and magnetic fields, =2 shifted and TM modes in the same resin  relative to each other, of the TE onator as in Fig. 2.

we put and , indicating that the equivalent-profile technique itself is not detrimental to the finite-difference modeling accuracy. B. Shielded Anisotropic Dielectric Resonator In this example, we benchmark our method against the rigorous mode-matching results of [15] for the resonant modes in a PEC-shielded uniaxially anisotropic dielectric cylinder resonator, reproduced with good accuracy in [16] employing, in anholonomic cylindric coordinates, the finite difference with simultaneous Chebyshev acceleration method. The diameter and height of the cavity shield are 15.6 and 13.0 mm, respectively, the diameter and the height of the sapphire cylinder centered in the cavity are 10.0 mm each, and its permittivity so that the anisotropy axis points along the axial or -direction. The dielectric supports of were also included as in [15] and [16]. The conand vergence graphs for the two lowest hybrid modes

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Fig. 4. Convergence for the EH -mode resonant frequency f to its exact value f = 8:827 GHz in the PEC-shielded uniaxially anisotropic high-indexcontrast dielectric resonator as in [15] and [16].

Fig. 5. Convergence for the HE -mode resonant frequency f to its exact value f = 9:121 GHz in the same PEC-shielded uniaxially anisotropic resonator as in Fig. 4.

are plotted in Figs. 4 and 5; the electric and magnetic field profiles of these modes shown in Fig. 6 can be compared to Figs. 3 and 4 in [16]. A common problem with the finite-difference grids, including the rectangular staggered Yee-type grid we use here, is how to represent material discontinuities running along the curved surfaces and curved lines on that grid. Defining the permittivity in a grid cell containing discontinuity as that at the cell center—the staircasing—appears too rough in many cases [24]. Two options follow: either and should be averaged carefully over the grid cells crossed by the discontinuities, or adaptive coordinates should be chosen such that the coordinate surfaces follow the material interfaces to avoid the very necessity for the averaging. Now we consider the permittivity averaging approach, leaving the adaptive coordinate transformations until Section III-C. In Figs. 4 and 5, one can compare the results obtained with of the the volume-weighted averaging

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Fig. 6. Axial half cross sections (=2 shifted in ) of electric and magnetic and HE modes for the same resonator as in Figs. 4 and fields of the EH 5, calculated with a resolution of 80 grid cells per each dimension.

mode for open dielectric sphere of permitFig. 7. Localization of the TE tivity  = 36 and radius a = 1:56 mm, as in [18], in the frequency (f ) versus quality factor (Q) coordinates.

permittivity, as proposed in [25], and supported in the authoritative book [2, Sec. 10.6.3] against the polarization-dependent homogenization [26, Sec. 14.5.2]

(19) for grid cells small enough to assume the discontinuity surface is the plane within each cell, being the unit vector normal and to that plane; the so-called Wiener limits are . (In fact, a more general formula [27] to account for material anisotropy is in order. Furthermore, the off-diagonal components of (19) were ignored in our computations. These details, however, are not pertinent to this discussion.) The systematic underestimation of the eigenfrequencies calculated with volume-weighted averaging, like that observed in [28]–[30] for the TM modes in 2-D photonic crystals or in [31, Sec. 3.3.1] for , the hybrid modes in microstructured fibers, is due to which can be checked via

(20) valid for any nonnegative function . The inadequacy of the simplistic volume-weighted averaging is especially well seen for the modes with an electric field vector directed predomimode nantly normally to material interfaces, as for the in our case. Taking together with into account in the finite-difference computations, as prescribed by (19), pointed out in [28] (see also [29]), and implemented in a widely used planewave modeling package MPB [32], improves the accuracy at no serious computation cost. C. Open Dielectric Sphere The resonant frequencies (free oscillations) and the radiation losses of an isolated sphere can be calculated by employing cylindric functions in the known manner [17], [18], [33], providing an etalon to test our equivalent-profile finite-difference

(a)

(b)

(c)

Fig. 8. Three different “physical” grids that we use. (a) Rectangular. (b) Polar. (c) Polar radially transformed to better represent the dielectric discontinuity.

method with the domain boundaries truncated by PMLs. Consider, for example, the TE modes that correspond to the (necessarily complex) zeros of (21)

and are the first Bessel and the second-kind where Hankel functions of the order and the complex wavenumber . We simulated the mode for the sphere of the dielectric permittivity and mm, as in [18]. It must be mentioned that the radius this mode, and not only this one, was inaccurately located nu-plane. In Fig. 7, a logarithm of merically in [18] in the is plotted in the vicinity of the -mode minimum. The location of the minimum found with the FindMinimum funcGHz and , tion in Mathematica2 gives the values supported also by brute-force 3-D time-domain simulations [34]. The order-of-magnitude difference in the slopes along the and coordinates justifies lower accuracy in determining the . The use of a rectangular grid, as in Fig. 8(a), together with the polarization-sensitive averaging of permittivity at the surface of dielectric discontinuity produce pretty good second-order convergence of the simulated results for both and with the errors being an order of magnitude higher at a fixed resolution 2[Online].

Available: www.wolfram.com

SHYROKI: EFFICIENT CARTESIAN-GRID-BASED MODELING OF ROTATIONALLY SYMMETRIC BODIES

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Fig. 9. Convergence for f and Q of the TE mode calculated on the 6 6 mm domain. The curves in each family are obtained with varying the PML parameters; polarization-dependent averaging of permittivity is applied.

Fig. 11. Convergence for f and Q (compared with Fig. 9) calculated on physically polar grids—unmodified and radially modified, as in Fig. 8, extending for 6 mm radially. No permittivity averaging applied.

Fig. 10. Sensitivity of f and Q to the computation domain width and height (increasing proportionally) for the resolutions of six (less accurate results in both f and Q), ten, and (best accurate) 14 grid cells per 1 mm.

Fig. 12. Sensitivity of f and Q to the domain radius at the same radial resolutions as in Fig. 10. The angular resolution is 30 grid cells per =2 for all the curves.

(Fig. 9). Simulations were done on the 6 6 mm domain representing a quarter of the sphere, with the PEC and perfect magnetic conducting (PMC) boundary conditions on the axis and in plane, respectively, according to the mode symmetry. the The 1-mm-wide PMLs were placed within the domain at the two outer boundaries, and a quadratic conductivity profile was specified with the maximum conductivity corresponding to the . Changing continuous-space reflection coefficient , the width of the PMLs the reflection coefficient to to mm, and the polynomial order to does not alter the overall convergence behavior for , as seen from this same figure, and exposes virtually no effect on the values. Fig. 10 indicates that, in coarse grid simulations, the values are sensitive to the domain width, exhibiting a slowly converging oscillatory behavior. Not so for sufficiently fine grids, however; in that case, one can place PMLs quite close to the simulated object. A natural alternative to calculations on the physically rectangular grid in this case is to use a polar grid, as in Fig. 8(b), with the equivalent permittivity and permeability profiles as given by

(13); or polar modified as in Fig. 8(c) to increase the density of grid nodes near dielectric discontinuity, e.g., via (22) with equivalent profiles modified accordingly. Our results for the and convergence and sensitivity to the domain radius and (Figs. 11 and 12) were obtained with when the radial transformation was applied. Noteworthy is that we neither performed the permittivity averaging, nor cared to place the discretized coordinate surfaces somehow specifically with respect to material boundaries, but the accuracy achieved on the modified polar grid is comparable to that in Figs. 9 and 10. Our point, however, is demonstrating not the superiority of adaptive meshing here, but the ease of swapping between different “physical” grids, while using one and the same “logically Cartesian” solver with different equivalent permittivity and permeability profiles.

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IV. CONCLUSION The equivalent-profile expressions (10) and (11) for an anisotropic inhomogeneous body of revolution have been derived for further use with the logically Cartesian-grid-based solvers. The following three reasons make our approach appeal. 1) Its computational efficiency is comparable to that of other body-of-revolution techniques which make use of rotational symmetry of the structure by casting Maxwell equations in anholonomic cylindric coordinates, but an arbitrary boundary-fitted grid in the body cross section is acceptable with our approach. 2) Its ease of program implementation is unsurpassed: actually, no or virtually no modifications to the widely available time- or frequency-domain Cartesian-grid-based Maxwell solvers are required, except the adjustments to the domain boundary conditions where necessary. 3) Its conceptual underpinning—the general covariance of Maxwell equations—has apparent applications beyond the case of rotational symmetry considered here. For example, homogeneous twists can be treated in a similar way by the separation of variables in the nonorthogonal helical coordinates. REFERENCES [1] M. G. Andreasen, “Scattering from bodies of revolution,” IEEE Trans. Antennas Propag., vol. AP-13, no. 3, pp. 303–310, Mar. 1965. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000. [3] D. B. Davidson and R. W. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 11, pp. 1471–1490, 1994. [4] A. D. Greenwood and J.-M. Jin, “A novel efficient algorithm for scattering from a complex BOR using mixed finite elements and cylindrical PML,” IEEE Trans. Antennas Propag., vol. 47, no. 4, pp. 620–629, Apr. 1999. [5] O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gen. Comput. Syst., vol. 21, pp. 1263–1274, 2005. [6] D. V. Redˇzic´ , “The operator in orthogonal curvilinear coordinates,” Eur. J. Phys., vol. 22, pp. 595–599, 2001. [7] J. A. Schouten, Tensor Analysis for Physicists. Oxford, U.K.: Clarendon, 1951, (reprinted by Dover, 1989). [8] E. J. Post, Formal Structure of Electromagnetics. Amsterdam, The Netherlands: North-Holland, 1962, (reprinted by Dover, 1997). [9] A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Modern Opt., vol. 43, pp. 773–793, 1996. [10] A. J. Ward and J. B. Pendry, “Calculating photonics Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B, Condens. Matter, vol. 58, pp. 7252–7259, 1998. [11] A. J. Ward and J. B. Pendry, “A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using non-orthogonal FDTD method,” Comput. Phys. Commun., vol. 128, pp. 590–621, 2000. [12] A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys., vol. 28, pp. 153–157, 2004. [13] J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt., vol. 11, pp. 235–241, 1980.

r

[14] D. M. Shyroki, “Squeezing of open boundaries by Maxwell-consistent real coordinate transformation,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 11, pp. 576–578, Nov. 2006. [15] Y. Kobayashi and T. Senju, “Resonant modes in shielded uniaxial anisotropic dielectric rod resonators,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 2198–2205, Oct. 1993. [16] J.-M. Guan and C.-C. Su, “Resonant frequencies and field distributions for the shielded uniaxially anisotropic dielectric resonator by the FD-SIC method,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1767–1777, Oct. 1997. [17] M. Gastine, L. Courtols, and J. L. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microw. Theory Tech., vol. MTT-15, no. 12, pp. 694–700, Dec. 1967. [18] A. Julien and P. Guillon, “Electromagnetic analysis of spherical dielectric shielded resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 6, pp. 723–729, Jun. 1986. [19] E. Madelung, Die Mathematischen Hilfsmittel des Physikers. Berlin, Germany: Springer, 1957. [20] F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett., vol. 7, no. 11, pp. 371–373, Nov. 1997. [21] P. G. Petropoulos, “Reflectionless sponge layers for the numerical solution of Maxwell’s equations in cylindrical and spherical coordinates,” Appl. Numer. Math., vol. 33, pp. 517–524, 2000. [22] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods. Philadelphia, PA: SIAM, 1998. [23] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998, ch. 8. [24] R. Holland, “Pitfalls of straircase meshing,” IEEE Trans. Electromagn. Compat., vol. 35, no. 4, pp. 434–439, Nov. 1993. [25] S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1737–1739, Sep. 1999. [26] M. Born and E. Wolf, Principles of Optics. Oxford, U.K.: Pergamon, 1968. [27] A. V. Lavrinenko and V. V. Zhilko, “Application of coordinate-free effective medium theory to periodically plane-stratified anisotropic media,” Microw. Opt. Technol. Lett., vol. 15, pp. 54–57, 1997. [28] R. D. Meede, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic bandgap materials,” Phys. Rev. B, Condens. Matter, vol. 48, pp. 8434–8437, 1993. [29] R. D. Meede, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Erratum to ‘Accurate theoretical analysis of photonic bandgap materials’,” Phys. Rev. B, Condens. Matter, vol. 55, p. 15 942, 2000. [30] C.-P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,” Opt. Express, vol. 12, pp. 1397–1408, 2004. [31] J. Riishede, “Modelling photonic crystal fibres with the finite difference method,” Ph.D. dissertation, Commun., Opt., Mater., Tech. Univ. Denmark, Lyngby, Denmark, 2005. [32] S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express, vol. 8, pp. 173–190, 2001. [33] H. C. van de Hulst, Light Scattering by Small Particles. New York: Wiley, 1957, sec. 10.5. [34] A. Ivinskaya, Oct. 2006, private communication. Dzmitry M. Shyroki was born in Minsk, Belarus, on September 24, 1981. He received the Specialist degree from Belarusian State University, Minsk, Belarus, in 2004, and is currently working toward the Ph.D. degree at the Technical University of Denmark, Lyngby, Denmark.

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Physical Interpretation and Implications of Similarity Transformations in Coupled Resonator Filter Design Smain Amari, Member, IEEE, and Maged Bekheit, Member, IEEE

Abstract—This paper discusses the physical interpretation and implications of similarity transformations for applications in coupled resonator filter design. It is shown that certain similarity transformations contain important “global” information that can be used to get insight into the workings of microwave bandpass filters. Instead of focusing on the effect a similarity transformation has on the original coupling matrix, the change of basis that the similarity transformation represents is considered the main operthat is designed as a set of coupled ation. For a filter of order resonators, it is shown that the transversal matrix is equivalent to using the global eigenmodes of the entire structure as a basis. The transversal matrix emerges as a universal and natural representation of coupled resonator bandpass filters of arbitrary orders, responses, and topologies. It can be used as an equivalent circuit in the optimization and diagnosis of bandpass coupled resonator filters. The results of this study have far reaching implications not only for the theory and design of microwave filters, but other circuits as well. Index Terms—Eigenmode, resonator filters, similarity transformation, synthesis.

I. INTRODUCTION

M

ICROWAVE coupled resonator bandpass filters find numerous applications in modern communication systems. A crucial step in the design of these components is the determination of the amount of coupling between the resonators, and the ports and the resonators as given by the coupling matrix. Except for certain configurations, such as canonical folded topologies, the final coupling matrix on which the actual design is based is not obtained from a direct extraction technique. Instead, the initial coupling matrix undergoes a series of similarity transformations to annihilate those coupling coefficients that cannot be implemented within the selected technology [1]–[10]. In other words, in all the publications in the extensive literature on this subject, the objective of the transformations, which are viewed as a mathematical tool, is only to force a coupling matrix to fit a realizable topology. Although this interpretation is very fruitful as a synthesis tool, it is only one aspect of a similarity transformation, and arguably not the most important one. A similarity

Manuscript received October 23, 2006; revised March 1, 2007. This work was supported in part by the Natural Science and Engineering Research Council of Canada. S. Amari is with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada K7K 7B4 (e-mail: [email protected]). M. Bekheit is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON, Canada K7K 5B3 (e-mail: bekheitm@ee. queensu.ca). 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.2007.897706

transformation is fundamentally a change of basis or representation. The linear system itself—the microwave filter—does not change under the transformation, it is only our view of it that does. This is analogous to taking two photographs of a person from two different angles. The photographs are different, but the person is still the same. In certain situations, the class of allowable transformations is restricted to those that preserve specific properties such as symmetry. In the case of coupled resonator filters, the transformations are limited to those that preserve the port parameters as function of frequency and in a very specific way. The transformation must not only preserve the port parameters, such as the scattering parameters, but do so by preserving the equations giving those parameters in terms of the state variables as well. For example, if the original model involves node voltages that are connected to ground by unit capacitors in parallel with constant susceptances, the transformation is required to allow the new voltages to be interpreted as node voltages that are also connected to ground by unit capacitors and possibly new constant susceptances. Consequently, the coupling matrix that results from an allowable similarity transformation can always be that of a set of coupled resonators that can be implemented, at least in principle. This is how similarity transformations have been understood and used in the area of coupled resonator filters. The complementary interpretation of a similarity transformation as a different view of the same system before and after the transformation has not been reported or exploited. It is the goal of this paper to examine in details this interpretation. More specifically, we focus on the following points in connection with coupled resonator filters. 1) What does a similarity transformation (rotation) mean or represent? 2) Can a similarity transformation be realized physically? 3) What happens to the resonators under a rotation with ? How does the annihilation of the coupling copivot with pivot really work? efficient 4) Does a unique coupling matrix exist for a given response? What is the topology and what are the resonators of such a coupling matrix? It is important to stress that both before and after the transformation, we are describing the same physical structure and not a different one. II. REVIEW OF SIMILARITY TRANSFORMATIONS We consider a set of coupled bandpass resonators, as shown schematically in Fig. 1. The dark circles are resonators, which are represented by unit capacitors in parallel with frequency-independent susceptances to account for the frequency shifts in the resonant frequencies with respect to the center

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and

excited. The relationship between the form

.. .

Fig. 1. Coupling scheme of pling.

N resonators with source/load-multiresonator cou-

of the passband of the filter. The light circles are the source and load, which are both set to unity . The lines connecting the resonators and the source and load are ). If the nodal admittance inverters (coupling coefficients of size , we voltages are grouped in a vector have [5]

where

is, therefore, of

(4)

.. .

is an

square matrix

.. .

.. .

Here,

is at this point an arbitrary real square matrix of size . It is obvious from the form of the matrix that and . To determine the class of allowed, we proceed as follows. Using the identity matrices , where is the inverse of , we can rewrite (1) as

(1) matrix such that and otherwise, is a diagonal , but and matrix such that is a symmetric real matrix of size , called the normalized coupling matrix. The excitation , of size , is since we assume that only one node is excited (node 1, the source). The normalized frequency is given by the standard low-pass to bandpass transformation Here,

is an

(6) Pre-multiplying by , we get

and using (4) and the fact that

(7) from (5) and the matrices Using the form of the matrix and defined in (1), we can show that and . Therefore, (7) becomes

(2) where is the center frequency of the passband and is the bandwidth of the filter. The response of the circuit at the two ports (input and output) and is fully specified by the voltages at these two nodes, i.e., . For example, the scattering parameters are given by

(5)

(8) Equation (8) can be interpreted as the node equations of another set of coupled resonators with a new coupling matrix . For this to be the coupling matrix of a physical system, it must be real and symmetric. This can be achieved if is orthogonal, i.e., . the transformation matrix in (5) is orthogThis condition also implies that the matrix through the onal. The transformation of the coupling matrix relationship

(3)

(9)

Equation (3) shows that the response of the structure is not changed as long as the voltages at the input and output nodes, and , are not changed and only the input remains i.e., excited and by the same source. In particular, the voltages at the internal nodes can be linearly combined amongst themselves without changing the response. We can, therefore define a new , which is related to the original nodal voltage voltage vector vector by a linear transformation with a matrix representaof size . To determine the subset tion , we enforce the conditions of acceptable matrices and in addition to the corresponding conditions when node , instead of node 1, is

where is an orthogonal matrix is referred to as a similarity transformation or rotation. It is now obvious from (8) that the are connected to ground by unit capacitors new voltages ). It is this and constant susceptances (diagonal elements of feature that is used in the current interpretation of similarity transformation in coupled resonator filters. III. INTERPRETATION In the area of coupled resonator filter theory, a similarity transformation is understood to refer to (9). Successive rotations (Given’s rotations) are applied to an original matrix simply to generate an easier-to-implement coupling matrix with the same

AMARI AND BEKHEIT: PHYSICAL INTERPRETATION AND IMPLICATIONS OF SIMILARITY TRANSFORMATIONS IN COUPLED RESONATOR FILTER DESIGN 1141

Over a narrowband of frequencies around the center of the passband, the response of the structure can be described by a coupling matrix of the form (10)

Fig. 2. Second-order Chebyshev filter as direct coupled resonators. (a) Coupling scheme. (b) -plane cavity realization.

H

response. For example, starting from a potentially full matrix, or a transversal matrix, a sequence of rotations is applied to achieve folded canonical topologies, as well as more sophisticated arrangements [2], [3]. According to this interpretation, the coupling matrix resulting from a sequence of similarity transformations is implemented as a set of coupled, but otherwise physically localized and distinct resonances. In particular, the coupling matrix to which the similarity transformation is applied is completely ignored; all that matters is the final coupling matrix. A similarity transformation as detailed in Section II is more than (9). It is fundamentally a change of basis or representation. The coupling matrix may be viewed as a representation of an abstract linear operator, i.e., the coupling operator. As such, the representation takes different forms depending on which basis is used as reflected by (9). The issue of representation of linear operators is central to many branches of theoretical physics such as quantum mechanics; the reader is referred to [11] and references therein for more details. In this paper, we present a different and complementary view. Instead of the prevailing interpretation, we view (9) only as a consequence of the more fundamental transformation in (4). In other words, the similarity transformation does not necessarily generate a new structure, but rather allows a different representation of the same structure. This view is admittedly more abstract, but has very important and far-reaching consequences. A second-order direct-coupled Chebyshev filter is used as an example to illustrate some of the key points. IV. SECOND-ORDER CAVITY FILTER We consider a second-order Chebyshev filter whose standard coupling and routing scheme is shown in Fig. 2(a). A simple implementation of this coupling scheme based on -plane cavities in WR75 rectangular waveguide technology is shown in resonance is used in each of the Fig. 2(b). The dominant two cavities. The coupling from the ports to the cavities and between the cavities is implemented by the -plane irises of and , respectively. The structhickness and apertures ture is symmetric with respect to its center.

The values of the nonzero entries of this coupling matrix for a given set of specifications are known analytically [12]. For example, for an in-band return loss of 20 dB, we get and . We now perform a similarity transformation to “annihilate” between the two resonators. This the coupling coefficient can be achieved by a similarity transformation with pivot [2], [3] since the diagonal elements of the coupling and angle in matrix are all zero [3]. Under these conditions, the matrix (5) takes the form

(11) Using (11) in (9), we get the new coupling matrix

(12) In existing applications of similarity transformations, the coupling matrix in (12) is implemented in its current form by using a dual-mode resonator, e.g., [13]. In other words, the design takes this transformed coupling matrix as the starting point and tries to find two modes, which are both coupled to the source and the load, but which are not coupled to each other. Here, we are interested in deciphering how the coupling matrix in (12) still describes the in-line Chebyshev filter shown in Fig. 2. In other words, where are the new resonances that resulted from the “an? How are they related to the resonihilation” of nances in the cavities? in (11). InThis information is contained in the matrix deed, the rows of this matrix are simply the projections of the unit vectors of the new basis over the old ones. For example, the second row states that resonance 1 in the new representation involves voltages in the cavities that are equal in magnitude, nor, and in phase. Similarly, the second resonance malized to involves the two cavities with voltages equal in magnitude, nor, but opposite in phase. The coupling matrix in malized to (12) describes the same filter, but with the even and odd modes

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In order to answer this question, we proceed as follows. We assume that the input and output are very weakly coupled to the system. By doing so, we allow the input and output to assess the intrinsic characteristics of the coupled-resonator filter. This condition amounts to assuming that all the coupling coefficients and in the coupling matrix are and . Although very weak, i.e., the coupling coefficients and are not equal in general, for simplicity, we assume that they take the same small value . Under these conditions, the scattering parameters at the two ports become

Fig. 3. Coupling scheme of second-order Chebyshev using the even and odd modes.

H -plane cavity filter

as resonances. The coupling coefficient that was “annihilated” is now given a different role: it affects the resonant frequencies of the even and odd modes. The iris that implements the couin the original coupling matrix is still in place and pling nothing has been physically annihilated from the structure. Another implication of (12) is that the coupling between the is related to the difference between the cavity resonators resonant frequencies of the even and odd modes by

V. WHAT DO THE PORTS SEE? that is described by a Consider a bandpass filter or order coupling matrix of arpotentially full square bitrary topology. We assume that the filter is obtained by implementing the coupling matrix of a set of physical microwave resonators such as cavities, dielectric resonators, and the like. Coupling elements, such apertures, loops or fringing fields, are used to implement the coupling coefficients. A pertinent question in relation to this model is the following: What do the ports see when they excite the filter? Do they see the individual resonators, specific combinations thereof or something else? The answer to this question is very important in engineering applications for it tells the designer of the fundamental parameters that determine the response of the system at the ports. Those parameters are in a way intrinsic to a given response. If two supposedly different structures have the same fundamental parameters, the input and output can not distinguish between them or, equivalently, their port responses are identical.

.. .

..

.

.. .

.. .

.. .

.. .

..

.

.. .

.. .

(14) and

(13) This equation is nothing other than the expression of the coupling bandwidth under the narrowband approximation ([14, p. 134]). Note that this result holds only when the coupling coefficients are assumed frequency independent. From this interpretation, it is clear that the original secondorder -plane cavity filter can be represented by the coupling scheme in Fig. 3 in which the even and odd modes are used as basic resonators. Finally, note that (13) becomes identical to the results given in [15] if the narrowband approximation is used in [15].

.. .

.. .

.. .

..

.

.. .

.. .

.. .

..

.

.. .

(15) denotes the determinant of a matrix . In the limit of Here, small values of , we can expand the numerators and denominator in (14) and (15) in successive powers of leading to .. .

..

.

.. .

.. .

..

.

.. . (16)

and

.. .

..

.

.. . (17)

where and are real constants, which depend on the specifics of the coupling matrix and are polynomials of the nor-

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malized frequency . However, the dependence of the constant on the normalized frequency is at least one order lower than that of the determinant appearing in these two equations. Equation (16) shows that as approaches 0, the reflection approaches 1. This is simply the statement coefficient that practically the totality of the incident power is reflected since only a vanishingly small amount of power is coupled to the system under the assumption of very weak couplings. This means that almost all the incident power is reflected due to the very weak coupling to the input. More important information is contained in (17). Since the coupling matrix is assumed real, we in (17) as can rewrite the magnitude of

.. .

..

.. .

.

(18) It is important to keep in mind that the strength of the coupling is weak, but nonzero. This equation shows that the magnitude of the transmission coefficient has maxima when the determinant in its denominator vanishes or, equivalently, at frequencies such that .. .

..

.

.. .

(19)

Equation (19) states that the maxima of the transmission coefficient occur at normalized frequencies that are the opposites of the eigenvalues of the original coupling matrix out of which the first and last columns and rows are eliminated. These frequencies are the resonant frequencies of the eigenmodes of the entire structure with the input and output removed. These are the resonances that the ports see. They are naturally combinations of the original resonators. More importantly, the ports do not see the original individual physical resonators upon which the initial design is usually based. Situations may occur where the positions of the transmission zeros are located very close to the resonant frequencies of the eigenmodes. This happens when a root of the parameter in (18) is close to a root of (19). Such cases should be easily identifiable from the skewed shape of the response. The case of two resonators is well known to filter designers. In order to determine the coupling between two resonators, the input and output are weakly coupled to the two-resonator structure (20-dB rule). Under these conditions, two peaks are observed in the transmission coefficient. The separation in frequency between these two peaks is the coupling bandwidth, which is related to the coupling coefficient between the two resonators, as explained, for example, in [14]. The important point is that the input and output see the two global resonances, the even and odd modes, but not the individual resonators. This paper presents not only a mathematical proof of this result, but a rigorous generalization to the case of an arbitrary number of resonators as well. The input and output never see the individual resonators, but rather the eigenmodes of the complete

filter. This makes sense since the main role of the filter is to selectively transport electromagnetic (EM) energy between the two ports. Only a solution to Maxwell’s equations, which satisfies the boundary conditions inside the structure, can carry energy between the two ports. Since a good lossless microwave bandpass filter is an electromagnetically closed structure, i.e., not radiating, the solutions of Maxwell’s equations form a discrete set [16]. The elements of the set are the resonances of the structure. These are the only separate “physical” paths that the EM energy can follow between the two ports. That the ports do not see the individual resonators is not surprising given the fact that these no longer exist once the coupling elements are introin an individual cavity is not strictly duced. Indeed, the a physical quantity once the coupling apertures are put in place; its boundary conditions are violated. The situation is similar to the case of strongly interacting quantum particles where global entities, such as quasi-particles, are used to describe the system instead of the individual particles, which no longer exist in a strict sense, or have a finite lifetime [17], [18]. VI. GLOBAL EIGENMODE REPRESENTATION Having shown that the input and output see the global eigenresonances and not the individual resonators, it is important to know how the filter is represented in this new set of resonance (new basis). We assume that the filter is initially described by a potentially . The multimode full coupling matrix of size resonator that remains after the input and output are eliminated of size , which is is represented by the sub-matrix obtained from by eliminating the first and last columns and rows, which contain the coupling coefficients from the source and load to the different resonators. This sub-matrix is real and symmetric for a lossless system and can always be diagonalized denote the th component of the th eigenvector [19]. Let of the sub-matrix

.. .

.. .

..

.

.. .

.. .

(20)

Since is real and symmetric (Hermitian), its eigenvectors are orthogonal, or can be orthogonalized if two or more are dein (20) is orthogonal. Let generate. Consequently, the matrix denote a diagonal matrix of size whose diagonal el. By using (20) in (9), we ements are the eigenvalues of get a new coupling matrix of the form

.. .

..

.

.. .

(21)

This coupling matrix is readily recognized as the transversal matrix introduced by Cameron [2]. The following points are worth mentioning.

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N

Fig. 4. Coupling scheme of th-order coupled resonator bandpass filter in the global eigenmode representation.

1) The eigenvalues of the sub-matrix are the opposites of the normalized resonant frequencies of the global eigenmodes. is not af2) The coupling between the source and load fected by the transformation that generated the coupling in (21) from the original coupling matrix. In matrix is zero in the original matrix it remains particular, if zero. This makes sense since the transformation involves represents a separate signal path only the resonators; that is not connected to any resonator. 3) The global eigenmodes are not coupled to each other. global 4) The source and load are coupled to each of the eigenmodes. 5) Any coupled resonator bandpass filter that is modeled by a coupling matrix is represented by a transversal coupling matrix in the global eigenmode basis. The coupling and routing scheme of a general filter in the global eigenmode representation is shown in Fig. 4. The representation of coupled resonator bandpass filters in terms of the transversal coupling matrix can also be obtained directly from Maxwell’s equations. The starting point is the generalized admittance matrix of the structure, which is fed by uniform sections of waveguide at the two ports. It can be shown that [19]–[23] (22) where and denote the ports, and and are real constants, which are related to the coupling integrals of the waveguide modes to the global eigenmodes. The resonant angular frequency of the th global eigenmode is . It is straightforward to show that the circuit shown in Fig. 5 has the admittance parameters given in (22). Each path between between the source and the input and output has an inverter th resonator and another inverter between the load and th resonator. The first term in (22) is represented by shunt elements

Fig. 5. Circuit representation of (22).

at the input and output and potentially a branch term between the is nonzero. As it stands, this circuit repinput the output if resentation is too detailed to be useful as an equivalent circuit, which is necessarily of lower order. It is a full-wave model of the structure. In order to extract an equivalent circuit, which is necessarily of lower order, we need to examine the roles played by the terms in (22). The resonances appearing in this equation do not play the same role, especially for a bandpass filter. Over the bandpass and its vicinity, only a finite number of resonances are responsible for the power transport between the input and output. The other resonances, higher order resonances in most designs, are responsible for the enforcement of the boundary conditions at the input and output. In an equivalent circuit, these are represented by reactive elements at the input and output nodes. Using nodal equations, the circuit in Fig. 5 is described by the following equation:

.. .

.. .

.. .. .

.. .

.

.. .

.. . .. .

.. . .. .

(23)

AMARI AND BEKHEIT: PHYSICAL INTERPRETATION AND IMPLICATIONS OF SIMILARITY TRANSFORMATIONS IN COUPLED RESONATOR FILTER DESIGN 1145

Here, and . Note that the cahas been set pacitor of resonator with resonant frequency for convenience. We now assume that only equal to resonances are responsible for practically the totality of power transport, the other ones appear far away from the passband of the filter. In order to eliminate the remaining resonances from (23), we use the fact that the voltage at the node of resonator is related to the voltages at the source and load by

loss of the filter. The result is a change in the resonant frequencies of the global eigenmodes, as well as the coupling coefficients. For broadband responses, this is not possible. Equation (28) must be used in its current form; it is then equivalent to the original admittance matrix in (22), which can handle wideband responses [21].

(24)

The representation of a bandpass microwave coupled resin terms of its global eigenmodes onator filter of order through a transversal coupling matrix is a natural consequence of the internal boundary conditions. The source that excites a fully operational bandpass filter sees a complex structure, which modes (resonances) within a limited frequency can support range around the passband of the filter. These modes satisfy all the internal boundary conditions and reflect the presence of all internal coupling and tuning elements. They still do not satisfy the boundary conditions at the input and output. A consequence of this is the failure of the model to predict the frequency shift that is observed when the strength of the coupling at the input and output is changed. However, the effect of this loading on the global modes is substantially less than the effect of the same coupling aperture on the first (last) resonator alone. The global modes simply capture more of the dominant physics of the problem. For example, it was found that representing a microstrip dual-mode filter in terms of the eigenmodes, which include the presence of the coupling elements, allows accurate prediction of the experimentally observed frequency shift of the passband when the transmission zeros are moved from the real onto the imaginary axis. A representation based on coupling individual resonators does not predict such a frequency shift [25]. However, designing a filter based on the global modes is far from simple, except for filters of order 1 or 2. It is, therefore, important to understand the conditions under which this representation offers and advantage over the one based on individual resonators. It is well known that no reliable and general direct design technique of coupled resonator filters is available. The word design is used here to mean the actual determination of all the dimensions of the structure. Available direct design techniques such as those dealing with direct-coupled resonator filters introduced by Cohn [26], TEM line filters, as described by Wenzel [27], and linear phase filter, as discussed by Rhodes [28], are applicable only to specific types of resonators and resonator arrangements. The design theory based on the expression of the

VII. DISCUSSION

When the voltages of the higher order resonances are eliminated from (23) by using (24), the only two equations that are affected are the ones for the source and load nodes. The result is the appearance of new reactive terms at these two nodes, as well as a source–load coupling term. The contribution of the higher order modes to this reactance at the source node is (25)

The additional loading reactive element at the load node is (26)

The source–load coupling term due to the combination of the higher order modes and the first term in (22) is then (27)

Once all the higher order resonances are eliminated, we are left with an matrix, which should be eventually compared to the transversal coupling matrix. Using (25)–(27) in combination with (23), we get the admittance matrix shown in (28) at the bottom of this page. Except for normalization, in the limit of narrrowband responses, this is the transversal coupling matrix. It should be obvious that the presence of the reactive elements at the source and load nodes will affect the resonant frequencies of the global eigenmodes. This is the loading effect that is missing from existing models. For narrowband filters, it is possible to eliminate the reactive elements at the source and load nodes by adding a phase term at the input and output [24]. The added phase term does not affect the insertion and return

.. .. .

.. .

.

.. .

(28)

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coupling coefficient between two resonators in terms of the resonant frequencies of the even and odd modes (coupling bandwidth) provides at best a starting point for optimization. In other words, it is safe to say that existing design methodologies of sophisticated coupled resonator filters require optimization in general. It is at this stage that the global eigenmode representation has a distinct advantage as the results presented in Section VII-A demonstrate. In addition, invaluable insight into the working of the filter can be gained by examining the field distributions of the global eigenmodes. Within the global eigenmode representation, one is no longer required to know the topology of the coupling matrix upon which the initial design is based. The topology is usually determined by examining the physical arrangements of the resonators. Although this intuitive approach works well, especially for narrowband filters, the authors are not aware of any systematic technique that yields the topology directly from the field equations (Maxwell’s equations). This becomes a crucial point in miniaturized filters and filter using higher order modes as well as 3-D arrangement of resonators such as in low-temperature co-fired ceramic (LTCC) technology, where stray couplings cannot be ignored. Without a rigorous technique to determine the topology, the best solution is simply to use a technique that does not require the knowledge of one. This is what the global eigenmode representation provides.

Fig. 6. Coupling scheme of a third-order Chebyshev filter. (a) Direct-coupled resonators. (b) Global eigenmode representation (transversal matrix).

A. Issue of Uniqueness Within the technique in [29], and the space-mapping technique in general [30], the issue of uniqueness of the equivalent circuit is crucial. It is well known that the process may fail to converge if multiple solutions for the equivalent circuit to reproduce a given response exist. The global eigenmode representation solves the uniqueness problem for coupled resonator filters. Indeed, if the scattering parameters of the structure can be approximated by rational functions of the complex variable , which are realizable by the structure, then the elements of the transversal coupling matrix in (21) are known analytically and uniquely [2]. B. Universality Any coupled resonator bandpass filter that is modeled by a coupling matrix is represented by a transversal coupling matrix in its global eigenmodes. This is a direct consequence of the fact that a real symmetric matrix can always be diagonalized. The universality of the transversal coupling matrix provides a very valuable tool in optimizing microwave filters. Once the initial design is set, it is no longer necessary to know the topology of the original coupling matrix in order to optimize the filter regardless of its order, symmetry, locations, and number of finite transmission zeros. C. Ease of Parameter Extraction A crucial step in the optimization of microwave filters through the technique in [29] is the extraction of the parameters of the equivalent circuit. If the response can be approximated by rational functions of the complex frequency, then the elements of the transversal coupling matrix are always known analytically [2]. This is not the case for many other coupling schemes.

Fig. 7. Geometry of a third-order

H -plane cavity Chebyshev filter.

VIII. APPLICATIONS A. Third Order

-Plane Waveguide Chebyshev Filter

In order to demonstrate the soundness of the global eigenmode representation, we choose the extreme case of a Chebyshev response where all the transmission zeros are at infinity. Such a response is commonly implemented as a set of directcoupled resonators where no resonator is bypassed according to the coupling scheme in Fig. 6(a). It may be surprising that by properly interpreting the effect of a similarity transformation, a Chebyshev microwave cavity filter is more accurately represented by a set of bypassed (cross-coupled) resonators, shown in Fig. 6(b), than by a set of direct-coupled ones. Note that this is not correct for lumped-element filters where the values of the ’s and ’s of the resonators do not depend on the coupling strength. We assume that the passband is centered at 12 GHz and is 200-MHz wide with a minimum in-band return loss of 20 dB. Although this filter can be readily designed by following the theory developed by Cohn [26], we purposely detune the response of the structure and then use the global eigenmode representation of the coupling matrix to retrieve the ideal response. rectangular An implementation of this filter in -plane cavities is shown in Fig. 7.

AMARI AND BEKHEIT: PHYSICAL INTERPRETATION AND IMPLICATIONS OF SIMILARITY TRANSFORMATIONS IN COUPLED RESONATOR FILTER DESIGN 1147

1) Ideal Coupling Matrix: The coupling matrix of a set of direct-coupled resonators is known analytically [12]. For the current example, we get

(29) The eigenvalues and eigenvectors of the 3 3 sub-matrix, which results when the first and last rows and columns are eliminated, are

(30)

The similarity transformation to the global eigenmode representation is

(31)

Applying (9) to the matrix in (29) with the transversal coupling matrix

given by (20), we get

(32) Note that this coupling matrix could have been obtained directly through the technique given by Cameron [2]. Here, and are the coupling coeffiand , recients of the input to the global eigenmodes spectively. The normalized resonant frequencies of these modes and , respecare tively. It is instructive to see how the negative coupling coeffiis brought about. By examining the components of cient the three eigenmodes given by (30), we see that all three vectors have positive first components. They couple to the source in phase. In contrast, the last component of the second vector is negative, while those of the two other vectors are positive. The second mode has, therefore, a negative coupling coefficient to

Fig. 8. Magnetic field distributions of the three lowest global eigenresonances in a three-resonator -plane filter. The input and output coupling apertures are covered by perfect conductors.

H

the load if the coupling coefficients of the first and third modes to the load are assumed positive. Here, we use the fact that the first and last components determine the relative signs of the coupling coefficients to the source and load, respectively. The magnetic field distributions of the three global eigenmodes are shown in Fig. 8. It should be clear that these modes can be used to describe how the filter functions. The corresponding coupling matrix is transversal. The relative signs of the coupling coefficients at the output (those at the input can always be chosen as positive) can be deduced from these plots. Indeed, at the output, two of the modes have their magnetic fields in the same direction, which is opposite to that of the third mode. This phase reversal is equivalent to a negative coupling coefficient at the output. 2) Parameter Extraction: To optimize the filter, we follow the technique used in [29] where the parameters of the circuit (nonzero entries of the coupling matrix) are extracted from the scattering parameters, which are obtained from a full-wave EM simulator. Since the gist of the technique is by now well documented, no further details are given here. The extraction is carried out by matching the full-wave response to that of the equivalent circuit through the minimization of a cost function. One of at least four options can be followed. 1) Extraction of the elements of the direct-coupled model in Fig. 6(a), which now includes potentially nonzero diagonal elements in the coupling matrix to account for the detuning of the resonators. 2) Extraction of the elements of the transversal coupling matrix in Fig. 6(b) directly, e.g., by using its nonzero entries as optimization variables. In this case, constraints to force the required number of transmission zeros at infinity must be first established as will be seen below.

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H

H

Fig. 9. -plane filter EM simulated response and the response from the extracted transverse coupling parameters extracted with no constraints.

Fig. 10. -plane filter EM simulated response and the response from the extracted transverse coupling parameters extracted with constraints.

3) The transversal matrix is obtained from the inline one by diagonalizing the sub-matrix. 4) Determine rational function approximations to the scattering parameters as a function of the complex frequency and then use the direct synthesis technique presented by Cameron [2]. The rational functions can be determined, for example, by following [31]. In principle, the four techniques should yield the same results although the last option might be more general. 3) Constrained Transversal Coupling Matrix: The optimization of this filter exploits a transversal coupling matrix of the form

interest, i.e., those with three transmission zeros at infinity. For the transversal coupling matrix to produce no transmission zeros and in at finite frequencies, we set the coefficients of , as shown in Appendix-A. For higher order the numerator of filters to avoid lengthy algebra, the coefficients of the numerator can be calculated from the Souriau–Frame algorithm of [32] or the closed-form equations given in Appendix-A. In the current example, we get the following relationships:

(34) (33)

where

and are positive. The frequency shifts and are of unknown signs. If the second option above is used to extract the nonzero elements of the transversal coupling matrix, it is more efficient to first impose constraints on its entries. These constraints are needed to ensure that all the three transmission zeros of the structure (Chebyshev filter) are located at infinity. Without these constraints, the extraction might converge to a solution, which is “too good,” and lies outside the set of target functions. Indeed, the unconstrained transversal coupling matrix can match almost perfectly the EM solution including its asymmetry. For example, Fig. 9 shows the response of the transversal coupling matrix (solid lines) that is extracted from a full-wave EM simulation (circles) of the current third-order filter when no constraints are imposed on its entries. The two results agree within plotting accuracy over the entire frequency range. However, the transversal coupling matrix achieves this match by placing two transmission zeros at finite frequencies far away from the passband. Such a response is not strictly a Chebyshev one and we need to force the transversal coupling matrix to yield only solutions within the class of functions of

Note that since there is no direct source–load coupling. Using these constraints, the independent parameters and are extracted by matching the response of the coupling matrix to the one obtained from the full-wave EM simulator. The response of the extracted constrained transversal coupling matrix is shown in Fig. 10 along with the EM response it is supposed to match. It is obvious that the agreement between the two responses is not as good as in Fig. 9, especially in the stopband. However, the constraints guarantee that the extracted solution falls within the desired class of functions. The extraction is repeated for slightly different values of the adjustable dimensions of the structure to establish a relationship between the elements of the coupling matrix and the geometrical dimensions. This relationship is then inverted to determine the next guess and the process is repeated until convergence is achieved. Here, the parabolic approximation introduced in [29] is used. If the transversal coupling matrix is extracted by optimization, it is important to weigh the passband more than the stopbands for two reasons. First, the stopbands can be very sensitive to errors in the coupling coefficients of the transversal coupling matrix. The passband, on the other hand, is not as sensitive to the same errors. Second, the model is inherently narrowband and is not expected to reproduce the wideband response of the actual filter.

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4) Results: For this structure, both the inline and the transversal coupling matrices have unique solutions (except for unconsequential sign changes). However, the extraction of the elements of the inline coupling matrix from a full-wave EM simulation is more demanding than those of the transversal coupling matrix for the same cost function and minimization algorithm, except possibly for extremely narrowband filters. This is attributed to the fact that the global eigenmodes do satisfy all the internal boundary conditions, as mentioned earlier. For small perturbations of a given adjustable geometrical parameter, such as the aperture of the first coupling iris, mainly the elements of the inline equivalent circuit that are directly connected to the given parameter are affected. This is equivalent to saying that the Jacobian of the inline coupling scheme is a sparse matrix. This is not the case for the transversal matrix where practically all the elements of the equivalent circuit are, in general, affected by any perturbation. Consequently, the Jacobian of the transversal coupling matrix is not necessarily sparse. Fortunately, the numerical effort required to evaluate the Jacobian (and the Hessian matrix if needed) are practically identical for both coupling schemes. As a numerical example, the following parameters are extracted to match the response mm, of the third-order -plane cavity filter when mm, mm, mm

H

Fig. 11. Optimization progress for the third order -plane filter from a detuned response. Solid line: initial response. Dashed line: iteration 1. Circles: iteration 3.

(35)

If the aperture of the first (and last) iris is changed by mm, we get the following parameters: Fig. 12. EM simulated response of optimized third-order lines: in-house package. Dotted lines: Wave Wizard.

H -plane filter. Solid

(36)

It is obvious from these two equations that the parameters of the transversal coupling matrix have all been affected by this perturbation. In contrast, only the coupling coefficient from the source (load) to the first (last) resonator and their resonant frequencies are affected in the inline coupling scheme. The progress of the optimization of this filter is shown in Fig. 11. The initial response, shown as the solid lines, is strongly detuned on purpose. Despite this, the process converges after three iterations. The first iteration uses the quadratic approximation with a target in-band return loss of 15 dB. The second and third iterations use a linear approximation through a Jacobian matrix with the desired response as a target. The EM simulation of this filter was based on the modematching technique within an in-house software package. To validate this package, the commercial software package Wave Wizard from Mician, Bremen, Germany, was used for comparison. The responses of this filter as obtained from these two tools

are shown in Fig. 12. The agreement between the two is excellent. Finally, the dimensions of the optimized filter are (in milmm and limeters) . The thickness of the irises is mm. For comparison, the same filter was optimized by using the inline coupling scheme in Fig. 6(a) starting from the same initial design and using exactly the same perturbation to calculate the Jacobian. The process converges after five iterations, as shown in Fig. 13. B. Third-Order Microstrip Filter With One Transmission Zero To further show the universality of the transversal coupling matrix, the same procedure is used to optimize a third-order microstrip filter with one transmission zero above the passband. The transmission zero is generated by a trisection, as shown in Fig. 14 [33]. The specifications of the filter are MHz, MHz, dB, and a transmission zero at MHz. A dielectric substrate of and thickness

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H

Fig. 13. Optimization progress for the third-order -plane filter from a detuned response using the inline coupling matrix. Solid line: initial response. Dashed line: iteration 1. Circles: iteration 5.

Fig. 15. Progress of optimization of trisection microstrip filter through the transversal coupling matrix. Solid lines: initial design. Dotted–dashed lines: iteration 1. Circles: iteration 3. Dotted lines: ideal response.

From the examination of the first and last components of these three vectors, we expect a transversal coupling matrix with only one negative coupling coefficient, the one coupling mode to the output. Indeed, by diagonalizing the sub-matrix and carrying out the similarity transformation, we get the transversal coupling matrix as

(39) Fig. 14. Geometry of microstrip trisection filter [33].

1.27 mm is used. The ideal coupling matrix of a trisection giving this response is found to be

(37)

The eigenmodes of the 3

It is clear from this equation that the coupling coefficient of the second mode to the output is negative. The actual optimization is still based on the coupling matrix in (33) in spite of the presence of a transmission zero at a finite frequency. To guarantee that the transversal coupling matrix have only one transmission zero at a finite frequency, we set the coefficient in to 0. The coefficient of is identically null since of . Only one constraint on the coupling it is equal to coefficients results, namely, (40)

3 sub-matrix are

The same optimization procedure was followed and the filter was optimized from a detuned response. Fig. 15 shows the progress of the optimization steps. It takes three quadratic iterations for the process to converge. The results shown in Fig. 15 were obtained from the commercial software package IE3D from Zeland Inc., Fremont, CA. The dimensions of the optimized filter are (in millimeters) (38) and

.

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IX. DESIGN STEPS The theory presented in this paper provides a reliable design technique of microwave coupled resonator bandpass filters. The three steps of the design are as follows. Step 1) Extract a coupling matrix with the desired topology and required number of transmission zeros at finite frequencies. This synthesis problem was recently solved in an exhaustive manner, and in all its generality, by Cameron et al. [34]. Naturally, other synthesis techniques may be used in this step. Step 2) Obtain an initial design based on well-established techniques [14], [33]. Step 3) Use the transversal coupling matrix in combination with the methods in [29], or other implementations of the space-mapping technique [30], to optimize the initial design. Constraints on the elements of this matrix should be imposed in order to force the extracted response to remain within the desired set of functions. However, if the response of the initial design deviates too strongly from the ideal response, the constraints should not be imposed in the first iteration, but only in subsequent ones. Although a rigorous proof that the process is guaranteed to converge is not available, in all cases examined thus far, at most few iterations were needed to reach a response that falls within the manufacturing tolerances of the filter and at most two iterations were needed for the detuned filter to reach a response in the neighborhood of the optimum response. It is important that the extracted coupling matrix matches the response accurately, especially in the passband and its immediate vicinity. This issue is even more important when the response is not close to the ideal response. The sophisticated techniques developed in [35] may be used to first extract an accurate rational function approximation from which the transversal coupling matrix follows directly and analytically. X. CONCLUSION A new interpretation of similarity transformations for application in coupled resonator bandpass filter design has been presented. The change of basis (coordinates) associated with a similarity transformation is considered the central operation instead of the effect of such a transformation on the coupling matrix. In this view, both the original and transformed coupling matrices are taken to represent the same filter and not a new one as in existing interpretations. It is shown that when the global eigenmodes of the entire structure with the input and output removed are used as a new basis, any coupling matrix takes the form of a transversal coupling matrix. The transversal coupling was shown to be a universal representation of coupled resonator bandpass filters of arbitrary orders, topologies, and number of positions of the transmission zeros. The uniqueness of the transversal coupling matrix for a given response (phase and magnitudes of the scattering parameters) provides a reliable design technique of microwave filters. The design uses a coupling matrix with an appropriate topology to obtain an initial design with the correct number of the finite transmission zeros. The transversal coupling matrix is then used to rapidly, and reliably, optimize the filter. To enhance the efficiency of the

optimization process, constraints are imposed on the elements of the transversal coupling matrix to force the solution to be an element of the desired set. The technique was applied to an “inline” third-order Chebyshev filter, as well as a pseudoelliptic third-order microstrip filter with excellent results. APPENDIX A. Derivation of Constraints on Transversal Coupling Matrix Entries As mentioned earlier, it is important to set constraints on the elements of the transversal coupling matrix in order to force the extracted values to represent a response that falls within the desired class of functions with the specified number of transmission zeros. is proportional to , therefore, the nuFrom (3), is proportional to the cofactor of and merator of can be written as

(A1)

Expanding (A1), the coefficients for different powers of be written as

can

(A2)

can take any values from , whereas can take any without any exceptions. Also, value from because there are different combinations out of a total of available , whereas of since there are different combinations of out of a total of available . It is obvious that for (coefficient of ), the first term is zero and only the second as . term contributes, yielding the coefficient of In order to find the constraints in closed form, the coefficient of the specified powers of are set to 0. For example, for the case of a third-order direct-coupled Chebyshev filter, the coeffiand should vanish in order to have all transcients of mission zeros at infinity. in (A2), it is obvious that . This First, setting is the first constraint. in (A2) and equating the coefficient of to 0 Setting gives where

, except for

(A3)

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Here, the relative signs between the coupling from the source and load were taken into consideration. in (A2) and equating the coefficient of to 0 Setting gives (A4) can be eliminated leading to the Substituting (A3) in (A4) second constraint, which takes the form

(A5)

Substituting (A5) in (A3) then gives rise to the third constraint, which is

(A6)

Equations (A5) and (A6) are identical to (34). It should be noted that the number of independent entries of the transversal coupling matrix (coupling coefficients and global eigenmode frequencies) is reduced by the number of constraints. For this example, there are seven elements ( and ) and three constraints so we end up with only four independent elements that are to be used as optimization and are not variables. It should also be noted that considered two different parameters because they are equal in magnitude and either of the same or different signs depending on how the mode couples to the input and output. REFERENCES [1] R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 4, pp. 433–442, Apr. 1999. [2] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 1–10, Jan. 2003. [3] R. J. Cameron, A. R. Harish, and C. J. Radcliffe, “Synthesis of advanced microwave filters without diagonal cross-coupling,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2862–2872, Dec. 2002. [4] G. Macchiarella, “Accurate synthesis of in-line prototype filter using cascaded triplet and quadruplet sections,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1779–1783, Jul. 2002. [5] A. E. Atia and A. E. Williams, “New types of bandpass filters for satellite transponders,” COMSAT Tech. Rev., vol. 1, pp. 21–43, Fall 1971. [6] H. C. Bell, “Canonical asymmetric coupled resonators filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1335–1340, Sep. 1982. [7] A. E. Atia and A. E. Williams, “Narrow bandpass waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 4, pp. 258–265, Apr. 1972. [8] A. E. Atia, A. E. Williams, and R. W. Newcomb, “Narrow-band multiple-coupled cavity synthesis,” IEEE Trans. Circuits Syst., vol. CAS-21, no. 9, pp. 649–655, Sep. 1974. [9] J. D. Rhodes and I. H. Zabalawi, “Synthesis of dual-mode in-line prototype network,” Int. J. Circuit Theory Applicat., vol. 8, pp. 145–160, 1980.

[10] R. J. Cameron and J. D. Rhodes, “Asymmetric realization for dual-mode bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 1, pp. 51–58, Jan. 1981. [11] J. J. Sakurai, Modern Quantum Mechanics. New York: Addison-Wesley, 1994. [12] G. L. Matthaei, L. Jones, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks and Coupling Structures. New York: McGraw-Hill, 1964. [13] S. Amari and M. Bekheit, “New dual-mode dielectric resonator filers,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 3, pp. 162–164, Mar. 2005. [14] I. C. Hunter, Theory and Design of Microwave Filters. London, U.K.: IEE Press, 2001. [15] K. A. Zaki and C. Chen, “Coupling of non-axially symmetric hybrid modes in dielectric resonators,” IEEE Trans. Microwave Theory Tech, vol. MTT-35, no. 12, pp. 1136–1142, Dec. 1987. [16] K. Kurokawa, An Introduction to the Theory of Microwave Circuits. New York: Academic, 1969. [17] A. A. Abrikosov, Methods of Quantum Field Theory in Statistical Physics. New York: Dover, 1975. [18] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems. New York: Dover, 2003. [19] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: The Johns Hopkins Press, 1989. [20] R. Muller, “Theory of cavity resonators,” in Electromagnetic Waveguides and Cavities, G. Goubeau, Ed. New York: Pergamon, 1961, ch. 2. [21] G. Conciauro, M. Guglielmi, and R. Sorrentino, Advanced Modal Analysis. New York: Wiley, 2000. [22] K. Kurokawa, “The expansion of electromagnetic fields in cavities,” IRE Trans. Microw. Theory Tech., vol. MTT-6, no. 4, pp. 178–187, Apr. 1958. [23] M. Dohlus, R. Schuhmann, and T. Weiland, “Calculation of frequency domain parameters using 3-D eigensolutions,” Int. J. Numer. Model., Electron., Devices, Fields, vol. 12, pp. 41–68, 1999. [24] H. C. Bell, “Cascaded singlets realized by node insertions,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 107–110. [25] S. Amari, “Comments on ‘Description of coupling between degenerate dual-mode microstrip loop resonator using a novel perturbation arrangement and its dual-mode bandpass filter application’,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2190–2192, Sep. 2004. [26] S. B. Cohn, “Direct coupled resonator filters,” Proc. IRE, vol. 45, pp. 187–196, Feb. 1957. [27] R. J. Wenzel, “Synthesis of combline and capacitively-loaded interdigital bandpass filters of arbitrary bandwidth,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 8, pp. 678–686, Aug. 1971. [28] J. D. Rhodes, “The generalized direct-coupled cavity linear phase filter,” IEEE Microw. Theory Tech., vol. MTT-18, no. 6, pp. 308–313, Jun. 1970. [29] S. Amari, C. LeDrew, and W. Menzel, “Space mapping optimization of planar coupled resonators filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2153–2159, May 2006. [30] J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 2874–2882, Dec. 1995. [31] G. Macchiarella and D. Traina, “A formulation of the Cauchy method suitable for the synthesis of lossless circuit models of microwave filters from lossy measurement,” IEEE Microw. Wireless Compon. Lett., vol. 16, pp. 243–245, May 2006. [32] L. O. Chua and P. M. Lin, Computer-Aided Analysis of Electronic Circuits. Englewood Cliffs, NJ: Prentice-Hall, 1975. [33] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [34] R. J. Cameron, J. C. Faugere, and F. Seyfert, “Coupling matrix synthesis for a new class of microwave filter configuration,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 119–122. [35] F. Seyfert, L. Baratchart, J. P. Marmorat, S. Bila, and J. Sombrin, “Extraction of coupling parameters for microwave filters: Determination of a stable rational model from scattering data,” in IEEE MTT-S Int. Microw. Symp Dig., Philadelphia, PA, Jun. 2003, pp. 24–28.

AMARI AND BEKHEIT: PHYSICAL INTERPRETATION AND IMPLICATIONS OF SIMILARITY TRANSFORMATIONS IN COUPLED RESONATOR FILTER DESIGN 1153

Smain Amari (M’98) received the D.E.S. degree in physics and electronics from Constantine University, Constantine, Algeria, in 1985, and the Masters degree in electrical engineering and Ph.D. degree in physics from Washington University, St. Louis, MO, in 1989 and 1994, respectively. From 1994 to 2000, he was with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada. From 1997 to 1999, he was a Visiting Scientist with the Swiss Federal Institute of Technology, Zurich, Switzerland, and a Visiting Professor in summer 2001. In 2006, he was a Visiting Professor with the University of Ulm, Ulm, Germany. Since November 2000, he has been with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada, where he is currently a Professor. He is interested in numerical analysis, numerical techniques in electromagnetics, applied physics, applied mathematics, wireless and optical communications, computer-aided design (CAD) of microwave components, and application of quantum field theory in quantum many-particle systems.

Maged Bekheit (M’03) received the B.Sc. degree from Ain Shams University, Cairo, Egypt, in 1999, the M.Sc. degree from Queen’s University, Kingston, ON, Canada, in 2005, and is currently working toward the Ph.D. degree at Queen’s University. Since 1999, he has been an RF Planning and Optimization Engineer with a number of cellular companies. His research is focused on the design and optimization techniques of microwave components. Mr. Bekheit was the recipient of the Natural Sciences and Engineering Council (NSERC) of Canada Graduate Scholarship (CGS) and an Ontario Graduate Scholarship (OGS) Award.

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A Reconfigurable Micromachined Switching Filter Using Periodic Structures Muhammad Faeyz Karim, Ai-Qun Liu, A. Alphones, and Aibin Yu

Abstract—In this paper, a reconfigurable filter using micromachined switches is designed, fabricated, and experimented. An equivalent-circuit model is derived for the reconfigurable cell structure. Extracted parameters show the characteristics of both bandpass and bandstop filters, which can be accurately analyzed using circuit analysis. Coplanar waveguide transmission lines for the reconfigurable filter are also analyzed. The bandpass filter is formed by cascading the unit cell structure. This bandpass filter can be switched to bandstop filter using the micromachined switches and p-i-n diode. Dispersion characteristics are obtained to investigate the electromagnetic wave behavior within the unit cell using Floquet’s theorem. Measurement results with micromachined switches show that insertion loss is 1.55 dB for the bandpass filter whereas band rejection level is 20 dB and the insertion loss in the passband is 1.2 dB for the bandstop filter. With the p-i-n diode, the insertion loss is 2.1 dB and the 3-dB bandwidth is 5.2 GHz for the bandpass filter, the 20-dB rejection bandwidth is 5.3 GHz, and the insertion loss in the passband is 1.6 dB for the bandstop filter. Index Terms—Bandpass filter, bandstop filter, coplanar waveguide (CPW), reconfigurable filter.

I. INTRODUCTION N MODERN communication systems, a single filter normally is not capable of meeting the requirements of all desired operating bands. Multiple filters, on the other hand, occupy a large surface area. Tunable filters, therefore, become an interesting solution to these problems. Typically tracking filters are either mechanically tuned by adjusting the cavity dimensions of the resonator or magnetically altering the resonance frequency of the ferromagnetic YIG element [1], [2]. None of these approaches can produce miniaturized products in large volumes for wireless communication systems. The filters must be custom machined, carefully assembled, tuned, and calibrated. The tunable filters that use semiconductor varactor diodes give a compact design, which is suitable for frequencies below 10 GHz [3]. The p-i-n diode and varactor diodes show with reduced cost, easier packaging, and lower bias voltage. Recently an increasing amount of research is found concentrating on the development of the tunable bandpass filters based on microelectromechanical systems (MEMS) switches [4]–[10]. RF MEMS switches were first demonstrated by Petersen as cantilever beams using electrostatic actuation [11].

I

Manuscript received November 27, 2006; revised February 19, 2007. The authors are with the School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]. sg). 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.2007.897670

MEMS switches consisting of a thin metal membrane (or beam) can be electrostatically actuated to an RF line using dc–bias voltage. Two major types of MEMS switches have come to the forefront; series metal to metal contact switches [12] and shunt capacitive switches [13]. The tunable filters based on the MEMS switching technique offer several advantages in terms of low loss, low power consumption, high linearity, and small size compared to other varactor devices. Electromagnetic bandgap (EBG) structures are periodic structures that exhibit frequency-selective behavior similar to carrier transport in semiconductors. EBG structures are useful for the construction of various filters such as bandstop filters and low-pass filters. EBG structures can be patterned on the microstrip transmission line on a ground plane [14] or coplanar waveguide (CPW) [15]–[20]. Additionally, it is also feasible to incorporate a tuning element into the CPW EBG structures [21]. Despite extensive research on tunable filters, filters that can be switched from bandpass to bandstop at the same frequency have not been reported. This paper shows a new reconfigurable filter using an EBG structure, which can be switched from the bandpass filter to bandstop filter at the same frequency using dc contact MEMS switches and a p-i-n diode [22]. An equivalent-circuit model has been derived based on circuit analysis theory. Dispersion characteristics are obtained to analyze the electromagnetic (EM) wave behavior within the unit cell for a bandpass and bandstop filter based on Floquet’s theorem. The remainder of this paper is organized as follows. Section II presents extensive work on the design of the reconfigurable filter. Section III shows the experiment results and provide discussions. Section IV offers conclusion. II. DESIGN OF THE RECONFIGURABLE FILTER A reconfigurable filter consists of three unit cells designed in a CPW configuration, as shown in Fig. 1. These unit EBG cells are placed at a period equal to half of the guided wavelength of the designed central frequency. Before analyzing the reconfigurable filter, the CPW transmission line and unit cell structure are investigated. A. Analysis of the CPW Structure A conventional CPW on a dielectric substrate consists of a central strip conductor with semi-infinite ground planes on either side. The dimensions of the central strip, gap, thickness, and permittivity of the dielectric substrate determine the effec, characteristic impedance , and tive dielectric constant the attenuation of the line [17]–[20]. This structure supports a quasi-TEM mode of propagation. The CPW structure firstly simplifies the fabrication, secondly it facilitates an easy shunt,

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Fig. 1. Schematic of the MEMS reconfigurable filter.

Fig. 2. Simulations result of the current distribution in CPW.

as well as series surface mounting of active and passive devices, thirdly it eliminates the need for wraparound and via holes, and fourthly, it reduces radiation loss. Furthermore, the character. Therefore, istic impedance is determined by the ratio of size reduction is possible without limit, the only penalty being higher losses. The simulation results of current distribution for the CPW is shown in Fig. 2, where the signal is passing without any attenuation. It can be seen that the electric field radiates from the center signal conductor to the two ground conductors, i.e., the dominant mode for this kind of transmission line is a quasi-TEM mode. There is strong penetration of the -field into the substrate. Therefore, the material properties of the substrate have significant effects on the performance of the CPW. The material properties affects the performance of the CPW structure in terms of its insertion loss or the energy that is transmitted to the other port. For a simple CPW structure, the insertion loss should be as low as possible so that the performance of the filters are not affected.

Fig. 3. Computed attenuation constant. (a) Conductor loss for different metal thickness tat 20 GHz. (b) Conductor loss at different frequencies.

The attenuation increases very rapidly once the conductor thickness becomes less than 2–3 times of the skin depth . The EM simulation result using Ansoft’s High Frequency Structure Simulator (HFSS) for analyzing the effect of the metal thickness on the insertion loss is shown in Fig. 3(a) and (b). The metal is gold with conductivity of 4.1 10 S/m, and when the thickness is 4 and 6 m, the insertion loss is 0.2 dB at 5 GHz and 0.45 dB at 15 GHz, respectively. When the thickness of gold (Au) is 2 m, it shows a higher insertion loss of 0.75 dB at 15 GHz. Therefore, the thicker the Au layer, the lower the attenuation of the transmission line. B. Design of Unit Reconfigurable Cell Structure and Modeling A lattice-shaped unit EBG structure is shown in Fig. 4. In the unit cell structure, a small square slot is etched in the ground plane with a side length . The smaller square etched slot is connected to the gap by a narrow transverse slot with a length of and width of , as shown in Fig. 4 via dotted lines. Another

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TABLE I EXTRACTED CIRCUIT PARAMETERS FOR MEMS SWITCH (HIGH FREQUENCY)

Fig. 4. Schematic of the unit reconfigurable cell structure.

TABLE II EXTRACTED CIRCUIT PARAMETERS FOR p-i-n DIODE (LOW FREQUENCY)

Fig. 5. Equivalent-circuit model of the unit reconfigurable cell structure [21].

bigger square slot and the transverse slot is etched on top of the small square slot. The resonance frequency depends on the transverse slot and the square etched hole on the ground plane. The comparison is made between the EM simulation and circuit simulation results using Agilent’s Advanced Design System (ADS) circuit and Momentum environment. The circuit modeling shows that phenomenon of bandpass is observed with the inclusion of the bigger square slot and the transverse slot, otherwise it behaves like a bandstop filter. An equivalent parallel LC circuit is used to model the EBG structure with dotted lines, as shown in Fig. 5 [22]. It consists of a series inductor, parallel capacitor, and resistor. In order to design an EBG cell structure circuit, it is necessary to extract the equivalent-circuit parameters, which can be obtained from the simulation results of the unit cell structure. is mainly contributed by the The lumped capacitance is transverse slot on the ground, while the inductance related to the magnetic flux passing through the small square apertures on the ground. The equivalent-circuit parameters are extracted for the high frequency at 19.6 GHz used in the MEMS reconfigurable filter, and low frequency at 7.8 GHz for the p-i-n diode-based reconfigurable filter are presented in Tables I and II, respectively. The circuit simulation results obtained using the equivalent-circuit

parameters are given in Fig. 6, and when compared with the EM simulation results, they are in good agreement. To validate the model at two different frequencies bands, the structure has been tested. At high frequency, the bandstop occurs at 19.6 GHz. For lower frequency, the bandstop is at 7.3 GHz [22]. As a result, the derived equivalent circuit for the EBG structure can be accurately adapted to design practical circuits. The whole unit cell structure, which includes both the smaller and bigger square slots, is now discussed. From the circuit simulation and EM simulation, as shown in Fig. 7, it demonstrates a bandpass characteristic at the same frequency of 19.6 GHz. and inFrom the circuit model, when another capacitor are included, the bandpass response is observed. ductor Fig. 8 shows the current distribution of the reconfigurable filter, where Fig. 8(a) presents the bandpass response as all the signals are passing through, and Fig. 8(b) shows the bandstop response as the energy is totally attenuated at the resonant frequency.

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Fig. 8. Simulation results for the current distribution for reconfigurable filter. (a) Bandpass response. (b) Bandstop response.

is increased. In other words, increased, and the transverse slot is only slightly changed. Therefore, the two transmission zeros are occurring at 15.2 and 22 GHz. Again, the equivalent-circuit simulation results are in good agreement with the EM simulation results. Fig. 6. Comparison of parameter extraction and EM simulation results for the unit reconfigurable cell (dotted line). (a) p-i-n diode (lower frequency) [22]. (b) MEMS switch (high frequency).

Fig. 7. Comparison of parameter extraction and EM simulation results for the unit reconfigurable cell (whole structure).

To improve the performance of the bandpass response, the relative size of the bigger square slot to the smaller square slot

C. Propagation Characteristics of Unit EBG Cell The propagation characteristic of the unit EBG is analyzed using Floquet’s theorem [23], [24]. Fig. 9(a) shows the dispersion diagram of the EBG with the smaller slot and transverse slot, and the bigger slot is also introduced in Fig. 9(b). In comparison, Fig. 9(a) shows typical bandstop propagation characteristics [20], and Fig. 9(b) validates that the periodic structure has shown the bandpass behavior at the same frequency of the bandgap. The curve follows a brillouin zone boundary as it starts at 3 GHz. From the simulation results of the unit cell, the bandgap occurs at 8.5 GHz. The attenuation curve indicates a normalized absolute value of 12. After that, the curves start to fall when the bandpass phenomenon is approaching. The attenuation value of the bandpass filter at the center frequency of 18 GHz is 0. The attenuation curve starts to rise again when it approaches the second pole. However its absolute value is lower due to the low rejection level. The phase response rises fast due to the bandgap at 4 GHz, and then it fluctuates within the bandgap and the bandpass region. In Fig. 10, the phase response of the reconfigurable filter is shown. It has a linear phase in the bandpass and bandstop regions, respectively.

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Fig. 9. Simulated dispersion characteristics of unit cell structure. (a) Bandstop filter. (b) Bandpass filter (shaded region indicates stopband).

Fig. 11. Simulation results for the change in the bigger slot width b. (a) S . (b) S .

Fig. 10. Simulated phase response of the reconfigurable filter.

D. Effect of the Bigger Slot Width The effect of the dimensions of the bigger slot on the resonant frequency is studied. Based on the equivalent circuit, the change

in the dimensions of the slots from 762, 1200, and 1500 m and subsequently shifts the changes the inductance value resonant frequency. The simulation results in Fig. 11(a) and (b) shows the effect in inductance will change the return loss and transmission response. When the slot width is 762 m, the resonant frequency is 20.33 GHz, but ripples are observed and the rejection level in the second zero is less than 21 dB. The inductance value using the equivalent circuit is 0.45 nH and the others parameters remain constant. When the dimension of the bigger slot is raised from 1200 to is 1 nH and the filter 1500 m, the value of the inductance resonates at the center frequency of 17.43 GHz. The insertion is smooth in the passband region and shows reloss curve jection greater than 35 dB at the upper and lower sides of the passband. The extracted parameters of the filter are listed in Table III.

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TABLE III EXTRACTED PARAMETERS FOR THE CHANGE IN THE BIGGER SLOT

TABLE IV EXTRACTED PARAMETERS FOR THE CHANGE IN THE TRANSVERSE SLOT

E. Effect of the Transverse Slot Width The change in the width of the transverse slot affect the capacitance of the filter, as depicted in the equivalent circuit. The change in the width perturbs the resonant frequency. However, this change is observed to be smaller than the change in the inductance. When the width of the transverse slot is changed from 120 to 300 m, the effective change of the capacitance is from 0.18 to 0.27 pF. The optimum result occurs at 180 m when the rejection on both sides of the passband is greater than 35 dB, and the insertion loss of the passband is minimum. The parametric values of the inductance, capacitance, and resistance are shown in Table IV. III. EXPERIMENTAL RESULTS AND DISCUSSIONS The experimental results of the CPW will be discussed before analyzing the performance of the reconfigurable filter. The CPW transmission line is fabricated on a high-resistivity silicon substrate to minimize losses. The RF performance of the silicon CPW structure are measured using the HP 8510C vector network analyzer with a gold-tip 150- m pitch from cascade microtech ground– signal–ground coplanar probes. The system is calibrated using a standard short-open-load-thru (SOLT) on-wafer calibration technique. A 5-mm plastic plate is placed between the probe chuck and the sample to remove higher order mode propagation. All experiments are performed in the room environment without any packaging.

Fig. 12. SEM micrograph of the reconfigurable filter with MEMS switches. (a) Overview of the reconfigurable filter. (b) Zoom view.

The reconfigurable filter is fabricated using gold material. The gold materials does not only provide long-term reliability, but also good RF properties. The reconfigurable filter structure with dc contact MEMS switches is fabricated for 50 of the CPW transmission line mm and the gapwidth with a signal line width of m, as shown in Fig. 12(a). Fig. 12(b) shows a zoom view of the switch part in the reconfigurable filter. The surface micromachining technology is used for fabrication [25]–[27]. High-resistivity ( 4000 cm) silicon wafers with a thickness of m are used as the substrate. A layer of 0.5- m-thick oxide is deposited on the substrate as a buffer layer. A layer of 1- m-thick gold is then sputtered and patterned for the CPW transmission line. A layer of photoresist with 1.5- m thickness is spun coated and patterned to fill in the CPW slot. A 2- m-thick photoresist sacrificial layer is spun coated and patterned. A 1.5- m-thick layer of gold thin film is evaporated and wet etched to form the bridge. Finally, the photoresist is etched away by oxygen plasma etching to free the metal bridge. In order to show the validity of the equivalent circuit and the extracted parameters for the proposed unit cell, the reconfigurable filter is formed by cascading three unit cells and the MEMS switches, as shown in Figs. 12 and 13. The extracted fF, and . The values of the MEMS switch are and of the filter are already presented in values of

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Fig. 13. Equivalent circuit of the reconfigurable filter with MEMS switch.

Fig. 14. Cross section of the MEMS dc contact switch.

Fig. 16. Measurement results of the reconfigurable filter using MEMS switches. (a) OFF state of switch, bandpass filter. (b) ON state of switch, bandstop filter.

Fig. 15. Simulation results of the reconfigurable filter using MEMS switches. (a) OFF state of switch, bandpass filter. (b) ON state of switch, bandstop filter.

Table I. Fig. 14 shows the cross-sectional view of the dc contact MEMS switch. The dc contact MEMS switches are placed at the transverse slot, as depicted in Fig. 12. The width and length of the bridges are 50 and 300 m. The height of the switches is 2 m. The switch resistance is 1 and the up-state capacitance is 50 fF. For

providing the dc voltage, the bias lines are designed and have a resistance of 1 k . The MEMS switch is used as a metal-tometal contact and is used for switching from the bandpass to bandstop filter. When the dc contact MEMS switches are at the up state, the filter behaves as a bandpass filter, while at the down state, the filter behaves as bandstop filter. The reason for the reconfigurable filter to behave as a bandspass or bandstop filter has already been mentioned in the Section II. A detailed analysis of the MEMS switch and the impact of its characteristics on the performance of the filter are discussed in [25]–[27]. The comparison between simulation and measurement results shows good agreement. The simulation results of the reconfigurable filter is shown in Fig. 15. It has a insertion loss of 0.9 dB at 19.8 GHz and the 3-dB bandwidth of 9.6 GHz. The bandstop filter has an insertion loss of 0.6 dB. The measurement result of the reconfigurable filter is shown in Fig. 16. When the dc contact MEMS switches are at the up state, i.e., not shortened, then the filter provides the function of the bandpass filter an insertion loss of 1.55 dB at 20.1 GHz. It has 3-dB bandwidth of 9.2 GHz and rejection greater than 20 dB. When the dc contact

KARIM et al.: RECONFIGURABLE MICROMACHINED SWITCHING FILTER USING PERIODIC STRUCTURES

MEMS switches are at the down state with an applied dc-bias voltage of 38 V, the top transverse slot and bigger square slot are shortened to ground, and it shows a bandstop characteristic. The measurement results of the bandstop show the resonant frequency of 19.8 GHz and the insertion loss is 1.2 dB. The 20-dB rejection bandwidth varies from 17 to 22.5 GHz. The detailed measurement results of the reconfigurable filter using a p-i-n diode are presented in [22]. When the diode is switched at its off state, then a bandpass filter is operating at 7.3 GHz with an insertion loss of 2.1 dB. It has 3-dB bandwidth of 5.2 GHz and rejection greater than 35 dB. When the diode is at its on state, it shows a bandstop filter with a resonant frequency of 7.3 GHz and the insertion loss is 1.6 dB. The 20-dB rejection bandwidth is 5.3 GHz. When comparing the different design of the reconfigurable filter using the micromachined switch and the p-i-n diode, the result shows that the MEMS switches have lower insertion loss even at higher frequency. Additionally, the size is more compact and easy to integrate on a single chip. IV. CONCLUSION A micromachined switching reconfigurable filter has been designed, fabricated, and experimented. A CPW structure has been studied based on the silicon wafer. A different type of parametric analysis was conducted to investigate the performance of the CPW transmission line. An equivalent-circuit model is derived for the reconfigurable circuit. Extracted parameters show both bandpass and bandgap characteristics. Dispersion characteristics are obtained to analyze the EM wave behavior within the unit cell of the bandpass filter and the bandstop filter by Floquet’s theorem. The measurement results of the CPW transmission line show an insertion loss at 5 GHz is 0.17 dB and at 15 GHz is 0.2 dB. Based on the measurement results, using the dc contact MEMS switches show that the insertion loss of the bandpass filter is 1.55 dB and, for the bandstop, the band rejection level is 20 dB, and the insertion loss in the passband is 1.2 dB. The insertion loss of the bandpass filter of the p-i-n diode is 2.1 dB, and for the bandstop, the band rejection level is 35 dB and the insertion loss in the passband is 1.6 dB. The MEMS reconfigurable filter shows better RF performance, as well as being easy to integrate compared to its p-i-n diode counterpart. REFERENCES [1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. New York: McGraw-Hill, 1964. [2] Y. Ishikawa, “Mechanically tunable MSW bandpass filter with combined magnetic units,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 143–146. [3] A. R. Brown and G. M. Rebeiz, “A varactor tuned RF filter,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 7, pp. 1157–1160, Jul. 2000. [4] E. Fourn, “MEMS switchable interdigital coplanar filter,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 320–324, Jan. 2003. [5] A. A. Tamijani., L. Dussopt, and G. M. Rebeiz., “Miniature and tunable filters using MEMS capacitors,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1878–1885, Jul. 2003. [6] Y. Liu, A. Borgioli, A. S. Nagra, and R. A. York, “Distributed MEMS transmission lines for tunable filter applications,” Int. J. RF Microw. Comput.-Aided Eng., pp. 254–259, 2001, Special issue.

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[7] G. Kraus, “A widely tunable RF MEMS end-coupled filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 429–432. [8] K. Entesari and G. M. Rebeiz, “A 12–18 GHz 3 pole RF MEMS Tunable Filter,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 8, pp. 2566–2571, Aug. 2005. [9] D. Peroulis, K. Saramandi, and L. Katehi, “Tunable components with application to reconfigurable filter,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 341–346. [10] S. Majumdar, J. Lampen, R. Morrison, and J. Maciel, “MEMS switches,” IEEE Instrum. Meas. Mag., pp. 12–15, Mar. 2003. [11] K. E. Petersen, “Micromechanical membrane switches on silicon,” IBM J. Res. Develop., vol. 23, pp. 376–385, Jul. 1971. [12] J. J. Yao and M. F. Chang, “A surface micromachined miniature switch for telecommunications applications with signal frequencies from DC up to 4 GHz,” in Int. Solid-State Sens. Actuators Conf. Dig., Stockholm, Sweden, Jun. 1996, pp. 384–387. [13] C. L. Goldsmith, “Characteristics of micromachined switches at microwave frequencies,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 1996, pp. 1141–1144. [14] N. C. Karmakar and M. N. Mollah, “Investigations into nonuniform photonic-bandgap microstripline low-pass filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 564–572, Feb. 2003. [15] Y. Yun and K. Chang, “Uniplanar one-dimensional photonic-bandgap structures and resonators,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 549–553, Mar. 2001. [16] C. P. Wen, “Coplanar waveguide: A surface strip transmission line suitable for nonreciprocal gyromagnetic device applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 12, pp. 1087–1090, Dec. 1969. [17] J. L. B. Walker, “A survey of European activity on coplanar waveguide,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, Jun. 14–18, 1993, vol. 2, pp. 693–696. [18] S. Gevorgian, L. P. Linner, and E. L. Kollberg, “CAD models for shielded multilayered CPW,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 4, pp. 772–779, Apr. 1995. [19] T. Sporkmann, “The evolution of coplanar MMICs over the past 30 years,” Microw. J., vol. 41, no. 7, pp. 96–111, Jul. 1998. [20] T. Sporkmann, “The current state of the art in coplanar MMICs,” Microw. J., vol. 41, no. 8, pp. 60–74, Aug. 1998. [21] M. F. Karim, A. Q. Liu, A. Alphones, and A. B. Yu, “A tunable bandstop filter via the capacitance change of micromachined switches,” J. Micromech. Microeng., vol. 16, no. 4, pp. 851–861, 2006. [22] M. F. Karim, A. Q. Liu, A. Alphones, and A. B. Yu, “A novel reconfigurable filter using periodic structures,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 11–16, 2006, pp. 943–946. [23] R. E. Collins, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992. [24] S. G. Mao and M. Y. Chen, “Propagation characteristics of finite-width conductor-backed coplanar waveguides with periodic electromagnetic bandgap cells,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2624–2628, Nov. 2002. [25] A. B. Yu, A. Q. Liu, Q. X. Zhang, A. Alphones, L. Zhu, and S. A. Peter, “Improvement of isolation for RF MEMS capacitive shunt switch via membrane planarization,” Sens. Actuators, Phys. A, vol. 19, pp. 206–213, 2004, 2005. [26] A. B. Yu, A. Q. Liu, Q. X. Zhang, and H. M. Hosseini, “Effects of surface roughness on electromagnetic characteristics of capacitive switches,” J. Micromech. Microeng., vol. 16, pp. 2157–2166, 2006. [27] A. B. Yu, A. Q. Liu, Q. X. Zhang;, A. Alphones, and H. M. Hosseini, “Micromachined DC contact capacitive switch on low-resistivity silicon substrate,” Sens. Actuators, Phys. A, vol. 127, pp. 24–30, 2006.

Muhammad Faeyz Karim received the B.Eng. degree from the National University of Science and Technology, Rawalpindi, Pakistan, in 2000, the Masters of Science degree from Nanyang Technological University, Singapore, in 2002, and is currently working toward the Ph.D. degree at Nanyang Technological University. His research interests includes RF MEMS, antennas, and EBG structures.

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Ai-Qun Liu received the Ph.D. degree from the National University of Singapore (NUS), Singapore, in 1994, the M.Sc. degree in applied physics from Beijing University of Posts and Telecommunications, Beijing, China, in 1988, and the B.Eng. degree from Xi’an Jiaotong University, Xi’an City, China, in 1982. He is currently an Associate Professor with the School of Electrical and Electronics Engineering, Nanyang Technological University (NTU), Singapore. He was a Guest Editor for Sensors and Actuators A, Physics. His research interests are MEMS design, simulation, and fabrication processes. Prof. Liu is an associate editor for the IEEE SENSOR JOURNAL.

A. Alphones received the B.Tech. degree from the Madras Institute of Technology, Madras, India, in 1982, the M.Tech. degree from the Indian Institute of Technology Kharagpur, Kharagpur, India, in 1984, and the Ph.D. degree in optically controlled millimeter wave circuits from the Kyoto Institute of Technology, Kyoto, Japan, in 1992. From 1996 to 1007, he was a Japan Society for the Promotion of Science (JSPS) Visiting Fellow. From 1997 to 2001, he was a Senior Member of Technical Staff with the Centre for Wireless Communications, National University of Singapore, where he was involved in research on optically controlled passive/active devices. He is currently an Associate Professor

with the School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore. He possesses 22 years of research experience. His research was cited in Millimeter Wave and Optical Integrated Guides and Circuits (Wiley, 1997). He has authored, coauthored, or presented over 130 technical papers in international journals/conferences. He authored the chapter “Microwave Measurements and Instrumentation” in the Wiley Encyclopedia of Electrical and Electronic Engineering (Wiley, 2002). He has delivered tutorials and short courses in international conferences. His current interests are EM analysis on planar RF circuits and integrated optics, microwave photonics, and hybrid fiber-radio systems. Dr. Alphones is on the Editorial Review Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was involved with the organization of the APMC’99, ICCS 2000, ICICS 2003, PIERS 2003, IWAT 2005, ISAP 2006, and ICICS 2007 conferences.

Aibin Yu received the B.Eng. degree in materials science and M.Eng. degree in electronic material and device from Shanghai Jiaotong University, Shanghai, China, and is currently working toward the Ph.D. degree at Nanyang Technological University, Singapore. He is currently a Research Associate with the Division of Microelectronics, School of Electrical and Electronics Engineering, Nanyang Technological University. His research interests include the microfabrication process and RF MEMS design.

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A Synthesis Method for Dual-Passband Microwave Filters Juseop Lee, Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—This paper describes a synthesis method for symmetric dual-passband microwave filters. The proposed method employs frequency transformation techniques for finding the locations of poles and zeros of a desired filter. This method can be used to design dual-passband filters with prescribed passbands and attenuation at stopbands directly without the need for any optimization processes. To validate the procedure a dual-passband stripline filter is designed and fabricated. The stripline dual-passband filter is designed with passbands at 3.90–3.95 and 4.05–4.10 GHz, and 30-dB attenuation at the stopband. This measured results show a good agreement with the theoretical ones. The frequency transformation for symmetric dual-passband filters is also extended to include asymmetric dual-passband responses. This flexible frequency transformation preserves the attenuation characteristics of the low-pass filter prototype. Examples are shown to discuss the flexibility of this transformation. Index Terms—Circuit synthesis, dual-passband filters, microwave filters.

I. INTRODUCTION

M

ODERN communication transceivers require high-performance microwave filters with low insertion loss, high frequency selectivity, and small group-delay variation. For high frequency selectivity, synthesis and design techniques for filters with transmission zeros near passband have been developed [1]–[3]. For those filters, flat group delay in the passband is accomplished using the external equalizer or the self-equalization design technique [4], [5]. Since modern communications systems, especially satellite communications systems, have a complex frequency and spatial coverage plan, noncontinuous channels might need to be amplified and transmitted through one beam. In this case, compared with the power divider/combiner configuration, circulator chain structures, or manifolds, multiple-passband filter can make the system simple. Dual-passband filters of a canonical structure with a single-mode technique [6], that of an in-line structure with a dual-mode technique [7], and that of a canonical structure with a dual-mode technique [8] have been designed and realized. A synthesis method of a self-equalized dual-passband filter has also been presented [9]. These deign methods are all based on the optimization techniques in the filter synthesis process.

Manuscript received December 14, 2006; revised March 7, 2007. The authors are with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]; saraband@eecs. umich.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.2007.897712

To avoid numerical optimization, a method known as the frequency transformation technique has been introduced in designing dual-passband filters [10]. Basically this synthesis for designing a dual-passband filter is accomplished by applying a frequency transformation to a low-pass filter prototype. Since the stopband response of the dual-passband filter obtained by using the frequency transformation is not identical to that of low-pass filter prototype, a few attempts are required to find a suitable placement of transmission zeros to acquire the desired attenuation value. One of the frequency transformation in [10] can be adopted for designing an asymmetric dual-passband filter. However, this transformation cannot provide the equiripple response in the stopband, which enables high-frequency selectivity. Filters with dual stopbands and the associated frequency transformation are presented in [11]. Similar to the transformation in [10], this transformation also requires the optimization to achieve an equiripple response in the stopband and its applications to asymmetric dual-passband or dual-stopband filters are not discussed. In this paper, we present a synthesis technique for symmetric dual-passband filters using frequency transformation without the need for optimization. Two frequency transformations are given and applied consecutively to the low-pass filter prototype to obtain the dual-passband filter response. With this method, the dual-passband filters can be designed with prescribed passbands and an attenuation value in the stopband since the transformation preserves the low-pass filter prototype characteristics. The frequency transformation in general form is also given for designing asymmetric dual-passband filters. This transformation is flexible enough to allow for the design of filters with two passbands of highly different bandwidths. This transformation also preserves low-pass filter prototype characteristics.

II. DESIGN THEORY Here, we introduce a procedure for designing dual-passband microwave filters using two frequency transformations. Fig. 1 shows the frequency response of the filter in three different frequency domains. The domain is the actual frequency domain where the filter operates and is a normalized frequency for the low-pass prototype. Generally, single-passband filters are designed in the domain and the frequency transformation is applied to make the filter operate in the domain. For dual-passis band filter design, an intermediate normalized frequency used. The frequency response in the domain can be obtained by applying two frequency transformations consecutively to the frequency response in the domain. The coupling matrix of the

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zero locations, we can rewrite the characteristic function as a rational function (5) where (6) which makes the magnitude of

in (5) 1 at

.

B. Frequency Transformation for Dual-Passband Filters

Fig. 1. Frequency response of the filter in the ; ; and ! domain. The coupling matrix of the dual-passband filter is obtained from the transfer function in the domain.

Let us assume that the dual-passband filter has two symmetric passbands and their passband regions are specified by and (Fig. 1). The coupling matrix is obtained from the frequency response in the normalized frequency . For dual-passband filter synthesis, the previous studies [8], [9] start domain by finding the locations of poles and zeros from the of the filters by direct optimization. In this paper, we obtain the poles and zeros of the dual-passband filter by the analytic frequency transformation technique. The frequency transformation from to can be expressed as follows: for

dual-passband filter is obtained from the transfer function of the domain. filter in the A. Low-Pass Filter Prototype Generally, the transfer function of an filter prototype can be expressed by

th-order low-pass

for

(7)

and . Since 1 and 1 in the domain where are transformed to 1 and in the domain for , respectively, and 1 and 1 in the domain are transformed to 1 and in for , respectively, we must enforce

(1) with the assumption that the characteristic funcwhere tion only has pure imaginary poles and zeros in the domain, and is a ripple constant related to the passband return loss by (2) Since here we do not deal with self-equalized dual-passband filters, pure imaginary poles and zeros are sufficient for our design. For the case of pure imaginary pole and zeros, the characteristic function of the elliptic function filter is given by [12]

(8) From (8), the constant in (7) can be expressed in terms of the band edge frequency of the dual-passband filter in the domain as follows:

(9) Similarly, the frequency transformation from narrow bandpass filters is expressed as

to

for

(3) (10) where

. Since band edge frequencies in the where are transformed to those in the domain as

domain

(4) In (4), is the location of the th transmission zero. Note that is 1 at for all . Once the the magnitude of transmission zeros are decided, the poles can be obtained easily by computing the return loss of the filter. Based on the pole and

(11)

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. This filter prototype has a pair of zeros at transmission zeros at finite frequencies and their locations are determined based on the attenuation requirement over the stopband. Using (1)–(3), we can find the locations of the poles with given transmission zeros, return loss, and the number of poles. Poles of the filter are found to be located at . Since band edge frequencies of the dual-passband filter are GHz, GHz, GHz, GHz, we have from (13). From and the frequency transformation in (7), the poles and zeros of the domain can be dual-passband filter at finite frequencies in determined as follows:

Fig. 2. Frequency response of a low-pass filter prototype with the transmission zeros at j 2:0.

6

therefore, the relationship between the band edge frequencies in the domain and the coefficients in (10) are as follows:

(12) From (12), we can express and frequencies of the dual-passband filter

(14)

in terms of band edge

(13) Using (9) and (13), the constant in (7) can also be expressed in terms of band edge frequencies of the dual-passband filter. Therefore, the frequency transformations can be expressed in terms of band edge frequencies of the dual-passband filter. Based on the prescribed narrow passbands of the dual-passband filter, the frequency transformations can be defined and, consequently, the transfer functions and coupling matrices can be determined. Section III describes the application of the frequency transformations and then direct applications for designing dual-passband filters.

Fig. 3 shows the frequency response of the dual-passband filter with poles and zeros in (14). With the given poles and zeros domain, we can obtain the of the dual-passband filter in the characteristic function in the form given by (5). Expanded form of the transfer function can be expressed as follows [4]: (15) where

III. FILTER DESIGN AND MEASUREMENT Here, the filter with two passbands is designed and realized to describe the presented filter synthesis theory. The passbands of the dual-passband filter are chosen to be 3.90–3.95 and 4.05–4.10 GHz. Each passband is set to have four poles and a maximum return loss of 20 dB. Minimum attenuation over stopbands is set to be 30 dB. Since we are seeking a four-pole passband response, we start from a four-pole low-pass prototype. Fig. 2 shows the frequency response of the low-pass filter prototype with the transmission

(16)

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Fig. 3. Frequency response of the dual-passband filter in the domain.

From the transfer function in the domain, the coupling matrix can be determined [3]. Since the eight-pole dual-passband filter has six zeros in finite frequencies, it can be of a canonical structure and its coupling matrix is given by (17), shown at the bottom of this page. The matrix similarity transformation can be applied to the coupling matrix to obtain different combinations of positive and negative inter-resonator coupling coefficients and/or different coupling structures (i.e., asymmetric canonical structure) [13], [14]. Applying the frequency transformation in (10), we can obtain the frequency response of the filter in the domain. Fig. 4 shows the frequency response of the filter. Note that the passbands of the filter are 3.90–3.95 and 4.05–4.10 GHz and the attenuation at the stopbands is 30 dB, which shows the validity of this proposed synthesis method for a dual-passband filter using frequency transformation with prescribed passbands and attenuation at stopbands. The coupling matrix in (17) can be realized for many types of filter structures. In this paper, we use a stripline structure for the filter. Fig. 5 shows the conductor layer of the dual-passband filter. This conductor layer is positioned in the middle of two metal-backed dielectric layers. The thickness, dielectric constant, and loss tangent of the dielectric layers are 1.574 mm, 2.2, and 0.0009, respectively. Open-loop resonators can provide both the electric and magnetic couplings between two resonators and these can be used for realizing both positive and negative coupling, as required by (17). The tapping position

Fig. 4. Frequency response of the dual-passband filter in the ! domain.

Fig. 5. Conductor layer of stripline structure for the dual-passband filter.

is determined by the external coupling coefficient and the distances between two resonators are determined by . The design bandinter-resonator coupling coefficients . The coupling coefficients can be calculated width is by the well-known method described in [15] and, hence, is not repeated in this paper. The full-wave electromagnetic simulator Zeland IE3D is used to calculate the coupling coefficients. The physical dimensions of the filter in Fig. 5 are summarized in Table I.

(17)

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TABLE I PHYSICAL DIMENSIONS OF THE DUAL-PASSBAND FILTER

Fig. 7. Fabricated conductor layer of the stripline structure for the dual-passband filter.

Fig. 6. Synthesized and simulated frequency response of the dual-passband filter.

The filter has also been simulated using IE3D and Fig. 6 compares the simulated and theoretical frequency response of the dual-passband filter. Since the filter synthesis does not take into account the losses of the filter, the loss factors are not included in the simulated response in Fig. 6 for clear comparison. A good agreement between theoretical and simulated frequency responses is shown. The simulated frequency response has somewhat lower attenuation at a higher stopband, which is due to asymmetric locations of transmission zeros. It has been reported that the asymmetric frequency response is attributed to the frequency-dependent couplings [16]. The measured response of the fabricated filter (Fig. 7) is compared to the simulation one in Fig. 8. The simulated response includes loss factors (conductor loss and dielectric loss) in order to take into account the losses of the fabricated filter. Measured response shows a reasonably good agreement with the simulated response. There is, however, a small frequency shift, which can be attributed to the fabrication error. IV. MORE EXAMPLES In Section III, we dealt with an eight-pole dual-passband filter . Here, a four-pole with repeated transmission zeros at dual-passband filter with repeated transmission zeros at

Fig. 8. Frequency response of the dual-passband filter. (a) S . (b) S .

and an eight-pole dual-passband filter with no transmission are briefly discussed. zeros at Fig. 9 shows a two-pole low-pass prototype filter with no transmission zeros at finite frequencies in the domain. The

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Fig. 9. Frequency response of a low-pass filter prototype with no transmission zeros at finite frequencies.

Fig. 10. Frequency response of a dual-passband filter whose low-pass prototype is shown in Fig. 9.

return loss and number of the poles of the low-pass prototype filter determines the location of the poles. Using the frequency , we can obtain the locatransformation in (7) with tion of the poles and zeros of the four-pole dual-passband filter domain and, therefore, its frequency response can be in the obtained as shown in Fig. 10. Based on the locations of the poles domain, the transfer function and coupling and zeros in the matrix can be obtained as described in Section III. The transfer function is (18) where Fig. 11. Frequency response of a low-pass filter prototype with no transmission zeros at infinite frequencies.

(19) and the coupling matrix is

(20) can also The filter with no transmission zeros at be synthesized. This kind of filter can be designed with the low-pass prototype filter having no transmission zeros at infidomain. Fig. 11 shows a four-pole nite frequencies in the low-pass prototype filter with the transmission zeros at and . Using the frequency transformation in (7), we can obtain the locations of the poles and zeros of the dual-passband filter in the domain and, therefore, its frequency response can be obtained as shown in Fig. 12. Based on the locations of the poles and zeros in the domain, the transfer function and coupling matrix can be obtained as described in Section III. Since

Fig. 12. Frequency response of a dual-passband filter whose low-pass prototype is shown in Fig. 11.

the eight-pole filter has eight zeros in finite frequencies, it might employ the coupling between source and load. V. ASYMMETRIC DUAL-PASSBAND FILTER DESIGN We have dealt with a synthesis method for designing symmetric dual-passband filters. The frequency transformation for

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obtain equiripple responses both in the passband and stopband. Fig. 13 shows the frequency transformation from the domain domain. In the domain, we have asymmetric to the passbands. The frequency transformation shown in Fig. 13 can be obtained by rewriting (7) in general form as follows: for for

Fig. 13. Frequency transformation from the domain to the domain using (21).

(21)

where and are determined by the band edge fredomain, which are also determined by arquencies in the bitrarily prescribed passbands of the filter. The upper frequency region and the lower one is bisected by , which can be chosen and . arbitrarily between Fig. 14 shows the frequency response in the domain using the frequency transformation in (21) and a low-pass filter prototype in Fig. 2. It should be noted that the frequency transformation in (21) is very flexible in designing asymmetric dual-passband filters while preserving the attenuation characteristics of the low-pass prototype. Based on the frequency transformation given in (21), the lodomain, cations of the poles and zeros can be found in the which makes it possible to obtain the transfer function and coupling matrix. VI. CONCLUSION This paper has described a synthesis method for a symmetric dual-passband filter. Frequency transformations have been established and applied to the low-pass filter prototype in order to obtain the frequency response of the dual-passband filter. For analytic filter synthesis, the frequency transformations have been given in terms of the prescribed passbands of the dual-passband filter. To validate the presented synthesis method, the eight-pole dual-passband filter with passbands of 3.90–3.95 and 4.05–4.10 GHz has been designed and measured. The frequency response of the designed filter has shown a good agreement with the synthesized frequency response. The frequency transformation for symmetric dual-passband filters has been generalized for asymmetric dual-passband filters. This transformation is found to be flexible enough to allow for designing bandpass filters with two passbands of significantly different bandwidths.

Fig. 14. Frequency responses of asymmetric dual-passband filters in the domain. (a)

= 0:1; = 0:7; = 0:5. (b) = 0:8;

0:2; = 0:3.

0

0

=

asymmetric dual-passband filters is given in [10], and it is possible to obtain two desired passband characteristics. However, it is not clear whether equiripple responses both in the passband and stopband, which enables high-frequency selectivity, can or cannot be achieved. Here, we briefly explain the frequency transformation for asymmetric dual-passband filters. The advantage of this frequency transformation is that the attenuation characteristics of the low-pass prototype is preserved. Therefore, it is possible to

REFERENCES [1] R. M. Kurzok, “General three-resonator filters in waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-14, no. 1, pp. 46–47, Jan. 1966. [2] R. M. Kurzok, “General four-resonator filters at microwave frequencies,” IEEE Trans. Microw. Theory Tech., vol. MTT-14, no. 6, pp. 295–296, Jun. 1966. [3] A. E. Williams, “A four-cavity elliptic waveguide filter,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 12, pp. 1109–1114, Dec. 1970. [4] C. M. Kudsia, “A generalized approach to the design and optimization of symmetrical microwave filters for communications systems,” Ph.D. dissertation, Dept. Eng., Concordia Univ., Quebec, QC, Canada, 1978. [5] G. Pfitzenmaier, “An exact solution for a six-cavity dual-mode elliptic bandpass filter,” in IEEE MTT-S Int. Microw. Symp. Dig., San Diego, CA, 1977, pp. 400–403. [6] D. R. Jachowski, “Folded multiple bandpass filter with various couplings,” U.S. Patent 5 410 284, Apr. 25, 1995.

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[7] S. Holme, “Multiple passband filters for satellite applications,” in 20th AIAA Int. Commun. Satellite Syst. Conf. and Exhibit, 2002, Paper AIAA-2002-1993. [8] J. Lee, M. S. Uhm, and I.-B. Yom, “A dual-passband filter of canonical structure for satellite applications,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 271–273, Jun. 2004. [9] J. Lee, M. S. Uhm, and J. H. Park, “Synthesis of self-equalized dualpassband filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 256–258, Apr. 2005. [10] G. Macchiarella and S. Tamiazzo, “Design techniques for dual-passband filters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3265–3271, Nov. 2005. [11] R. Cameron, M. Yu, and Y. Wang, “Direct-coupled microwave filters with single and dual stopbands,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3288–3297, Nov. 2005. [12] S. Amari, “Synthesis of cross-coupled resonator filters using an analytical gradient-based optimization technique,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1559–1564, Sep. 2000. [13] R. J. Cameron and J. D. Rhodes, “Asymmetric realizations for dual-mode bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 1, pp. 51–58, Jan. 1981. [14] C.-S. Ahn, J. Lee, and Y.-S. Kim, “Design flexibility of an open-loop resonator filter using similarity transformation of coupling matrix,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 262–264, Apr. 2005. [15] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [16] J. S. Hong and M. J. Lancaster, “Theory and experiment of novel microstrip slow-wave open-loop resonator filters,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2358–2365, Dec. 1997.

Juseop Lee (A’02–M’03) received the B.E. and M.E. degrees in radio science and engineering from Korea University, Seoul, Korea, in 1997 and 1999, respectively, and is currently working toward the Ph.D. degree at The University of Michigan at Ann Arbor. In 1999, he joined LG Electronics (formerly LG information and Communications), where his research activities included reliability analysis of RF components for code-division multiple-access (CDMA) cellular systems. In 2001, he joined Electronics and Telecommunications Research Institute (ETRI), where he was involved in designing passive microwave equipment for - and -band communications satellites. In 2005, he joined The University of Michigan at Ann Arbor, where he is currently a Research Assistant with the Radiation Laboratory. His research interests include RF and microwave components, satellite transponders, and electromagnetic theories.

Ku

Ka

Kamal Saranbandi (S’87–M’90–SM’92–F’00) received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1980, and the M.S. degree in electrical engineering and the M.S. degree in mathematics and Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1986, 1989, and 1989, respectively. He is currently Director of the Radiation Laboratory and a Professor with the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor. He possesses 22 years of experience with wave propagation in random media, communication channel modeling, microwave sensors, and radar systems, and is leading a large research group including two research scientists, 12 doctoral students, and two masters students. He has graduated 24 doctoral students and has supervised numerous postdoctoral students. He has served as the Principal Investigator on numerous projects sponsored by the National Aeronautics and Space Administration (NASA), Jet Propulsion Laboratory (JPL), Army Research Office (ARO), Office of Naval Research (ONR), Army Research Laboratory (ARL), National Science Foundation (NSF), Defence Advanced Research Projects Agency (DARPA), and numerous industries. He has authored or coauthored numerous book chapters and over 145 papers in refereed journals on miniaturized and on-chip antennas, metamaterials, electromagnetic scattering, wireless channel modeling, random media modeling, microwave measurement techniques, radar calibration, inverse scattering problems, and microwave sensors. He also has had over 340 papers and invited presentations in numerous national and international conferences and symposia on similar subjects. He is listed in American Men and Women of Science, Who’s Who in America, and Who’s Who in Science and Engineering. His research areas include microwave and millimeter-wave radar remote sensing, metamaterials, electromagnetic wave propagation, and antenna miniaturization. Dr. Sarabandi is a member of the NASA Advisory Council appointed by the NASA Administrator. He has also served as a vice president of the IEEE Geoscience and Remote Sensing Society (GRSS) and as a member of the IEEE Technical Activities Board Awards Committee. He serves as an associate editor for PROCEEDINGS OF THE IEEE and has served as an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE SENSORS JOURNAL. He is also a member of Commissions F and D of URSI. He was the recipient of the Henry Russel Award presented by the Regent of The University of Michigan at Ann Arbor. He was the recipient of the 1999 German American Academic Council (GAAC) Distinguished Lecturer Award presented by the German Federal Ministry for Education, Science, and Technology, which is given to approximately ten individuals worldwide in all areas of engineering, science, medicine, and law. He was a recipient of a 1996 Electrical Engineering and Computer Science (EECS) Department Teaching Excellence Award and a 2004 College of Engineering Research Excellence Award. He was a recipient of the IEEE Geoscience and Remote Sensing Distinguished Achievement Award and The University of Michigan at Ann Arbor Faculty Recognition Award, both in 2005. He was also a recipient of the Best Paper Award presented at the 2006 Army Science Conference. Over the past several years, joint papers presented by his students at numerous international symposia (IEEE APS’95,’97,’00,’01,’03,’05,’06; IEEE IGARSS’99,’02, IEEE MTT-S IMS’01, USNC URSI’04,’05,’06, AMTA’06) have received Student Paper Awards.

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Theoretical and Experimental Studies of Flip-Chip Assembled High-Q Suspended MEMS Inductors Jun Zeng, Changhai Wang, and Alan J. Sangster

Abstract—This paper reports the theoretical and experimental studies of high- suspended microinductors produced by flip-chip assembly for multigigahertz RF integrated-circuit applications. The effects of device and substrate parameters on the factor of the inductor devices are studied by numerical simulation using Ansoft’s High Frequency Structure Simulator electromagnetic field simulation package. Suspended inductor devices are realized using a flip-chip assembly method in which the inductor structures with the supporting pillars are fabricated on a low-cost polyimide thin-film carrier and then assembled onto a low resistivity (3–4 cm) silicon substrate by flip-chip bonding. Individual and 2 2 arrays of meander and spiral inductor designs have been successfully fabricated with air gap heights ranging from 15 to factors of 15 and 13 at 1 GHz have 31 m. Maximum been achieved for meander and spiral suspended inductor devices before pad deembedding. It is shown that the optimal air gap between the inductor and substrate surface is 15 m beyond which no further enhancement in the factor can be obtained for devices on low-resistivity substrates. The experimental results are in excellent agreement with that of theoretical simulation. The inductor assembly method requires minimal chip/wafer processing for integration of high- inductors.



Index Terms—Flip-chip assembly, high , microelectromechanical systems (MEMS), modeling, passive inductor, polyimide film, silicon RF integrated circuit (RFIC).

I. INTRODUCTION NTEGRATION of passive inductors into integrated circuits (ICs) is attractive for RF integrated circuits (RFICs) and monolithic microwave integrated circuits (MMICs) since it provides many benefits in cost, size, performance, and power consumption. Conventional on-chip inductors in RFICs are planar conductor coils fabricated directly on the IC substrate along with active components using the standard IC fabrication technology. The inductors and, hence, the RFICs suffer from poor performance due to the parasitic effects of the low-resistivity silicon substrate. The magnetic energy stored in the inductor is severely dissipated by the substrate, resulting in a low quality ( ) factor of the inductor. To achieve high- inductors on low resistive silicon IC substrates, advanced multilevel IC technologies capable of processing multiple metallic and dielectric layers on the substrate have been employed to produce stacked inductors [1], [2], air bridge inductors [3], inductors with ground shields

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Manuscript received August 18, 2006; revised December 13, 2006. The authors are with the School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. (e-mail: [email protected]. uk). 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.2007.897716

[4], [5], and inductors with thick dielectric isolation [6], [7]. Some improvement in factor has been obtained by reducing either the substrate loss or the ohmic loss of the inductors. An alternative approach is to use novel micromachining techniques to integrate 3-D microelectromechanical systems (MEMS) inductors on the RFIC substrates. Bulk micromachining (bulk etching) techniques are used to selectively remove the lossy silicon substrate underneath the inductor coils, which results in membrane supported high- microinductors [8], [9]. Surface micromachining techniques based on the thick-film photolithography and conductor electro-deposition has been used to fabricate suspended inductors on RFIC substrates for high-performance operation [10]–[12]. Out-of-plane inductors have been developed to reduce the substrate loss and, factor of the devices [13]–[15]. To hence, to improve the obtain out-of-plane devices, planar inductor structures have been produced on the substrate by surface micromachining. The inductor coils are rotated away from the substrate using solder-based surface tension self-assembly [13], interlayer stress self-assembly [14], and the plastic deformation magnetic assembly techniques [15]. Significant improvements in the factors have been achieved using these approaches, but all of the methods involve significant post-processing of the RFIC chips/wafers for inductor integration that not only increases the fabrication cost and time, but also increases the risk of damage to the RFIC wafers. A novel approach for integrating high- inductors on silicon substrates using flip-chip assembly and MEMS fabrication techniques has recently been developed and is reported in [16]. The inductor structure and the supporting pillars are fabricated on a low-cost carrier substrate and then assembled onto a silicon substrate using thermocompression flip-chip bonding. The pillars (bumps) are used to produce an air gap between the inductor and the substrate to reduce the substrate effect and, hence, to achieve a high- factor. In this paper, we present the theoretical and experimental studies of flip-chip assembled high- suspended inductors for RFIC applications. II. MODELING AND SIMULATION A. Inductor Design Suspended meander and spiral inductors have been designed for electromagnetic (EM) simulation and experimental investigation. Fig. 1 shows the schematics of the suspended inductor devices. The spiral design consists of 2.5 turns of conductor track and the meander design consists of 2.5 periods of conductor track. Six and five metallic pillars are used to support the inductor structures for the spiral and meander designs, respectively. Two of the pillars also provide signal pass between

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Fig. 1. Schematic drawing of suspended microinductors. (a) Spiral inductor design. (b) Meander inductor design. TABLE I SUMMARY OF SPIRAL AND MEANDER INDUCTOR DESIGNS SHOWN IN FIG. 1

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Fig. 2. Simulated factors of the suspended spiral inductor on the silicon substrates of different resistivity.

the suspended inductors and the signal lines on the surface of the substrate. The supporting pillars are anchored on the contact pads on the substrate. The meander design is primarily used as a test structure for the investigation of the effect of the air gap factor. Meander between the inductor and substrate on the inductors are seldom used in ICs since they occupy more of the limited chip area than their spiral counterparts for the same inductance. The design parameters for the inductor structures shown in Fig. 1 are summarized in Table I. The substrate material is silicon with a layer of silicon dioxide on the surface. The oxide layer is used to provide electrical insulation between an inductor and the semiconducting silicon substrate. The thicknesses of the substrate and the oxide layer are 380 and 1 m, respectively. The performances of suspended inductors on GaAs substrates have also been studied through EM simulation as GaAs is also a commonly used substrate material for RFICs and MMICs. B. EM Simulation A commercial 3-D EM field solver, Ansoft’s High Frequency Structure Simulator (HFSS), was used to study the RF performance of the suspended inductor devices. The software tool is based on the finite-element method and has been reported to have high accuracy in modeling conventional on-chip IC inductors [17]. The effects of substrate resistivity, air gap thickness, factor conductor material, and thickness on the inductor’s have been studied. In the EM simulation, the models of the meander and spiral inductor designs previously described were built up by using

the graphical user interface of Ansoft’s HFSS. The chosen materials were then assigned to the corresponding objects in the models. Simulation boundaries were defined and ports were assigned to both ends of the inductor devices. The ground plane is on the bottom side of the substrate. The frequency sweep was specified as from 0.2 to 15 GHz with a step size of 0.1 GHz. During the field analysis, metals in the model were meshed automatically and the ports were excited with a voltage potential. The EM analysis engine of Ansoft’s HFSS computed the EM field using the finite-element method at each frequency sweep point. After the analysis was completed, the scattering parameters ( -parameters) of the inductor model were extracted from the field solution. The scattering parameters were then converted into impedance parameters ( -parameters) and admittance parameters ( -parameters). The factor of the inductor at each -pafrequency point was calculated from its characteristic -parameter. rameter or C. Results and Analysis 1) Effect of Substrate Resistivity: The spiral inductor design was used to study the effect of substrate resistivity on the factor. The air gap and conductor thicknesses were chosen to be 20 and 10 m, respectively. The air gap is defined as the distance from the substrate surface to the lower surface of the inductor. The conductor material is gold. The conductivity of gold, 4.1 10 S/m, was obtained from the built-in library of the software package. For each value of substrate resistivity, the factor was obtained using the previously described procedure. Fig. 2 shows the results of the factor as a function of frequency for resistivity values from 4 to 10 k cm. It can be seen that as the substrate resistivity increases, both the inductor’s factor and the peak- frequency increase significantly. The peakfrequency is the frequency at which the factor has a maximum value. This shows the significant detrimental effect of the low substrate resistivity on the inductor’s factor, as previously reported [18]. The peak- factor is 55 at 6.5 GHz for the substrate resistivity of 10 k cm and is 16 at 1 GHz when the substrate resistivity is 4 cm.

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Fig. 3. Simplified one-port lumped element model of on-chip inductor [19].

The frequency behavior of the inductor can be explained from its one-port equivalent circuit, as shown in Fig. 3 [19]. and are the series resistance, inductance, and series capacand are the itance of the inductor device, respectively. resistance and capacitance associated with the substrate. The factor of the inductor is given by [20]

(1) The expression of the factor consists of three multiplied terms. At low frequencies, the first term dominates and the factor of the inductor simply increases with the operating frequency. This term is determined by the frequency, the inductance, and the series resistance of the inductor; therefore, the substrate resistivity has a negligible effect on the factor. However, as the frequency increases, the RF signal starts to couple through the series capacitance to the output end, and shunts through the dielectric capacitance, substrate resistance, and substrate capacitance to the ground. The second and third terms represent the degrading effects of the series capacitance factor to deviate and substrate parasitics, which cause the . These terms start to take effect at higher frefrom quencies and, thus, the increase of the factor with frequency slows down. When the degrading effects of the second and third terms overcome the increasing effect of the first term, the factor of the inductor starts to drop. As in the second term represents the substrate resistance, it is, therefore, a function of the substrate resistivity. Lower substrate resistivity results in a more evident degrading effect of the second term. Thus, as the substrate resistivity decreases, the inductor shows a smaller factor at high frequencies. In the very high-frequency range, the RF signal can couple to the ground easily through the parasitic capacitances without passing though the inductor, causing the decrease of the factor. The third term illustrates the effect of the inductor’s parasitic capacitances. It severely degrades the factor in the very high-frequency band near to the self-resonant frequency at which the factor is 0. By setting the third term to 0, the self-resonance frequency can be obtained as (2)

Q factors of meander suspended inductors with different

Fig. 4. Simulated air-gap thicknesses.

When the frequency is high enough, dominates over . Therefore, the self-resonant frequency is just determined by the series resistance, inductance, and the parasitic capacitances and is independent of the substrate resistivity, as shown in Fig. 2. 2) Effect of Air Gap Thickness: The effect of the air gap on the factor was studied through the simulations on both meander and spiral inductor designs. This was carried out to find out if the factor could be enhanced by introducing an air gap between the inductor and substrate. Fig. 4 shows the factor as a function of frequency for a range of air-gap values for the meander design. The increase in air gap is achieved by varying the height of the supporting pillars. The substrate resistivity and cm and 10 m, thickness of the gold conductor track were 4 respectively. It can be seen that the air gap significantly improves the factor of the inductor at frequencies above 1 GHz. factor is apFor the inductor without an air gap, the peak proximately 9 at 0.5 GHz and then drops rapidly to 0. When an air gap of even 1 m of thickness is inserted between the inductor and substrate, a significant improvement in the factor is achieved. Further increase of the air gap thickness results in a maximum factor of 19 at 1 GHz. A usable factor of 4 is still available even at 5 GHz, while the factor is less than 1 for the inductor without any air gap at the same frequency. The enhancement in the factor at the higher frequency region ( 0.5 GHz) is primarily due to the improved isolation of the inductor from the substrate by the air gap. At lower frequencies, the factor of the inductor is largely determined by its series resistance and, therefore, the air gap has less effect, but as the frequency rises to above 0.5 GHz, the RF signal starts to couple into the substrate through the dielectric layer. The air gap in the suspended inductor acts as an additional dielectric layer to the oxide layer, thus increasing the isolation of the planar inductor track from the lossy substrate. From the equivalent-circuit point of view, the air gap reduces the overall dielectric capacitance since air has the lowest dielectric constant among the IC dielectrics. The reduction of the dielectric capacitance results in the increase of its impedance, which effectively reduces the current leakage from the inductor tracks into the substrate through

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Fig. 5. Simulated factors of spiral suspended inductors with different air-gap heights on a low-resistivity silicon substrate.

the dielectric layer and, hence, improves the factor of the inductor. The maximum enhancement in the factor is achieved for an air-gap value of approximately 20 m, beyond which no further improvement is obtained. This saturation effect with the air gap thickness is associated with the substrate loss due to the segments of the signal lines that are in contact with the substrate. For example, when the signal line segments are removed, a maximum factor of 34 can be obtained at 3 GHz for the air-gap value of 40 m. Similar results were obtained for the spiral inductor design and are shown in Fig. 5. In order to compare the performance of the suspended designs with that of the design with the coil in contact with the oxide layer, the spiral design shown in Fig. 1(a) was inverted (i.e., with an over pass) and used to obtain the characteristics, as presented in Fig. 5, but the suspended design in Fig. 1(a) is convenient for practical implementation as the pillars have the same height and, therefore, can be fabricated in one processing step. As can be seen from Fig. 5, an air-gap height of 3 m is sufficient to produce considerable enhancement in the factor of the inductor device. The optimal air-gap height is approximately 15 m, beyond which no further improvement in the factor can be obtained. The reason that the spiral design is less sensitive to the air gap is that the total area of the conductor track is less than that of the meander design. The substrate loss is proportional to the area occupied by the inductor track [19]. Theoretical simulation has also been carried out for suspended inductors on GaAs substrates to investigate the effect of the air gap on the factor. GaAs is an alternative substrate material to silicon for RFIC and MMIC applications due to its high electron mobility and high substrate resistivity, making it a highly desirable substrate for high-frequency electronic circuits. Fig. 6 shows the factor as a function of frequency for suspended inductors on a semi-insulating GaAs substrate cm. It can be seen that although with a resistivity of 1 10 the substrate is semi-insulating (high resistivity), considerable improvement in the factor can be gained by inserting an air gap between the inductor coil and the substrate. An increase of the factor by a factor of 1.7 can be achieved by increasing the air-gap height from 3 to 40 m. It should be noted that this

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Fig. 6. Simulated factors of suspended spiral inductors on semi-insulating GaAs substrate with different air-gap heights.

Fig. 7. Simulated ductor thicknesses.

Q factors of spiral suspended inductors with different con-

improvement in the factor is due to the reduced capacitive coupling between the inductor and substrate. This is characterized by a much higher peak frequency even for a smaller air gap, in contrast to that shown in Fig. 5 for a spiral inductor on a low-resistivity substrate. 3) Effect of Conductor Thickness: The effect of conductor thickness on the factor of the suspended inductor was studied using the spiral inductor design shown in Fig. 1(a). Numerical simulation was carried out to determine the factor as a function of frequency for different values of the thickness of the gold conductor track. The substrate resistivity and air-gap thickness cm and 25 m, respectively. The rewere maintained at 4 sults of simulation are shown in Fig. 7. The factor increases with the increase of conductor thickness below 3 GHz. This is simply due to the decrease of the series resistance when the conductor thickness is increased. However, thicker conductor tracks have a negligible effect on the factor beyond 3 GHz and this is due to the skin effect at higher frequencies. The optimal conductor thickness is in the region of 7–10 m. Further increase in conductor thickness results in insignificant improvement in

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inductor and substrate, leading to significant enhancement in RF performance, particularly in the multigigahertz frequency region. The optimal air gap height and conductor thickness were found to be 15 and 10 m for achieving the best percm) formance for a suspended inductor on low resistivity (4 substrate. The simulation results also show that considerable improvement in the factor can also be obtained for inductors on semi-insulating GaAs substrates. III. FABRICATION AND ASSEMBLY

Fig. 8. Simulated ductor materials.

Q factors of spiral suspended inductors of different con-

the factor while substantially increasing the manufacturing time and cost. The figure of optimal conductor thickness is also similar to the recommended conductor thickness for RF devices (three to five times of the skin depth at the frequency of interest) [18], [21]. 4) Effect of Conductor Material: Aluminum has been the main conductor material used in IC manufacture due to its compatibility with the IC fabrication technology and its good conductivity. However, with the advent of copper interconnect technology in IC manufacture and MEMS technology for inductor fabrication, copper and gold can be used as the conductor materials for the fabrication of IC-compatible high- microinductors for RFIC applications [2], [12], [22], [23]. To compare the performance of suspended inductors made of different conductors, simulation has been carried out for the suspended spiral inductor design. The substrate resistivity, air gap, and conductor thickcm, 25 m, and 10 m, respectively. Fig. 8 nesses were 4 shows the results of the factor for aluminum, gold, and copper inductor devices. The factors of the aluminum and gold inductors have the same characteristics. This is not surprising since the conductivities of aluminum and gold are almost the same, i.e., 3.8 10 and 4.1 10 S/m, respectively. The factor of the copper inductor is larger than those of the aluminum and gold devices at the frequencies below 2 GHz, as expected, since the conductivity of copper, i.e., 5.96 10 S/m, is approximately a factor of 1.5 larger than those of aluminum and gold. For operating frequencies above 2 GHz, there is no difference in the performance of the inductors due to the skin effect. D. Summary of RF Simulation Theoretical studies have been conducted to investigate the effects of substrate resistivity, air gap, conductor thickness, factor of suspended and the conductor material on the inductor devices. It has been shown that the conduction loss in the low-resistivity silicon substrate is the dominant factor of on-chip inductors. The detrimental governing the low effect of substrate loss on the factor of an inductor can be significantly reduced by inserting an air gap between the

Suspended spiral and meander inductors with the layouts shown in Fig. 1 have been fabricated. To facilitate one-port RF characterization using coplanar probes, ground tracks were added to the designs used for modeling. The original designs allow the modeling of inductor performance without the parasitic effect of a coplanar ground plane. The suspended inductor devices were produced using flip-chip assembly [16]. The inductor structures with the supporting pillars and the coplanar ground planes with signal feedthroughs and bond pads are fabricated separately and then assembled together using flip-chip bonding to obtain suspended inductor devices on silicon substrates. This approach for inductor integration on semiconductor substrates minimizes the requirement of post-processing of RFIC wafers/chips for producing high-performance on-chip microinductors. It allows batch fabrication and selective transfer of the inductor devices onto RFICs. It is also compatible with wafer-level integration. A. Fabrication of Inductor Structures The inductor structures and the supporting pillars were fabricated on polyimide carrier of 125- m thickness using thick-film UV lithography and electroforming. A thin titanium film was deposited onto the polyimide carrier followed by a thin nickel layer, both by electron beam evaporation. The titanium layer was used as an adhesion layer, while the nickel layer as the sacrificial layer for carrier release after inductor assembly. The nickel layer also acts as a seed layer for electroforming of the inductor structure. A photoresist layer (AZ9260, Clariant, Muttenz, Switzerland) was then spin coated onto the carrier and baked on a hotplate at 80 C to produce a photoresist film. The photoresist layer was patterned using UV photolithography to produce a polymer mould for the electrodeposition of conductors. Electroforming was used to deposit gold into the openings of the photoresist mould to produce an inductor structure of similar thickness to that of the photoresist layer. This was followed by deposition and patterning of another layer of photoresist of chosen thickness for the fabrication of the supporting pillars. The thickness of the second photoresist layer was determined by the height requirement for the air gap. After the second electroforming of gold pillars in the photoresist mould, the photoresist and the seed layer were removed by wet etching and a gold inductor structure with supporting pillars for flip-chip assembly was obtained. Fig. 9 shows the micrographs of a fabricated spiral inductor structure and a meander inductor structure with supporting pillars. This process was repeated to achieve the inductor structures having tracks of different thicknesses and pillars of different heights for the specific designs.

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Fig. 9. SEM micrographs of fabricated inductor structures with supporting pillars. (a) Spiral design. (b) Meander design.

Fig. 11. SEM micrographs of the fabricated ground planes with bonding pads: (a) for spiral design and (b) for meander design.

Fig. 10. SEM micrograph of a single copper pillar made with gold and nickel layers on a copper inductor structure. TABLE II SUMMARY OF GROUND-PLANE DESIGNS

The fabrication procedure for a copper inductor device is similar to that previously described. After the fabrication of the photoresist mould, copper electroforming was used to produce the copper inductor structure and the pillars. Thin layers of nickel and gold were then deposited on top of the pillars by electroforming. The thicknesses of the nickel and gold layers were 2 and 6 m, respectively. The gold layer is for thermocompression bonding of the inductor onto the bond pads on the silicon substrate, while the nickel layer is used to prevent the inter-diffusion between the copper pillars and gold layers. Fig. 10 shows the detail of a copper pillar with nickel and gold layers. B. Fabrication of Ground Plane As previously described, coplanar ground planes are required for measurement of the factor and inductance at RF frequencies. The ground planes were designed for one-port probing RF measurements. The design parameters of the coplanar ground planes for the inductor designs shown in Fig. 1 are summarized in Table II. The fabrication process is similar to that for inductor fabrication. Fig. 11 shows scanning electron microscope (SEM) micrographs of the fabricated ground planes with

Fig. 12. Illustration of flip-chip assembly procedure.

bond pads for the spiral and meander inductor designs. The ground-plane design is a rectangular conductor surround with a ground–signal–ground electrode layout. The conductor track is a gold layer of 6- m thickness with titanium and copper thin-film layers ( 200 nm) as the adhesion layer to the substrate and the seed layer for electroforming, respectively. The substrate is a silicon wafer of 380- m thickness with a pre-deposited oxide layer of 1- m thickness. The resistivity of the subcm. strate is approximately 4 C. Flip-Chip Assembly Flip-chip assembly of the inductor structures onto the silicon chips was carried out using a flip-chip bonder (FC6, SUSS MicroTec, Garching, Germany). The schematic of the assembly process is illustrated in Fig. 12. After the alignment of the supporting pillars on an inductor structure to the corresponding bond pads on the silicon chip, the pillars were brought into contact with the pads and bonded together using gold–gold thermocompression bonding. After bonding, the inductor carrier was removed by wet etching the nickel layer to obtain suspended microinductors on the silicon substrate. Fig. 13 shows microscopy pictures of the flip-chip assembled suspended inductors on silicon substrates. The air gap between the inductor and substrate is 31 m for the meander inductor and 16 m for the spiral inductor. The air-gap values were obtained using a Zygo white light interferometer after the carrier was removed. The bonding temperature, force, and time were 240 C, 40 gf/bump, and 30 s, respectively. Similar conditions were used to assemble the 2 2

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Fig. 14. Comparison of simulated and measured factors and inductances for a suspended spiral inductor with 16 m of air gap.

of the characteristic input impedance ( -parameter) of the devices and the inductance were obtained from the Smith chart on the network analyzer over a frequency range from 200 MHz to 10 GHz. The factor was determined using the ratio of the imaginary part and real part of the characteristic impedance (3)

V. ANALYSIS OF MEASURED RESULTS A. Validation of RF Modeling

Fig. 13 (a) SEM micrograph of a suspended meander inductor. (b) Optical picture of a 2 2 array of suspended spiral inductors with polyimide carrier. (c) SEM micrograph of a 2 2 array of spiral inductors without polyimide carrier.

2

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array of spiral inductors. The thicknesses of the conductor tracks are 7 and 10 m for the meander and spiral devices, respectively. The ability to assemble multiple inductor devices onto the substrate simultaneously demonstrates the potential of the technology for parallel integration of inductors onto an RFIC chip/wafer.

In order to validate the EM simulation of suspended microinductors using Ansoft’s HFSS, which was presented in Section II, theoretical simulation was carried out for the fabricated devices and the results are compared with the measured data. Fig. 14 shows the simulated and measured results for a spiral inductor shown in Fig. 13(c). The experimental results factor and inductance were obtained by following of the the above-described procedures, while the simulation results were obtained using Ansoft’s HFSS. It can be seen that the simulation results are in good agreement with the measured results. This illustrates that numerical simulation using Ansoft’s HFSS can accurately predict the RF performance of on-chip microinductors. The results presented below will provide further evidence that the EM simulation can accurately predict the performance of MEMS inductors.

IV. RF CHARACTERIZATION RF characterization of the suspended inductors was carried out by one-port RF measurements using an HP8720 network analyzer and an air coplanar probe (ACP40, Cascade Microtech, Beaverton, OR). Errors associated with the parasitics in the cable, probe, and connectors were removed by using the standard open, short and load calibration on a line, reflect, and match (LRM) impedance standard substrate (ISS 101-190, Cascade Microtech). The values of the real and imaginary parts

B. Effect of Air-Gap Thickness In order to investigate the influence of the air gap between the inductor and substrate on the inductor’s factor, meander devices with different air-gap heights were fabricated to compare the performances. A device without an air gap was also produced for comparison with the suspended devices. The suspended inductors were produced using the fabrication and assembly processes described in Section III. Suspended devices

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Fig. 15. Measured factors and inductances of the meander inductors devices with air-gap heights of 0, 15, and 31 m, respectively.

with air gaps of 15 and 31 m have been assembled. The thickness of the gold conductor track is approximately 10 m for all of the devices. The measured RF characteristics with the corresponding results of simulation are presented in Fig. 15. The air gap has a negligible effect on the inductance, as expected, since the increase in conductor track length for the air-gap devices is insignificant compared to the device without an air gap. The inductance is approximately 1.7 nH for all of the devices. However, there is a considerable enhancement in the factor for the suspended devices due to the reduction of the substrate effect. The inductor suspended 15 m over the substrate shows a maximum factor of 14.6 at 1.2 GHz, while the maximum factor of the inductor fabricated directly on substrate is approximately 9 at 1 GHz. An improvement of a factor of 1.7 in the peak of the meander device was achieved by creating an air gap between the inductor and substrate surface. In the frequency range of 1–5 GHz, significant improvement in the factor due to the insertion of an air gap can also be achieved, particularly in the frequency band of 2.4 GHz that is widely used for wireless communications such as Bluetooth, WiFi, and Zigbee. The factor of the suspended inductor device is a factor of 2 larger than that of the device without an air gap. At the frequency of 4 GHz, the suspended inductors still have a factor of approximately 3. This figure is the same as the maximum factor of a conventional silicon IC inductor at 0.9 GHz [24]. The results also show that the optimal air gap is approximately 15 m, beyond which no further enhancement in the factor can be obtained. This is in agreement with that predicted by theoretical simulation. The experimental results are in good agreement with the corresponding simulation results, further illustrating that the RF modeling is valid. Fig. 16 shows the measured and simulated factors and inductances of the spiral suspended inductors with air-gap thicknesses of 16 and 31 m, respectively. The inductance of the devices is approximately 2 nH for both devices, and the peakfactors are about 13 and 14. As for the meander devices, the experimental results of the spiral devices are also in good agreement the simulation results.

Fig. 16. Measured and simulated Q factors and inductances of spiral inductor devices with air gaps of 16 and 31 m, respectively.

Fig. 17. Measured and simulated Q factors of spiral inductor devices having the conductor thicknesses of 2, 7, and 14 m.

C. Effect of Conductor Thickness In order to study the effect of conductor thickness on the factor of the suspended inductor device, spiral inductors with conductor thicknesses of 2, 7, and 14 m were fabricated and assembled onto the silicon chips. The conductor coils and the supporting pillars were all made of electroplated gold. An air-gap height of 26 m was used for all of the devices to allow the comparison of the inductor performance on variation of conductor thickness. Fig. 17 shows the measured factors as a function of frequency for the devices. It can be seen that the peak- factor of the inductor is increased from 8 to 13 as the conductor thickness is increased from 2 to 7 m. However, on further increasing the conductor thickness from 7 to 14 m, the improvement in the factor is less significant, as predicted by the results of the theoretical investigation. As previously analyzed, the effective cross section for current passing through in a conductor is restricted by skin depth. When the thickness of inductor track is much thicker than the skin depth, further increasing the thickness cannot effectively

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Fig. 18. Comparison of measured and simulated copper and gold inductor devices.

Q factors and inductances of

reduce the series resistance and increase the inductor’s factor. The skin depth of the gold conductor is calculated to be approximately 2.36 m at 1 GHz of frequency. The optimal thickness of the effective cross section for the current flowing in the top surface and bottom surface of a conductor is approximately 4.7 m (the width of the conductor is much larger than its thickness). The track thickness of 7 m is already thicker than the effective thickness of gold conductor of approximately 5 m. Therefore, it is not surprising that no significant improvement in the factor was obtained when the thickness of the inductor track was increased to 14 m. D. Effect of Conductor Material The inductor devices discussed thus far were fabricated using a gold conductor material. However, it is desirable to use copper as conductor material for practical applications. Copper has better conductivity than gold and it is a low-cost conductor material. The theoretical studies also predicted a better factor for the copper inductor than gold- and aluminum-based devices, as shown in Fig. 8. To determine the performance of the copper inductors, copper devices were produced as described in Section III. A spiral suspended inductor with a copper coil and Cu/Ni/Au pillars was assembled for measurement of RF performance. The thickness of the copper coil is approximately 6 m and the thickness of the air gap is approximately 27 m. The thicknesses of the nickel barrier and gold bonding layers were 2 and 6 m, respectively. Fig. 18 shows the comparison of the factor of the copper device and that of a gold inductor with a similar air gap height. The corresponding simulation results are also shown for the gold and copper inductors and are in excellent agreement with the experimental results. The factor is approximately 20% by improvement in the peak using a copper inductor, as predicted by the theoretical simulation shown in Fig. 8. The inductance is approximately 2 nH for both devices. E. Effect of Polyimide Carrier The basic function of the polyimide carrier is only to facilitate the fabrication and assembly of suspended inductor devices,

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Fig. 19. Measured factor and inductance for a suspended microinductor with and without a polyimide carrier and the corresponding theoretical results for the inductor with the carrier.

after that it is removed to obtain freestanding inductors on a substrate. However, it may be desirable to leave the polyimide film in place after the assembly process so that the film can improve the mechanical integrity of the inductor structures, particularly when a thin conductor track is used to reduce the manufacturing time and material cost. Therefore, it is necessary to determine any detrimental effect of the polyimide carrier on the RF performance of the inductor. This effect was studied using the spiral inductor devices shown in Fig. 13(b) and (c). The factor and inductance of a device in the 2 2 array was measured with the polyimide carrier in place [see Fig. 13(b)] and after it was removed [see Fig. 13(c)]. The results are shown in Fig. 19 along with the corresponding simulation results of the device with the carrier film. It can be seen that the polyimide film has no effect on the factor of the device. Therefore, the parasitic effect of the polyimide film is negligible. The film can, therefore, be left in place after inductor assembly in order to improve the mechanical reliability of the inductor structure, e.g., to prevent deformation. F. Pad Deembedding The factor obtained from RF measurement includes the parasitic effects of the ground ring and bond pads. These include the capacitive effect associated with the pads and the conductive and capacitive effects associated with the ground ring. To obtain the actual factor of the inductor device, a two-step deembedding method has been widely used to remove the parasitic effects of the ground ring and probe pads [12]. In the deembedding process, the RF measurement is firstly made for the inductor device with the ground ring and then the same measurement is repeated for a ground ring without the signal lines, bond pads, and inductor. The measured impedances of both the inductor device and ground ring were then converted into admittances. After subtracting the ground ring admittance from the admittance of the inductor device, the resultant admittance is used factor of the inductor device to calculate the deembedded using (3). Fig. 20 shows the experimental and corresponding simulation results of the factor for a spiral inductor before

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VI. CONCLUSION

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Fig. 20. Comparison of the simulated and measured factors of a spiral inductor with and without ground plane and the pad deembedded results.

Fig. 21. Simplified one-port lumped element model of the ground plane, inductor, and signal lines [25].

pad deembedding and the deembedded results. As previously shown, the experimental and simulation results agree well before pad deembedding. On the other hand, as shown in Section II. it is possible to predict the actual factor of the inductor device by numerical simulation that does not require the presence of a coplanar ground ring in the inductor design. The results of the simulation for the spiral inductor are also shown in Fig. 20. It can be seen that the results are significantly different from that obtained using the deembedding method. as described above. The deembedding procedure is based on the assumption that the parasitics of the ground ring with probe pads are in parallel with the lumped elements of the inductor. As shown in Fig. 21, the capacitive ef, and the substrate effect fect of the pads is represented by , and [25]. of the ground ring is represented by However, this deembedding approach, in which the ground ring and pad parasitics are isolated by performing a second scattering measurement on the ground ring on its own, is flawed. While at low frequencies, the this measurement possibly isolates other ground ring parasitics, which are associated with stray fields between the inductor perimeter and ground ring, are not accommodated by the measurement. Since these parasitics will become influential at high frequencies, the gross deviation between the theoretical and deembedded results is not unexpected.

Theoretical and experimental studies have been carried out for flip-chip assembled high- MEMS inductors for RFIC and MMIC applications. The effects of air gap, substrate resistivity, conductor material, and track thickness and the polyimide carrier on the factor of the inductors have been investigated. It factor has been shown that significant enhancement in the of an on-chip inductor can be achieved by inserting an air gap between the inductor and substrate. The improvement in the factor for devices on a low-resistivity substrate is a result of the reduced substrate conduction loss, while for a high-resistivity substrate, it is the reduced capacitive coupling between the inductor and substrate. In addition, using a thick conductor track ( 7 m) or employing copper as conductor material can also improve the factor of the inductor device significantly below the frequency of 3 GHz as a result of the reduction of the series resistance of the inductor. An improvement by a factor of 1.6 was achieved when the conductor track thickness was increased from 2 to 7 m. Individual and 2 2 arrays of high suspended microinductors have been successfully fabricated on silicon substrates using flip-chip assembly. Peak factors of 15 and 13 at the frequency of 1 GHz have been achieved for meander and spiral inductor devices, respectively, before deembedding. The corresponding inductances are 1.7 and 2 nH. The performance of a suspended inductor is better than the IC inductors fabricated by industry adopted copper damascene techniques [26], [27]. The excellent agreement between the experimental and simulation results shows that Ansoft’s HFSS can accurately predict the RF performance of the high- suspended on-chip inductors. It has been shown that the effect of the polyimide carrier on the inductor performance is negligible. This illustrates that it is not necessary to remove the polymer carrier after inductor assembly onto a silicon substrate, simplifying the inductor integration process and reducing manufacturing cost. The light flexible polyimide carrier will also improve the mechanical reliability of the devices. This represents an advantage over similar devices fabricated directly on silicon substrates [12]. Inductor integration by parallel flip-chip assembly reduces post-processing requirement of an RFIC chip/wafer since it only requires the processing of the contact pads for inductor integration. The high- inductors and the associated method of integration onto semiconductor substrates are expected to find applications in next-generation wireless and microwave communications systems. REFERENCES [1] J. N. Burghartz, K. A. Jenkins, and M. Soyuer, “Multilevel-spiral inductors using VLSI interconnect technology,” IEEE Electron Device Lett., vol. 17, no. 9, pp. 428–430, Sep. 1996. [2] M. Soyuer, J. N. Burghartz, K. A. Jenkins, S. Ponnapalli, J. F. Ewen, and W. E. Pence, “Multilevel monolithic inductors in silicon technology,” Electron. Lett., vol. 31, pp. 359–360, Mar. 1995. [3] M. E. Goldfarb and V. K. Tripathi, “The effect of air bridge height on the propagation characteristics of microstrip,” IEEE Microw. Guided Wave Lett., vol. 1, no. 10, pp. 273–274, Oct. 1991. [4] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Si-based RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, May 1998. [5] A. Rofougaran, J. Y. C. Chang, M. Rofougaran, and A. A. Abidi, “A 1 GHz CMOS RF front-end IC for a direct-conversion wireless receiver,” IEEE J. Solid-State Circuits, vol. 31, no. 7, pp. 880–889, Jul. 1996.

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[6] N. Choong-Mo and K. Young-Se, “High-performance planar inductor on thick oxidized porous silicon (OPS) substrate,” IEEE Microw. Guided Wave Lett., vol. 7, no. 8, pp. 236–238, Aug. 1997. [7] K. Bon-Kee, K. Beom-Kyu, L. Kwyro, J. Ji-Won, L. Kun-Sang, and K. Seong-Chan, “Monolithic planar RF inductor and waveguide structures on silicon with performance comparable to those in GaAs MMIC,” in Int. Electron. Device Meeting, 1995, pp. 717–720. [8] J. Y. C. Chang, A. A. Abidi, and M. Gaitan, “Large suspended inductors on silicon and their use in a 2-m CMOS RF amplifier,” IEEE Electron Device Lett., vol. 14, no. 5, pp. 246–248, May 1993. [9] Y. Sun, H. Van Zejl, J. L. Tauritz, and R. G. F. Baets, “Suspended membrane inductors and capacitors for application in silicon MMIC’s,” in Microw. Millimeter-Wave Monolithic Circuits Symp. Dig., San Francisco, CA, Jun. 1996, pp. 99–102. [10] J.-Y. Park and M. G. Allen, “High Q spiral-type microinductors on silicon substrates,” IEEE Trans. Magn., vol. 35, no. 9, pp. 3544–3546, Sep. 1999. [11] L. Fan, R. T. Chen, A. Nespola, and M. C. Wu, “Universal MEMS platforms for passive RF components: Suspended inductors and variable capacitors,” in Proc. MEMS’98, Heidelberg, Germany, Jan. 1998, pp. 29–33. [12] Y. Jun-Bo, C. Yun-Seok, K. Byeong-Il, E. Yunseong, and Y. Euisik, “CMOS-compatible surface-micromachined suspended-spiral inductors for multi-GHz silicon RF ICs,” IEEE Electron Device Lett., vol. 23, no. 10, pp. 591–593, Oct. 2002. [13] G. W. Dahlmann and E. M. Yeatman, “High Q microwave inductors on silicon by surface tension self-assembly,” Electron. Lett., vol. 36, pp. 1707–1708, Sep. 2000. [14] V. M. Lubecke, B. Barber, E. Chan, D. Lopez, M. E. Gross, and P. Gammel, “Self-assembling MEMS variable and fixed RF inductors,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 11, pp. 2093–2098, Nov. 2001. [15] Z. Jun, L. Chang, D. R. Trainor, J. Chen, J. E. Schutt-Aine, and P. L. Chapman, “Development of three-dimensional inductors using plastic deformation magnetic assembly (PDMA),” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1067–1075, Apr. 2003. [16] J. Zeng, A. J. Pang, C. H. Wang, and A. J. Sangster, “Flip chip assembled MEMS inductors,” Electron. Lett., vol. 41, pp. 480–481, Apr. 2005. [17] L. Feng, S. Jiming, K. Telesphor, Y. Yingying, W. Blood, M. Petras, and T. Myers, “Systematic analysis of inductors on silicon using EM simulations,” in Proc. Electron. Compon. Technol. Conf., May 2002, pp. 484–489. [18] C. P. Yue and S. S. Wong, “Physical modeling of spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 47, no. 3, pp. 560–568, Mar. 2000. [19] C. P. Yue, C. Ryu, J. Lau, T. H. Lee, and S. S. Wong, “A physical model for planar spiral inductors on silicon,” in Int. Electron. Device Meeting, 1996, pp. 155–158. [20] J. E. Post, “Optimizing the design of spiral inductors on silicon,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 1, pp. 15–17, Jan. 2000. [21] J.-Y. Park and M. G. Allen, “Packaging-compatible high Q microinductors and microfilters for wireless applications,” IEEE Trans. Adv. Packag., vol. 22, no. 2, pp. 207–213, May 1999. [22] G. Watanabe, H. Lau, T. Schultz, C. Dozier, C. Denig, and H. Fu, “High performance RF front-end circuits for CDMA receivers utilizing BiCMOS and copper technologies,” in RAWCON, Sep. 2000, pp. 211–214. [23] J. N. Burghartz, D. C. Edelstein, K. A. Jenkins, C. Jahnes, C. Uzoh, E. J. O’Sullivan, K. K. Chan, M. Soyuer, P. Roper, and S. Cordes, “Monolithic spiral inductors fabricated using a VLSI Cu-damascene interconnect technology and low-loss substrates,” in Int. Electron. Device Meeting, 1996, pp. 99–102. [24] N. M. Nguyen and R. G. Meyer, “Si IC-compatible inductors and LC passive filters,” IEEE J. Solid-State Circuits, vol. 25, no. 8, pp. 1028–1031, Aug. 1990.

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[25] Y. Jun-Bo, H. Chui-Hi, Y. Euisik, and K. Choong-Ki, “Monolithic high-Q overhang inductors fabricated on silicon and glass substrates,” in Int. Electron. Device Meeting, 1999, pp. 753–756. [26] B. H. W. Toh, D. W. McNeil, and H. S. Gamble, “Characterisation of copper inductors fabricated by dual damascene and electroplating techniques,” J. Mater. Sci.: Mater. Electron., vol. 16, pp. 233–238, May 2005. [27] D. C. Edelstein and J. N. Burghartz, “Spiral and solenoidal inductor structures on silicon using Cu-damascene interconnects,” in Proc. IEEE Interconnect Technol. Conf., Jun. 1998, pp. 18–20.

Jun Zeng received the Bachelor’s degree in electrical engineering from National Huaqiao University, Quanzhou, China, in 2001, the M.Sc. degree in microsystems engineering from Heriot-Watt University, Edinburgh, U.K., in 2003, and is currently working toward the Ph.D. degree at Heriot-Watt University. His current research interests include modeling, fabrication and packaging of MEMS devices.

Changhai Wang received the B.Sc. degree in semiconductor physics and devices from Jilin University, Changchun, China, in 1985, and the M.Sc. degree in opto-eletronic and laser devices and Ph.D. degree in low power all-optical switching devices from HeriotWatt University, Edinburgh, U.K., in 1988 and 1991, respectively. He is currently a Lecturer of electrical and electronic engineering with the School of Engineering and Physical Sciences, Heriot-Watt University. He has authored or coauthored over 60 conference and journal papers. His current research interests include fabrication and assembly of microstructures, MEMS devices and sensors, MEMS packaging, laser-assisted processes for MEMS, and electronics manufacture and biochip technology. Dr. Wang was the recipient of a 1997/1998 Royal Society of Edinburgh Enterprise Fellowship in Optoelectronics.

Alan J. Sangster received the B.Sc. degree in electrical and electronic engineering and M.Sc. and Ph.D. degrees from the University of Aberdeen, Aberdeen, U.K., in 1963, 1964, and 1967, respectively. He spent four years with Ferranti plc, Edinburgh, U.K., where he was involved with research on wideband traveling-wave tubes. He then spent three years with Plessey Radar Ltd., Cowes, U.K., where he investigated and developed microwave devices and antennas for microwave landing systems and frequency scanned radar systems. Since 1972, he has been with Heriot-Watt University, Edinburgh, U.K., where he became Professor of electromagnetic engineering in 1989. He has authored over 200 papers and presentations. His current research interests are microwave antennas, millimeter-wave sensing, electrostatically driven micromotors and microactuators, electromagnetic levitation, medical applications of microwaves, and the numerical solution of electromagnetic radiation and scattering problems. Dr. Sangster is a Fellow of the Institution of Electrical Engineers (IEE), U.K. He has been a member of the Electromagnetics Academy of New York since 1993.

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Mutual Synthesis of Combined Microwave Circuits Applied to the Design of a Filter-Antenna Subsystem Michaël Troubat, Stéphane Bila, Marc Thévenot, Dominique Baillargeat, Member, IEEE, Thierry Monédière, Serge Verdeyme, Member, IEEE, and Bernard Jecko, Member, IEEE

Abstract—A mutual-synthesis approach is presented for the design of subsystems combining two microwave circuits. In this paper, we are focused on subsystems combining a filter and an antenna. In a first step, the methodology consists of synthesizing the global subsystem, which realizes both filtering and radiation functions, with respect to overall specifications. Considering circuits interactions within the subsystem, the resulting constraints are distributed optimally between the two circuits. The latter mutual synthesis then leads, on one hand, to a set of optimal impedances for their connection and, on the other hand, to the corresponding ideal characteristics of each circuit. A solution can then be selected by carrying out a compromise between the simplicity of achieving the characteristics of each circuit and the performances and compactness of the subsystem. In a second step, the circuits are designed independently with respect to the synthesized characteristics. Compared to a classical synthesis approach, where circuits are synthesized independently with respect to a common reference impedance (generally 50 or 75 ), the proposed mutual synthesis allows to simplify the global subsystem since interactions between the two circuits are considered and used optimally. The technique is illustrated with the design of a filter-antenna at -band.



Index Terms—Antennas, filters, synthesis.

I. INTRODUCTION

T

HE DESIGN of microwave and millimeter-wave systems has to face constraints of electrical performances, weight, size, and cost. These constraints are reported on each circuit of the system, defining particular specifications for each of them. Defining a common reference impedance (50 or 75 ) at the input/output ports for connecting the circuits, each of them can be designed separately. When two circuits, which have been designed separately, are then connected together, if their respective input/output ports are not perfectly matched with the common reference impedance, the maximum achievable return loss at the input/output ports of the subsystem is the worst of the two circuits. Otherwise, for achieving input/output impedances suited with the reference impedance, matching networks are often used with the circuits, increasing the complexity of the overall system, together with its weight, size, and losses. Moreover, it is well known that the matching condition for connecting two circuits is to present conjugate input/output impedances at connecting ports. As a consequence, after the first circuit is designed, the second one can be matched to its port Manuscript received November 7, 2006; revised January 22, 2007. The authors are with the XLIM, Unité Mixte de Recherche, Center National de la Recherche Scientifique 6172, University of Limoges, 87060 Limoges, France (e-mail: [email protected]; [email protected]; marc.thevenot@ unilim.fr; [email protected]; [email protected]; verdeyme@xlim. fr; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.897719

impedance. Such an approach is used for manifold multiplexers [1], [2], where a channel filter needs to see a conjugate match into the manifold. However, applying such a strategy may be uncomfortable since all the matching constraints are reported on the design of the second circuit. This paper aims to show that, using optimal input/output impedances between two circuits, the design of each circuit is simplified and the overall subsystem can also attain better performances. The main difference with a classical approach is that the definition of the connection impedance results in a compromise between the circuits. In particular, we are interested in designing a subsystem, which combines a filter and an antenna. Generally, matching an antenna to a classical 50- impedance all over the radiation frequency band is a difficult task, especially when a high gain is required. Generally the input impedance is just transformed using a matching network as in the real frequency technique pioneered by Yarman and Carlin [3] and Newman [4]. In this study, the main differences with the latter technique are that the frequency filtering characteristics (selectivity, bandwidth, etc.) are incorporated in the synthesis approach, and that both the impedances presented by the antenna and the filter are optimized for simplifying the architecture of the subsystem. The possibility of designing subsystems that integrate filtering and radiating functions has been already proven [5], [6]. In this paper, we focus on optimally associating the circuits by synthesizing the overall subsystem. Interactions between the filter and antenna are considered in this way and the electrical constraints are distributed optimally between the two circuits. The connection impedance is chosen so as to reduce the design constraints for both circuits. Consequently, the design and architecture of the resulting subsystem are simplified, leading to more compact devices. Applying this mutual-synthesis procedure, both the optimal connection impedance and the ideal characteristics of each circuit can be derived. Section II describes the electrical characteristics of a filter-antenna subsystem and the mutual synthesis methodology. Section III deals with the application of the methodology on an electromagnetic bandgap (EBG) antenna associated with a cavity filter. The overall performance of separately optimized filters and antennas is then compared to the performance of the jointly optimized filter antenna. Finally, Section V presents experimental verification. II. ELECTRICAL CHARACTERISTICS OF A FILTER-ANTENNA SUBSYSTEM The proposed method for synthesizing filter-antenna subsystems consists of four steps.

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Fig. 3. Ideal reflection function.

Fig. 1. Mutual synthesis procedure.

Fig. 2. Filter-antenna subsystem.

The first step is the definition of subsystem characteristics and specifications. The second step is devoted to the antenna synthesis. The goal is to attain the radiation specifications and to define the admissible domain of input impedance for matching the antenna. The third step is related to the filter synthesis. In this step, the goal is to attain the required selectivity and to define the domain of output impedance for matching the filter. The final step is to define an optimal impedance for matching the two circuits, considering their admissible domains, and then to optimize the subsystem. A flowchart, given in Fig. 1, summarizes the mutual synthesis procedure. The proposed approach is now detailed in the following Sections II-A–D. A. Characteristics of the Subsystem A filter-antenna subsystem allows to radiate an incident wave, as depicted in Fig. 2. The reflected power at the input of the subsystem is related to its scattering parameter (1) where is the power at the input of the subsystem and is the radian frequency. Considering a lossless subsystem, the total accepted power is also related to by (2) The total accepted power can be written in the following integral form: (3)

Fig. 4. Normalized ideal radiation function

H.

where is the radiated power and and are the spherical coordinates. The radiated power in one direction, i.e., , is related to the directivity in the same direction, i.e., , and to the total accepted power (4) Now the normalized radiated power function by

is defined

(5) Combining (2), (4), and (5), the radiation function and to the -parameter related to the directivity

is then (6)

The ideal reflection function pattern is presented in Fig. 3. The function is characterized by the bandwidth, return loss level, and selectivity. The ideal radiation function is given in Fig. 4. This function is determined by three electrical characteristics of the subsystem: the gain, bandwidth, and selectivity. Each characteristic of the subsystem implies constraints, which can be reported either on the filter or antenna. Table I presents the distribution of constraints between the two circuits. The constraints now have to be reported on the two circuits with respect to Table I, and the circuits have to be matched to each other. In order to realize the matching between the two

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TABLE I REPARTITION OF CONSTRAINTS ON SUBSYSTEM CIRCUITS

circuits, the solution is to present conjugated impedances at the common port. The design characteristics related to each circuit are presented below in order to introduce the mutual synthesis approach. B. Antenna Characteristics An antenna is characterized by its gain, sidelobes, and input matching. The gain can be tuned by adjusting the dimensions of the radiating aperture. Classically, antenna designers are looking for important gains, e.g., antennas with gains higher than 20 dB. Moreover, the input matching is directly due to the excitation system. Thus, by adjusting the excitation system, it is possible to improve the matching of the antenna. Fig. 5. Chebyshev filtering functions with the same out-band selectivity.

C. Filter Characteristics A microwave filter is characterized by its transfer function . Classically, the order of the transfer function is selected in order to fulfill specifications both on the in-band return-loss level and on out-band selectivity. Considering a given selectivity, the return-loss level can be improved for matching the standard reference impedance by increasing the order of the transfer function, as shown in Fig. 5 for Chebyshev functions. Considering that out-band selectivity is mainly due to the filter, the approach is to find a transfer function respecting the requested global selectivity. However, in our case, the return-loss level has to be matched with the input impedance presented by the antenna.

of the filtering function, as explained before. On the other hand, the antenna return loss can be adjusted by the excitation system. The mutual synthesis approach then allows to choose a common return loss level, i.e., a common reference impedance for connecting the two circuits. This common return-loss level is selected in order to optimize the design of each circuit, consequently reducing the complexity of the overall subsystem. The mutual synthesis is now illustrated with the design of a subsystem combining an EBG antenna and a rectangular cavity filter. III. APPLICATION

D. Mutual Synthesis The critical point in associating a filter and an antenna is the return loss at the input/output port of each circuit. With the proposed mutual synthesis, the matching problem will be reported both on the filter and antenna. The condition for matching the circuits is to obtain conjugated input/output impedances at connection ports. As a consequence, in term of scattering parameters, the circuits will have to satisfy

and (7) where and are scattering parameters of the circuits at connection ports. For obtaining the previous matching condition, the filter return loss can be adjusted by choosing appropriately the order

A. Description of the Subsystem The chosen structure is presented in Fig. 6. The subsystem combines an EBG antenna [7] and a cavity filter, which are circuits currently used in space telecommunication systems. The circuits are connected directly together through a waveguide without any matching network. The EBG antenna is composed of three dielectric slabs combined to a metallic plane, arranged so that a leaky defect mode is created at 12 GHz. The metallic plane allows the application of image theory. Moreover, a nonstandard waveguide supplies two slots for exciting the antenna through the metallic plane. The half-wavelength distance between the slots forbids the horizontal radiation [8]. Indeed, the slot network allows destructive interferences in far field in the -plane. The filter is composed of rectangular waveguide cavities mode. Cavities are coupled together working on the

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Fig. 8. Antenna return loss tuned by slot dimensions. TABLE II DIFFERENT FILTER RETURN LOSSES Fig. 6. Filter-antenna subsystem.

Slot dimensions can then be modified in order to tune the antenna return loss. Fig. 8 shows three cases of antenna returnloss characteristics corresponding to three slot dimensions. By tuning the slot dimensions, it is possible to obtain return losses between 2–10 dB over the whole bandwidth. C. Pre-Design of the Filter

Fig. 7. Antenna principal plane patterns at 12 GHz.

and to a standard waveguide at the input of the subsystem with rectangular irises. For designing the subsystem, global specifications have been defined as follows: • radiation in the free space with a gain of 20 dB; • sidelobes lower than 10 dB from the maximum; • 600-MHz bandwidth around 12 GHz (5%); • attenuation of 20 dB in the stopband at 450 MHz from the center frequency; • acceptable return loss at the input of the subsystem in the bandwidth (typically 15 dB). Since the global specifications are defined, a pre-design of the two circuits can be realized. During these pre-designs, the matching impedance domain of each circuit will be defined. A common value intersecting the two domains will then be chosen for synthesizing the subsystem. B. Pre-Design of the Antenna Two specifications are reported on the antenna: the maximum gain to be reached and the maximum sidelobe’s level. In order to reach the 20-dB gain, the dielectric slab dimensions have to be tuned. 3.7- Plexiglas slabs seem to be a good compromise to obtain the desired directivity [8]. Fig. 7 shows the radiation of the antenna in free space, in planes and . We note a 15-dB isolation with the secondary lobes.

As the objective selectivity is set, the filtering function can be selected. As explained before, the selectivity can be obtained with many orders and classes of filtering functions. The only difference between the functions is the ripple obtained in the passband, i.e., the achievable return-loss level. The filtering function selection then has a direct impact on the design of the antenna: e.g., if the filtering function is chosen to be a Chebyshev function of order 6, as illustrated in Fig. 5, the antenna will have to present a return loss around 18 dB. Since it is easier to design an antenna with a lower return loss, a Chebyshev function with a lower order will be selected for the filter. Moreover, a lower order is interesting since it is smaller and easier to design. Table II presents several return-loss characteristics that have been synthesized from different orders of filtering functions. All filtering functions allow to obtain the required selectivity. Return-loss levels are found to be from 2 to 18 dB for filtering functions of orders 3–6. D. Mutual Synthesis Applying the mutual synthesis approach, a common returnloss level has to be defined for combining the two circuits. Selecting the common return-loss level is critical in order to simplify the design of the circuits since, on one hand, a lower return-loss level allows to reduce the number of resonators for the filter and, on the other hand, an almost constant return-loss value can be obtained all over the passband at the input of the antenna. However, the common return-loss level should not be too weak, otherwise matching the two circuits becomes impossible. 5-dB return loss seems to be a good compromise for matching the circuits. Of course, matching the circuits at 12 dB is also

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Fig. 9. Return loss of the subsystem (hybrid EM-circuit model). Fig. 10. Return loss of the subsystem (EM model).

possible, but a fifth-order filtering function, i.e., a filter with five cavities, will be necessary, increasing the size of the subsystem. With respect to the previous specifications, the fourth-order Chebyshev filter and the antenna with 10.9-mm slots are synthesized. A coupling matrix can be synthesized from the previous filtering function applying a classical method [9]

The latter coupling matrix is directly related to the equivalent lumped-element circuit of the filter. Models of each circuit are then combined on a circuit software, defining a hybrid electromagnetic-circuit (EM-circuit) model of the subsystem: the filter is modeled by its equivalent lumped-element circuit, the waveguide with an analytical model, and the antenna with finite-difference time-domain (FDTD) simulation data. For optimizing the subsystem, the lumped elements (i.e., the coupling matrix) and the waveguide length between the filter and antenna have to be adjusted. A 20-dB return loss is finally obtained in the passband of the subsystem. Fig. 9 shows the return-loss characteristic obtained after optimization with the hybrid model of the subsystem. The optimized coupling matrix is then

output coupling after optimization (0.573) is still close to its initial value (0.617), justifying the choice for the initial coupling matrix. An electromagnetic (EM) model now has to be tuned in order to determine the optimal dimensions of the filter. The initial dimensions of the filter are computed applying an EM synthesis of each distributed element [11]. The EM model of the filter is then optimized applying a parameter-extraction procedure: after each simulation, the coupling matrix is extracted from the scattering parameters. The geometrical dimensions are then adjusted in order to converge toward the coupling matrix optimized with the hybrid model of the subsystem. Once the extracted coupling matrix is sufficiently close to the previous one, the EM model is connected to the model of the antenna defining a full EM model of the subsystem. The waveguide length is adjusted in order to tune the phase between the circuits. Fig. 10 shows the return loss obtained with the full EM model of the subsystem. Five resonances can be seen in the passband of the subsystem. The fifth resonance is brought by the transition waveguide between the two circuits. This waveguide becomes a resonating cavity when the antenna is connected. Indeed, its length is approximately a half-wavelength. Fig. 11 compares return losses, respectively, at the output port of the filter and at the input port of the antenna. One can observe that scattering parameters are conjugated, satisfying (7). Theoretically, the synthesis method is validated. Indeed, the filtering specifications are respected, and concerning the radiation specifications, the subsystem presents the objective gain and objective selectivity, as shown in Fig. 12. E. Comparison With a Classical Synthesis Approach

One can observe that input and output normalized couplings (1.037 and 0.573, respectively) are asymmetrical, similar to singly terminated filters [10], due to the difference between input and output impedances of the circuit. Moreover, the

In order to prove the efficiency of the proposed approach, a classical synthesis approach is applied. In a classical approach, the antenna with the best achievable return loss is selected. The antenna with 12.59-mm slots leads to a return loss around 12 dB, as shown in Fig. 8. Three filters of orders 4–6 are synthesized with respect to a 50- reference and associated to this antenna. The filters are synthesized in order to respect the out-band selectivity of the

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Fig. 11. Filter and antenna S -parameters at connection ports.

Fig. 13. Combined and separate methods. (a) S -parameter. (b) Gain.

Fig. 12. Global radiation function of the subsystem.

normalized radiated power and to obtain an equiripple transfer function in the passband. The filters are then connected directly to the antenna, optimizing the waveguide section between the circuits. Fig. 13 compares the performances (radiated power and return loss) obtained applying the classical approach and the mutual synthesis. For an equivalent radiated selectivity, the mutual synthesis leads to a global return loss better than with a classical approach. Indeed, the two subsystems designed with the classical approach only reach 8- and 10-dB return losses. Furthermore, if the return loss is improved at the input of the antenna, using, for example, a matching network, a filter with at least six coupled resonators will be necessary in order to achieve theoretically more than 15 dB of return loss at the input of the subsystem with a classical approach. The latter comparison shows the efficiency of the mutual synthesis approach, which is based on the selection of the optimal impedance for connecting two circuits.

Fig. 14. Antenna gain.

IV. MEASUREMENTS The synthesized antenna has been first manufactured for char. Far-field acterizing its gain and its scattering parameter measurements have been performed in an anechoic chamber. Fig. 14 shows the gain of the antenna. The measured gain is approximately 17 dB, i.e., 3 dB lower than expected. The discrepancy is mainly due to losses in Plexiglas slabs (the material loss tangent is estimated around 8.10 ). Fig. 15 presents the scattering parameter measured at the input port of the antenna. The measurement shows good agreement with the simulation, even if a little shift of the slots resonance modifies the return-loss level and phase. The EM model of the filter has been adjusted in order to fit the measured data. However, the model shows that only a 15-dB return-loss level can be attained in this case with the subsystem.

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The filter has been built and connected to the antenna. The behavior of the subsystem is slightly adjusted thanks to tuning screws in filter cavities. Subsystem measurements are given in Fig. 16. The subsystem measurements present globally good agreement with simulations. Indeed, the bandwidth is approximately 5.5% around a center frequency at 11.9 GHz, the return loss is approximately 13 dB, and the sidelobes are better than 15 dB. Moreover, the selectivity in the upper frequency band is lower than expected. These differences can be explained by fabrication tolerances and by losses that have not been taken into account during the of the cavities is apdesign. The estimated quality factor proximately 4000, and this implies a slight impedance mismatch at the antenna–filter interface. Nevertheless, the principle of designing a subsystem applying mutual synthesis approach has been verified. Indeed, it is possible to synthesize a subsystem with good radiating and filtering characteristics by selecting an optimal matching impedance for connecting the two circuits. Fig. 15. Antenna S -parameter.

V. CONCLUSION A mutual synthesis approach has been proposed in this paper for the design of subsystems combining radiating and filtering functions. The approach is based on the selection of an optimal matching impedance for connecting the circuits. The mutual synthesis allows to attain electrical specifications while reducing the complexity of both circuits. The method has been applied to the design of a filter antenna -band. The design has been verified subsystem working at by measurements. The objective of this study is to synthesize subsystems with higher gains. Indeed, with this synthesis approach, the antenna return loss is no longer a crucial problem. By synthesizing the appropriate filter, the transmission channel can be well matched and the subsystem can be more compact. REFERENCES

Fig. 16. Filter-antenna measurements. (a) S -parameter. (b) Gain. (c) Radiation diagram.

[1] J. D. Rhodes and R. Levy, “Design of general manifold multiplexers,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 2, pp. 111–123, Feb. 1979. [2] M. Chen, F. Assal, and C. Mahle, “A contiguous band multiplexer,” COMSAT Tech. Rev., vol. 6, no. 2, pp. 285–306, Fall 1976. [3] B. S. Yarman and H. J. Carlin, “A simplified ‘real frequency’ technique applied to broadband multistage microwave amplifiers,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 12, pp. 2216–2222, Dec. 1982. [4] E. H. Newman, “Real frequency wideband impedance matching with non minimum reactance equalizers,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3597–3603, Nov. 2005. [5] F. Queudet, I. Pele, Y. Mahe, and S. Toutain, “Integration of pass band filter in patch antenna,” presented at the Eur. Microw. Conf., Oct. 2002. [6] H. Blondeaux, D. Baillargeat, P. Leveque, S. Verdeyme, P. Vaudon, P. Guillon, A. Carlier, Y. Cailloce, and E. Royeaux, “Microwave device combining filtering and radiating functions with circular polarization for telecommunication satellites,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 3, pp. 1721–1724. [7] M. Thévenot, C. Cheype, A. Reinex, and B. Jecko, “Directive photonic bandgap antennas,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2115–2122, Nov. 1999. [8] R. Chantalat, P. Dumon, M. Thévenot, T. Monédière, and B. Jecko, “Interlaced feeds design for a multibeam reflector antenna using a 1-D dielectric PBG resonator,” in IEEE AP-S Int. Symp./USNC/CNC/URSI North Amer. Radio Sci., Jun. 2003, pp. 867–869.

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[9] R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 4, pp. 433–442, Apr. 1999. [10] M. H. Chen, “Singly terminated pseudo-elliptic function filter,” COMSAT Tech. Rev., vol. 7, pp. 527–541, 1977. [11] S. Bila, D. Baillargeat, S. Verdeyme, M. Aubourg, P. Guillon, F. Seyfert, J. Grimm, L. Baratchart, C. Zanchi, and J. Sombrin, “Direct electromagnetic optimization of microwave filter,” IEEE Micro, vol. 2, no. 1, pp. 46–51, Mar. 2001.

Michaël Troubat was born in Rochefort sur mer, France, in March 1979. He received the Ph.D. degree from the University of Limoges, Limoges, France, in 2006. His current research interests includes the design of multifunction (radiating and filtering functions) devices and subsystems for RF applications.

Stéphane Bila was born in Paris, France, in September 1973. He received the Ph.D. degree from the University of Limoges, Limoges, France, in 1999. He then held a post-doctoral position for one year with the Centre National d’Etudes Spatiales (CNES), Toulouse, France. In 2000, he became a Researcher with the National de la Recherche Scientifique (CNRS), Limoges, France. He then joined the Microwave Circuits and Devices Team with the Institut de Recherche en Communications Optiques et Microondes (IRCOM), Limoges, France. His research interests include numerical modeling and computer-aided techniques for the advanced synthesis and design of microwave components and circuits.

Marc Thévenot was born in Limoges, France, in February 1971. He received the B.S. and M.Sc. degrees in microwaves and Doctor degree in electronic from the University of Limoges, Limoges, France, in 1995, 1995, and 1999, respectively. In 2001, he joined the National de la Recherche Scientifique (CNRS), Limoges, France. His current research interest is microwave electromagnetism applied to the antenna domain and photonic bandgap (PBG) materials for microwaves.

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Dominique Baillargeat (M’04) was born in Le Blanc, France, in 1967. He received the Ph.D. degree from the Institut de Recherche en Communications Optiques et Microondes (IRCOM), University of Limoges, Limoges, France, in 1995. From 1995 to 2006, he was an Associate Professor with the Micro et Nanotechnologies pour Composants Optoélectroniques et Microondes (MINACOM) Department, XLIM Laboratory, University of Limoges, where he is currently a Professor. His field of research concerns the development of methods of design for microwave devices. These methods include computer-aided design (CAD) techniques based on hybrid approach coupling electromagnetics, circuits and thermal analysis, synthesis and EM optimization techniques, etc. He is mainly dedicated to the packaging of millimeter-wave and opto-electronics modules and to the design of millimeter original filters based on new topologies, concepts (EBG, etc.) and/or technologies (silicon, low-temperature co-fired ceramic (LTCC), etc.).

Thierry Monédière was born in Tulle, France, in 1964. He received the Ph.D. degree in electronic from the University of Limoges, Limoges, France, in 1990. His main research interest is the study of multifunctions antennas. He is also involved with EBG material and their application to antennas and also on beamforming using the Butler matrix.

Serge Verdeyme (M’99) was born in Meilhards, France, in June 1963. He received the Doctorat degree from the University of Limoges, Limoges, France, in 1989. He is currently Professor with the XLIM Laboratory, University of Limoges, and Head of the Micro et Nanotechnologies pour Composants Opto-électroniques et Microondes (MINACOM) Department. His main area of interest concerns the design and optimization of microwave devices.

Bernard Jecko (M’91) was born in Trelissac, France, in May 1944. He received the Science Physic Doctor degree in electronics from the University of Limoges, Limoges, France, in 1979. He is currently a Professor with the University of Limoges, where he manages the Ondes et Systèmes Associés (OSA) Department, XLIM. His main interest is the study of scattering problems of EM waves, particularly in the time domain.

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Analysis of Multiconductor Coupled-Line Marchand Baluns for Miniature MMIC Design Chin-Shen Lin, Student Member, IEEE, Pei-Si Wu, Student Member, IEEE, Mei-Chao Yeh, Jia-Shiang Fu, Student Member, IEEE, Hong-Yeh Chang, Member, IEEE, Kun-You Lin, Member, IEEE, and Huei Wang, Fellow, IEEE

Abstract—The analysis and systematic design procedure for multiconductor coupled-line Marchand baluns are presented in this paper. A simple two-conductor coupled-line model is used to analyze the Marchand balun and simplify the analysis significantly. Two monolithic balanced frequency doublers with miniature Marchand baluns are implemented to verify the design procedure. Both the chips achieve the smallest chip sizes at their operating frequencies with comparable performance. Index Terms—Balun, diode, frequency doubler, Marchand balun, monolithic microwave integrated circuit (MMIC).

I. INTRODUCTION

B

ALUNS ARE important components for many microwave and millimeter-wave circuits such as balanced mixers, frequency doublers, and balanced amplifiers. There are various balun configurations reported for monolithic microwave integrated circuit (MMIC) applications. Marchand baluns have broadband performance and can be easily implemented on planar structures and, therefore, are suitable for MMIC design. The planar Marchand baluns, which are composed of two coupled-line sections, have been realized by using multilayer coupled structure [1], two-conductor edge-coupled lines [2], three-conductor edge-coupled lines [3], multiconductor edge-coupled lines (i.e., Lange couplers) [4], and spiral-shaped coupled lines [5]. In order to broaden the bandwidth of the Marchand balun, the coupled-line sections with a high coupling coefficient are desired. Multiconductor coupled lines were often used to achieve a high coupling coefficient in the planar structure. The bandwidth analysis for three-conductor coupled-line Marchand baluns has Manuscript received October 17, 2006; revised February 5, 2007. This work was supported in part by the National Science Council of Taiwan, R.O.C., under Grant NSC 95-2752-E-002-003-PAE, Grant NSC 95-2219-E-002-009, and Grant NSC 95-2218-E-002-057, and by National Taiwan University under the Excellent Research Project 95R0062-AE00-00. C.-S. Lin, K.-Y. Lin, and H. Wang are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]. ntu.edu.tw; [email protected]). P.-S. Wu and M.-C. Yeh were with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. They are now with the Realtek Semiconductor Corporation, Hsinchu 300, Taiwan, R.O.C.. J.-S. Fu was with the Department of Electrical Engineering and Computer Science, National Taiwan University, Taipei 10617, Taiwan, R.O.C. He is now with the The University of Michigan at Ann Arbor, Ann Arbor, MI 48109 USA. H.-Y. Chang is with the Department of Electrical Engineering, National Central University, Jhongli City, Taoyuan County 32001, Taiwan, R.O.C. Digital Object Identifier 10.1109/TMTT.2007.897689

been reported [6], but there is lack of analysis for multiconductor coupled-line Marchand baluns. The multiconductor coupled-line Marchand baluns are usually designed using circuit simulators. However, it is not easy to tune the multiple parameters of the balun to achieve desired performance. A computer-aided analysis and design procedure for a planar multilayer Marchand balun was reported in [1] with complex design equations, but there is no systematic design procedure to design multiconductor coupled-line Marchand baluns. In this paper, the relation between the Marchand balun bandwidth and two-conductor-coupled-line coupling coefficient is investigated. We also demonstrate that the multiconductor coupled lines can be simplified as a pair of equivalent two-conductor coupled lines. The equivalent two-conductor coupled lines are generally asymmetrical coupled lines and can be analyzed with and modes [9]. By assuming the phase velocities of the and modes to be the same, the impedances of the equivalent two-conductor coupled line can be extracted to calculate the coupling coefficient. A systematic design procedure for the multiconductor coupled-line Marchand balun is then proposed, and the design parameters can be determined easily. Using this design procedure, two MMIC balanced frequency doublers are demonstrated and achieve small chip sizes with comparable performance. II. ANALYSIS AND DESIGN PROCEDURE OF MULTICONDUCTOR COUPLED-LINE MARCHAND BALUNS A. Marchand Balun Analysis If the termination impedances of an asymmetrical coupled line are chosen to be equal to the characteristic impedances of the individual uncoupled lines, the propagation constant of the and modes are assumed to be equal. This assumption will be examined later. The four-port -parameters of the asymmetric two-conductor coupled lines, which has the electrical length of , is given by (1)

where

and and is the coupling coefficient [8]. The block diagram of a Marchand balun, which consists of two identical coupled-line sections, is shown in Fig. 1 [7]. The termination impedances of port 1–3 are and , which

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LIN et al.: ANALYSIS OF MULTICONDUCTOR COUPLED-LINE MARCHAND BALUNS FOR MINIATURE MMIC DESIGN

Fig. 2. L

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S and S of the Marchand balun with different coupling coefficients. denotes the maximum desired insertion loss.

Fig. 1. Block diagram of a Marchand balun with two identical coupled-line sections. The characteristic impedances of the individual uncoupled lines in the coupled-line section are Z and Z , respectively. The termination impedances of ports 1–3 are chosen as Z ; Z ; and Z .

are characteristic impedances of the individual uncoupled lines in the coupled-line section, respectively. The -parameters of the Marchand balun in Fig. 1 can be derived as (2) where Fig. 3. Normalized bandwidth versus coupling coefficient for different insertion-loss ripple.

and in (2) are functions of and with the same magnitude and 180 out of phase for all and . To analyze the relation between bandwidth and , the frequency responses for different are plotted in Fig. 2. It is observed that a higher coupling coefficient results in wider bandwidth of the Marchand balun. The flattest frequency response occurs for between 0.62–0.63. If is smaller than 0.62, the lowest insertion loss occurs at center frequency . When is greater than 0.63, the insertion loss at decreases. The lowest insertion loss will occur at higher and lower frequencies than . Suppose there is a desired maximum insertion loss , as shown in Fig. 2. If is 4 dB, e.g., intersects the insertion loss with at and , the insertion-loss ripple is defined as the difference between and the lowest insertion loss at with a bandwidth of . For a large ,

the insertion loss at is greater than , and intersects the insertion loss with at and , both below . The insertion-loss ripple is also defined as the difference between and the lowest insertion loss at with a bandwidth of . Fig. 3 shows the normalized bandwidth versus for different insertion-loss ripples. When the insertion loss at is lower than the desired , the frequencies from to can be selected. When the insertion loss at is greater than the desired , only the frequency range from to can be used. For certain desired , the lowest useful frequency of the Marchand balun is denoted as . Fig. 4 illustrates versus for different insertion-loss ripples. It is observed that when increases, the Marchand balun with the same can cover lower frequencies. B. Multiconductor Coupled Lines The above-mentioned analysis is based on the two-conductor coupled lines with coupling coefficient . The coupling coefficient between two lines increases as the mutual capacitance increases. The mutual capacitance can be increased by decreasing the spacing between the lines. However, the minimum

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Fig. 4. f =f versus coupling coefficient for different insertion-loss ripple.

Fig. 5. Block diagram of a five-conductor coupled line.

line spacing in the implementation of the circuits limits the coupling coefficients of the planar two-conductor coupled lines. To resolve the issue in planar transmission lines, multiconductor coupled lines, e.g., Lange couplers, can be used to achieve higher coupling with the same conductor spacing [4]. Fig. 5 shows the schematic of a set of five-conductor coupled lines. Lines 1, 3, and 5 are connected together with short lines and lines 2 and 4 are connected together. The cross section of the five-conductor coupled lines is shown in Fig. 6(a), where denotes the capacitance between the conductor and ground stands for the mutual capacitance between adjaplane, and cent lines. If the connections between odd-number lines (lines 1, 3, and 5) and even-number lines (lines 2 and 4) are very short compared to wavelength, the connections can be neglected, and thus, Fig. 6(a) can be simplified as Fig. 6(b). This figure can be further simplified as a pair of asymmetrical coupled lines, , , shown in Fig. 6(c), with equivalent capacitances and . Another way to increase the mutual capacitances in multiconductor coupled lines is to employ broadside coupling among the coupled lines [17]. Fig. 7 shows the cross section of a set of three-conductor coupled lines with an air bridge to increase the coupling between odd and even lines. is parallel connected with the other two mutual capacitances, and thus, increases the total equivalent mutual capacitance . The three-conductor coupled lines with an air bridge can also be simplified to a pair of equivalent two-conductor coupled lines, as shown in Fig. 6(c). In general, even and odd modes are no longer the normal modes of the asymmetrical coupled lines [9]. The normal modes

Fig. 6. (a) Cross section of the five-conductor coupled line. (b) Simplified model of the five-conductor coupled line. (c) Equivalent two-conductor coupled-line model of the five-conductor coupled line.

Fig. 7. Cross section of a three-conductor coupled line with air bridge to enhance coupling coefficient.

of the asymmetrical coupled lines are designated as and modes. For lossless TEM-mode coupled lines, the propagation constant for the and modes are the same. The analysis below is based on this assumption. From [9], the coupling coefficient of Fig. 6(c) can be written as (3) where If

and and are fixed, increases when is fixed, increases when and

. If increases. decrease.

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The - and -mode impedances

and

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can be written as

(4) (5) where and are the characteristic impedances of equivalent and lines 1 and 2 [see Fig. 6(c)], respectively. The ratio of is (6) The coupling coefficient increases when Equation (6) can be also expressed as

increases.

Fig. 8. Calculated normalized propagation constant ( = ) for the c and  modes of the seven-conductor coupled line with linewidth and line spacing of 5 m.

(7) The coupling coefficient of the coupled line can be determined after the - and -mode impedances are selected. C. Impedance Extraction and Design Procedure WIN Semiconductors’ standard 0.15- m high-power InGaAs/AlGaAs/GaAs pseudomorphic HEMT (pHEMT) process is used to demonstrate the design procedure. This MMIC process employs a hybrid lithographic approach using direct-write electron beam lithography for the submicrometer gate definition and optical lithography for the other steps. Other passive components including a thin-film resistor, metal–insulator–metal capacitors, spiral inductors, and air bridges are all available. There are two metal layers available in this process. The wafer is thinned to 4 mil for the gold planting of the backside, and reactive ion etching via-holes are used for dc grounding. The minimum conductor width and spacing are both 5 m. The diode is realized by connecting the drain and source pads of an HEMT device to form the cathode. The cutoff frequency of the two-finger 15- m Schottky diode is approximately 381 GHz. To examine the equal propagation constant assumption in the analysis, the seven-conductor coupled line in this 4-mil GaAs process are used to calculate the propagation constant for and modes. Both linewidth and line spacing are 5 m. The effective dielectric constants of these two modes are calculated by the full-wave electromagnetic simulator SONNET [19]. The normalized propagation constant can be expressed as (8) where is the propagation constant of the transmission line in the dielectric, is the propagation constant in free space, and is the effective dielectric constant for the transmission line. The real parts of the calculated normalized propagation constants for the two modes are shown in Fig. 8. The small difference between the propagation constants of the two modes leads to little magnitude and phase imbalance of the balun. For example, if there are

Fig. 9. Extracted Z and Z for planar coupled lines with different conductor width (W ) and spacing (S ).

two in-phase signals with equal magnitude injecting into ports 2 and 3 of an ideal balun, the signals should cancel out at port 1, but when the balun has magnitude and phase imbalance, the signals will not completely cancel out at port 1. The signal rejection ratio is defined as the ratio between the total in-phase input powers at ports 2 and 3 and the power at port 1. If the magnitude and phase imbalance are less than 1 dB and 10 , the signal rejection ratio can be greater than 20.1 dB. These values are acceptable for initial design, and thus, the assumption is valid. In order to determine the conductor number , conductor width , and conductor spacing from the desired coupling coefficient, the relation between them can be found from (7). If and for different and can be found, we can use to find proper and . and of the multiconductor coupled lines are extracted by a full-wave electromagnetic simulator [19]. The coupled from 2 to 7 are selected to extract and lines with to form the database for and from 5 to 30 m. The extracted and for different and with equal to 2 and 7 are plotted in Fig. 9. If is larger, the equiva-

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Fig. 11. Chip photograph of the seven-conductor coupled-line Marchand balun from 21 to 41 GHz with both the conductor width and spacing of 5 m.

Fig. 10. Extracted Z and Z for different W and S with N equal to 3. The solid line is the planar three-conductor coupled lines without air bridge, and the dashed line is the three-conductor coupled lines with air bridge.

lent conductor-to-ground capacitance also becomes larger, thus decreasing the -mode impedance. Although the -mode impedance decreases when increases, the -mode impedance also decreases due to the increasing of the equivalent mutual capacitance. The coupling coefficient still increases when increases. and for the three-conductor coupled lines with and without air bridges are extracted and plotted in Fig. 10. The solid lines denote the three-conductor coupled lines without air bridges, and the dash lines are the three-conductor coupled lines with air bridges. With the air bridges, decreases a lot, but only decreases slightly, and thus, the coupling coefficient increases. By applying (7), can be plotted in the charts of versus (Figs. 9 and 10). In Fig. 9, the curves for and are plotted, while in Fig. 10, the curve for is plotted. Most applications of the microwave baluns are balanced mixers and doublers. In these applications, the input impedance is chosen to be close to 50 . In all integrated circuits, chip area is always an important concern. Consequently, the width, spacing, and number of conductors are chosen to be the smallest values for the desired . The output impedances of the balun can be matched by properly choosing the diode size with simple matching networks. The design procedure of multiconductor coupled-line Marchand balun is summarized as follows. 1) For a selected MMIC process, and for different and are extracted to create the database. 2) Calculate the quarter-wavelength of the desired center frequency. 3) From Figs. 3 and 4, find the proper coupling coefficient for the desired bandwidth or . 4) Plot on the chart of versus and find and for the initial design. 5) Use full-wave electromagnetic simulator to verify the performance of the complete Marchand balun.

Fig. 12. Comparison of the measured and simulated insertion losses and S of the seven-conductor coupled-line Marchand balun.

III. DESIGN EXAMPLES One seven-conductor coupled-line Marchand balun and two MMIC balanced doublers with a multiconductor coupled-line Marchand balun using our design procedure are presented. The two doublers are both fabricated using WIN Semiconductors’ standard 0.15- m high-power InGaAs/AlGaAs/GaAs pHEMT process. The design procedure and circuit performances are described as follows. A. Seven-Conductor Coupled-Line Marchand Balun In this design, the desired lowest frequency is 30 GHz. In order to reduce the chip area, the 300- m coupled line, which is a quarter-wavelength line at 92 GHz, is selected to form the Marchand balun. From Fig. 4, the coupling coefficient to achieve the frequency ratio (30 GHz/92 GHz) with the insertion-loss ripple of 1 dB is 0.78. The next step is to plot for in Fig. 9 to find the proper conductor number, width, and spacing. The chosen is 7 and, and are both 5 m. close to 50 in the The balun with these parameters has desired frequencies and also occupies the smallest chip area with the desired . The chip photograph of the seven-conductor coupled-line Marchand balun from 21 to 41 GHz is shown in Fig. 11. This balun is simulated using a full-wave electromagnetic simulator. The simulated and measured insertion and return losses of port 1 are shown in Fig. 12, and the simulated and measured return losses of ports 2 and 3 and phase difference are shown in Fig. 13. The insertion losses are less than 6 dB from 21 to

LIN et al.: ANALYSIS OF MULTICONDUCTOR COUPLED-LINE MARCHAND BALUNS FOR MINIATURE MMIC DESIGN

Fig. 13. Comparison of the measured and simulated phase difference S S of the seven-conductor coupled-line Marchand balun.

and

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Fig. 15. Simulation results of the Marchand balun with seven-conductor coupled lines in frequency doubler 1. The conductor width and spacing are both 5 m.

TABLE I MODEL PARAMETERS OF THE DIODE, WHICH HAS FOUR FINGERS WITH TOTAL GATEWIDTH OF 30 m

Fig. 14. Circuit schematic of the frequency doubler 1.

41 GHz and the return loss of port 1 is greater than 8 dB from 24 to 34 GHz. The magnitude and phase imbalances are within 1 dB and 5 from 5 to 70 GHz, respectively. B. Frequency Doubler 1 The circuit topology of this frequency doubler is shown in Fig. 14. This circuit consists of a Marchand balun and two diodes. The input power are divided into two parts to feed two antiparallel diodes. Due to the diodes being antiparallel, the generated second-harmonic signals are 180 out-of-phase. The second-harmonic signals are combined through the Marchand balun at the output. In this design, the desired output frequency range is from 30 to 50 GHz. The design procedure in Section III-C can be apis 7, and plied to design the Marchand balun. The chosen and are both 5 m for the 300- m coupled lines. Using these design parameters, the Marchand balun is simulated by the full-wave electromagnetic simulator [19]. The simulated results of the Marchand balun is shown in Fig. 15. The simulated insertion losses from 30 to 50 GHz are lower than 5 dB. The amplitude imbalance and phase imbalance are within 1 dB and 10 , respectively.

Next, the diode size is selected and matched to the port impedance of the Marchand balun. The diode has four gate fingers with a total finger width of 30 m. The model used in circuit simulation is a general nonlinear diode model by fitting the measured direct-current current–voltage (DC-IV) curves and -parameters. The model parameters of this diode are shown in Table I. In Fig. 14, two microstrip lines between diodes and the balun are used as the matching networks. These two microstrip lines can transform the output impedances of the Marchand balun conjugately matched to the diode impedance under 10-dBm driven power near the center frequency. Finally, the -parameters of the balun and matching networks from SONNET results are used in the circuit simulator with diode models to simulate the entire circuit. A chip photograph is shown in Fig. 16 with a chip size of 0.85 mm 0.66 mm. This seven-conductor coupled-line Marchand balun is a planar balun without an air bridge. This circuit is measured via on-wafer probing for input and output ports through ground–signal–ground RF probes. Fig. 17 shows the comparison of simulated and measured conversion gains with a 10-dBm input power. The conversion loss is less than 12.5 dB for input frequency from 11 to 25 GHz. The fundamental rejection ratio is shown in Fig. 18. The fundamental rejection ratio is greater than 19 dB for input frequency from 10 to 25 GHz. C. Frequency Doubler 2 The circuit topology of this frequency doubler is the same with frequency doubler 1, except for the Marchand balun. The

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Fig. 16. Chip photograph of frequency doubler 1. The chip size of this circuit is 0.85 mm 0.66 mm.

Fig. 19. Simulated and measured insertion loss of the Marchand balun with three-conductor coupled line in frequency doubler 2. Air bridges are used to increase the coupling coefficient.

Fig. 17. Simulated and measured conversion gain of frequency doubler 1 with a 10-dBm input power.

Fig. 20. Simulated and measured phase difference of the Marchand balun with three-conductor coupled line in frequency doubler 2. Air bridges are used to increase the coupling coefficient.

2

Fig. 18. Fundamental rejection ratio for frequency doubler 1.

desired output frequency range is from 80 to 120 GHz. The couat 166 GHz, is sepled line with length of 155 m, which is lected to form the Marchand balun. From Fig. 4, the coupling coefficient to achieve the frequency ratio (80 GHz/166 GHz) with is the insertion-loss ripple of 1 dB is 0.64. From Fig. 10,

selected with and are both 5 m. Air bridges are used to increase the coupling coefficient. In order to further increase the coupling coefficient, the center conductor is broaden to 15 m in Fig. 7. to enhance The simulated and measured results of the Marchand balun are shown in Figs. 19 and 20. The -parameters are measured by using the port reduction methods [20]. The simulated insertion loss from 78 to 135 GHz is lower than 4 dB. The magnitude imbalance and phase imbalance are smaller than 1 dB and 10 , respectively. A chip photograph is shown in Fig. 21 with a chip size of 0.56 mm 0.42 mm. This circuit is also measured via on-wafer probing through ground–signal–ground RF probes. Fig. 22 shows the comparison of simulated and measured conversion gains with a 12.5-dBm input power. The measured conversion loss is less than 12.5 dB for input frequency from 25 to 64 GHz. The fundamental rejection ratio is shown in Fig. 23. The measured fundamental rejection ratio is greater than 20 dB for input frequency from 15 to 67 GHz. Table II shows the comparison of reported frequency doublers in the millimeter-wave region with our designs. These two frequency doublers all have the smallest chip sizes compared

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TABLE II REPORTED PERFORMANCE OF WIDEBAND MILLIMETER-WAVE FREQUENCY DOUBLERS

Fig. 21. Chip photograph of frequency doubler 2. The chip size of this frequency doubler is 0.56 mm 0.42 mm.

2

IV. CONCLUSION A systematic analysis and design procedure for broadband multiconductor coupled-line Marchand baluns has been presented in this paper. By using this design procedure, multiconductor coupled-line Marchand baluns can be easily designed with miniature sizes. A 21-41-GHz balun and two MMIC frequency doublers have been designed and implemented to verify the design procedure. The frequency doublers achieved the smallest chip sizes with good performances among their operating frequencies.

ACKNOWLEDGMENT Fig. 22. Simulated and measured conversion gain of frequency doubler 2 with 12.5-dBm input power.

The authors would like to thank C.-H. Wang, Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., C.-H. Tseng, Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., and M.-F. Lei, Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, for their helpful suggestions. The chips were fabricated by WIN Semiconductors, Taoyuan, Taiwan, R.O.C., through the Chip Implementation Center (CIC) of Taiwan, R.O.C. REFERENCES

Fig. 23. Fundamental rejection ratio for frequency doubler 2.

with the MMIC doublers in the same operating frequencies. Frequency doubler 2 also achieves the widest bandwidth among the reported diode frequency doublers.

[1] R. Schwindt and C. Nguyen, “Computer-aided analysis and design of a planar multilayer Marchand balun,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 7, pp. 1429–1434, Jul. 1994. [2] C. Y. Ng, M. Chongcheawchamnan, and I. D. Robertson, “Analysis and design of a high-performance planar Marchand balun,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 1, pp. 113–116. [3] S. A. Maas and K. W. Chang, “A broadband, planar, doubly balanced monolithic -band diode mixer,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2330–2335, Dec. 1993. [4] M. C. Tsai, “A new compact wideband balun,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1993, vol. 1, pp. 141–143.

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[5] P.-S. Wu, C.-H. Wang, T.-W. Huang, and H. Wang, “Compact and broadband millimeter-wave monolithic transformer balanced mixers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 10, pp. 3106–3114, Oct. 2005. [6] J.-W. Lee and K. J. Webb, “Analysis and design of low-loss planar microwave baluns having three symmetric coupled lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 1, pp. 117–120. [7] K. S. Ang and I. D. Robertson, “Analysis and design of impedancetransforming planar Marchand baluns,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 402–406, Feb. 2001. [8] K. Sachse, “The scattering parameters and directional coupler analysis of characteristically terminated asymmetric coupled transmission lines in an inhomogeneous medium,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 4, pp. 417–425, Apr. 1990. [9] R. Mongia, I. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA: Artech House, 1999. [10] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [11] C.-M. Tsai and K. C. Gupta, “A generalized model for coupled lines and its applications to two-layer planar circuits,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 12, pp. 2190–2199, Dec. 1992. [12] M. Shimozawa, K. Itoh, Y. Sasaki, H. Kawano, Y. Isota, and O. Ishida, “A parallel connected Marchand balun using spiral shaped equal length coupled lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1999, vol. 4, pp. 1737–1740. [13] M. N. Tutt, H. Q. Tserng, and A. Ketterson, “A low loss, 5.5 GHz–20 GHz monolithic balun,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1997, vol. 2, pp. 933–936. [14] Y. C. Leong, K. S. Ang, and C. H. Lee, “A derivation of a class of 3-port baluns from symmetrical 4-port networks,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 2, pp. 1165–1168. [15] K. Nishikawa, I. Toyoda, and T. Tokumitsu, “Compact and broadband three-dimensional MMIC balun,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 1, pp. 96–98, Jan. 1999. [16] J.-S. Sun and T.-L. Lee, “Design of a planar balun,” in Asia–Pacific Microw. Conf. Dig., Dec. 2001, vol. 2, pp. 535–538. [17] P.-S. Wu, C.-S. Lin, T.-W. Huang, H. Wang, Y.-C. Wang, and C.-S. Wu, “A millimeter-wave ultra-compact broadband diode mixer using modified Marchand balun,” in Gallium Arsenide and Other Semiconduct. Applicat. Symp. Dig., Oct. 2005, pp. 349–352. [18] MATLAB. ver. 6.5, The MathWorks Inc., Natick, MA, 2002. [19] SONNET. ver. 10.52, Sonnet Software Inc., North Syracuse, NY, 2004. [20] H.-C. Lu and T.-H. Chu, “Port reduction methods for scattering matrix measurement of an -port network,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 6, pp. 959–968, Jun. 2000. [21] P.-S. Wu, C.-H. Tseng, M.-F. Lei, T.-W. Huang, H. Wang, and P. Liao, “Three-dimensional -band new transformer balun configuration using the multilayer ceramic technologies,” in Eur. Microw. Conf. Dig., Oct. 2004, vol. 1, pp. 385–388. [22] V. Puyal, A. Konczykowska, P. Nouet, S. Bernard, S. Blayac, F. Jorge, M. Rjet, and J. Godin, “A DC–100 GHz frequency doubler in InP DHBT technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 1, pp. 167–170. [23] Y.-L. Tang, P.-Y. Chen, and H. Wang, “A broadband pHEMT MMIC distributed doubler using high-pass drain line topology,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 5, pp. 201–203, May 2004. [24] K.-L. Deng and H. Wang, “A miniature broadband pHEMT MMIC balanced distributed doubler,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1257–1261, Apr. 2003. [25] Y. Lee, J. R. East, and L. P. B. Katehi, “High-efficiency -band GaAs monolithic frequency multipliers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 529–535, Feb. 2004. [26] G. P. Ermak and P. V. Kuprijanov, “Development of a planar multiplier circuit for millimeter-wave frequency multipliers,” in 4th Int. Phys. Eng. Millimeter Sub-Millimeter Waves Symp., Jun. 2001, vol. 2, pp. 696–698. [27] G.-L. Tan and G. M. Rebeiz, “High-power millimeter-wave planar doublers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, vol. 3, pp. 1601–1604. [28] S. A. Maas and Y. Ryu, “A broadband, planar, monolithic resistive frequency doubler,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1994, vol. 1, pp. 443–446.

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Chin-Shen Lin (S’03) was born in Hsinchu, Taiwan, R.O.C., in 1979. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2001, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University. His research interests include monolithic microwave/millimeter-wave circuit design.

Pei-Si Wu (S’02) was born in Changhua, Taiwan, R.O.C., in 1980. He received the B.S. degree in electric engineering and Ph.D. degree from the Graduate Institute of Communication Engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2002 and 2006, respectively. He is currently a Senior Engineer with the Realtek Semiconductor Corporation, Hsinchu, Taiwan, R.O.C. His research interests include the microwave and millimeter-wave circuit designs.

Mei-Chao Yeh was born in Kaohsiung, Taiwan, R.O.C., on January 10, 1981. She received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003, and the M.S. degree from the Graduate Institute of Communication Engineering, National Taiwan University Taipei, Taiwan, R.O.C., in 2005. She is currently an engineer with the Realtek Semiconductor Corporation, Hsinchu, Taiwan, R.O.C. Her research interests are in the areas of RF and millimeter-wave integrated circuits in CMOS technologies.

Jia-Shiang Fu (S’06) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003, the M.S. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2005, and is currently working toward the Ph.D. degree at The University of Michigan at Ann Arbor. His research interests include reconfigurable microwave circuits and linearization of power amplifiers.

Hong-Yeh Chang (S’02–M’05) was born in Kinmen, R.O.C., in 1973. He received the B.S. and M.S. degrees in electric engineering from National Central University, Chung-Li, Taiwan, R.O.C., in 1996 and 1998, respectively, and Ph.D. degree from the Graduate Institute of Communication Engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004. From 1998 to 1999, he was with Chunghwa Telecom Laboratories, Taoyuan, Taiwan, R.O.C., where he was involved in the research and development of code division multiple access (CDMA) cellular phone system. In 2004, he was a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University, where he was involved with research on advanced millimeter-wave integrated circuits. In February 2006, he joined the faculty of the Department of Electrical Engineering, National Central University, Jhongli City, Taiwan, R.O.C., as an Assistant Professor. His research interests are microwave and millimeter-wave circuit and system designs.

LIN et al.: ANALYSIS OF MULTICONDUCTOR COUPLED-LINE MARCHAND BALUNS FOR MINIATURE MMIC DESIGN

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., Hsinchu, Taiwan, R.O.C. In July 2006, he joined the faculty of the Department of Electrical Engineering and 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.

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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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Submillimeter-Wave Phasor Beam-Pattern Measurement Based on Two-Stage Heterodyne Mixing With Unitary Harmonic Difference Yuh-Jing Hwang, Member, IEEE, Ramprasad Rao, Rob Christensen, Ming-Tang Chen, and Tah-Hsiung Chu, Member, IEEE

Abstract—A near-field phasor beam measurement system is developed for the characterization of heterodyne receiver optics at submillimeter-wave frequencies. The system synthesizes a pair of submillimeter-wave signals as the RF and local oscillator (LO) sources from common reference sources. The synthesized harmonic numbers of the RF and LO sources are arranged with difference by one, which makes this a new configuration with a unitary harmonic difference. The coherent RF and LO signal are down-converted by the receiver under test, then mixed with the microwave-frequency common reference signal to generate the second-order IF signal around 100 MHz for amplitude and phase comparison. The amplitude and phase fluctuation of the measurement system at 683 GHz is within 0.2 dB and 4 in a 1-h period, respectively. The system dynamic range at 683 and 250 GHz can be as high as 43 and 47 dB, respectively. The system is then used to measure the receiver beam patterns at 683 and 250 GHz with different RF transmitting probe antennas. Index Terms—Frequency conversion, Gaussian beams, phase measurement, submillimeter-wave receivers.

I. INTRODUCTION ITH THE increasing demands on atmospheric observing and radio astronomical instruments, the development of the submillimeter-wave heterodyne receivers with high sensitivity and accurate pointing is required [1]–[3]. The highly sensitive heterodyne receiver can be implemented by using a nearly quantum limited superconductor–insulator–superconductor (SIS) mixer or a hot-electron bolometer mixer as the front-end [4]–[6]. For accurate pointing of the receiver optics, a useful and efficient diagnostic instrument is required to measure the beam pattern to analyze the optical parameters of the receiver at the operating frequency.

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Manuscript received May 18, 2006; revised February 16, 2007. This work was supported by Academia Sinica under the Submillimeter Array of Taiwan Project and by the R.O.C. National Science Council under Grant NSC95-2752E002-004-PAE and Grant NSC95-2221-E002-086-MY3. Y.-J. Hwang, R. Rao, and M.-T. Chen are with the Institute of Astronomy and Astrophysics, Academia Sinica Taipei 10617, Taiwan, R.O.C. (e-mail: yjhwang @asiaa.sinica.edu.tw; [email protected]; [email protected]). R. Christensen is with the Submillimeter Array, Hilo, HI 96720 USA (e-mail: [email protected]). T.-H. Chu is with the Graduate Institute of Communication Engineering and the Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C., and also with the Institute of Astronomy and Astrophysics, Academia Sinica Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]. ntu.edu.tw). 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.2007.897843

A typical phasor antenna pattern measurement system consists of a coherent transmitter–receiver module, or so-called phasor network analyzer, combined with a precision scanner to compare the amplitude and phase between transmitted and received signals at different relative positions. For frequencies below 500 GHz, a commercially available millimeter-wave phasor network analyzer was introduced recently [7]. For higher frequencies below 1 THz, a millimeter-wave vector network analyzer is also available with user-reconfigurable test sets.1 Previous research on antenna or telescope-optics pattern measurement using commercially available phasor network analyzers has also been published [8]–[17], and the amplitude dynamic range can be as high as 60 dB at 860 GHz. However, these commercially available phasor network analyzers are based on the harmonic mixer as the down-converting device, which limits its application to the integrated measurement of the antenna system with a heterodyne receiver. To solve this problem, some research for frequency ranging from 200 to 1500 GHz has also been published [18]–[22]. These receiver beam pattern measurement configurations typically require complicated frequency synthesis and conversion. Phase stability is an important factor in the measurement system. For a system with phase sensitive to the environmental temperature, the scanner carried RF transmitter requires an axis along the beam propagation to scan the beam pattern at different propagation positions. However, once the measurement configuration exhibits stable phase, the measured amplitude and phase data secured by scanning at a fix plan perpendicular to the beam axis are sufficient to extract the beam parameters. In this paper, the simple, but stable measurement configuration for submillimeter-wave phasor beam pattern and receiver optics alignment described in [23] for 690 GHz is adapted to multiple frequency measurement in Section II. The configuration only needs two frequency standards—one microwave swept frequency source and one fixed reference frequency source—to synthesize and convert the frequencies required for the measurement. When measuring the receiver systems equipped with multiple channels for different frequency bands, this configuration can be used for measuring beam pattern by replacing the transmitting antenna, rotating the polarization of the RF transmitting module, and switching the active receiver channel. A theoretical formulation of the phase stability of the measurement configuration is derived in Section III. Both the theoretical prediction and 1Type MVNA-8-350 Millimeter-Wave Network Analyzer, AB Millimetre, Paris, France.

0018-9480/$25.00 © 2007 IEEE

HWANG et al.: SUBMILLIMETER-WAVE PHASOR BEAM-PATTERN MEASUREMENT

the measurement show that this method can reduce the system phase variation to as low as 4 during a 1-h period. In Section IV, a multichannel SIS-based heterodyne receiver for the Submillimeter Array (SMA) telescope is measured as the receiver under test. Operating frequencies around 250, 349, 409, and 683 GHz are chosen to measure the receiver front-end quasi-optics beam pattern. The measurement results show a dynamic range around 47 dB for the 230-GHz band receiver and 43 dB for the 690-GHz band. A formulation for extracting the optical parameters of the truncated Gaussian Beam is also derived. The extracted parameters are compatible with the theoretical prediction. II. MEASUREMENT ARRANGEMENT In order to configure the submillimeter-wave coherent transmitter–receiver module in a simpler approach with better performance, the frequency synthesis and conversion approaches of the measurement arrangements in [7]–[22] are investigated to find a method for improvement. The millimeter-wave phasor network analyzer in [7] requires two microwave synthesizers to generate RF and local oscillator (LO) signal. An additional 20-MHz signal source is used as phase and amplitude comparison reference. The receiver inside the test sets uses harmonic mixers as down-converters, but it is still possible to adapt it for heterodyne receiver measurement. However, even if one can apply this configuration to the receiver beam pattern measurement, the disadvantage on the coherence between RF and LO signals due to the locking frequency at only 10 MHz limits its operation at higher submillimeter-wave frequencies. Typical operation of this kind of millimeter-wave phasor network analyzer requires 128 times averaging to reduce the random fluctuation, which leads to longer measurement time. The other popular phasor network analyzers described in [8]–[17] provide flexible and different frequency synthesis and conversion approaches. When operating at a lower frequency, its utilizes a frequency multiplier to generate RF source and down conversion of the received RF signal to IF by harmonic mixer, which is similar to [7], for higher submillimeter-wave frequency it relies on one swept microwave synthesizer and two distinct low-frequency signal sources for phase and amplitude comparison [11]. Similar to the case in [7], this network analyzer is quite suitable for measurement related to an antenna without an integrated receiver, but for the measurement related to antenna–receiver integration, suitable adoption of the frequency synthesis approach is required. The measurement configuration of [21] is another approach aiming to solve the antenna–receiver integrated measurement. The basic idea is to generate a pair of coherent RF and LO sources by using a harmonic-mixer-based phase-locked loop (PLL), followed by a cascading frequency multiplier to terahertz output frequency. A microwave synthesizer is used to provide the pumping of the harmonic mixers in a PLL pair, and the 10-MHz reference signal is multiplied to generate the reference signal for the LO PLL, the RF PLL, second down-conversion pumping, and phasor comparison. The arrangement of different reference signals for frequency synthesis is mainly due to the limited harmonic number suitable for frequency multiplication

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to terahertz frequency. The main disadvantage of this configuration is the opposite sidebands of the RF and LO signals in the PLL, and the resulting IF leads to additive phase drifting, which will be discussed later. The measurement configurations in [18]–[20] are basically the hybrid types of [7] and [21]. Based on the above discussion, the principle to simplify the frequency synthesis and down-conversion to fulfill the antenna–receiver integration measurement can be summarized as follows. 1) Try to synthesize both RF and LO signal frequencies by the same signal source. For - or -band signal sources, which need a harmonic mixer in the PLL, a microwave synthesizer to pump the harmonic mixer with frequency and an 80–110-MHz reference signal with frequency distributed by power dividers are required. 2) Arrange the RF and LO PLL on the same sideband to avoid phase drifting enlargement. 3) For the receiver under test, the frequency of the IF signal may be too high to compare the amplitude and phase directly, second down conversion is required. To ensure coherence, the second-stage mixing pumping frequency and the and , respectively. second IF frequency should be Based on the above principle, a new measurement arrangement is proposed, as shown in Fig. 1. In the configuration, the SIS mixer based heterodyne receiver, including the front-end optics and the LO, is the device-under-test. - or -band Gunn oscillators are used as the fundamental frequency sources to generate RF signal and LO signal for the receiver. The frequency relationship of this measurement arrangement is described as follows. and be the frequencies of the phaseLet locked oscillators for RF and LO sources, and let and be the frequencies of the synthesizers for the PLL harmonicmixing pumping LO source and reference source. One can then express the frequencies as (1a) (1b) (2a) (2b) In (1), and are the harmonic numbers of the harmonic mixers in the RF and LO PLLs, respectively. In (2), and are the multiplying factors of the frequency multipliers for the RF and LO sources, respectively. Please note that, in (1), the PLL is arranged at the same sideband, and the upper sideband is expressed in the formulation. This arrangement is to minimize the harmonic difference in the IF output of the receiver under test. The RF and LO frequency difference should be within the heterodyne receiver IF frequency passband. From (1) and (2), one can get

(3) Since , , , and are all integers, and the IF signal of the receiver is only down-converted once, the best

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Fig. 1. System block diagram of the submillimeter-wave heterodyne receiver field pattern measurement arrangement.

way is to choose the IF frequency to be equal to . The reference frequency based on the above-described principle is much lower than the IF frequency. Therefore, (4) and

manual rotation stage is used for installing the RF transmitter. For the application to other situations, for example, measuring the large-aperture reflector antenna, the proposed frequency synthesis approach can be used along with an innovative hologram-based CATR for large-aperture antenna measurement [13]–[16]. III. PHASE STABILITY CONSIDERATION

(5) (6) and in (4) are the lower and higher edges of the IF passband. Since (5) and (6) represent the harmonic difference of the synthesizer and the phase-locked reference signal sources in the heterodyne receiver output, and these differences are both equal to 1, one can call this approach the “unitary harmonic difference” method. In operation, one of the four frequency bands in the receiver system to be measured can be selected by the microwave switches and the rotating optical mirror. An open-ended waveguide antenna with suitable aperture to keep single mode propagation is installed at the output of the RF transmitter when measuring the different frequency bands. Concerning the mapping approaches, we choose convenscanning. Considering the 90 rotation on tional motorized the polarization of the different frequency bands, an additional

The phase fluctuation of the measurement system can be analyzed as the following. Consider the signal frequencies , , , , existing in the beam pattern system as with the corresponded phase errors , , and , , , and . In addition, the phase variations induced by the cable length variation due to am, , bient temperature change are denoted as , , , and . The phase errors introduced by the phase-locked circuits of the RF and and . LO modules are denoted as According to (1), since both the RF and LO sources are phase locked to the upper sideband of the harmonic mixer, the phase fluctuations of RF and LO sources can be expressed as

(7)

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From (5) and (6), the phase fluctuation of the IF signal is

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TABLE I HARMONIC NUMBERS AND SIGNAL FREQUENCIES FOR UNITARY HARMONIC DIFFERENCE BEAM PATTERN MEASUREMENT

(8) Similarly, from (5) and (6), the phase fluctuation for the second IF mixer output signal can be expressed as

(9) In the measurement arrangement, the networks distributed to RF and LO modules for share a common 96-in-long coaxial cable connected to a power divider and followed by a pair of to the cables with the same length of 24 in. The link for second IF mixer is directly connected after the first-stage power is ignored. Since all the cables divider, which means are operated under the same environment with the same ambient . temperature variation, one can assume In addition, the cable lengths of RF and LO reference signals . The second IF phase are the same, or variation is then

(10) From the phase-noise measurement on the phase-locked Gunn oscillator, the phase noise of the phase-locked circuit is approximately 2.0 . The phase stability is approximately 3.0 at 26.5 GHz for a 36-in-long cable,2 which corresponds to 2.15 at 5.74 GHz for a 120-in-long cable. Therefore, from (10), the second IF output phase fluctuation is approximately 4.97 . For the amplitude fluctuation, it is basically due to the bias voltage variation controlled by the PLL and the size of the Gunn oscillator cavity resonator changing with temperature. It is believed the amplitude fluctuation can be minimized by proper temperature control, and it should be small if the Gunn oscillator is turned on for a long time to be thermally balanced to the ambient temperature. For comparative purposes, the measurement arrangement is analyzed such as that in [21]. The arrangement requires RF and LO sources phase locked at two opposite sidebands, and the reference signals for RF, LO, and IF are 120, 200, and 240 MHz, in the RF respectively. The microwave synthesizer signal and LO signals is cancelled by the submillimeter-wave mixer is and the IF frequency (11) It is clear that the phase fluctuation at IF in [21] is additive. In addition, the 3-GHz LO signal for the second IF mixer, the reference signals of RF and LO, and the reference signal for the 2W.

L. Gore PHASEFLEX series microwave cable assembly

vector voltmeter comparison are through frequency multiplying of a 10-MHz signal. If this 10-MHz signal drifts by temperature even with a very small value, then the second IF output phase fluctuation will be amplified. Accurate environmental temperature control is, therefore, required. When scanning the field of the beam pattern, there are necessarily moving or flexing cables. The phase variation of these flexing cables may introduce phase error between the LO and RF source in the measured data. Considering the extreme case of , the phase change is 1.0 for one bend the cable carrying around 2.25-in (57 mm) radius, which corresponds to 45.0 for measurement at 250 GHz or 120.0 at 685 GHz. However, during the measurement, the flexing cable is bent with a radius much larger than 57 mm, and the scanning distance is not larger than 150 mm, thus the corresponding phase change can be negligible. For the Gaussian beam measurement far from its beam waist, the phase variation at the scanning plane will be quite large, which will also make this cable bend effect much smaller. For measuring the antenna patterns with expected flat phase profile or with large scanning area such as a 1–2-m size, the pilot signal phase correction approach introduced in [24] will be very helpful to enhance the measurement accuracy. IV. MEASUREMENT RESULTS In our experiment, the multiple channel receiver system of the SMA telescope is used as the receiver under test. Considering all the criteria given in (4)–(6) and the fact of the fixed harmonic number of the LO sources, the allowable harmonic numbers and its corresponding frequencies are listed in Table I. A. Phase Stability Measurement To verify the phase stability formulation described in (10), the 690-GHz heterodyne receiver is tested with the operating frequency listed in Table I. The amplitude and phase fluctuations over a reasonably long time are measured with the RF transmitter mechanical position fixed. The measurement results, as shown in Fig. 2, indicate the amplitude and phase fluctuation is within 0.2 dB and 4.0 over a 1-h period, respectively. To be noted is that the testing environment is not air-conditioned. Table II shows the stability performance comparison of this measurement arrangement for other published results.

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Fig. 2. Measured results of: (a) amplitude and (b) phase fluctuation of the field pattern measurement system at 690 GHz.

TABLE II STABILITY PERFORMANCE COMPARISON OF VECTOR BEAM PATTERN MEASUREMENT SYSTEMS IN SUBMILLIMETER-WAVE FREQUENCIES

B. 2-D Beam Pattern Contour After ensuring the stability of the system, the system is further scanned at the frequency listed in Table I. The optics of the lower frequency receiver bands (176–256 and 250–354 GHz) incorporates the wire grid polarizer, LO coupling mesh, and the turning mirrors. The optics of the higher frequency receiver band in 600–696 GHz is equipped with a Martin–Puplett diplexer (MPD) for LO–RF combining. Details of the receiver optics design are described in [25]. The RF source, which is composed of an 84–110-GHz Gunn oscillator, a WR-8 isolator, a crossguide coupler, and a diode frequency multiplier is mounted on a precision – scanner. An open-ended waveguide with proper aperture size and integrated taper to WR-3 is used as the RF transmitting probe antenna. The frequency multiplier is originally equipped with a single-ended Schottky diode and the original multiplying factor is three. However, all the other harmonics can be generated by the diode [26]. As in [27] and [28], the radiation pattern of the probe antenna is regarded to be isotropic in the scanning area.

TABLE III SCANNING PARAMETERS OF THE BEAM PATTERN MEASUREMENT

The total optical path from the RF transmitting probe antenna to the receiver horn, including the reflection within the MPD, is approximately 1600 mm for the 690-GHz band. The scanning step and points for the RF source are listed in Table III. Fig. 3 shows the amplitude and phase contour plots of the field pattern of the 600–696-GHz SIS receiver. The MPD and cryogenic optical axis of the system are aligned mechanically prior to the installation. It shows that the system has a 43.5-dB dynamic range for the 690-GHz band, which is comparable to 43 dB given in [23]. The phase contour is matched with the amplitude contour.

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Fig. 3. Measured results of: (a) amplitude and (b) phase contour of the 600–696-GHz SIS receiver beam pattern.

Fig. 4. Measured results of: (a) amplitude and (b) phase contour of the 176–256-GHz SIS receiver beam pattern.

Fig. 4 shows the amplitude and phase contour plots of the field pattern of the 176–256-GHz SIS receiver. With sufficient RF power level, the system dynamic range is approximately 47 dB at 250 GHz. It is worth noting that the contour plots at lower frequencies are not as smooth as the measured results of 690 GHz, which indicates: 1) stronger background scattering from the metallic mechanical structures of the optics; 2) limited isolation on position-dependent impedance mismatch along the microwave distribution network; and 3) a larger machining error on the receiver corrugated horn antenna and Teflon receiver lens due to larger device size.

paraxial propagated electromagnetic (EM) wave is expressed as [29]

C. Gaussian Optics Parameter Analysis In order to ensure that the optics are satisfactory with respect to the design specification, the Gaussian beam parameters are extracted from the 690-GHz measured beam pattern. The detailed formulation for parameter extraction is described as follows. For a fundamental Gaussian beam without offset and shifting, the amplitude distribution of the vector potential of a

where

(12) , , beam waist radius , and is the confocal parameter. , At the beam center of , the relative amplitude pattern is given as

, ,

(13)

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Fig. 5. Measured results and the Gaussian fit of: (a) amplitude and (b) phase profile of E - and H -plane beam pattern. TABLE IV COMPARISON ON THE EXTRACTED AND THEORETICAL GAUSSIAN PARAMETERS OF THE 600–696-GHz RECEIVER IN TEST DEWAR

Note: Theoretical calculation (I) is based on thin lens assumption and the theoretical calculation (II) is based on thick lens assumption.

The beam radius at the scanning plane can then be calculated from the measured relative amplitude pattern as

positions for the amplitude and phase peaks. Thus, the truncated Gaussian beam through an MPD then becomes

(14a) (17a) (14b) (17b)

From (12), the relative phase pattern is (15) hence, the phase front curvature can be calculated from the measured relative phase pattern as (16) For practical situations, the beam axis shift and offset should be considered. In other words, the term becomes , where , , and , is the beam center position, and and are the beam offsets in - and -axis. For the receiver system characterization demonstrated here, the optical beam propagation paths along the - and -plan are different due to the roof mirror pair, which leads to the different truncation of the beam in an orthogonal plan. As shown in Fig. 4, the amplitude pattern is elliptical and the phase pattern is almost circular. The slight misalignment introduced by the multiple reflection within the MPD also leads to a slightly difference

, , , , In (17), ) is the position of the amplitude peak and ( , and ( , ) is the position of the phase peak. Therefore, solving and from the measured pattern, the shift peak positions , , , and are determined directly from the measured data, , , , and are extracted by and the tilt angles

(18) The measured Gaussian parameters can then be extracted from (12)–(18). Please note that the - and - planes are along the - and -axis, respectively. With the dimensions of the receiver lens–horn combination, one can calculate the theoretical Gaussian beam results of the fundamental mode. Fig. 5 shows the measured beam patterns from [23] of the - and -planes, with 20- m scanning resolution. The extracted and theoretical Gaussian beam parameters are listed in Table IV. From this

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table, one can see the difference between the theoretical values and measured results of the beam waist radius and positions. It can be regarded as the result of the truncation introduced by the MPD. V. CONCLUSION In this paper, the optics of a multiple-channel SIS heterodyne receiver has been measured by a new submillimeter-wave phase stable phasor beam pattern measurement configuration. The phase fluctuation at 690 GHz is as small as 4 within 1 h. The measurement results indicate this new setup is an efficient tool to characterize and align the optics system of the sub-terahertz SIS receiver. With suitable frequency selection and hardware configuration, this method may be applied to other frequencies as high as 1 THz. ACKNOWLEDGMENT The authors would like to thank Dr. R. Blundell, Dr. C.-Y. E. Tong, Dr. S. Paine, and C. Papa, all with the Smithsonian Astrophysical Observatory, Cambridge, MA, for kindly providing the receiver design of the SMA, Dr. M. J. Wang, Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), Taipei, Taiwan, R.O.C., for his contributions of the SIS junctions and mixers, F. Patt, European South Observatory, Garching bei Munich, Germany, for useful discussion, T. S. Wei, ASIAA, Q. Yao, Purple Mountain Observatory, Nanjing, China, S.-H. Chang, ASIAA, S.-W. Chang, ASIAA, and C.-C. Chen, ASIAA, for their technical contributions. REFERENCES [1] P. T. P. Ho, J. M. Moran, and K. Y. Lo, “The Submillimeter Array,” Astrophys. J. Lett., vol. 616, no. 1, pt. 2, pp. L1–L6, Nov. 2004. [2] A. Wooten, Ed., Science With the Acatama Large Millimeter Array. San Francisco, CA: Astronom. Soc. of the Pacific, 2001, pp. 11–24. [3] J. Inatani, H. Ozeki, R. Satoh, T. Nishibori, N. Ikeda, Y. Fujii, T. Nakajima, Y. Iida, T. Iida, K. Kikuchi, H. Masuko, T. Manabe, S. Ochiai, M. Seta, Y. Irimajiri, Y. Kasai, M. Suzuki, T. Shirai, S. Tsujimaru, K. Shibasaki, and M. Shiotani, “Submillimeter limb-emission sounder JEM/SMILES aboard the Space Station,” in Proc. SPIE, Sendai, Japan, Dec. 2000, vol. 4152, Microw. Remote Sens. Atmosphere Environ. II, pp. 243–254. [4] J. R. Tucker and M. J. Feldman, “Quantum detection at millimetre wavelengths,” Rev. Modern Phys., vol. 57, no. 4, pp. 1055–1112, Oct. 1985. [5] M. J. Wengler and D. P. Woody, “Quantum noise in heterodyne detection,” IEEE J. Quantum Electron., vol. QE-23, no. 5, pp. 613–622, May 1987. [6] B. S. Karasik, M. C. Gaidis, W. R. McGrath, B. Bumble, and H. G. LeDuc, “Low noise in a diffusion-cooled hot-electron mixer at 2.5 THz,” Appl. Phys. Lett., vol. 71, no. 11, pp. 1567–1569, Sep. 1997. [7] A. Feng, D. Dawson, L. Samoska, K. Lee, T. Gaier, P. Kangaslahti, C. Olison, A. Denning, Y. Lau, and G. Boll, “Two-port vector network analyzer measurements in the 218-344- and 356–500 GHz frequency bands,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4507–4512, Dec. 2006. [8] T. Lüthi, D. Rabanus, U. U. Graf, C. Granet, and A. Murk, “Expandable fully reflective focal-plane optics for millimeter- and submillimeterwave array receivers,” Rev. Scientific Instrum., vol. 77, no. 1, Jan. 2006, Paper 014702, 5 pp. [9] D. Rabanus, C. Granet, A. Murk, and T. Tils, “Measurement of properties of a smooth-walled spline-profile feed horn around 840 GHz,” Infrared Phys. Technol., vol. 48, no. 3, pp. 181–186, 2006. [10] A. Murk, P. Fürholz, P. Yagoubov, M. Birk, and G. Wagner, “Near-field antenna measurements for the terahertz limb sounder TELIS,” in Proc. Eur. AMTA Symp., 2006, pp. 94–99. [11] A. Murk, “Characterization of optical components with a 100–800 GHz network analyzer,” presented at the IEEE MTT-S Int. Microw. Symp. Workshop, San Francisco, CA, Jun. 2006.

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[12] A. Murk, N. Kämpfer, R. Wylde, T. Manabe, M. Seta, and J. Inatani, “Beam pattern measurements of the Submillimeter Limb-Sounder SMILES ambient temperature optics,” in 3rd ESA Millimetre Wave Technol. Applicat. Workshop, Espoo, Finland, May 2003, pp. 597–602. [13] T. Hirvonen, J. Ala-Laurinaho, J. Tuovinen, and A. V. Räisänen, “A compact antenna test range based on a hologram,” IEEE Trans. Antenna Propag., vol. 45, no. 8, pp. 1270–1276, Aug. 1997. [14] J. Mallat, “Vector measurements in MilliLab,” Millimetre Wave Lab. Finland, Espoo, Finland, Tech. Note, Aug. 2000. [15] T. Koskinen, J. Ala-Laurinaho, J. Säily, A. Lönnqvist, J. Häkli, J. Mallat, J. Tuovinen, and A. V. Räisänen, “Experimental study on a hologram-based compact antenna test range at 650 GHz,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2999–3006, Sep. 2005. [16] A. Lönnqvist, T. Koskinen, J. Häki, J. Säily, J. Ala-Laurinaho, J. Mallat, V. Viikari, J. Tuovinen, and A. V. Räisänen, “Hologram-based compact range for submillimeter-wave antenna testing,” IEEE Trans. Antenna Propag., vol. 53, no. 10, pp. 3151–3159, Oct. 2005. [17] P. H. Siegel, R. J. Dengler, T. Tsai, P. Goy, and H. Javadi, “Multiple frequency submillimeter-wave heterodyne imaging using an AB millimetre MVNA,” in Proc. IRMMW-THz, Sep. 2005, vol. 2, pp. 576–577. [18] C.-Y. E. Tong, S. Paine, and R. Blundell, “Near-field characterization of 2-D beam patterns of submillimeter superconducting receivers,” in Proc. 5th Int. Space Terahertz Technol. Symp., Ann Arbor, MI, Apr. 1994, pp. 660–673. [19] M. T. Chen, C. Y.-E. Tong, L. Chen, S. Paine, and R. Blundell, “Fullwave numerical modeling of near-field beam profiles at 200 and 700 GHz,” in Proc. 7th Int. Space Terahertz Tech. Symp., Charlottesville, VA, Mar. 1996, pp. 369–378. [20] M. T. Chen, C. E. Tong, S. Paine, and R. Blundell, “Characterization of corrugated feed horns at 216 and 300 GHz,” Int. J. Infrared Millim. Waves, vol. 18, no. 9, pp. 1697–1711, Sep. 1997. [21] C.-Y. E. Tong, D. V. Meledin, D. P. Marrone, S. Paine, H. Gibson, and R. Blundell, “Near-field vector beam measurement at 1 THz,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 6, pp. 235–237, Jun. 2003. [22] C.-Y. E. Tong, D. N. Loudkov, S. N. Paine, D. P. Marrone, and R. Blundell, “Vector measurement of the beam pattern of a 1.5 THz superconducting HEB receiver,” in Proc. 16th Int. Space Terahertz Tech. Symp., Göteborg, Sweden, May 2005, pp. 453–456. [23] Y.-J. Hwang, M.-T. Chen, E. Chung, and T.-H. Chu, “A novel nearfield vector beam pattern measurement system at 690 GHz,” in Proc. 34th Eur. Microw. Conf., Amsterdam, The Netherlands, Oct. 2004, pp. 557–560. [24] J. Säily, P. Eskelinen, and A. V. Räisänen, “Pilot signal based real-time measurement and correction of phase errors caused by microwave cable flexing in planar near-field tests,” IEEE Trans. Antenna Propag., vol. 51, no. 2, pp. 195–200, Feb. 2003. [25] S. Paine, D. C. Papa, R. L. Leombruno, X. Zhang, and R. Blundell, “Beam waveguide and receiver optics for SMA,” in Proc. 5th Int. Space Terahertz Technol. Symp., Ann Arbor, MI, Apr. 1994, pp. 811–823. [26] C.-Y. E. Tong, 1998, private communication. [27] A. D. Yaghjian, “Approximate formulas for the far field and gain of open-ended rectangular waveguide,” IEEE Trans. Antennas Propag., vol. AP-32, no. 4, pp. 378–384, Apr. 1984. [28] Y. Fujino and C.-Y. E. Tong, “Analysis of an open-end waveguide as a probe for near field antenna measurement by using TLM method,” IEICE Trans. Commun., vol. E77-B, no. 8, pp. 1048–1055, Aug. 1994. [29] H. A. Haus, “Hermite–Gaussian beams and their transformations,” in Waves and Fields in Optoelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1984, ch. 5, pp. 108–157.

Yuh-Jing Hwang (S’03–M’05) was born in ChangHua, Taiwan, R.O.C., in 1969. He received the B.S., M.S. degree in electrical engineering and Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1991, 1993 and 2005, respectively. From 1993 to 1995, he served in the Navy of the R.O.C., as a reserve officer. Since 1995, he has been with the Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), Taipei, Taiwan, R.O.C., as a Microwave Engineer, where he is currently an Assistant Research Fellow. His current research interest is the system integration of low-noise submillimeter-wave SIS heterodyne receivers, antenna measurement techniques in millimeter waves, and millimeter-wave monolithic integrated circuit design.

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Ramprasad Rao received the B.S.E.E. from the Indian Institute of Technology, Madras, Chennai, India, in 1991, and the Ph.D. degree in astronomy from the University of Illinois at Urbana-Champaign, in 1999. He is currently an Engineer and Scientist with the Academia Sinica Institute of Astronomy and Astrophysics, Hilo, HI, where he is involved with the Submillimeter Array. His current research involves the study of magnetic fields in the star formation process in astrophysics and the development of techniques that enable radio telescopes to detect such magnetic fields using polarimetric techniques. Dr. Rao was the recipient of a 1999 Grainger Fellowship to pursue post-doctoral studies with the Department of Physics, University of Chicago, Chicago, IL. He was also the recipient of the Submillimeter Array Post Doctoral Fellowship presented by the Harvard–Smithsonian Center for Astrophysics.

Rob Christensen, photograph and biography not available at time of publication.

Ming-Tang Chen was born in Tainan, Taiwan, R.O.C. He received the B.S. degree in physics from National Cheng Kung University, Tainan, R.O.C., in 1986, and the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1990 and 1993, respectively. From 1993 to 1995, he was Post-Doctoral Fellow with Case Western Reserve University, Cleveland, OH, where he studied pattern formation in helium isotope mixtures in low temperature. In 1995, he shifted his field to join the Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan, R.O.C., where he was involved with the development and construction of two antenna elements for the Sub-Millimeter Array. He is currently involved with a cosmology project: the Array for Microwave Background Anisotropy. He has broad interests in various scientific fields in experimental physics ranging from quantum dot to the comic microwave background. His expertise is in the cryogenic technique, quasi-optics, microwave and millimeter-wave instrumentations, and system integration of all kinds.

Tah-Hsiung Chu (M’87) received the B.S. degree in electrical engineering 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 a faculty member with 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 circuit and subsystems, microwave measurements, and calibration techniques.

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Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators Mark Oxborrow

Abstract—This paper explains how a popular commercially available software package for solving partial-differential-equations (PDEs), as based on the finite-element method, can be configured to efficiently calculate the frequencies and fields of the whispering-gallery (WG) modes of axisymmetric dielectric resonators. The approach is traceable; it exploits the PDE-solver’s ability to accept the definition of solutions to Maxwell’s equations in so-called weak form. Associated expressions and methods for estimating a WG mode’s volume, filling factor(s), and in the case of closed (open) resonators, its wall (radiation) loss, are provided. As no transverse approximation is imposed, the approach remains accurate even for quasi-transverse-magnetic/electric modes of low finite azimuthal mode order. The approach’s generality and utility are demonstrated by modeling several nontrivial structures, i.e., 1) two different optical microcavities (one toroidal made of silica, the other an AlGaAs microdisk), 2) a third-order sapphire:air Bragg cavity, and 3) two different cryogenic sapphire WG-mode resonators; both 2) and 3) operate in the microwave -band. By fitting one of 3) to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated. Index Terms—Finite-element methods (FEMs), microwave, optical, resonators, rotational symmetry, sapphire, simulation, whispering gallery modes (WGMs).

I. INTRODUCTION ONTRIVIAL electromagnetic structures can be modeled through computer-aided design (CAD) tools in conjunction with programs for numerically solving Maxwell’s equations. Though alternatives abound [1]–[3], the latter often use the finite-element method (FEM) [4], [5]. Within such a scheme, a problem frequently encountered when attempting to determine the values of electromagnetic parameters from experimental data is a lack of traceability: significant dependencies between the data, the model’s configurational settings, and the inferred values of parameters cannot be adequately isolated, understood, or quantified. Traceability demands that both the model’s definition and its solution remain amenable to complete explicit description. Furthermore, convenience

N

Manuscript received November 21, 2006; revised February 19, 2007. This work was supported by the U.K. National Measurement System under the Quantum Metrology Programme. The author is with the National Physical Laboratory, Teddington, Middlesex TW11 0LW, 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.2007.897850

requires that the representations adopted for this purpose be concise—yet wholly unambiguous. A. Whispering-Gallery (WG)-Mode Resonators Certain compact electromagnetic structures support closed WG modes. Though elliptical [6] or even nonplanar (“crinkled” [7] or “spooled” [8]) WG morphologies exhibit advantageous features with respect to certain applications, this paper considers only (closed, planar) WG modes with circular trajectories, as supported by axisymmetric resonators within the class depicted in Fig. 1, upon which various recent scientific innovations [9]–[12] are based. Many of the current commercial software packages for modeling electromagnetic resonators suffer, however, from a “blind spot” when it comes to calculating efficiently (hence, accurately) such resonators’ WG modes. The popular MAFIA/CST package [13] is a case in point: as Basu et al. [14] and no doubt others have experienced, it cannot be configured to take advantage of a circular WG mode’s known az, where (an integer 0) imuthal dependence viz. is the mode’s azimuthal mode order, and is the azimuthal coordinate. Though frequencies and field patterns can be obtained (at least for WG modes of low azimuthal mode order), the computationally advantageous reduction of the problem from 3-D to 2-D that the resonator’s rotational symmetry affords is, consequently, precluded;1 and the ability to simulate high-order WG modes with sufficient accuracy (for metrological purposes) is, exasperatingly, lost. B. Brief Selected History of WG-Mode Simulation The method of “separating the variables” provides analytical expressions for the WG modes of right-cylindrical uniform dielectric cavities (or shells) [15]–[17]. By matching expressions across certain boundaries, approximate WG-mode solutions for composite cylindrical cavities can be obtained [9], [18], [19], whose discrete/integer indexes (related to symmetries) provide a nomenclature [20] for classifying the lower order WG modes of all similar structures. Extensions of the basic mode-matching method encompassing spatially nonuniform field polarizations have been developed [3]. The accurate solution of arbitrarily shaped axisymmetrical dielectric resonators requires numerical methods of which there are several relevant classes and variants. Apart from the FEM itself, the most developed and (thus) immediately exploitable 1About

the best one can do is simulate a “wedge” (over an azimuthal domain

1 = =(2M ) wide) between radial electric and magnetic walls.

0018-9480/$25.00 © 2007 IEEE

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Fig. 1. Generic axisymmetric resonator in cross section (medial half-plane). A dielectric volume (in 3-D) or “domain” (in 2-D) is enclosed by an electric wall (E1)—or one subject to some different boundary condition, as per Section III-E.2. This domain comprises several subdomains (D1, D2, and D3), each containing a spatially uniform dielectric (that could be just free space). Some of these subdomains (D2 and D3) are bounded internally by electric walls (E2, E2 , and E3). The resonator has (optionally) a mirror symmetry through its (horizontal) equatorial plane (dashed line M1); on imposing an electric or magnetic wall over this plane, only either the upper or lower half of the resonator need be simulated.

spurious solutions [30], [31], associated with the local gauge invariance, or “null space” [31], that is a feature of the equations’ “curl” operators. At least two research groups have nevertheless developed in-house software tools for calculating the WG modes of axisymmetric dielectric resonators, that: 1) solve for all field components (i.e., no transverse approximation is invoked); 2) are 2-D (and thus, numerically efficient); and 3) effectively suppress spurious solutions (without detrimental side effects) [21], [22], [30], [32], [33]. The method described in this paper sports these same three attributes. With regard to 3), the approach adopted by Auborg et al. was to use different finite elements (viz. a mixture of “Nedelec” and “Lagrange”—both second order) for different components of the electric and magnetic fields; Osegueda et al. [32], on the other hand, used a so-called “penalty term” to suppress (spurious) divergence of the magnetic field. Stripping away its motivating remarks, applications, and illustrations, this paper, in essence, translates the latter approach into explicit weak-form expressions that can be directly and openly ported to any PDE solver (most notably COMSOL/FEMLAB [27]) capable of accepting such. II. METHOD OF SOLUTION A. Weak Forms

alternatives include (given here only for reference—not considered in any greater detail): 1) the Ritz–Rayleigh variational or “moment” methods [21]–[23]; 2) the finite-difference time-domain (FDTD) method [1], [24]; and 3) the boundary-integral method [2] or boundary-element method (BEM) (including FEM-based hybridizations thereof [25]). Zienkiewicz and Taylor [4] (particularly their Table 3.2) indicate various commonalities between them (and the FEM).2 The application of the FEM to the solving of Maxwell’s equations has a history [26], and is now an industry [27], [28]; [4] supplies FEM’s theoretical underpinnings. Though the method can solve for all three of a WG mode’s field components, the statement of Maxwell’s (coupled partial differential) equations in component form can be onerous if not excluded outright by the equation-solving software’s lack of configurability. With circular WG modes, the configurational effort can be significantly reduced by invoking a so-called “transverse” approximation [14], [29], wherein only a single (scalar) partial differential equation (PDE) is solved (in 2-D). Here, either the magnetic or electric field is assumed to lie everywhere parallel to the resonator’s axis of rotational symmetry (see [29, Fig. B.1]). This approximation is, however, uncontrolled.3 This paper demonstrates that, through only a modicum of extra configurational effort, the transverse approximation and its associated doubts can be wholly obviated. A problem that besets the direct application of the FEM to the solving of Maxwell’s equations is the generation of (many) 2It

is remarked here that the FDTD may be regarded as a variant of FEM employing local discontinuous shape functions. It is perhaps also worth acknowledging that, for resonators comprising just a few, large domains of uniform dielectric, the boundary-integral methods (based on Green functions), which—in a nutshell—exploit such uniformity to reduce the problem’s dimensionality by one, will generally be more computationally efficient than FEM. 3It is noted that basic mode matching [9] also invokes the same transverse approximation and is thus equally uncontrolled.

Scope: The resonators treated below comprise volumes of dielectric space bounded by either electric or magnetic walls (or both)—see again Fig. 1; the restriction to axisymmetric resonators is only invoked at the beginning of Section II-B. The resonator’s dielectric space comprises voids (i.e., free space) and pieces of (sufficiently low-loss) dielectric material; its (default) enclosing surfaces will generally be metallic, corresponding to electric walls. When modeling resonators whose forms exhibit reflection symmetries, where the magnetic and electric fields of their solutions transform either symmetrically or antisymmetrically through each mirror plane, perfect magnetic and electric walls can be alternatively imposed over these planes to solve for different “sectors” of solutions. The electromagnetic fields within the dielectric volumes of the resonator obey Maxwell’s equations [17], [34], as they are applied to continuous macroscopic media [35]. Assuming the resonator’s constituent dielectric elements have negligible (or at least the same) magnetic susceptibility, the magnetic field will be continuous across interfaces.4 This propstrength erty makes it advantageous to solve for (or, equivalently, the magnetic flux density —with a constant global magnetic permeability ), as opposed to the electric field strength (or displacement ). Upon substituting one of Maxwell’s curl equations into the another, the problem reduces to that of solving a (modified) vector Helmholtz equation (1) subject to appropriate boundary conditions (read on). Here, is the speed of light and is the inverse relative permittivity tensor; one assumes that the resonator’s dielectric elements are is a (tensorial) constant—i.e., independent linear such that 4The method described in this paper could be to extended to treat resonators containing dielectrics with different magnetic susceptibilities by setting up (within the PDE solver—i.e., COMSOL) “coupling variables” at interfaces.

OXBORROW: TRACEABLE 2-D FINITE-ELEMENT SIMULATION OF WG MODES OF AXISYMMETRIC ELECTROMAGNETIC RESONATORS

of field strength. Providing no magnetic monopoles lurk inside be free of the resonator, Maxwell’s equations demand that . The middle so-called “penalty” term divergence, i.e., on the left-hand side of (1) acts to suppress spurious solutions, ; it has exactly the same form as for which (in general) that used by Osegueda et al. [32].5 The constant controls the penalty term’s weight with respect to its Maxwellian neighbors; was taken for every simulation presented in Sections IV and V below. Reference [17] (particular Sect. 1.3 thereof) supplies a primer on the electromagnetic boundary conditions stated here. The magnetic flux density at any point on a (perfect) electric wall , where denotes the wall’s surface normal satisfies vector. Providing the magnetic susceptibility of the dielectric medium bounded by the electric wall is not anisotropic, this condition is equivalent to

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where

denotes a volume integral over the resonator and , where are the components of the inverse relative permittivity tensor. The three terms appearing in the integrand correspond directly to the three weak-form terms required to define an appropriate model within a PDE solver. Assuming that the physical dimensions and electromagnetic properties of the resonator’s components are temporally invariant (or at least “quasi-static”), solutions or “modes” take , where is the vector the form of spatial position, is the time, and is the mode’s resonance frequency. The last “temporal” term in (6)’s integrand can , where thereupon be re-expressed as ; this re-expression reveals the integrand’s complete dual symmetry between and . B. Axisymmetric Resonators

(2) The electric field strength at the electric wall obeys (3) These two equations ensure that the magnetic (electric) field strength is oriented tangential (normal) to the electric wall. As is pointed out in [32], (3) is a so-called “natural” (or, synonymously, a “naturally satisfied”) boundary condition within the FEM—see [4]. The boundary conditions corresponding to a perfect magnetic wall (dual to those for an electric wall) are

The analysis is now restricted to axisymmetric resonators, where a system of cylindrical coordinates (see top right of Fig. 1), aligned with respect to the resonator’s axis of rotational symmetry, has components `rad(ial)' `azi(muthal)' `axi(ial)' . The aim is to calculate the resonance frequencies and field patterns of the resonator’s circular WG modes, whose phase varies as , with being the mode’s azimuthal mode order.6 Viewed as a three-component vector field over (for the moment) a 3-D space, the time-independent part of the magnetic field strength now takes the form (7)

(4) and (5) These two equations ensure that the electric displacement (magnetic field strength) is oriented tangential (normal) to the magnetic wall. Again, the latter equation is naturally satisfied. One now invokes Galerkin’s method of weighted residuals [4]; [31] provides an analogous treatment when solving for . Both sides of (1) are multiplied the electric field strength (scalar-product contraction) by the complex conjugate of a , then integrated over the “test” magnetic field strength resonator’s dielectric volume. Upon expanding the permittivity-modified “curl of a curl” operator (to extract a similarly modified Laplacian operator), then integrating by parts (spatially), then disposing of surface terms through the electric- or magnetic-wall boundary conditions stated above, one arrives (equivalent to [32, eq. (2)]) at

where an has been inserted into the field’s azimuthal component to allow, in subsequent solutions, all three to be each expressible as component amplitudes a real amplitude multiplied by a common complex phase factor. The relative permittivity tensor of an axisymmetric dielectric material is diagonal with entries (running down the diagonal) , where is the material’s relative permittivity in the axial direction (in the transverse or “perpendicular” plane—as spanned by its radial and azimuthal directions). One now substitutes (7) into (6)’s integrand; textbooks provide the required explicit expressions for the vector differential operators in cylindrical coordinates [16], [17]. A radial factor is included here from the volume integral’s measure (the common factor of is dropped from all expressions below.) The first “Laplacian” weak term is given by (8) where

(6) 5Though COMSOL can implement mixed (“Nedelec” plus “Lagrange”) finite elements [21], it was found that (1)’s penalty term (with its weighting factor somewhere in the range 0:01  10) could, in conjunction with secondorder Lagrange finite elements (applied to all three components of ) always satisfactorily suppress the spurious modes.

 

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(9) 6The method does not require M to be large; even modes that are themselves axisymmetric, corresponding to M = 0, such as the one shown in Fig. 6(b), can be calculated.

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and

gives both (22)

(10) and (11) Notation: denotes the partial derivative of (the azimuthal component of the magnetic field strength) with respect to (the radial component of displacement), etc. Here, the individual factors and terms have been ordered and grouped so as to display the dual symmetry. Similarly, the weak penalty term is given by

(23) connects Note that the transformation (17) with (22) and (20) with (21). The above weak-form expressions and boundary conditions, viz. (8)–(23) are the key results of this paper: once inserted into a PDE solver, the WG modes of axisymmetric dielectric resonators can be readily calculated.

(12)

III. POSTPROCESSING OF SOLUTIONS

(13)

Having determined for each mode its frequency and all three components of its magnetic field strength as a function of position, other relevant fields and parameters can be derived from this information.

where

A. Other Fields (Related Through Maxwell’s Equations) (14) (15) and the temporal weak-form (“dweak”) term is given by

(16) where denotes the double partial derivative of w.r.t. time, etc. Note that none of the terms on the right-hand sides of (8) through (16) depend on the azimuthal coordinate ; the problem has been reduced from 3-D to 2-D.

Straightaway, the magnetic flux density . As no real (“nondisplacement”) current flows within a dielectric, , thus , and , where is the (diagonal) inverse permittivity tensor, as already discussed above in connection with (6). B. Mode Volume Accepting various caveats (most fundamentally, the problem of mode-volume divergence—see footnote 8), as addressed by Kippenberg [29], the volume of a mode is defined as [12] (24)

C. Axisymmetric Boundary Conditions An axisymmetric boundary’s unit normal in cylindrical components can be expressed as —note the vanishing azimuthal component. The electric-wall boundary conditions, in gives cylindrical components, are as follows: (17) and

gives both (18)

where denotes the maximum value of its functional denotes integration over and argument and around the mode’s “bright spot”—where its electromagnetic field energy is concentrated. C. Filling Factor The resonator’s electric filling factor, for a given dielectric component (labeled “diel.”), a given mode, and a given field direction ` ' radial, azimuthal, axial is defined as (25)

and (19) When the dielectric permittivity of the medium bounded by the electric wall is isotropic, the last condition reduces to (20) The magnetic-wall boundary conditions, in cylindrical components, are as follows: gives (21)

where denotes integration over for the component’s volume and radial or azimuthal, axial . The numerators and denominators of (24) and (25) can be readily evaluated using the PDE solver’s post-processing features. D. Wall Loss (Closed Resonators) Real resonators suffer losses that render the ’s of their modes finite. The energy stored in a mode’s electromag. For axisymmetric netic field is

OXBORROW: TRACEABLE 2-D FINITE-ELEMENT SIMULATION OF WG MODES OF AXISYMMETRIC ELECTROMAGNETIC RESONATORS

resonators, the 3-D volume integral reduces to over the resonator’s medial a 2-D integral half-plane. The surface current induced in the resonator’s ; enclosing electric wall is (see, e.g., [17, p. 205]) the averaged-over-a-cycle power dissipated in the wall is , where is the tangential (with is the respect to the wall) component of , wall’s surface resistivity (see [17] and [34]), is the wall’s (bulk) electrically conductivity,7 and is the mode’s frequency. For axisymmetric resonators, the 2-D surface integral reduces to a 1-D integral around the periphery of the resonator’s medial ( – ) half-plane. The quality factor, defined , due to the wall loss can thus be expressed as as

(26) where , which has the dimensions of a length, is defined as

(27)

Again, both integrals (numerator and denominator) (hence, itself) can be readily evaluated through the PDE solver’s post-processing features.

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induction theorem [39], [41], the tangential magnetic field of , at any point just inside the closed resonator’s this mode electric wall, can be related to that of the corresponding open at the same point, through resonator’s radiation . The radiation loss can be evaluated by integrating the cycle-averaged Poynting vector over the electric wall, i.e., , where is the impedance of free space. A bound on the mode’s radiative factor can thus be expressed as (28) approaching equality when the WG mode’s bright spot lies (in effect) at an antinode of the mode’s standing wave; here, is , only now the exactly that given by (27) with enclosing electric wall is in the radiation zone. 2) Overestimator of Loss Via Outward-Going Free-Space Impedance Matching: A complementary bound can be constructed by replacing the above closed resonator’s electric wall with one, of the same form, that attempts to match, impedance-wise, the open resonator’s radiation—and thus absorb it. For transverse locally plane-wave radiation in the radiation zone (in free space) sufficiently far from the resonator, the required impedance-matching boundary condition on the ,9 where is the wall’s wall is inward-pointing normal. Upon differentiating with respect to time and using Maxwell’s displacement-current equation, this condition can, for a given mode, be generalized to

(29)

E. Radiation Loss (Open Resonators) Preliminary Remarks: With open WG-mode resonators (either microwave [38] or optical [10], [12]), the otherwise highly localized WG mode spreads throughout free space;8 energy flows away from the mode’s bright spot (where the electricand magnetic-field amplitudes are greatest) as radiation. The tangential electric and magnetic fields on any closed surface surrounding the bright spot constitute, by the “Field Equivalence Principle” [39], [40] (as a formalization of Huygen’s picture), a secondary source of this radiation. 1) Underestimator of Loss Via Retro-Reflection: Consider a closed equivalent of the open resonator, with an enclosing electric wall in the WG-mode’s radiation zone. The wall’s form is chosen such that—as far as possible—the open resonator’s radiation propagates (as a predominantly transverse and locally plane wave) in a direction that is locally normal to the wall. The electric wall then reflects the otherwise open resonator’s radiation back onto itself—thus creating a standing wave, i.e., a lossless mode. Through an argument reminiscent of Schelkunoff’s

is the mode’s frequency, as before, where and is a “mixing angle”;10 for impedance matching (with . Unless respect to an outward-going radiation), for integer , the in (29) breaks the Hermitian-ness of the matrix that the PDE solver is required to eigensolve, leading to decaying modes with complex eigenfre, and corresponding quality factors equal [12] quencies , where and denote real to and imaginary parts. Without fine tuning, the enclosing wall’s shape will not lie everywhere exactly orthogonal to the direction of propagation of the WG mode’s radiation; thus, even for , the radiation will not be completely absorbed at the wall. A bound on the mode’s radiative factor can thus be expressed as

7It is pointed out here that, at liquid-helium temperatures, the bulk and surface resistances of metals can greatly depend on the levels of (magnetic) impurities within them [36], and the textbook f dependence of surface resistance on frequency is often violated [37].

9Note that the direction (polarization) of or in the wall’s plane is not constrained; the two fields need only be orthogonal with their relative amplitudes in the ratio of the impedance of free space z . 10The two terms on the left-hand side of (29) can be viewed as implementing electric wall [cf. (3)] and magnetic wall [cf. (5)] boundary conditions, respectively. The (composite) boundary condition can be continuously adjusted between these two cases by varying the mixing angle  . ( = =4 corresponds to impedance matching inward-coming (as opposed to an outwardgoing) radiation.)

8As understood by Kippenberg [29], this observation implies that the support . . . dV integral (spanning the WG mode’s bright spot) must of (24)’s be somehow limited, spatially, or otherwise (asymptotically) rolled off, lest the integral diverge. (The so-called “quantization volume” associated with the radiation extends to infinity.)

(30) approaching equality on perfect absorption (no reflections).

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TABLE I ELECTRIC FILLING FACTORS FOR THE WGE MODE OF UWA’s SLOPING-SHOULDERS RESONATOR

Fig. 2. UWA’s sloping-shoulder cryogenic sapphire resonator. (a) Medial cross section through its geometry; the grey (white) shading corresponds to sapphire (vacuum); S and S indicate the sapphire piece’s upper and lower “shoulders.” (b) Mesh of the resonator’s model structure generated by the FEM-based PDE solver; for clarity, only every other meshing line is drawn (i.e., (b) displays the “half-mesh”); within (c) and (d), the (logarithmic) gray scale reflects the absolute value of the vectorial magnetic and electric fields, respectively; white arrows indicate the magnitude and direction of each field’s medial component.

H

E

IV. EXAMPLE APPLICATIONS The source codes and configuration scripts used to implement the simulations presented here and in Section IV-B are freely available from the author. A. “Sloping-Shoulders” Cryogenic Sapphire Microwave Resonator (University of Western Australia) This axisymmetric resonator [42] comprises a piece of monocrystalline sapphire mounted (coaxially) within a cylindrical metal can—see Fig. 2(a); the crystal’s optical (or “ ”) axis lies parallel to the resonator’s geometric axis. The piece’s sloping shoulders ( and ) make accurate simulation via mode-matching less straightforward. The resonator’s form, as encoded into the PDE solver, is taken from [9, Fig. 3],11 with the piece’s outer diameter, the length of its outer axial sidewall, the axial extent of each sloping shoulder, and the radius of each of its two spindles equal to, at liquid-helium temperature (i.e., including cryogenic shrinkages—see Section V) 49.97, 19.986, 4.996, and 19.988 mm, respectively. The sapphire crystal’s cryogenic permittivities were taken to , as given in [20]. Since be the sapphire piece and its surrounding metal do not share a common transverse (“equatorial”) mirror plane, the speeding up of the simulation through the imposition of a magnetic or electric wall on such a plane (so halving the 2-D region to be analyzed) is precluded. 11It

is remarked here that the drawn shape of the sapphire piece in [9, Fig. 3] is not wholly consistent with its given dimensions: its outer axial sidewall is too long and the slope of its shoulders is too slight.

Fig. 2(b) displays the FEM-based PDE solver’s meshing of the model resonator’s structure; in COMSOL’s vernacular,12 the mesh comprises 7296 base-mesh elements and 88587 DOFs. It took typically 75 s to obtain the resonator’s lowest (in frequency) 16 modes, for a single given azimuthal mode order at [with respect to Fig. 2(b)] full mesh density, on a standard 2004-vintage personal computer (2.4-GHz Intel Xeo CPU) without altering the PDE solver’s default eigensolver settings. , the model resWith the azimuthal mode order set at mode was found to lie at 11.925 GHz, to onator’s be compared with 11.932 GHz found experimentally [9]. Wall Loss: This mode’s characteristic length , as evaluated with respect to the can wall’s enclosing surface, was determined to be 2.6 10 m. Based on [37], one estimates the surface resistance of copper at liquid-helium temperature to be 7 10 per square at 11.9 GHz, leading to a wall-loss of 3.5 10 for the mode. Filling Factor: Using (25), the electric filling factors for the mode were evaluated. These factors, presented in Table I, are in good agreement with those labeled “FE” in [9, Table IV], that were obtained via a wholly independent FEM simulation of the same resonator. B. Toroidal Silica Optical Resonator (California Institute of Technology) The resonator modeled here, based on [11], comprises a silica toroid, supported above a substrate by an integral interior “web”; its geometry is shown in Fig. 3(a). The toroid’s principal and minor diameters are m, respectively. The silica dielectric is presumed to be wholly isotropic (i.e., no significant residual stress) with a relative per, corresponding to a refractive index of mittivity of at the resonator’s operating wavelength (around 852 nm) and temperature. The FEM model’s pseudorandom triangular mesh covered an 8 8 m square [shown in dashed outlined on the right-hand side of Fig. 3(a)] with an enhanced meshing density over the silica circle and its immediately surrounding free space; in total, the mesh comprised . Temporarily 5990 (base-mesh) elements with adopting Spillane et al.’s terminology, the resonator’s fundamental TE-polarized 93rd-azimuthal-mode-order mode (where “TE” implies here that the polarization of the mode’s electric field is predominantly aligned with the toroid’s principal axis—not transverse to it) was found to have a frequency of 3.532667 10 Hz, corresponding to a free-space wavelength nm (thus close to 852 nm). Using (24), this of mode’s volume was evaluated to be 34.587 m ; if, instead, 12The size/complexity of a finite-element mesh is quantified, within COMSOL Multiphysics, by: 1) the number of elements that go to compose its so-called “base mesh” and 2) its total number of degrees of freedom (DOFs)—as associated with its so-called “extended mesh.”

OXBORROW: TRACEABLE 2-D FINITE-ELEMENT SIMULATION OF WG MODES OF AXISYMMETRIC ELECTROMAGNETIC RESONATORS

Fig. 3. (a) Geometry (medial cross section) and dimensions of a model toroidal silica microcavity resonator—after [11]; the torus’ principal diameter D m; the central vertical dashed line inm and its minor diameter d dicates the resonator’s axis of continuous rotational symmetry. (b) False-color surface plot of the (logarithmic) electric-field intensity over the dashed box WG mode; white arrows inappearing in (a) for this resonator’s dicate the electric field’s magnitude and direction in the medial plane.

16

=

=3

TE

j Ej

the definition stated in [11, eq. 5] is used, the volume becomes greater. These two values 72.288 m —i.e., a factor of straddle the volume of 55 m for the same dimensions of silica toroid and (TE) mode polarization, as inferred by the eye from [11, Fig. 4].

C. Conical Microdisk Optical Resonator (California Institute of Technology) The mode volume can be reduced by going to smaller resonators, which, unless the optical wavelength is commensurately reduced, implies lower azimuthal mode orders. The model “microdisk” resonator analyzed here, as depicted in Fig. 4(a), duplicates the structure defined in [12, Fig. 1(a)]; the disk’s constituent dielectric (alternate layers of GaAs and GaAlAs) is approximated as a spatially uniform isotropic dielectric with . The FEM-modeled doa refractive index equal to main comprised 4928 quadrilateral base-mesh elements with . Adopting Srinivasan et al.’s notation, the resWG mode, as shown in Fig. 4(b), was onator’s found at 2.372517 10 Hz, equating to nm; for comparison, Srinivasan et al. found an equivalent mode at 1265.41 nm (as depicted in [12, Fig. 1(b)]). It is pointed out here that the white electric-field arrows in Fig. 4(b) [and also, though to a lesser extent, in Fig. 3(b)] are not perfectly vertical—as the transverse approximation taken in [9] and [29] would assume them to be; the quasiness of the true mode’s transverse-electric polarization is thus revealed. Mode Volume: Using (24), but with the mode excited as a standing-wave (doubling the numerator while quadrupling the denominator), the mode volume is determined to be

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Fig. 4. (a) Geometry (medial cross section) and (half-) meshing of model GaAlAs microdisk resonator—after [12]; the disk’s median diameter is : m and its thickness (axial height) t nm; its conical external D sidewall subtends an angle  to the disk’s (vertical) axis; for clarity, only every other line of the true (full) mesh is drawn. The modeled domain in the medial half-plane is a rectangle stretching from 0.02 to 1.5 m in the radial direction and from 0.5 to 0.5 m in the axial direction. (b) False-color for the resonator’s surface plot of the (logarithmic) electric-field intensity mode at  : nm; again, white arrows indicate the electric field’s magnitude and direction in the medial plane.

= 2 12

= 26

0

TE

+ = 1263 6

= 255

jE j

m , which is still in good agreement with Srinivasan et al.’s . mode’s radiation loss was Radiation Loss: The estimated by implementing both the upper and lower bounding estimators described in Section III-E. Here, the microdisk and its mode were modeled over an approximate sphere, equating to a half-disk in 2-D (medial plane). The half-disk’s diameter was 12 m and different electromagnetic conditions were imposed on its semicircular boundary—see Fig. 5.13 With an electric-wall condition (i.e., (2) and (3) or, equivalently, 17 and {18, 20}), imposed on the half-disk’s entire boundary [as per Fig. 5(b)], the right-hand side of (28) was evaluated. With the condition [viz. (3)] on the boundary’s semicircle replaced by the outward-going plane-wave (in free space) ), impedance-matching condition (viz. (29) with condition [see (2)] is maintained, the while the right-hand side of (30) was evaluated for the radiation pattern displayed in Fig. 5(c). For a pseudorandom triangulation , the mesh comprising 4104 elements, with PDE solver took, on the author’s office computer, 6.55 and 13.05 s, corresponding to Fig. 5(b) and (c), respectively,14 to calculate ten eigenmodes around 2.373 10 Hz, of which the mode was one. Together, the resultant estimate mode’s radiative-loss quality factor is on the 13It is acknowledged that, in reality, the microdisk’s substrate would occupy a considerable part of the half-disk’s lower quadrant. 14The complex arithmetic associated with the impedance-matching boundary condition meant that the PDE solver’s eigensolution took approximately twice as long to run with this condition imposed, as compared to pure electric or magnetic wall boundary conditions that do not involve complex arithmetic.

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Fig. 5. Radiation associated with the same (TE ,  = 1263:6 nm) WG mode as presented in Fig. 4 (all three maps use the same absolute false-color scale). (a) Standing-wave (equal outward and inward going) radiation with the outer semicircular boundary set as a magnetic wall. (b) Same, but now with the boundary set as an electric wall. (c) Somewhat traveling (more outward than inward going) radiation with the boundary’s impedance set to that of an outward-going plane wave in free space (and with the normal magnetic field constrained to vanish). That (c)’s radiation field is somewhat dimmer than (b)’s is consistent with the two different estimates of the resonator’s radiative Q corresponding to (b) and (c) [see text].

Fig. 6. (a) Geometry (medial cross section) of a third-order DBR resonator within a cylindrical metallic can (hence, electric interior walls—represented by a solid black rectangle); as per [43], the can’s interior diameter is 10.98 cm and its interior height is 13.53 cm; the horizontal and vertical gray (pink in online version) stripes denote cylindrical plates and barrels of sapphire; white rectangles correspond to right cylinders/annuli of free space; the vertical arrow indicates the resonator’s axis of rotational symmetry, with which the sapphire crystal’s c-axis is aligned; a magnetic wall is imposed over the resonator’s equatorial plane of mirror symmetry (dashed horizontal line; cf. M1 in Fig. 1). (b) Falseover the top-right mecolor plot of the (logarithmic) electric-field intensity dial quadrant of the rotationally invariant (M = 0) TE mode; note how the mode is strongly localized within the resonator’s central cylinder of free space.

j Ej

of 9.8

, to be compared with the estimate 10 (at 1265 nm) reported in [12, Table 1].

TABLE II NPL’s CRYOGENIC SAPPHIRE RESONATOR: SIMULATED AND EXPERIMENTAL WG MODES COMPARED

D. Distributed Bragg Reflector (DBR) Microwave Resonator The method’s ability to simulate axisymmetric resonators of arbitrary cross-sectional complexity is demonstrated here by mode of a DBR resonator as anasimulating the 10-GHz lyzed by Flory and Taber (F&T) [43] through mode matching. The resonator’s model geometry was generated through an auxiliary script written in MATLAB. Its corresponding mesh comprised 5476 base-mesh elements, with 66603 DOFs, with eight layer of sapphire. Based on [44]’s edge vertices for each quartic fitting polynomials, the sapphire crystal’s dielectric perK, were set to mittivities, at a temperature (consistent with [43]) and . The mode shown in Fig. 6(b) was found to lie at 10.00183 GHz, which is in good agreement with F&T’s “precisely 10.00 GHz.” Using (25), the mode’s electric filling factor for the resonator’s sapphire parts was 0.1270, which, assuming an (isotropic) loss tangent of 5.9 10 as per F&T, corresponds to a dielectric-loss factor of 1.334 million. Through (26) and (27), and assuming a surface resistance of 0.026 per square as per F&T, the wall-loss factor was determined to be 29.736 million, leading to a composite (for dielectric and wall losses operating in tandem) of 1.278 million. These three values are consistent with F&T’s stated (composite) of “1.3 million, and limited entirely by dielectric losses.” V. DETERMINATION OF THE PERMITTIVITIES OF CRYOGENIC SAPPHIRE The method is applied here to determine the two dielectric constants of monocrystalline sapphire (HEMEX grade [45]) at 4.2 K from a set of experimental data, listed in the four right-most columns of Table II, and corresponding to the resonator whose innards are shown in Fig. 7(a). Allowance was made for the shrinkages of the resonator’s constituent materials

the nomenclature of [7] is used for this column. FWHM ( 3 dB). the difference in frequency between the orthogonal pair of standing-wave resonances (akin to a “Kramers doublet” in atomic physics) associated with the WG mode; the experimental parameters stated in other columns correspond to the strongest resonance (greatest S at line center) of the pair.

0

from room to liquid-helium temperature15 and the values of the sapphire’s two dielectric constants ( and ) were initially set equal to those specified in [20]. Fig. 7(b)’s geometry was meshed with quadrilaterals over the medial half-plane with . For 8944 elements in its base mesh, and with 15By integrating up published linear-thermal-expansion data (viz. [46, Table 4], [47, Table I]), sapphire’s two cryo-shrinkages were estimated to be (1.0 7.21 10 ) and (1.0 5.99 10 ) for directions parallel and perpendicular to the sapphire’s c-axis, respectively. From [48, Table F], the cryo-shrinkage of (isotropic) copper was taken to be (1.0 3.26 10 ).

2

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2

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Fig. 8. Plot used to aid the identification of experimental with simulated WG modes. Solid horizontal lines (16 in total) indicate the center frequencies of the former. Solid circles indicate the identification of a simulated mode with an experimental one (the difference in their frequencies corresponds to much less than a circle’s radius in all cases); hollow circles indicate simulated modes that were not identified with any experimentally measured one. Quasi-transversemagnetic (q-TM) and quasi-transverse-electric (q-TE) WG modes of the same family are joined by dashed and dotted lines respectively; a few of the lowest lying mode families are labeled using standard notation [7].

Fig. 7. (a) Close-up of NPL’s cryo-sapphire resonator, with the main body of its outer copper removed. The resonator’s chamfered HEMEX sapphire ring has an outer diameter of 46.0 mm and an axial height of 25.1 mm. The ring’s integral interior “web,” 3-mm thick, lies parallel to, and is centered (axially) on, the ring’s equatorial plane, and is supported through a central copper post. Optical refraction at the ring’s outer surface falsely exaggerates its internal diameter. Above the ring are two loop probes for coupling electromagnetically to the resonator’s operational WG mode. (b) Geometry of the resonator in medial cross section; gray indicates sapphire, white free space; bounding these dielectric domains, and shown as solid black lines, are copper surfaces belonging to the resonator’s can and web-supporting post [the dashed vertical line (lower right) runs along the resonator’s cylindrical axis (r = 0)]. (c) False-color for the resonator’s 11th-azimuthal-mode-order map (logarithmic scale) of fundamental quasi-transverse-magnetic (N1 in [7]’s notation) WG mode at 9.146177 GHz (simulated), as detailed on the sixth row of Table II. The white arrows indicate the magnitude and direction of this mode’s electric field strength ( ) in the medial plane.





j Hj

E

a given azimuthal mode order , calculating the lowest 16 eigenmodes took around 3 min on the author’s office PC (as previously specified). With Fig. 8 as a guide, each of the 16 experimental resonances was identified to a particular simulated WG mode, as specified in the fourth column of Table II, lying near to it in frequency; these identifications were influenced by requiring that the measured attributes (e.g., the full width of half maximum (FWHM) linewidths) of the experimental resonances belonging—as per their identifications—to the same “family” of WG modes (e.g., S1 or N2) varied smoothly with . Filling factors were then calculated to quantify the differential change and in the frequency of each identified mode with respect to . The two dielectric constants were then adjusted to minimize variance of the residual (simulated minus measured) the frequency differences. The resultant best fit values, to which the simulated data occupying the three left-most columns in Table II correspond were (31) (32)

The nominal error assigned to each reflects uncertainties in the identifications of certain experimental resonances, each lying almost equally close (in both frequency and other attributes) to two or more different simulated WG modes. Errors resulting from a finite meshing density [33], or those from the finite dimensional/geometric tolerances to which the resonator’s shape was known, were estimated to be small in comparison. ACKNOWLEDGMENT The author thanks A. Laporte and D. Cros, both with XLIM, Limoges, France, for an independent (and corroborating) 2-D FEM simulation of the resonator considered in Section V, and J. Breeze, Imperial College, London, U.K., for suggesting the DBR resonator analyzed in Section IV-D. The author also thanks three colleagues at the National Physical Laboratory (NPL), Teddington, Middlesex, U.K., namely, G. Marra, for some of the experimental data presented in Table II, C. Langham for his estimated values for the cryo-shrinkages of sapphire, and L. Wright, for a detailed review of an early manuscript of this paper. REFERENCES [1] K. S. Kunz, The Finite Difference Time Domain Method for Electromagnetics. Boca Raton, FL: CRC, 1993. [2] S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient design of specially engineered whispering-gallery-mode laser resonators,” Opt. Quantum Electron. vol. 35, pp. 545–559, 2003 [Online]. Available: http://www.nottingham.ac.uk/ggiemr/Project/boriskina2.htm [3] J. Ctyroky, L. Prkna, and M. Hubalek, “Rigorous vectorial modelling of microresonators,” in Proc. 6th Int. Transparent Opt. Netw. Conf., Jul. 4–8, 2004, vol. 2., pp. 281–286. [4] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 5th ed. New York: Butterworth Heinemann, 2000, vol. 1, The Basis. [5] M. M. Taheri and D. Mirshekar-Syahkal, “Accurate determination of modes in dielectric-loaded cylindrical cavities using a one-dimensional finite element method,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 10, pp. 1536–1541, Oct. 1989.

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[6] J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonator optical cavities,” Nature, vol. 385, pp. 45–47, 1997. [7] M. E. Tobar, J. G. Hartnett, E. N. Ivanov, P. Blondy, and D. Cros, “Whispering gallery method of measuring complex permittivity in highly anisotropic materials: Discovery of a new type of mode in anisotropic dielectric resonators,” IEEE Trans. Instrum. Meas., vol. 50, no. 2, pp. 522–525, Apr. 2001. [8] M. Sumetsky, “Whispering-gallery-bottle microcavities: The three-dimensional etalon,” Opt. Lett., vol. 29, pp. 8–10, 2004. [9] P. Wolf, M. E. Tobar, S. Bize, A. Clairon, A. Luiten, and G. Santarelli, “Whispering gallery resonators and tests of Lorentz invariance,” Gen. Relativity and Gravitation, vol. 36, no. 10, pp. 2351–2372, 2004,  = 9:272 and  = 11:349 are stated for sapphire at 4 K; preprint version: arXiv:gr-qc/0401017 v1. [10] H. Rokhsari, T. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express, vol. 13, pp. 5293–5301, 2005. [11] S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A, Gen. Phys., vol. 71, 2005, 013817. [12] K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express, vol. 14, pp. 1094–1105, 2006. [13] MAFIA. CST GmbH, Darmstadt, Germany. [Online]. Available: http:// www.cst.de/Content/-Products/MAFIA/Overview.aspx [14] R. Basu, T. Schnipper, and J. Mygind, “10 GHz oscillator with ultra low phase noise,” in Proc. 8th Int. Jubilee ‘From Andreev Reflection to the International Space Station’ Workshop, Björkliden, Kiruna, Sweden, Mar. 20–27, 2004. [Online]. Available: http://fy.chalmers.se/~f4agro/ BJ2004/FILES/Bjorkliden_art.pdf [15] I. G. Wilson, C. W. Schramm, and J. P. Kinzer, “High Q resonant cavities for microwave testing,” Bell Syst. Tech. J., vol. 25, pp. 408–434, 1946. [16] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communications Electronics, 2nd ed. New York: Wiley, 1984. [17] U. S. Inan and A. S. Inan, Electromagnetic Waves. Englewood Cliffs, NJ: Prentice-Hall, 2000, particularly Subsection 5.3.2 (pp. 391–400). [18] M. E. Tobar, “Resonant frequencies of higher order modes of cylindrical anisotropic dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 12, pp. 2077–2082, Dec. 1991. [19] J. G. Harnett and M. E. Tobar, “Determination of whispering gallery modes in a uniaxial cylindrical sapphire crystal,” Mathematica, 2004, code/notebook, private correspondence. [20] J. Krupka, K. Derzakowski, A. Abramowicz, M. E. Tobar, and R. G. Geyer, “Use of whispering-gallery modes for complex permittivity determinations of ultra-low-loss dielectric materials,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 752–759, Jun. 1999, this paper provides  = 9:2725 and  = 11:3486 for sapphire at 4.2 K. [21] J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study of whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 1, pp. 56–61, Jan. 1994. [22] J. Krupka, D. Cros, A. Luiten, and M. Tobar, “Design of very high Q sapphire resonators,” Electron. Lett., vol. 32, no. 7, pp. 670–671, 1996. [23] J. A. Monsoriu, M. V. Andrés, E. Silvestre, A. Ferrando, and B. Gimeno, “Analysis of dielectric-loaded cavities using an orthonormalbasis method,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2545–2552, Nov. 2002. [24] N. M. Alford, J. Breeze, S. J. Penn, and M. Poole, “Layered Al O TiO composite dielectric resonators with tunable temperature coefficient for microwave applications,” Proc. Inst. Elect. Eng.–Sci., Meas., Technol., vol. 47, pp. 269–273, 2000. [25] J. P. Wolf, The Scaled Boundary Finite Element Method. New York: Wiley, 2003. [26] B. M. A. Rahman, F. A. Fernandez, and J. B. Davies, “Review of finite element methods for microwave and optical waveguides,” Proc. IEEE, vol. 79, no. 10, pp. 1442–1448, Oct. 1991. [27] COMSOL Multiphysics. COMSOL Ab., Stockholm, Sweden. [Online]. Available: http://www.comsol.com/, the work described in this paper used ver. 3.2(.0.224) [28] ANSYS Multiphysics. ANSYS Inc., Canonsburg, PA. [Online]. Available: http://www.ansys.com/ [29] T. J. A. Kippenberg, “Nonlinear optics in ultra-high-Q whispering-gallery optical microcavities” Ph.D. dissertation, Dept. Appl. Phys., California Inst. Technol., Pasadena, CA, 2004 [Online]. Available: http://www.its.caltech.edu/~tjk/TJKippenbergThesis.pdf, particularly Appendix B

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[30] A. Auborg and P. Guillon, “A mixed finite element formulation for microwave device problems: Application to MIS structure,” J. Electromagn. Wave Applicat., vol. 5, pp. 371–386, 1991. [31] J.-F. Lee, G. M. Wilkins, and R. Mittra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 11, pp. 1981–1987, Nov. 1993. [32] R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villava, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” Univ. Texas, El Paso, and Jet Propulsion Lab., California Inst. Technol., Pasadena, CA, Tech. Rep., 1994. [Online]. Available: http://trs-new.jpl.nasa. gov/dspace/bitstream/2014/32335/1/94-0066.pdf [Online]. Available: http://hdl.handle.net/2014/32335 [33] D. G. Santiago, R. T. Wang, G. J. Dick, R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, and G. J. Villalva, “Experimental test and application of a 2-D finite element calculation for whispering gallery sapphire resonators,” in IEEE 48th Int. Freq. Control Symp., Boston, MA, 1994, pp. 482–485. [Online]. Available: http://trs-new.jpl.nasa. gov/dspace/bitstream/2014/33066/1/94-1000.pdf [Online]. Available: http://hdl.handle.net/2014/33066 [34] B. I. Bleaney and B. Bleaney, Electricity and Magnetism, 3rd ed. Oxford, U.K.: Oxford Univ. Press, 1976. [35] F. N. H. Robinson, Macroscopic Electromagnetism, ser. Int. Ser. Monographs in Nat. Philos., D. T. Haar, Ed. New York: Pergamon, 1973, vol. 57. [36] F. Pobell, Matter and Methods at Low Temperatures. Berlin, Germany: Springer, 1992. [37] R. Fletcher and J. Cook, “Measurement of surface impedance versus temperature using a generalized sapphire resonator technique,” Rev. Sci. Instrum., vol. 65, pp. 2658–2666, 1994. [38] P.-Y. Bourgeois, F. Lardet-Vieudrin, Y. Kersale, N. Bazin, M. Chaubet, and V. Giordano, “Ultra-low drift microwave cryogenic oscillator,” Electron. Lett., vol. 40, pp. 605–606, 2004. [39] S. A. Schelkunoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J., vol. 15, pp. 92–112, 1936. [40] C. A. Balanis, Antenna Theory. New York: Wiley, 1997, particularly Ch. 12. [41] S. A. Schelkunoff, “On diffraction and radiation of electromagnetic waves,” Phys. Rev., vol. 56, no. 4, pp. 308–316, 1939. [42] A. N. Luiten, A. G. Mann, N. J. McDonald, and D. G. Blair, “Latest results of the U.W.A. cryogenic sapphire oscillator,” in Proc. 49th Int. IEEE Freq. Control Symp., San Francisco, CA, 1995, pp. 433–437. [43] C. A. Flory and R. C. Taber, “High performance distributed Bragg reflector microwave resonator,” IEEE Trans. Ultrason. Ferroelect. Freq. Control, vol. 44, no. 2, pp. 486–495, Mar. 1997, specifically the mode corresponding to Figs. 1, 2, and 7, and the last three paragraphs of Sec. IV. [44] C. A. Flory and R. C. Taber, “Microwave oscillators incorporating cryogenic sapphire dielectric resonators,” in IEEE Int. Freq. Control Symp., 1993, pp. 763–773. [45] HEM Sapphire Crystal Systems Inc.. Salem, MA, 2007. [Online]. Available: http://www.crystalsystems.com/sapprop.html [46] G. K. White and R. B. Roberts, “Thermal expansion of reference materials: Tungsten and  Al O ,” High Temperatures—High Pressures, vol. 15, pp. 321–328, 1983. [47] G. K. White, “Reference materials for thermal expansion: Certified or not?,” Thermochimica Acta, vol. 218, pp. 83–99, 1993. [48] G. K. White, Experimental Techniques in Low-Temperature Physics, 3rd ed. Oxford, U.K.: Clarendon, 1979.

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Mark Oxborrow was born near Salisbury, U.K., in 1967. He received the B.A. degree in physics from the University of Oxford, Oxford, U.K., in 1988, and the Ph.D. degree in theoretical condensed-matter physics from Cornell University, Ithaca, NY, in 1993. His thesis concerned random-tiling models of quasi-crystals. During subsequent post-doctoral appointments with the Niels Bohr Institute, Copenhagen, Denmark, and University of Oxford, he investigated acoustic analogs of quantum wave-chaos. In 1998, he joined the National Physical Laboratory, Teddington, U.K., where his eclectic project-based research has included the design and construction of ultra-frequency-stable microwave and optical oscillators, the development of single-photon sources, and the applications of carbon nanotubes to metrology.

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Multilevel Modulated Signal Transmission Over Serial Single-Mode and Multimode Fiber Links Using Vertical-Cavity Surface-Emitting Lasers for Millimeter-Wave Wireless Communications Anthony Nkansah, Anjali Das, Nathan J. Gomes, Senior Member, IEEE, and Pengbo Shen

Abstract—Quadrature phase-shift keying, 16 quadrature amplitude modulation (QAM), and 64-QAM data transmission—Worldwide Interoperability for Microwave Access modulation schemes—at 6 and 20 MS/s is demonstrated for a link that emulates a cost-effective 1.55-m vertical-cavity surface-emitting laser based radio over fiber millimeter-wave indoor picocellular system. The system consists of a concatenation of 20-km single-mode fiber and 300-m multimode fiber links between a central office and remote antenna unit and employs remote 30-GHz local oscillator delivery. Successful transmission over both optical and wireless paths is achieved with good error vector magnitude performance recorded for both uplink and downlink. The performance is compared to other demonstrations of multilevel signal transmission in millimeter-wave over fiber systems. Index Terms—Millimeter-wave communications, picocellular systems, radio over fiber system, vertical-cavity surface-emitting laser (VCSEL), Worldwide Interoperability for Microwave Access (WiMAX).

I. INTRODUCTION HE increasing demand on current wireless consumer links has led to proposals for the use of millimeter-wave frequencies for broadband wireless [1]–[3], and to topology proposals where a central office is connected to remote antenna units via optical fiber links [4]–[13]. At millimeter-wave frequencies, the high path loss and attenuation through man-made and natural objects favors the use of well-defined picocells. As a result, however, larger numbers of remote antenna units will be required for a given coverage area. For cost reduction reasons, there is then a requirement for the remote antenna unit to be of low complexity, while a highly centralized central office is equipped with the more expensive (shared) optical and millimeter-wave components. The millimeter-wave generation techniques [12], transportation schemes [4]–[15], and architectural topologies [4], [5], [9], [10], [12], [13] of these systems

will have a significant influence on the final cost of their deployment. A comparison of systems consisting of star/tree and ring architectural topologies was performed in [4]. Various systems based on either single-mode fiber [5]–[7], [11]–[13], [15] or multimode fiber [7], [8], [14] links have been proposed and experimentally demonstrated for millimeter-wave over fiber applications. Note that only [5], [10], and [11] conducted full-duplex experiments and the other systems concentrated on downlink transmissions, except for [9], where separate downlink and uplink experiments were carried out. Only [6], [10], [13], and [15] incorporated a wireless path in their experiments. These experiments will be reviewed in Section IV. The aim of this study is to provide a fuller experimental verification of the system proposed in [9], including the wireless path. The experimental setup for the proposed system is described in Section II, while the results are presented in Section III. A performance comparison of the system presented here with previously reported experiments is presented in Section IV, and this is followed by conclusions in Section V.

T

Manuscript received November 20, 2006; revised March 8, 2007. This work was supported in part by the European Union under the “ISIS” Network of Excellence. The authors are with the Broadband and Wireless Communications Group, Department of Electronics, University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [email protected]; [email protected]; [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.2007.897728

II. EXPERIMENTAL SETUP A. Proposed Star-Tree Architecture In the proposed star-tree architecture [9], the indoor picocells are generally located many kilometers from the central office. As shown in Fig. 1, the central office possesses multiple arms, where each arm is connected to a cluster. Each arm from the central office consists of a number of single-mode fiber cable bundles, where each bundle is assigned to a premises. Each bundle is terminated at a fiber distribution unit (patch panel) from which individual single-mode fibers service a “region.” The single-mode fibers are terminated at a remote antenna termination unit from which multimode fibers service individual remote antenna units. The architecture has been described in detail in [9]. In this study, an attempt has been made to match the experimental setup as closely as possible to the architecture proposed in [9]. Note that the experimental setups for both uplink and downlink do not have complete central offices, remote antenna units, etc.; rather a number of components that provide the relevant sets of functions (but not all functions, and not for simultaneous uplink and downlink operation) were implemented in

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Fig. 1. Topology of proposed millimeter-wave radio over fiber system from [9]. Mobile unit (MU), remote antenna unit (RAU), remote antenna termination unit (RATU), fiber distribution unit (FDU), single-mode fiber (SMF), multimode fiber (MMF).

the experiments. The experiments for both downlink and uplink have been carried out for a single remote antenna unit. B. Downlink Setup The experimental setup for the downlink transmission including the wireless path is shown in Fig. 2. A vector signal generator was used to emulate the Worldwide Interoperability for Microwave Access (WiMAX) modulation schemes at an IF of 1.14 GHz. The modulation formats specified in WiMAX 802.16c (10–66 GHz) [2] consist of quadrature phase-shift keying (QPSK), 16 quadrature amplitude modulation (QAM), and 64-QAM schemes. A Nyquist square-root raised cosine pulse-shaping filter with a roll-off factor of 0.25 is used in WiMAX for all modulation schemes. Note that there are only three symbol rates (16, 20, and 22.4 MS/s) specified in the WiMAX 802.16c standards. A prototype 1550-nm single-mode vertical-cavity surfaceemitting laser (VCSEL) was used as the optical transmitter at the central office. The emulated WiMAX IF signals (QPSK, 16 QAM, and 64 QAM) were used to modulate the VCSEL. The attenuator used after the VCSEL is used to represent the insertion loss of a wavelength multiplexer, as proposed in [9], and further attenuation is incorporated into the link to emulate the losses from circulators proposed in [9]. To generate the optical millimeter-wave reference signal, the well-known frequency-doubling technique using a Mach–Zehnder modulator biased at its null is employed [12], producing an optical double-sideband suppressed carrier (DSB-SC) signal. An uncooled NEC distributed feedback (DFB) laser provides the optical signal to the Mach–Zehnder modulator. The optical millimeter-wave reference and the attenuated VCSEL signal were coupled together into a single-mode fiber of 20-km length using

a 3-dB coupler. The 20-km single-mode fiber was terminated by a regional splitter (which would be located at the destination premises) consisting only of the remote antenna termination unit function. The (low) insertion loss that would be caused by the fiber distribution unit was not included in the regional splitter. The system proposed is cost effective as the Mach–Zehnder modulator is shared by eight remote antenna units (as is the uncooled DFB laser). As these components are located in the central office, stable operating temperatures are more readily achievable. The remote antenna termination unit was made up of erbium-doped fiber amplifier (EDFA) and an eight-port power splitter with an insertion loss of 10 dB. A 300-m length of multimode fiber (of 600 MHz km bandwidth-distance product at 1300 nm) was connected to one of the output ports of the power splitter via a single-mode fiber patch cord using approximately center launch conditions. The multimode fiber terminates at the remote antenna unit where the optical signal was split using a two-way single-mode power splitter, as a multimode power divider was unavailable to us. For the IF data detection, a single-mode tunable optical bandpass filter was inserted between one of the outputs of the two-way power splitter and a 3-GHz bandwidth single-mode fiber-pigtailed photodiode. The filter passband was set to select only the IF data wavelength. The detected IF signal was amplified and fed to a passive millimeter-wave mixer. For the millimeter-wave detection, a 45-GHz bandwidth single-mode fiber-pigtailed photodiode with a responsivity of 0.4 A/W was used. Due to the lower responsivity of the millimeter-wave photodiode, more optical power was required than was for the IF photodiode. Therefore, to avoid additional insertion loss, no optical filter was used. Although this meant that the IF signal was also detected in the millimeter-wave photodiode, the IF

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Fig. 2. Downlink experimental setup including wireless path. Photodiode (PD), erbium-doped fiber amplifier (EDFA), Mach–Zehnder modulator (MZM), optical bandpass filter (OBP), single-mode fiber (SMF), multimode fiber (MMF), remote antenna termination unit (RATU), vertical-cavity surface-emitting laser (VCSEL), distributed feedback laser (DFBL), effective isotropic radiated power (EIRP)

was effectively filtered out by the following millimeter-wave amplifiers, which had low-frequency cutoffs at 20 GHz. It is suggested that the system proposed in [9] could also be modified to eliminate the optical filter prior to the millimeter-wave photodiode. The amplified output electrical signals were fed to the mixer. The 30-GHz millimeter-wave signal acts as the local oscillator (LO) for the mixer, which up-converts the IF signal to the millimeter-wave band at 31.14 GHz. The up-converted millimeter-wave signal is then amplified and fed to a 20-dBi horn antenna for wireless transmission. The downlink effective isotropic radiated power (EIRP) was 20 dBm. The mobile receiver consisted of a 21-dBi in-house horn antenna, 15-dB millimeter-wave amplifier, a passive mixer, millimeter-wave LO, 20-dB IF amplifiers, and a vector signal

analyzer for error vector magnitude (EVM) measurements, as shown in Fig. 2. Note that EVM is specified in the IEEE 802.16c standard as a valid performance measurement. C. Uplink Setup The uplink experimental setup is shown in Fig. 3. At the central office, a photodiode and amplifier receive the uplink optical signal. The millimeter-wave reference optical generation technique and the remote LO delivery to the remote antenna unit millimeter-wave photodiode is the same as for the downlink. However, an optical 3-dB coupler was inserted at the remote antenna termination unit prior to the EDFA to enable the uplink signal to be transported over the same single-mode fiber

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Fig. 3. Uplink experimental setup including the wireless path. Photodiode (PD), erbium-doped fiber amplifier (EDFA), Mach–Zehnder modulator (MZM), bandpass filter (BP), single-mode fiber (SMF), multimode fiber (MMF), remote antenna termination unit (RATU), vertical-cavity surface-emitting laser (VCSEL), distributed feedback laser (DFBL), effective isotropic radiated power (EIRP)

as the reference LO signal. In the original architecture, as described in [9], an optical circulator is used rather than a coupler due to lower insertion loss. Again, some component losses in the uplink architecture of [9] were not emulated as the prototype VCSEL at the remote antenna unit had low output power. A lack of enough suitable connectors resulted in approximately 9.5-dB excess optical power loss in connecting the multimode fiber to the single-mode fiber patch cord of the power splitter. To compensate for this loss, no 3-dB coupler at the remote antenna unit prior to the millimeter-wave photodiode was used, as would be required in the system configuration of [9]. For the uplink wireless transmission, a mobile transmitter with the 21-dB in-house horn antenna transmitted emulated WiMAX signals at 31.14 GHz wirelessly to a remote antenna unit receiver using the 20-dB horn antenna. At the remote antenna unit, a passive mixer (similar to the one used in the downlink) is used to down-convert the incoming uplink signal to IF, using the remotely delivered millimeter-wave LO. The resulting IF signal is amplified and is used to modulate the VCSEL. This modulated optical signal is then coupled via an optical filter into a separate uplink multimode fiber of length 300 m (same specifications as the downlink multimode fiber) for transport to the regional splitter, as described in [9]. The optical filter is used to protect the VCSEL from the incoming downlink optical millimeter-wave signal, as shown in Fig. 3, where a circulator would be used in the system proposal [9].

The signal exiting the uplink multimode fiber is coupled into a 20-km single-mode fiber via a 3-dB coupler in the regional splitter and transported to the central office. Thus, the 20-km single-mode fiber link simultaneously transported both the uplink VCSEL signal and the optical millimeter-wave reference signal (in opposite directions). At the central office, the optical uplink signal is detected by a photodiode and amplified. The EVM of the amplified signal was measured using a vector signal analyzer for different experimental wireless distances. III. MEASUREMENT RESULTS A. Phase Noise of Millimeter-Wave Reference The linewidth and phase noise of LOs in up- or down-conversion configurations can degrade data modulated signals. For this reason, it is essential that the millimeter-wave over fiber link has little impact on the remotely delivered millimeter-wave LO [16], [17]. The frequency of the signal generator at the central office was set at 15 GHz and its phase noise was measured with a spectrum analyzer with an external mixer (26.5–40 GHz). At 10and 100-kHz offsets, the phase noise was measured to be 87 and 107 dBc/Hz, respectively. After optical generation of the millimeter-wave reference signal at 30 GHz, the linewidth and phase noise were measured prior to and after transmission over the optical link (single-mode fiber and multimode fiber). At the

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TABLE I EVM REQUIREMENTS FOR WiMAX BASE-STATION TRANSMITTER

TABLE II EVM REQUIREMENTS FOR WiMAX RECEIVER

input to the optical link, the linewidth was measured as 600 Hz (using 300-Hz resolution bandwidth) and the phase noise was recorded as 80 dBc/Hz (10-kHz offset) and 101 dBc/Hz (100-kHz offset). The phase noise is expected to increase as (in decibels) for the th harmonic. Hence, the phase noise increase is close to the theoretical 6 dB. The phase noise at the remote antenna unit was measured to be 80 dBc/Hz (10-kHz offset) and 101 dBc/Hz (100-kHz offset) and the linewidth also remained unchanged. Thus, the optical link has a negligible impact on the linewidth and phase noise of the remotely delivered millimeter-wave signal. B. WiMAX EVM Requirements The EVM requirements for a WiMAX base-station transmitter and for a WiMAX receiver are shown in Tables I and II, respectively [2]. The minimum symbol rate specified in the WiMAX 802.16c standards is 16 MS/s. However, the two EVM measuring systems used in this study supported maximum symbol rates of 6 and 20 MS/s, respectively. Thus, the measurements have been carried out at these transmission rates only. The different WiMAX modulation schemes (QPSK, 16 QAM, and 64 QAM) were emulated using a vector signal generator at 1.14 GHz for both 6- and 20-MS/s transmission rates. The quality of the signal from the vector signal generator, at both 6 and 20 MS/s, was checked using root mean square (rms) EVM measurements for different input power levels. The IF signal was then fed to a VCSEL connected to a photodiode by a short single-mode fiber patchcord (back to back) and rms EVM and output power were recorded at the photodiode, again for different input power levels. This enables an investigation of the dynamic range and signal degradation in the back-to-back optical configuration for each modulation scheme. Note that all EVM measurements were conducted without using an equalizer. Each rms EVM reading was measured over 2500 symbols on both systems. For the 6-MS/s symbol rate, five sets of 100 rms EVM readings, with a 20-s interval between each set, were recorded using a macro program. The average and standard deviation was then calculated over the 500 rms EVM readings. For the 20-MS/s rate, only two sets of 20 rms EVM readings, with a 5-min interval between the two sets, were recorded manually.

Fig. 4. Average rms EVM and power measurement for WiMAX emulated schemes at 20 MS/s after the VCSEL and photodiode link (back to back) for different input power levels.

Fig. 4 shows the results for average rms EVM both at the output of the vector signal generator and after passing through the back-to-back configuration for the different modulation schemes at 20 MS/s. The results show that the back-to-back configuration only meets the 64-QAM WiMAX transmitter requirements, shown in Table I, over the input power level range from 15 to 5 dBm. For 16 QAM, the input power has a threshold upper limit of 5 dBm, below which it satisfies the WiMAX transmitter EVM of 6%, but no lower limit within the measured power range. The QPSK signal was within the WiMAX transmitter EVM specifications of 12% for all power levels even in the VCSEL saturation region, which can be seen from the plot of the IF signal power measured at the photodiode. Multilevel modulation schemes are very sensitive to intermodulation distortion. Thus, the modulation of the VCSEL in its nonlinear region with multilevel modulation schemes results in poor EVM, as shown in Figs. 4 and 5. The best average rms EVM readings are obtained at input power levels between 12 and 7 dBm for all modulation schemes. The average rms EVM and power levels for the back-to-back configuration for the different modulation schemes at 6 MS/s are shown in Fig. 5. As with the 20-MS/s signals, the 6-MS/s signals experience poor EVM when driven into the VCSEL nonlinear region. However, all modulation schemes do meet their associated WiMAX transmitter EVM requirements when driven in the VCSEL linear region. QPSK allows drive signals well into the laser’s saturation region while still meeting the EVM limit. C. Downlink Performance The experimental results for the wireless downlink millimeter-wave over fiber transmission for the different modulation schemes at both 6 and 20 MS/s are shown in Fig. 6. The results show the average rms EVM measured for different wireless distances. The distance was limited to 4 m due to the size of the measurement laboratory. Distance zero corresponds to the average EVM at the input of the remote antenna unit horn antenna (downlink transmitter). The power to the VCSEL at the

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Fig. 5. Average rms EVM and power measurement for WiMAX emulated schemes at 6 MS/s after the VCSEL and photodiode link (back to back) for different input power levels.

Fig. 6. Downlink average rms EVM measurements of emulated WiMAX schemes for different wireless distances. The lines between symbols are only an aid for viewing the results and do not represent predicted trends.

central office was set to 12 dBm for all modulation schemes. The EVM requirements for the WiMAX transmitter, as listed in Table I, set the upper limit for the downlink transmitted signal at the remote antenna unit. On the other hand, WiMAX receiver requirements, as stated in Table II, are used as the limit for the mobile receiver for the downlink and at the central office receiver for the uplink. Only the 64-QAM signal at 20 MS/s fails the very tight WiMAX requirements at the downlink transmitter (remote antenna unit). All the systems pass the receiver EVM specifications. Fig. 7 shows the constellation and eye diagrams for a 1-m wireless downlink for the different modulation schemes. The clear and well-defined constellation and eye diagrams are evidence of the achievement of successful transmission. D. Uplink Performance The results for the uplink signal transmission are shown in Fig. 8. At 20 MS/s, 16-QAM signals meet the WiMAX receiver

Fig. 7. Constellation diagram and eye diagram of 1-m wireless downlink for: (a) 20-MS/s 64-QAM data, (b) 20-MS/s 16-QAM data, and (c) 20-MS/s QPSK data.

Fig. 8. Uplink rms EVM measurements of emulated WiMAX schemes for different wireless distances. The lines between symbols are only an aid for viewing the results and do not represent predicted trends.

requirement up to a wireless range of 2 m. The 20-MS/s QPSK signal has an average EVM of 11.1% at the maximum measurement distance of 3 m. As the WiMAX receiver limit for QPSK modulation is 32%, a much greater wireless distance can clearly be achieved for the 20-MS/s QPSK signal. The 20-MS/s 64 QAM signal, however, fails the WiMAX receiver requirements. With 6-MS/s modulation, only the 64-QAM signal fails the WiMAX receiver limit, and only beyond 2 m. To increase the wireless distance by compensating for some wireless path loss, a 15-dB millimeter-wave amplifier was inserted after the remote antenna unit horn antenna of Fig. 3. A comparison of the averaged rms EVM measurement results at 6 MS/s for the two cases of with and without the 15-dB millimeter-wave amplifier at the remote antenna unit for a wireless distance of 4 m is shown in Table III. The results show that the WiMAX receiver EVM requirement is met for all modulation schemes. At smaller dis-

NKANSAH et al.: MULTILEVEL MODULATED SIGNAL TRANSMISSION OVER SERIAL SINGLE-MODE AND MULTIMODE FIBER LINKS USING VCSELs

TABLE III UPLINK EVM MEASUREMENTS OF EMULATED WiMAX FIBER-WIRELESS TRANSMISSION AT 6 MS/s (4 m)

tances, however, the 15-dB additional amplification will cause the VCSEL to be driven into its nonlinear region. As can be observed from Fig. 5, this will result in poor EVM for all modulation schemes, except QPSK at 6 MS/s. This dynamic range problem, in a real system, could be solved by the central office using dynamic power control techniques to reduce the transmit power of closer mobile units. E. Discussion The 20-MS/s/120-Mb/s 64-QAM uplink transmission did not meet the WiMAX requirements and the 20-MS/s/80-Mb/s 16-QAM uplink transmission only met the requirement for shorter wireless range. The inclusion of a millimeter-wave amplifier would enable longer distance transmission at the cost of making nearer mobile transmissions fail due to distortion from overdriving the VCSEL. However, this dynamic range problem is evident in the back-to-back measurements of Figs. 4 and 5; in Fig. 4, there is a sudden increase in EVM as the input power is reduced to 15 dBm, and in Fig. 5, there is a peak in the measured EVM at around 15 dBm. Separate measurements confirmed that these artefacts were particular to the IF photodiode unit being used (probably due to bias circuit resonances); they were not apparent when using the millimeter-wave photodiode to detect the IF signals. Thus, improved results should be obtainable with a different photodiode for the IF modulated signal detection. For the complete millimeter-wave over fiber link, core mismatch occurs when the signal from the large core multimode fiber is coupled into a single-mode fiber or a photodiode with a small active area. In the case of coupling to a single mode fiber, as only the fundamental mode from the multimode fiber is coupled, power penalties will occur. Due to the larger diameter of the signal beam exiting the multimode fiber, there will be a coupling loss (and modal noise) problem when using a photodiode with a small active area. When a core mismatch problem is combined with a link with a poor dynamic range, the performance of the overall link degrades dramatically, as observed in the results for the 64-QAM modulation scheme. A more detailed discussion of the core mismatch problem, along with suggestions of components to mitigate this, has been provided in [9]. However, such components have been unavailable to us while conducting these experiments. IV. PERFORMANCE COMPARISON OF PROPOSED SYSTEMS Here, the results obtained are compared with those of other experiments, representing other system architectures. Millimeter-wave over fiber systems consist of three elements, i.e.: 1) the millimeter-wave generation scheme; 2) the transportation scheme; and 3) the topology. We confine ourselves to

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comparisons between experiments using multilevel modulation schemes for which EVM results are available (or can be deduced), as shown in Table IV. Thus, the experiments reported in [10] and [11], which demonstrated successful transmission of high bit-rate data (up to 155 Mb/s) over full-duplex single mode fiber links (including a wireless path in [10]) at around 37 and 60 GHz, respectively, are not included as binary phase-shift keying (BPSK)/differential phase-shift keying (DPSK) modulation was used. In addition, the experiment of [12], in which simultaneous transmission of three downlink signals on separate wavelengths, and an uplink transmission, and with single-mode fiber and wireless paths included for both directions of transmission, is not included, as no EVM values are given; clear constellation diagrams are reported, as has been achieved in this study (see Fig. 7). Similarly, in [15], 100-Mb/s QPSK signals were transported over a single-mode fiber link, up-converted to the 60-GHz band at the remote antenna unit, and then transmitted to the mobile unit over a wireless distance of 4–12 m; but only eye diagrams and no EVM measurements were reported. Although none of the schemes compared are identical to that presented here, there are elements that are similar. The aim for all is to also transmit millimeter-wave signals to/from mobile units via remote antenna units and fiber, thus a comparison is useful. Separate fibers were used for the full-duplex transportation of downlink and uplink signals in [13] in the emulation of a bus topology proposal. Three IF modulated optical signals, each at a different wavelength, were up-converted into the millimeterwave band, by transmission though a Mach–Zehnder modulator driven by a microwave signal and biased at its null. The up-converted optical signal was transported to a remote antenna unit where a tunable optical filter could select any given wavelength. The electrical signal exiting the photodetector thus consisted of a millimeter-wave carrier and two sidebands. One of the sidebands was filtered and emitted to the mobile unit wirelessly at a distance of 5 m, while the millimeter-wave carrier was used to down-convert the uplink signal to IF before transportation to the central office (in a similar manner to the uplink transportation of this study). Although it was demonstrated that error correction coding can relax signal-to-noise ratio and EVM requirements for the same bit error rate (BER), the results most comparable with this study are those without the error correction coding, from which an EVM of around 17% can be inferred for uplink and downlink transmissions over both single-mode fiber and 5-m wireless paths. For the experiments reported in [7] and [8], no topology was proposed, but the transportation schemes both involved millimeter-wave signal generation at the central office followed by delivery to the remote antenna unit. In [7], the transportation scheme for both downlink and uplink is the same as in [13], except that an optical frequency multiplication technique is used to generate the millimeter-wave signal. The frequency multiplication technique is based on harmonic generation using frequency modulation to intensity modulation conversion [7], in this case, in a Mach–Zehnder interferometer located at the central office. Experimentally, a carrier at 3 GHz and a 24-Mb/s 16-QAM signal at 200 MHz were transported over different fibers to a remote antenna unit. At the output of the photodiode at the

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TABLE IV PERFORMANCE COMPARISON OF PROPOSED SYSTEM WITH OTHER MEASURED SYSTEMS

remote antenna unit, RF components at every harmonic of the 3-GHz 200-MHz signal are obtained. An IF filter selects the lower sideband of the sixth harmonic at 17.8 GHz. The fibers used in the experiments were a 4.4-km length of multimode fiber, and 12.5- and 25-km lengths of single-mode fiber; rms EVM recordings of 4.5%, 4.6%, and 5.9%, respectively, were recorded. The values are slightly higher than the equivalent measurement in this study (3.6%); however, for the multimode fiber link, a longer length was used in [7] (4.4 km rather than 300 m). Thus, [7] suggests that it may be possible to use a longer length of multimode fiber for the system in this paper. In [8], an electrical millimeter-wave LO at the central office is used to up-convert the IF signal before it is optically transported over the single-mode fiber link. The signals transported were 10-Mb/s QPSK and 54-Mb/s 64-QAM orthogonal frequency division modulation (OFDM) at 20 GHz, over different fibers: single-mode fiber of length 2 m (back-to-back), and multimode fiber of lengths 575 m and 1 km; the rms EVM results, taking into account bit rates, were of the same order as those recorded for the fiber link only in this study. As in [7], the results suggest it may be possible to use a longer multimode fiber length for the system here. It should be noted that for the systems of [7] and [8], in which multimode fiber was employed, no uplink or wireless transmissions were demonstrated.

The topology proposed in [5] is of a star tree, with a ring at the end of each arm of the star tree interconnecting several remote antenna units. The transportation scheme used is similar to that used here, where IF and millimeter-wave reference signals are delivered to the remote antenna units. In [5], experimental results were obtained for the “first” remote antenna unit in the ring of such a topology; thus, the downlink distance corresponds to transmission over a 12.8-km star-tree arm only, whereas the uplink includes a 2.2-km ring (all single-mode fiber). The rms EVM values obtained are well within requirements, but are higher than those reported here for a fiber link only. No wireless transmission was demonstrated in [5]. In [6], there was no particular proposed topology. Wireless transmission was carried out. The transportation scheme differs slightly from that used here as the millimeter-wave signal is electrically generated at the remote antenna unit. A 2.5-GHz 16-QAM IF signal at 155 Mb/s was transported over a singlemode fiber link of length 25 km to a remote antenna unit. At the remote antenna unit, a mixer with an LO frequency of 57.2 GHz was used to up-convert the IF signal to the millimeter-wave band at 59.7 GHz. The up-converted signal was wirelessly emitted to a mobile unit at a distance of 2.6 m. The rms EVM obtained at the mobile unit was between 4.7%–8.4%. The results are largely consistent with the 16-QAM measurements presented in

NKANSAH et al.: MULTILEVEL MODULATED SIGNAL TRANSMISSION OVER SERIAL SINGLE-MODE AND MULTIMODE FIBER LINKS USING VCSELs

this study; our marginally lower rms EVM values having been achieved for lower data rate and carrier frequency. In [14], a similar transportation scheme to [6] was used with an IF signal transported over the optical link and up-converted using an electrical LO located at the remote antenna unit. However, [14] included an multimode fiber rather than single-mode fiber link. An inexpensive multimode VCSEL was used to transmit a 50-Mb/s QPSK signal at an IF of 0.5 GHz over 300 m of multimode fiber to a remote antenna unit. A 57-GHz LO was used to up-convert the IF signal at the remote antenna unit to 57.5 GHz. The up-converted signal was wirelessly transmitted to a mobile unit at a distance of 5 m. The experiment was repeated with a 100-Mb/s 16-QAM signal at the same IF frequency of 0.5 GHz. The EVM reported for the 50-Mb/s QPSK and 100-Mb/s 16-QAM signals were 4.4% and 4.6%, respectively. These EVM values are slightly lower than the equivalent measurement in this paper (4.7% and 5% for 40-Mb/s QPSK and 80-Mb/s 16 QAM, respectively). Note that [14] operates at a higher frequency (57 GHz) and longer wireless distance (5 m) than the system in this paper. However, the system here operates over a combined 20-km single-mode fiber and 300-m multimode fiber path. In [14], transmission over longer 600-m lengths of multimode fiber was carried out with no noticeable degradation in performance. V. CONCLUSION A cost-effective VCSEL-based millimeter-wave radio over fiber star-tree architecture has been investigated. The architecture comprises a central office providing indoor coverage to large premises with the signal distribution between central office and indoor remote antenna units performed using both singlemode fiber, for the longer distance distribution, and multimode fiber, for the in-building distribution. The physical layer performance of such a system has been experimentally verified by successful transmission of QPSK, 16-QAM, and 64-QAM signals (as used in the WiMAX standard) at 20 and 6 MS/s over the millimeter-wave optical and wireless links for both downlink and uplink paths. Good signal quality was received for both symbol rates, as evidenced by the low rms EVM values obtained, indicating the feasibility of the architecture. The performance experimentally obtained has been shown to compare favorably with that obtained for other millimeter-wave over fiber system proposals.

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[3] ETSI DVB EN300 748 v1.1.2, ETSI TM4 EN301 213-4 v1.1.1 (2001–08), 1997–1998. [4] A. Nkansah and N. J. Gomes, “A WDM/SCM star/tree fibre-feed architecture for picocellular broadband systems,” in Convergence of Telecommun. Networking and Broadcasting (PGNET) Postgraduate Symp., Liverpool, U.K., Jun. 2004, pp. 272–276. [5] T. Ismail, C. P. Liu, J. E. Mitchell, A. J. Seeds, X. Qian, A. Wonfor, R. V. Penty, and I. H. White, “Transmission of 37.6-GHz QPSK wireless data over 12.8-km fiber with remote millimeter-wave local oscillator delivery using a bi-directional SOA in a full-duplex system with 2.2-km CWDM fiber ring architecture,” IEEE Photon. Technol. Lett., vol. 17, no. 9, pp. 1989–1991, Sep. 2005. [6] A. Kim, Y. H. Joo, and Y. Kim, “60 GHz wireless communication systems with radio over fiber links for indoor wireless LANs,” IEEE Trans. Consumer. Electron., vol. 50, no. 2, pp. 517–520, May 2004. [7] M. Garcia Larrode, A. M. J. Koonen, J. J. Vegas Olmos, and A. Ng’Oma, “Bidirectional radio over fiber link employing optical frequency multiplication,” IEEE Photon. Technol. Lett., vol. 18, no. 1, pp. 241–243, Jan. 2006. [8] P. Hartmann, X. Qian, A. Wonfor, R. V. Penty, and I. H. White, “1–20 GHz directly modulated radio over MMF link,” in Int. Microw. Photon. Top. Meeting, Seoul, Korea, Oct. 2005, pp. 95–98. [9] A. Nkansah, A. Das, N. J. Gomes, P. Shen, and D. Wake, “VCSEL based single-mode and multimode fiber star//tree distribution network for millimeter-wave wireless systems,” in Int. Microw. Photon. Top. Meeting, Grenoble, France, Oct. 2006, Paper P6. [10] G. H. Smith, D. Novak, and C. Lim, “A millimeter-wave full-duplex fiber-radio star-tree architecture incorporating WDM and SCM,” IEEE Photon. Technol. Lett., vol. 10, no. 11, pp. 1650–1652, Nov. 1998. [11] T. Kuri, K. Kitayama, and Y. Takahashi, “A single light source configuration for full-duplex 60-GHz-band radio-on-fiber system,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 431–439, Feb. 2003. [12] J. J. O’Reilly, P. M. Lane, J. Attard, and R. Griffin, “Broadband wireless systems and networks: An enabling role for radio-over-fibre,” Philos. Trans. R. Soc. London A, Math. Phys. Sci., vol. 358, pp. 2297–2308, Aug. 2000. [13] K. Kojucharow, M. Sauer, H. Kaluzni, D. Sommer, F. Poegel, W. Nowak, A. Finger, and D. Ferling, “Simultaneous electrooptical upconversion, remote oscillator generation, and air transmission of multiple optical WDM optical wdm channels for a 60-GHz high-capacity indoor system,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2249–2255, Dec. 1999. [14] C. Loyez, C. Lethien, R. Kassi, J. P. Vilcot, D. Decoster, N. Rolland, and P. A. Rolland, “Subcarrier radio signal transmission over multimode fibre for 60 GHz WLAN using a phase noise cancellation technique,” Electron. Lett., vol. 41, pp. 91–92, Jan. 2005. [15] S. Dupont, C. Loyez, N. Rolland, P. A. Rolland, O. Lafond, and J. F. Cadiou, “60 GHz fiber-radio communication system for indoor ATM network,” Microw. Opt. Technol. Lett., vol. 30, pp. 307–310, Sep. 2001. [16] M. Iqbal, J. Lee, and K. Kim, “Performance comparison of digital modulation schemes with respect to phase noise spectral shape,” in Proc. Can. Elect. Comput. Eng. Conf., Halifax, NS, Canada, Mar. 2000, vol. 2, pp. 856–860. [17] A. Hajimiri, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 3, pp. 179–194, Mar. 1998.

ACKNOWLEDGMENT The authors wish to acknowledge the advice and assistance of their colleagues Prof. M. Sobhy, B. Sanz-Izquierdo, M. Mjeku, R. Davis, Dr. D. Wake, and A. Jastrzebski, all with the University of Kent, Canterbury, U.K. The authors are grateful to Anritsu EMEA for the loan of a Signature vector signal analyzer. REFERENCES [1] Working Group for Wireless Personal Area Networks , TG3c Selection Criteria, IEEE Standard 802.15, 2006, (draft). [2] Working Group on Broadband Wireless Access Standards, IEEE Standard 802.16-2004, , 2004, includes IEEE Standard 802.16c.

Anthony Nkansah received the B.Eng. (Hons.) degree in electronic engineering and the M.Sc. degree in broadband and mobile communication networks from the University of Kent, Canterbury, U.K., in 2000 and 2001, respectively, and is currently working toward the Ph.D. degree in electronic engineering at the University of Kent. His research interests include low-cost microwave and millimeter-wave radio over fiber networks and their deployment within premises.

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Anjali Das received the B.Sc. (Honors) and M.Sc. degrees in electronic science from Delhi University, Delhi, India, in 1999 and 2001, respectively, the M.Sc. degree in information and communication engineering from the University of Leicester, Leicester, U.K., in 2003, and is currently working toward the Ph.D. degree in electronic engineering at the University of Kent, Canterbury, U.K. Her research interests include low-cost radio over fiber systems and their deployment within buildings for improving coverage.

Nathan J. Gomes (M’92–SM’06) received the B.Sc. degree from the University of Sussex, Brighton, U.K., in 1984, and the Ph.D. degree from University College London, London, U.K., in 1988, both in electronic engineering. From 1988 to 1989, he held a Royal Society European Exchange Fellowship with the Ecole Nationale Supérieure des Télécommunicaitons (ENST), Paris, France. Since late 1989, he has been a Lecturer, and since 1999, a Senior Lecturer with the Electronics Department, University of Kent, Canterbury, U.K. His current research interests include radio over fiber systems and networks, the photonic generation and transport of millimeter-wave signals, and photoreceivers for such applications.

Pengbo Shen received the B.Eng. degree from Shanghai Jiaotong University, Shanghai, China, in 1996. Upon graduation, he joined Shanghai Jiaotong University, as a Research Engineer involved with optical communications. In 1999, he joined the University of Kent, Canterbury, U.K., where he is involved with the development of the photonic LO for the Actama Large Millimeter-Wave Array. His research interests are in the field of microwave photonics and communication, including the generation and distribution of high-quality millimeter-wave signals.

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Thermally Actuated Multiport RF MEMS Switches and Their Performance in a Vacuumed Environment Mojgan Daneshmand, Student Member, IEEE, Winter D. Yan, Student Member, IEEE, and Raafat. R. Mansour, Fellow, IEEE

Abstract—In this paper, unique multiport RF microelectromechanical system switches are proposed as building blocks of redundancy switch matrices. Novel single-pole single-throw (SPST), single-pole double-throw (SPDT), and C-type (transfer) switches are developed, fabricated, and tested. These switches are integrated with thermal actuators that require low actuation voltage and result in high contact force. The RF measurement results reveal an excellent performance with extremely low loss. Up to 20 GHz, the SPST switch has less than 0.3 dB, the SPDT switch has less than 0.5 dB, and the C-type switch has less than 1-dB insertion loss. In addition, the vacuum behavior of these switches is evaluated, which shows a 75% reduced power consumption with an identical RF performance. These switches are excellent candidates for integrating in the form of redundancy switch matrices. Index Terms—Redundancy switches, RF microelectromechanical systems (MEMS) switches, switch matrices, thermal actuators. Fig. 1. (a) SPDT switch schematic: state I port 1 is connected to port 2, and in state II, port 1 is connected to port 3. (b) C-type switch schematic: state I ports 1 and 2 and 3 and 4 are connected, and in state II, ports 1–4 and 2 and 3 are connected.

I. INTRODUCTION OPHISTICATED redundancy switch matrices are employed in satellite payloads to provide redundancy schemes and improve the reliability of both receive and transmit subsystems. The building blocks of such switch matrices are multiport switches such as single-pole double-throw (SPDT) and C-type switches depicted in Fig. 1. Compact and energy-efficient thermal microactuators with a low actuation voltage that can result in a high contact force are very desirable for switch applications. Although they have relatively low switching speed (microseconds range [1]), they find wide spread use in applications where a low actuation voltage is required. Among the existing thermal actuators, the ones developed from metallic materials are superior due to their lower power consumption, smaller actuating voltage, and larger displacements. Over the last five years, numerous papers on thermally actuated RF microelectromechanical systems (RF MEMS) switches have been published [2], [3]. However, most of the research reported in the literature has been directed toward the development of single-pole single-throw (SPST) switches. The use of C-type

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Manuscript received September 25, 2006; revised January 28, 2007. This work was supported in part by the Natural Science and Engineering Research Council of Canada and by COM DEV. The authors are with the Center for Integrated RF Engineering, Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (e-mail: [email protected]; d2yan@engmail. uwaterloo.ca; [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.2007.897740

switches, rather than SPST switches, as the building blocks, considerably simplifies the integration problem of large size redundancy switch matrices. Daneshmand and Mansour have recently reported an electrostatic C-type switch with a high actuation voltage of 65 V [4]. To our knowledge, there are no reports on thermally actuated C-type switches. In this study, an entirely new type of multiport switches that use metallic thermal actuators is introduced. The actuators are based on thermal expansion devices (TEDs) [5] and utilize a nitride layer to isolate the dc from the RF circuitry. This idea is used to develop novel SPST, SPDT, and C-type switches with an extremely good RF performance and a very low actuation voltage. The behavior of the proposed structures in a vacuum is studied to resemble vacuumed hermetic packaged switches. This shows an identical RF performance with 75% less dc power consumption. These switches are extremely good candidates for integration in the form of redundancy switch matrices. II. OPERATION PRINCIPLE OF THE PROPOSED THERMALLY ACTUATED MEMS SWITCH The TED actuator consists of two arms with different widths, as portrayed in Fig. 2. The actuator deflection is produced from the nonequal heating of the cold (thick) and hot (thin) arms. Since the arms consist of the same material, the thinner hot arm has a relatively larger electrical resistance than the cold arm. When current passes through the cold and hot arms, the hot arm is resistively heated to a higher temperature. Hence, the hot arm expands in length more than the cold arm. The difference in

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Fig. 2. Proposed actuator based on the TED.

Fig. 3. SEM of the proposed SPST switch.

the expansion between the cold and hot arms causes a forward movement. This movement is transferred to an RF contact element by a dielectric layer (silicon nitride), which helps to isolate the RF from the dc. Fig. 3 depicts the structure of the proposed SPST switch fabricated by Metal-MUMPs process [6]. The process allows the use of a thick Ni metal of 20 m as the structural layer overlaid with gold at the contact points. The switch is based on finite ground CPW lines on silicon substrate. The signal linewidth is 80 m, the spacing is 75 m, and the ground-plane width is 80 m. All the lines have a thickness of 20 m, which creates a 50- line. Under the transmission lines, a 25- m trench is etched. A dielectric layer of nitride is used to transfer the actuator movement to the contact element. This layer is released and located under the ground conductor. The use of an electric isolator, rather than a conductor, provides flexibility in the design of the proposed switch, and renders the RF performance independent of the actuators structure. The contact element consists of a small gold-plated nickel pad with a 3- m gap with the signal line. By applying a dc voltage, the actuator moves forward and shorts out the input port to the output port, turning the switch ON. On removal of the dc bias, the actuator returns to its initial position and the switch turns OFF. At room condition (room

Fig. 4. (a) Simulation and (b) measurement results for the proposed design shown in Fig. 3.

temperature and air environment), a dc-bias voltage as low as 1.64 V and a current of 360 mA is required to close the existing gap between the contact metal and signal line. The switch is simulated with HFSS [7] and the results are presented in Fig. 4. A two-port on-wafer measurement is also performed. The measured results correlate well with the simulation results. This figure reveal a good return loss of better than 20 dB and an insertion loss of less than 0.9 dB up to 10 GHz. The slightly high insertion loss of the switch is mainly due to the low resistivity of the silicon substrate, as will be explained in Section V. The insertion loss can be considerably improved with the use of a high-resistivity silicon substrate. The switch exhibits a good isolation of more than 25 dB for the entire frequency range. The proposed structure promises to handle higher power levels in comparison to the conventional MEMS switches as it has thick metal conductors (20 m). In addition, the switch avoids the “self-biasing” problem [8] due to the large stiffness of the thermal actuator.

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Fig. 6. Proposed switch configuration after initial assembly in vacuum.

Fig. 5. Comparison of the maximum temperature of the proposed actuator with and without considering the effect of heat radiation.

III. PERFORMANCE OF THE PROPOSED THERMALLY ACTUATED SWITCH IN A VACUUM ENVIRONMENT Vacuum hermetic packaging offers handling and performance advantages for RF MEMS switches and considerably enhances the reliability of the switches. The need for such packaging is even more pronounced for resistive contact switches, which require a stable vacuum condition to achieve a long life time. Here, we investigate the performance of the proposed switch in a vacuum environment and show that the vacuum packaging can reduce the dc power consumption as much as 75%. To have a good understanding of the switch operation, finite-element analysis is performed. There are three mechanisms of heat flow, i.e., conduction, convection, and radiation. At room condition, the heat dissipation through radiation to the ambient can be neglected in comparison to the heat losses through the conduction of the anchors and convection to the air [5]. This assumption is verified by finite-element analysis and use of commercial COMSOL software [9]. The results are presented in Fig. 5. By applying the same input voltage, the maximum temperature of the thermal actuator, with and without the radiation boundary condition, is almost identical up to 1800 K, which is well beyond the melting point of nickel. When the heat transfer is under steady-state condition, the resistive heating power generated in the element is equal to the heat conduction and convection from the element. This is written as [10] (1) where is the generated heat from the electric current, the dissipated conduction heat through the anchors, and the power lost through convection, which is expressed as

is is

(2) is the device surface area that is exposed to air, is the thermal conductivity of the ambient (air), and and are the actuators and the ambient temperature, respectively. At room condition, when current passes through the thin and thick arms, the actuator generates a forward motion due to the asymmetric resistive

Fig. 7. Temperature distribution of the thermal actuator: (top) with convection and (bottom) without convection (vacuum).

heating of the arms. Furthermore, the thin arm has smaller exposed surface area, i.e., , and thus dissipates less heat through convection (2). Hence, the thin arm expands in length more than the thick arm and produces a forward movement. When the thermal actuator is operated in a vacuum, the actuator initially produces a forward movement, and on removing the current, the actuator remains in a new position (ON state), as shown in Fig. 6. This is due to the fact that during the transient state in vacuum with the lack of convection, the temperature of the thin arm rises faster than the thick arm since the thin arm has less thermal capacity. Thus, the thin arm goes under plastic deformation, while the thick arm is still under elastic deformation and the switch stays in the ON state. On the application of the current, the actuator moves backward and turns OFF the switch. This behavior can be explained as follows: in the vacuum condition, the heat transfer reaches the steady state at a much higher temperature than that of the room condition. The thick arm temperature is also much higher than the thin arm. Fig. 7 shows the comparison of steady-state temperature distribution of the thermal actuator with and without convection when the same input voltage is applied. This phenomenon produces a larger thermal expansion of the thick arm in comparison to the thin arm (the thick arm is still under the elastic deformation while the thin arm is under plastic deformation). This fact is verified by finite-element analysis (COMSOL software). Fig. 8(a) displays a solid model of a plastic deformed actuator. By applying

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Fig. 9. SUSS Cryo RF prober at the Center for Integrated RF Engineering Laboratory, Waterloo, ON, Canada.

Fig. 8. (a) Plastically deformed thermal actuator after initial assembly. (b) Backward movement of the thermal actuator by applying voltage.

voltage to this actuator, a backward movement is observed, as illustrated in Fig. 8(b). The vacuum operation of the switch resembles its operation when the switch is hermetically packaged in vacuum. In this case, the efficiency of the switch is enhanced and the power dissipation is drastically reduced due to the absence of convection. Several experiments are carried out in the Cryo vacuum prober, shown in Fig. 9. The results reveal that the required voltage for switching is 0.8 V and the current is 170 mA. This represents a power consumption of 136 mW, which is 75% less than that at room condition (590 mW). Additionally, the RF performance of the switch in all the aforementioned conditions is evaluated and the results are recorded in Fig. 10. As is expected, both in the room condition and vacuum, the switch has very similar RF performance. IV. THERMALLY ACTUATED MULTIPORT SWITCHES: SPDT AND C-TYPE (TRANSFER) RF MEMS SWITCHES The proposed concept is a good candidate for integration in large numbers and thus suitable for the basic building blocks of the switch matrices. This idea is adopted to develop SPDT and C-type or transfer switches as the main building blocks for redundancy switch matrices. The above concept is used to design a SPDT switch that is presented in Fig. 11. The switch makes use of TED actuators with an extended rigid silicon nitride arm to transfer the force to the contact point. A bridge over the nitride layer is used to connect the ground lines of the adjacent ports and eliminate the unwanted parasitic modes of the coplanar waveguide (CPW) lines. When the thermal actuator is in the rest position, the associated ports are OFF representing two series capacitors [Port 2 in

Fig. 11(b)]. When voltage is applied, the thermal actuator moves forward, shorting the input port to the output port, as illustrated in Fig. 11(b) for Port 3. To verify the concept, a measurement is performed, and the results up to 10 GHz are illustrated in Fig. 12. The return loss of the switch is less than 20 dB for the entire frequency range. The insertion loss is better than 1 dB up to 7 GHz. The slightly high insertion loss is attributed to the substrate loss, as well as the discontinuity of the ground plane. The use of the high-resistivity silicon as a base substrate and also air bridges on the ground planes can significantly improve the performance. The isolation of the switch is also measured, which is more than 27 dB for the entire frequency band. The C-type or transfer switch is another type of multiport switches that is investigated in this study. This switch has four ports and two operational states, as denoted in Fig. 1(b). In state I, the connection is established between ports 1 and 2, and between ports 3 and 4; in state II, ports 1 and 4, and ports 2 and 3 are connected. A new configuration, as shown in Fig. 13, is proposed that is based on TED actuators and finite ground CPW lines. A nickel-based actuator is used to move a gold covered Ni contact element to turn the switch ON and OFF. In state I, actuators 1 and 3 are excited and move forward to connect ports 1 and 2 and ports 3 and 4. As is illustrated in Fig. 13(c), in this state, the connection between ON-state ports is modeled by two series contact resistance, whereas the OFF-state ports are separated by two series capacitors. The ground planes of the adjacent ports are connected to reduce unwanted asymmetry CPW modes. Besides, the continuity of the ground lines is maintained to avoid unwanted parasitic CPW modes and obtain wideband performance. In the same way, actuators 2 and 4 are biased to represent state II of the switch. In this structure, the actuators are electrically isolated from the RF section of the switch [see Fig. 13(b)], which renders the switch more suitable for integration in switch matrices. For example, a 5–7 redundancy switch matrix, as is explained in [4],

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Fig. 11. (a) Scanning electron microscope (SEM) photograph of the proposed SPDT thermally actuated RF MEMS switch. (b) Operation model when port 2 is OFF and Port 3 is ON.

Fig. 12. Measured results for the proposed design shown in Fig. 11.

Fig. 10. (a) Insertion loss, (b) return loss, and (c) isolation of the proposed switch at room condition and vacuum.

would require ten of the proposed C-type switches. For the applications with large numbers of switches, a latching mechanism is beneficial. Latching will help not only to increase the

life time of the switches, but also reduce the power consumption considerably. A two-port measurement is performed and the results are plotted in Fig. 14. It conveys a return loss of better than 17 dB up to 10 GHz and the insertion loss is less than 1 dB at 8 GHz. Although there is a trench under the CPW lines, the slightly high insertion loss at higher frequencies is attributed to the silicon substrate loss. In addition, the discontinuity of the ground

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Fig. 13. (a) Proposed modified C-type switch. (b) Its RF section. (c) Operational model for state I, while ports 1 and 2 and ports 3 and 4 are connected.

Fig. 15. (a) Simulation and (b) measurement results for the proposed design shown in Fig. 3.

V. IMPROVED SPDT AND C-TYPE (TRANSFER) RF MEMS SWITCHES

Fig. 14. Measured Fig. 13.

S -parameters

for the proposed C-type switch shown in

planes induces parasitic odd modes and limits the operating frequency band. The addition of more air bridges helps to improve the performance. The switch isolation exceeds 25 dB for all the frequency bands of interest. Due to the symmetric structure of the switch, the results for state II are identical to those of state I.

Although the presented switches demonstrate a good mechanical performance, their RF response can be further improved by using a higher resistivity substrate. The idea is initially verified by realizing an SPST switch by using a high-resistivity silicon substrate. The design is similar to that shown in Fig 3. The RF results in Fig. 15 indicate an excellent performance. The insertion loss is less than 0.3 dB at 20 GHz and less than 0.5 dB for all the frequency range up to 40 GHz with a return loss of better than 15 dB. Compared to the previous thermal SPST switch in Fig. 3, not only is the loss drastically decreased, but the operating frequency band is also significantly improved. The switch maintains an isolation of better than approximately 20 dB for all the frequency range up to 40 GHz. The proposed SPDT switch shown in Fig. 11 has a contact metal that is round. This provides the 90 bend required to connect the two adjacent ports. However, the amount of the applied

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Fig. 16. Modified design of the proposed SPDT thermally actuated RF MEMS switch.

Fig. 18. (a) Proposed modified C-type switch. (b) Its RF section.

Fig. 17. Measured results for the proposed SPDT switch shown in Fig. 16.

force at the contact points is reduced due to the 45 angle between the actual generated force of the actuators and the contact surfaces. To overcome this problem and enhance the switch performance, a new design, shown in Fig. 16, is proposed. In this configuration, the entire force of the actuators is transferred to the contact points, thus significantly reducing the contact resistant. Besides, the use of high-resistivity silicon substrate further improves the performance and reduces the RF loss. The results that are illustrated in Fig. 17 indicate an insertion loss of less than 0.35 dB at 10 GHz and 0.5 dB at 20 GHz. The return loss is better than 20 dB at 10 GHz and the isolation of the switch is higher than about 35 dB for the frequency band of interest. This concept is also implemented on the C-type switch in Fig. 18. The force generated on the actuators is directly transferred to the contact points and results in very low contact resistance. The continuity of the ground lines is also maintained to avoid unwanted parasitic CPW modes and obtain wideband performance. The switch is comprised of two different states, while ports 1 and 2 and port 3 and 4 are connected, and in the other state, ports 1 and 4 and ports 2 and 3 are linked. The results

Fig. 19. Measured results for the C-type switch illustrated in Fig. 18.

are shown in Fig. 19, which represent a significant improvement compared to the previously discussed C-type switch. The insertion loss is as low as 1 dB at 20 GHz and the isolation is better than 30 dB for all the frequency bands of interest. VI. SUMMARY In this paper, novel multiport thermally actuated switches for redundancy switch matrix applications have been designed, analyzed, and fabricated. The switches demonstrate an extremely low loss. The SPST has 0.3 dB, SPDT has 0.5 dB, and the C-type switch has less than 1-dB loss at 20 GHz. These switches also

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provide a high isolation that is more than 30 dB for the frequency range of interest. The prototype units are fabricated monolithically using well-known metal thermal actuators (i.e., TEDs). These actuators have a very low dc power consumption, which is even further reduced in vacuum hermetic packages. These switches are superb nominees for the basic building blocks of redundancy switch matrices in satellite communications.

circuits. She has authored or coauthored several scientific papers. She holds one patent with two pending in the area of RF MEMS switches. Dr. Daneshmand is vice chair of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Kitchener/Waterloo (KW) Chapter. She is the current recipient of a Natural Sciences and Engineering Research Council of Canada (NSERC) and a Canadian Space Agency (CSA) Postdoctoral Fellowship (since January 2006).

REFERENCES [1] J. Qin, J. H. Lang, A. H. Slocum, and A. C. Weber, “A bulk micromachined bistable relay with U-shaped thermal actuators,” J. Microelectromech. Syst., vol. 14, no. 5, pp. 1099–1109, Oct. 2005. [2] D. Girbau, A. Lazaro, and L. Pradell, “RF MEMS switches based on the buckle beam thermal actuator,” in Proc. 33rd Eur. Microw. Conf., 2003, pp. 651–654. [3] P. Robert, B. Saias, C. Billard, S. Boret, N. Sillon, C. Maeder-Pachurka, P. L. Charvet, G. Bouche, P. Ancey, and P. Berruyer, “Integrated RF-MEMS switch based on a combination of thermal and electrostatic actuation,” presented at the 12th Int. Solid State Sens., Actuators, Microsyst. Conf., Boston, Jun. 8–12, 2003. [4] M. Daneshmand and R. R. Mansour, “C-type and R-type RF MEMS switches for redundancy switch matrix applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2006, pp. 144–147. [5] N. D. Mankame and G. K. Ananthasuresh, “Comprehensive thermal modeling and characterization of an electro-thermal-compliant microactuator,” J. Micromech. Microeng., vol. 11, pp. 452–462, 2001. [6] A. Cowen, B. Dudley, E. Hill, M. Walters, R. Wood, S. Johnson, and H. Wynads, “,” in Metal MUMPs Design, Handbook. Durham, NC: Cronos Integrated Microsyst., 2002. [7] High Frequency Structure Simulator (HFSS). Ansoft Corporation, Pittsburgh, PA, 2006. [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [8] G. M. Rebeiz, RF MEMS, Theory, Design, and Technology. New York: Wiley, 2003. [9] Comsol. FEA Commercial Softw., Burlington, MA, 1997–2007. [Online]. Available: http://www.comsol.com [10] F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer. New York: Wiley, 1985. Mojgan Daneshmand (S’00) received the B.Sc. degree from the Iran University of Science and Technology (IUST), Tehran, Iran, in 1999, the M.Sc. degree from the University of Manitoba, Winnipeg, MB, Canada, in 2001, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 2005, all in electrical engineering. She is currently with the Center for Integrated RF Engineering lab (CIRFE), University of Waterloo, where she is involved with RF MEMS device fabrication and characterization and microwave/antenna

Winter D. Yan (S’04) received the B.Sc. degree in automatic control from the Beijing Institute of Technology, Beijing, China, in 2000, the M.Sc. degree in mechanical engineering from the University of Waterloo, Waterloo, ON, Canada in 2002, and is currently working toward the Ph.D. in electrical engineering at the University of Waterloo. He is currently a Research Assistant with the Center for Integrated RF Engineering (CIRFE), where he is involved in the area of MEMS actuator and latching mechanism development for RF/microwave applications.

Raafat R. Mansour (S’84–M’86–SM’90–F’01) was born in Cairo, Egypt, on March 31, 1955. He received the B.Sc. (with honors) and M.Sc. degrees from Ain Shams University, Cairo, Egypt, in 1977 and 1981, respectively, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 1986, all in electrical engineering. In 1981, he was a Research Fellow with the Laboratoire d’Electromagnetisme, Institut National Polytechnique, Grenoble, Grenoble, France. From 1983 to 1986, he was a Research and Teaching Assistant with the Department of Electrical Engineering, University of Waterloo. In 1986, he joined COM DEV Ltd. Cambridge, ON, Canada, where he held several technical and management positions with the Corporate Research and Development Department. In 1998, he was promoted to Scientist. In January 2000, he joined the University of Waterloo as a Professor with the Electrical and Computer Engineering Department. He holds a Natural Sciences and Engineering Research Council of Canada (NSERC) Industrial Research Chair in RF Engineering with the University of Waterloo. He has authored or coauthored numerous publications in the areas of filters and multiplexers and high-temperature superconductivity. He holds several patents related to microwave filter design for satellite applications. His current research interests include superconductive technology, MEMS technology, and computer-aided design (CAD) of RF circuits for wireless and satellite applications.

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High-Performance CMOS-Compatible Solenoidal Transformers With a Concave-Suspended Configuration Lei Gu, Student Member, IEEE, and Xinxin Li

Abstract—Reported are high-performance solenoid transformers concave suspended in standard (low resistivity) silicon wafers by using a post-CMOS microelectromechanical systems process. The concave-suspended structure of the transformers effectively depress the substrate effects including eddy current and capacitive coupling between the solenoids and substrate, thereby achieving both high- factor and broad usable frequency band. has been measured as high as 0.89 Maximum available gain for the transformer. Within the wide usable frequency band of value is always higher approximately 16 GHz, the measured than 0.75. An equivalent-circuit model has been established for optimal design of the high-performance transformers. By using the equivalent-circuit model and ADS software, the transformer parameters are calculated and simulated, resulting in a satisfactory agreement with the measured data. The CMOS-compatible concave-suspended solenoidal microtransformers are promising to be post-CMOS integrated with high-performance RF integrated circuits. Index Terms—Available gain, post-CMOS process, RF integrated circuits (RF ICs), transformers.

I. INTRODUCTION

O

N-CHIP integrated high-performance transformers have recently become highly demanded in RF integrated circuits (RF ICs) like low-noise amplifiers (LNAs), filters, power amplifiers, and mixers [1]–[5]. High-performance transformers are difficult to be monolithically integrated in standard CMOS (low resistivity) silicon wafers due to the substrate effects of both electromagnetic and capacitive coupling between the passive components and substrate [6]–[8]. Firstly, eddy–current induced energy loss in the substrate significantly lowers the factor of the transformers. This effect is especially serious at high frequency. Secondly, the parasitic capacitance between the transformer coil and substrate lowers the resonant frequency and further causes energy loss by electric coupling with the substrate. For improving the performance of on-chip transformers, efforts have been made to depress the substrate effects

Manuscript received November 23, 2006; revised February 15, 2007. This work was supported in part under the Chinese 973 Program (2006CB300405). The authors are with the State Key Laboratory of Transducer Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China, and also with the Graduate University, Chinese Academy of Sciences, Beijing 100049, China (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.2007.897853

by using thick silicon dioxide [8] or porous silicon [9] to enhance the substrate isolation. With this kind of methods utilized, the substrate wafers have to be modified at the regions where the RF components will be accommodated. However, the substrate-modifying methods will, more often than not, bother the integrated circuit (IC) fabrication and make difficulties in compatibility with the CMOS process. Micromechanically suspended structures were used to suppress the substrate effects of microinductors. A type of reported suspended inductors used either a terminal-anchored spiral configuration [10] or a solenoid structure whose figure is protruding above the wafer surface by approximately 70 m in height [11]. The former structure is mechanically flexible and weak in its robustness, thereby generally suffering a coil deflection problem under environmental vibration. For the latter, the vertical protruding configuration does not facilitate the following packaging process such as solder-bump flip-chip. In general, solenoidal inductors feature better performance than its planar spiral counterparts. As a tradeoff, the 3-D solenoids are more difficult to be fabricated and integrated with ICs. It is no doubt that solenoidal transformers are more complicated in structure compared to spiral transformers. Therefore, new techniques are highly demanded for the formation of high-performance CMOS-compatible RF transformers. In this study, we develop a post-CMOS concavely suspending microelectromechanical systems (MEMS) process to fabricate high-performance transformers with a solenoid-DNA-like configuration. With the schematic shown in Fig. 1, the transformer is concavely embedded into the silicon wafer. The silicon–dioxide film is laterally protruding at both sides of the transformer to sustain the suspension of the solenoid, while the substrate surrounding to the transformer is removed to form an air-gap distance for depressing the substrate effects. The post-CMOS fabrication is concisely processed with only three layers of mask required. According to the component construction, an equivalent-circuit model for designing the transformer is established, with the simulated transformer parameters agreeing well with the measurement results. Several key parameters can be considered as the figure-ofmerit for evaluating the transformer performance, such as available gain, insert loss, and characteristic resistance [12]. Insert loss is a popular parameter that has been widely used in many types of RF devices and circuits. For evaluation of a transformer, characteristic resistance and available gain are based on maximizing the available output power and minimizing the transformer power loss, respectively. Power transfer is the

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Fig. 2. Main steps of the fabrication processes with the cross-sectional views at the left side and the top views at the right side.

Fig. 1. Schematic of the concave-suspended solenoidal transformer, with the dashed cross-sectional cutting line A–A denoted for the process steps described in Fig. 2.

primary objective of a transformer. The transformer efficiency can be defined as the ratio between the power delivered to the load and the power input to the network. In view of the power has be widely considered transfer efficiency, available gain as an important figure-of-merit for overall evaluation of the RF transformers [9], [13]–[15]. For comparing the performance of present concave-suspending transformer with those reported in previous publications, herein we still use the available gain value as the figure-of-merit of the transformers. A higher means a lower loss in view of both RF power and signal. In portable wireless applications, the power efficiency becomes especially important. From the previous results published in was in the range [9] and [13]–[15], the achieved maximum of 0.45–0.78. For the concave-suspended transformers in the value can be improved to as current research, the maximum high as 0.89. In addition, the current transformer can remain as performance for a wide frequency band. the high II. FORMATION OF THE TRANSFORMERS As shown in Fig. 1, the transformer is concavely embedded in a silicon cavity and suspended by the double-sided protruding SiO film. An air-gap distance is formed between the solenoids and silicon substrate to depress the substrate effects. In this way, a high- factor is expected to be achieved. On the other hand, the solenoids are concavely embedded in the silicon wafer so that the wafer surface still remains a flat plane. The planar chip surface facilitates the following packaging process such as solder-bump flip-chip. The 3-D solenoids are constructed by jointing the top and bottom copper strips at the wafer surface where the joints also serve as the anchors to the SiO sustaining film. The cross-sectional view of the solenoid shows a trapezoid shape. The shape is defined by a concave mould that is formed by using a specific metal-protecting TMAH anisotropic etching [16]. The primary coil and secondary coil wind together, with a 5–10- m air gap in between. Of course, the smaller the gap,

the higher the mutual coupling effect between the two coils. The minimum gap distance is limited by the linewidth resolution of the microfabrication process. The three-mask post-CMOS process steps are sketched in Fig. 2 and described as follows. At the left side of the topview schematics, the cross-sectional views are all cut along the dashed line of – shown in Fig. 1. 8 cm silicon (a) The process starts from 1 wafers, with either n- or p-type doping. 2–3- m-thick silicon dioxide thin film is deposited at 350 C by plasma enhanced chemical vapor deposition (PECVD). A 200-nm-thick TiW/Cu seed layer is sputtered that is followed by 10- m-thick photoresist spin coating and patterning. After the 8- m-thick 15- m-wide copper strips are formed by electroplating, a 200-nm-thick Au surface coating layer is continually electroplated for long-term antioxidation of Cu in ambient air. The gap distance between the primary coil and secondary coil is set as 7 m. After the photoresist striped by acetone, the Cu and TiW seed layers are sequentially removed by H SO H O and pure aqueous H O . (b) The SiO film in the solenoid region is patterned into a rectangular window. Then anisotropic etching is processed for approximately 70 m deep by using an aluminum-protecting TMAH anisotropic etching [16]. The etching opening is along 110 orientation, while the topcopper strips are with an inclined angle from 110 orientation. Within the range from 10 to 17 , the detailed angle can be flexibly chosen according to the designed turn density of the solenoid. Perpendicular to this inclined direction, lateral under-etch will excavate the silicon beneath the top-copper strips into through and, finally, the whole trapezoid cavity is formed for accommodating the solenoidal transformer. (c) The second TiW/Cu seed layer (200-nm thick) is sputtered on both the sidewalls and the bottom of the cavity despite that the top-copper strips are existing there. The existing top-copper strips cannot prevent the cavity-bottom areas from the seed-layer sputtering. The reason lies in the scattering of the sputtered metal by the aid of the spatial gap distance under the top-copper strips. A 10- m-thick photoresist layer is spray coated on all the surfaces of the cavity bottom, sidewalls, and front surface of the top-copper strips. A commercial EVG-101 spray coater is used to implement the photoresist spray

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Fig. 4. Multisectional quasi-distributed equivalent circuit for the concave-suspended transformers.

Fig. 3. SEM images of the fabricated transformers. (a) and (b) 9.5-turn 1 : 1 solenoidal transformer and its close-up view, respectively. (c) Transformer is broken for illustrating the cross section of the concave-suspended solenoidal structure. (d) 2 : 1 transformer is fabricated that demonstrates the design flexibility of the developed technique.

coating. Attributed to the similar above-described reason, the cavity bottom can be fully coated with the photoresist. The bottom-metal strips for the transformer are then patterned on the surface of the cavity bottom and sidewalls. As the pattern feature size of 7 m, i.e., the gap between the primary coil and secondary coil, is not very critical, standard contacting exposure equipment (e.g., Karlsuss MA6) is sufficient for this usage. A 8- m-thick copper plus 200-nm-thick Au antioxidation surface layer is sequentially electroplated. With the photoresist and seed layer sequentially removed, the copper solenoids are formed. gaseous isotropic etching is used to excavate the (d) surrounding substrate silicon to form a 40–50- m air-gap distance. By now the transformer has been suspended in the silicon cavity with the laterally protruding silicon–oxide membrane sustaining at both sides. A fabricated 1 : 1 transformer is shown in Fig. 3(a) with its close-up view shown in Fig. 3(b). In view of the chip area occupation, the whole solenoid transformer is 1.2 mm in length and 200 m in width. Fig. 3(c) shows the cross section of the suspended transformer. With this flexible micromachining techtransformers can be designed and nique, various types of formed. As an example, a fabricated 2 : 1 transformer is shown in Fig. 3(d).

and are the series resistance, and and In the model, are the inductance of the primary and the secondary coils, respectively. For the solenoid turn number of , both and should be equally distributed into parts into the equivalent circuit. The mutual inductance between the two , where is the mutual reactive coils is models the coupling feed-forward coupling coefficient. capacitance between the primary coil and secondary coil. The substrate parasitic effects are modeled by three couples of pa, , and , which represent rameters of the parasitic capacitance from the air gap plus the SiO layer at the suspending anchor regions, substrate capacitive, and resistive effects, respectively. All the inductance value, resistive value, and parasitic effects are considered identical between the primary solenoid and the secondary one, and all the -sectional elements are equally distributed in the equivalent circuit of Fig. 4. A. Self-Inductance and Mutual Inductance The series inductance of a transformer is built on the concept of the self-inductance of a wire and the mutual inductance between a pair of wires. The self-inductance of a straight conductor can be expressed as [17], [18] (1) For the trapezoid-shaped solenoid in the current research, can be considered the length of one straight conductor segment in one coil turn, i.e., one straight side of the four sides in the trapezoid. The mutual inductance between the two parallel wires can be calculated using (2)

III. EQUIVALENT-CIRCUIT MODEL The DNA-like 1 : 1 transformer consists of two identical solenoids inter-wound with each other. For modeling the component, a multisectional quasi-distributed element model is established, with the equivalent circuit illustrated in Fig. 4.

where is the mutual-inductance parameter that can be expressed as

(3)

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Here, denotes the geometric mean distance between the wires that approximately equals to the pitch of the wires. A polynomial expression of can be given as

(4) where denotes the pitch of the wires. For a spiral inductor comprising multiple segments, the total inductance of the whole spiral inductor can be deduced from the self-inductance of every segments and the mutual inductance between every segment and all other segments in the spiral inductor. The inductance of the spiral inductor can be calculated with the following equation:

Fig. 5. Simplified substrate capacitor structure of C

.

in the conductor. The proximity effect takes place when the conductor is influenced by a time-varying field that is induced by the time-varying current in nearby conductors. The classical 1-D can be expressed as approximation of the series-resistance

where (5) where is the total segment number of the spiral, is the is the self-inductance of the th segment of the inductor, and . mutual inductance between the th and th segments For solenoid inductors, the calculation can be simplified since every turn in the solenoid has identical geometric dimensions. Besides, the mutual inductance of one turn of coil with all other turns can be simply obtained by calculating the mutual inductance between this turn and its two adjacent turns since the mutual inductive coupling with the farther turns is comparably much weaker and can be ignored. In this way, the total inductance of the solenoid inductor can be obtained by calculating one turn inductance and its mutual inductance with the adjacent turns, and then multiplied by the total turn number of the soleand can be calculated noid. Therefore, the inductance of by

and

This simple expression reflects the physics properties and facilitates optimal layout design of the transformer. It is also with a satisfactory accuracy in a not very high-frequency range ( 8 GHz). However, owing to the proximity effect, (8) is not suitable for series-resistance modeling at high frequency. For precisely modeling the series resistance within a wide frequency range, the empirical equations reported in [19] can be used, which are

(9)

and (10)

(6) where is the total turn number of the solenoid. is the inductance of one turn that includes the algebraic sum of the self-inductance values of the four straight segments in one trapezoid turn and the sum of the mutual inductance between every couple of the adjacent segments in one turn. Accordingly, is the mutual inductance between two adjacent turns. The inductance model for the solenoidal transformer has been verified with a satisfactory accuracy by the testing results to be given in Section III-B. , between the primary and secFor the mutual inductance ondary solenoids, it can also be calculated by still using (2)–(4). , the mutual reactive couAfter calculation of , , and pling coefficient of the transformer can be obtained with (7) B. Series Resistance Along with the frequency increases, the effective resistance will increase owing to of either one of the two solenoids skin and proximity effects. With the skin effect, the time-varying current-flow induced magnetic field will cause an eddy current

(8)

In (8)–(10), , , , and are the thickness, length, crosssectional area of the copper wire, and skin depth, respectively. , , , and represent the resistivity of the copper, permeability, operation frequency, and self-resonance frequency, respectively. C. Capacitive Effect The coupling feed-forward capacitance indicates the capacitance between the primary and secondary solenoids. It can be expressed as (11) where is the dielectric constant of air and and are the overlapping area and the gap between the primary and the secondary solenoids, respectively. The substrate effects of planar spiral inductors or transformers have been extensively investigated [7], [18]–[22]. In current concave-suspended solenoid inductors, however, the substrate parasitics becomes much more complicated. Therefore, some reasonable assumptions for approximate calculation represents the capacitance between the have to be used. solenoid and substrate, whereas and are the silicon

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TABLE I GEOMETRIC AND ELECTRIC PARAMETERS OF THE CONCAVE-SUSPENDED SOLENOIDAL TRANSFORMER

Note: The data with superscript “3” are calculated by using (14) and (15) with the C and G value extracted from the measurement results.

substrate resistance and capacitance, respectively. comprises two parts that are the capacitance between the copper strips and the surrounding silicon with air as the dielectric material, as well as, the parasitic capacitance at the suspending anchor regions with SiO as the dielectric layer. Therefore, (12) Based on the suspended structure, we have built a model to . In the model, is constructed between a ficalculate nite length conductor and an equivalent planar silicon ground [20]. Since the capacitive effect of the top conductor can be ignored due to the relatively large distance from the substrate, the value can be obtained by calculating the capacitance between the three-segment bottom conductor and substrate. The suspended three-segment capacitive structure is reasonably reshaped into an equivalent straight conductor parallel to the silicon ground (see Fig. 5). The corresponding capacitance can then be calculated by

Fig. 6. For the 200-m-wide transformer, the: (a) magnitude and (b) phase of the measured S -parameters are compared with the extracted and simulated results.

(13)

where is the turn number of the solenoid, is the equivalent length of the bottom conductor, and is the air gap between the equivalent conductor and substrate. The calculation of the planar is quite straightforward with the suspending capacitance of SiO as the dielectric film. The substrate capacitance and resistance are approximately proportional to the transformer area occupation as (14) (15) and indicate the length and width of the where coil occupying region of the transformer. and are the capacitance and conductance per unit area for the silicon substrate. Even for a conventional planar spiral inductor, the analytic caland is very complicated [7], [24], [25]. In culation of fact, and are the functions of the substrate doping level and can be extracted from measurement results. For different solenoids fabricated in the same substrate with the same techand cannot vary significantly. As a result, niques, and can only be scaled by and . Practically, many researchers have used the parameter extracting method and values of both spiral and solenoidal to obtain the inductors [8], [18], [21], [27]. In the current research, we also

Fig. 7. For the 260-m-wide transformer, the: (a) magnitude and (b) phase of the measured S -parameters are compared with the extracted and simulated results.

used the measurement-fitting and parameter-extracting method to obtain the values of and . Using the established physics model and the software of ADS, the parameters in the equivalent circuit of the concave-suspended transformer can be calculated by substituting the detailed dimensions of the transformer into the equations listed above. Alternatively, based on the measurement results, the fitted parameters in the circuit model can also be extracted from the professional software of ADS. The details of the parameter-extracting method have been described in [24] and [25]. In the current research, the simulated and extracted lumped parameters are obtained and listed in Table I for comparison with each other. IV. RESULTS AND DISCUSSION Two-port -parameters of the transformers are measured using an HP8722D network analyzer. The pad parasitic effect is eliminated by a normal open-pad deembedding method. Two types of 9.5-turn 1 : 1 transformers are tested. One type of transformer is with 200- m solenoid lateral width. Another type

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is with an identical structure, except for its 260- m solenoid lateral width. Figs. 6 and 7 show the measured -parameters of the two types of transformers compared with both the extracted parameters by fitting the measured data into the circuit model and the simulated parameters directly from the equivalent-circuit model. The software of ADS is used for both the parameter extraction and simulation. Based on the measurement, the factors of the primary and secondary solenoids can be obtained with and , respectively. When or becomes purely resistive, the resonant frequency of can be worked out. Beyond , the component will no longer function as an inductive component. With the model, the factor of the transformer to express , as described can be obtained by using in [27]

Fig. 8. Measured, extracted, and model-simulated inductance and Q-factor values versus frequency with the results of the: (a) 200- and (b) 260-m-wide transformer.

or deIt can be seen clearly from (16) that increasing creasing helps to enhance the factor. In addition to the factor, can be calculated by

simulated peak factor could then agree well with the measurement. In future research, we will try to evaluate the resistivity of the electroplated copper and further control the electroplating process. Attributed to the effectively depressed substrate effects with the suspended structure, the peak factor has been measured with a much higher value than that of the planar spiral transformers [9], [13]–[15]. Besides, the depressed substrate loss leads the transformers to feature high resonant frequencies of 17 and 15 GHz, respectively. Thanks to the suspended structure, both the eddy current and shunt capacitance have been effectively suppressed, which contributes to the high- factor, especially at the frequency beyond 3 GHz where the substrate loss dominates. Based on the measurement results, the mutual reactive coupling factor of the transformer can be calculated with the following definition:

(19)

(22)

(16) where (17) (18)

where (20)

(21) Fig. 8 shows the factor and inductance values for both the 200- and 260- m-wide solenoidal transformers, respectively. Agreeing well with the extracted and simulated results, the measured self-inductance values are 2.30 and 2.99 nH, respectively, at 2.3 GHz. Accordingly, the measured peak -factor values are 14.3 at 2.4 GHz and 13.6 at 2.3 GHz, respectively. The measured peak -factor values are slightly lower compared with the ADS simulation results. In the simulation of series resistance, m that we use the copper resistivity value of 1.694 10 has been widely accepted for bulk copper material. In the current research, however, the electroplated copper should possess a resistivity value somewhat higher than that of the bulk conductive copper, as the electroplated copper still cannot compare with the bulk material made by conventional process. We tried to use a 10% higher resistivity value for the -factor simulation. The

The mutual reactive coupling represents the interaction between the primary solenoid and secondary solenoid by means of the time-varying magnetic or electric flux into them. The mutual reactive coupling factor equals to the mutual magnetic coupling factor at low frequencies. At high frequency, however, will deviate from the purely magnetic coupling factor, as the parasitic capacitance between the primary coil and secondary coil and the parasitic capacitance between the coils and the substrate begin to take effect. With the measured data comparing with the extracted and simulated results, Fig. 9 shows the obtained value in terms of frequency. For 200- and 260- m-wide solenoids, the values at low frequency are 0.736 and 0.702, respectively. The usable bandwidths are broad, approximately 16.5 and 14 GHz, respectively. There would be no power loss in an ideal transformer. For practical transformers, however, the transforming efficiency can be evaluated by available gain , which is defined as the power delivered to the load over the input power. High- factor and high both lead to a high available-gain value. Based on the can be expressed in terms of the -parameters measurement, as [13], [14] (23)

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TABLE II PERFORMANCE COMPARISON AMONG THE SOLENOIDAL TRANSFORMERS

Fig. 9. Measured, extracted, and simulated mutual reactive coupling coefficient versus frequency with the results of the: (a) 200- and (b) 260-m-wide transformer.

Fig. 11 Simulated contour map of the concave-suspended solenoid transformer under a 55 C range of temperature change. Fig. 10. Measured, extracted, and simulated available gain G versus frequency with the results of the: (a) 200- and (b) 260-m-wide transformer.

where (24) With the measurement results shown in Fig. 10, the developed transformers feature a very high maximum of 0.89 and 0.85 at approximately 2 GHz for 200- and 260- m-wide solenoids, result is ranked respectively. To our best knowledge, this the highest among those published air-core integrated transvalue always remains formers [9], [13]–[15]. Moreover, the higher than 0.7 within a very wide frequency band of 16.5 and 14 GHz for 200- and 260- m-wide solenoids, respectively. It means that both the eddy current and capacitive power loss have been significantly reduced by using the CMOS-compatible concave-suspending technique. Therefore, the broadband performance has been achieved by the developed hightransformers. The high performance of the transformer is helpful for the designers to develop high-performance RF ICs such as low-voltage-supply transformer-feedback LNAs and voltage-controlled oscillators (VCOs). As shown in Figs. 6–10, there is a satisfactory agreement between the equivalent-circuit model and the measured data within the usable frequency band. With this model, the concave-suspended solenoid transformers can be designed and optimized to meet the requirements of high-performance RF ICs. The comparison of measured solenoid-transformer performance between the developed 200- m-wide transformer in this paper and those reported in previous publications is shown in Table II. The transformer developed in this study indeed behaves as a higher factor and with a broader usable

frequency range, thereby, achieving a much higher maximum available-gain value. Mechanical robustness is an important issue for suspended RF components [28]. In the current study, SiO film is used to double-side sustain the concave-suspended solenoids. This double-sided anchoring method has been verified with a satisfactory mechanic stability and robustness. Firstly, we evaluate the mechanical strength of the transformer using ANSYS’ finite-element simulation. 100-g acceleration is sequentially applied to the -, -, and -axes [denoted in Fig. 3(a)]. The maximum warpage of the solenoid is smaller than 0.15 nm. A shock experiment is then carried out. Reliability of the transformers is physically tested by applying 10 000-g shocking acceleration along the three directions. The experimental results show that the transformers survive the high shock well. After process optimization for copper electroplating, the residual stress of the electroplated copper has been well controlled. Possessing a structure different from the suspended inductor reported in [28], the current solenoid transformer is with its every turn of coil clamped–clamped supported, thereby with mechanical stability and robustness behaving satisfactorily. The thermal stress effect on the solenoid is also simulated by using ANSYS’ software. Since copper features about an order of magnitude larger thermal expansion coefficient 17 10 C compared to silicon 2.6 10 C and silicon dioxide 5.4 10 C , we mainly consider the effect from the copper thermal expansion with the thermal effect from the silicon substrate and the SiO film reasonably neglected. With every turn of the coil supported by the silicon–dioxide film from the double sides, thermal deformation of the copper solenoid is simulated, corresponding to the temperature change from 20 C to 75 C. The resultant thermal displacement is shown in Fig. 11. The average

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nodal displacements along -, -, and -axes are simulated as 8.08, 1.30, and 4.25 nm, respectively. Compared with the suspended spiral inductor in [28], the thermal deformation of the current concave-suspended solenoid transformer is approximately one order of magnitude lower. Within the 55 C range of temperature change, the simulation results in [28] showed a more than 5% inductance variation. In contrast, our simulation results for the current solenoid transformer demonstrate a less than 1% variation in inductance. V. CONCLUSION High-performance solenoid transformers for RF ICs have been fabricated with post-CMOS concave-suspending MEMS technology. The concave-suspended solenoidal structure effectively depresses substrate effects, thereby achieving a high- factor, broad usable frequency band, and, especially, within the usable frequency very high available gain band. An equivalent-circuit model is established for the transformer. Based on the model and ADS software, the simulated transformer parameters agreed well with the measured ones. As a CMOS-compatible on-chip solution for RF passives, the developed transformers are promising in applications of high-performance RF ICs. ACKNOWLEDGMENT The authors would like to thank Prof. X. Sun, Ms. R. Qian, and Prof. Y. Wang, all with the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China, for their help with the -parameters test and academic discussions. REFERENCES [1] J. R. Long, “Monolithic transformers for silicon RF IC design,” IEEE J. Solid-State Circuits, vol. 35, no. 9, pp. 1368–1382, Sep. 2000. [2] N. Fong, J. Plouchart, N. Zamdmer, J. Kim, K. Jenkins, C. Plett, and G. Tarr, “High-performance and area-efficient stacked transformers for RF CMOS integrated circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 2, pp. 967–970. [3] A. H. Aly, D. W. Beishline, and B. E. Sharawy, “Filter integration using on-chip transformers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 3, pp. 1975–1978. [4] D. J. Cassan and J. R. Long, “A 1-V transformer-feedback low-noise amplifier for 5-GHz wireless LAN in 0.18- m CMOS,” IEEE J. SolidState Circuits, vol. 38, no. 3, pp. 427–435, Mar. 2003. [5] Y. Zhuang, M. Vroubel, B. Rejaei, and J. N. Burghartz, “Ferromagnetic RF inductors and transformers for standard CMOS/BiCMOS,” in IEEE Int. Electron Device Meeting, Dec. 2002, pp. 475–478. [6] M. Y. Bohsali and A. M. Niknejad, “Microwave performance of monolithic silicon passive transformers,” in IEEE RFIC Symp., Jun. 2004, pp. 647–650. [7] A. M. Niknejad and R. G. Meyer, “Analysis of eddy–current losses over conductive substrates with applications to monolithic inductors and transformers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 166–176, Jan. 2001. [8] H. Jiang, Z. Li, and N. Tien, “Reducing silicon-substrate parasitics of on-chip transformers,” in IEEE Int. Microelectromech. Syst. Conf. Tech. Dig., Jan. 2002, pp. 649–652. [9] K. Chong and Y. Xie, “High-performance on-chip transformers,” IEEE Electron Device Lett., vol. 26, no. 8, pp. 557–559, Aug. 2005. [10] J. Yoon, Y. Choi, B. Kim, Y. Eo, and E. Yoon, “CMOS-compatible surface-micromachined suspended-spiral inductors for multi-GHz silicon RF ICs,” IEEE Electron Device Lett., vol. 23, no. 10, pp. 591–593, Oct. 2002.

[11] Y. J. Kim and M. G. Allen, “Surface micromachined solenoid inductors for high frequency application,” IEEE Trans. Compon. Packag. Technol., vol. 21, no. 1, pp. 26–33, Jan. 1998. [12] A. Italia, F. Carrara, E. Ragonese, T. Biondi, A. Scuderi, and G. Palmisano, “The transformer characteristic resistance and its application to the performance analysis of silicon integrated transformers,” in IEEE RFIC Symp., Jun. 2005, pp. 597–600. [13] K. T. Ng, B. Rejaei, and J. N. Burghartz, “Substrate effects in monolithic RF transformers on silicon,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 377–383, Jan. 2002. [14] D. C. Laney, L. E. Larson, P. Chan, J. Malinowski, D. Harame, S. Subbanna, R. Volant, and M. Case, “Lateral microwave transformers and inductors implemented in a Si/SiGe HBT process,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1999, vol. 3, pp. 855–858. [15] Y. S. Lin, H. B. Liang, T. Wang, and S. S. Lu, “Temperature-dependance of noise figure of monolithic RF transformers on a thin (20 m) silicon substrate,” IEEE Electron Device Lett., vol. 26, no. 3, pp. 208–211, Mar. 2005. [16] G. Yan, P. Chan, I. Hsing, R. Sharma, J. Sin, and Y. Wang, “An improved TMAH Si-etching solution without attacking exposed aluminum,” Sens. Actuators A, Phys., vol. 89, pp. 135–141, 2001. [17] H. M. Greenhouse, “Design of planar rectangular microelectronic inductors,” IEEE Trans. Parts, Hybrids, Packag., vol. PHP-10, no. 2, pp. 101–109, Jun. 1974. [18] C. P. Yue and S. S. Wong, “Physical modeling of spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 47, no. 3, pp. 560–568, Mar. 2000. [19] A. Scuderi, T. Biondi, E. Ragonese, and G. Palmisano, “A lumped scalable model for silicon integrated spiral inductors,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 6, pp. 1203–1209, Jun. 2004. [20] T. Biondi, A. Scuderi, E. Ragonese, and G. Palmisano, “Analysis and modeling of layout scaling in silicon integrated stacked transformers,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2203–2210, May 2006. [21] S. S. Mohan, “The design, modeling and optimization of on-chip inductor and transformer circuits,” Ph.D. dissertation, Dept. Elect. Eng., Stanford Univ., Standford, CA, 1999, pp. 16. [22] N. A. Talwalkar, A. P. Yue, and S. S. Wong, “Analysis and synthesis of on-chip spiral inductors,” IEEE Trans. Electron Devices, vol. 52, no. 2, pp. 176–182, Feb. 2006. [23] T. L. Xue and X. M. Meng, “Using equivalent charge to calculate the conductor bar’s capacitance,” (in Chinese) J. Elect. Power, vol. 18, pp. 8–10, Jan. 2003. [24] W. Gao and Z. P. Yu, “Scalable compact circuit model and synthesis for RF CMOS inductors,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1055–1064, Mar. 2006. [25] J. C. Guo and T. Y. Tan, “A broadband and scalable model for on-chip inductors incorporating substrate and conductor loss effects,” IEEE Trans. Electron Devices, vol. 53, no. 3, pp. 413–421, Mar. 2006. [26] C. P. Yue, C. Ryu, J. Lau, T. H. Lee, and S. S. Wong, “A physical model for planar spiral inductors on silicon,” in IEEE Int. Electron Device Meeting, Dec. 1996, pp. 155–158. [27] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Si-based RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, May 1998. [28] J. W. Lin, C. C. Chen, and Y. T. Cheng, “A robust high-Q micromachined RF inductor for RFIC applications,” IEEE Trans. Electron Devices, vol. 52, no. 7, pp. 1489–1496, Jul. 2005.

Lei Gu (S’06) received the B.S. and M.S. degrees in electrical and electronic engineering from Southeast University, Nanjing, China, in 2001 and 2004, respectively, and is currently working toward the Ph.D. degree in microelectronics at the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China He is currently with the State Key Laboratory of Transducer Technology, Shanghai Institute of Microsystem and Information Technology. His current research interest is integrated RF passives components and CMOS-compatible MEMS processes.

GU AND LI: HIGH-PERFORMANCE CMOS-COMPATIBLE SOLENOIDAL TRANSFORMERS

Xinxin Li was born in Liaoning, China, in 1965. He received the B.S. degree in semiconductor physics and devices from Tsinghua University, Beijing, China, in 1987, and the M.S. and Ph.D. degree in microelectronics from Fudan University, Shanghai, China, in 1995 and 1998, respectively. He was a Research Engineer with the Shenyang Institute of Instrumentation Technology for five years. He was sequentially a Research Associate with the Hong Kong University of Science and Technology, Hong Kong and a Research Fellow with Nanyang Technological University, Singapore. He then joined Tohoku University, Tohoku, Japan, as a nonpermanent Lecturer under a COE Research Fellowship. Since 2001, he has been a Professor and Executive Director of

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the State Key Laboratory of Transducer Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China. He is also an Adjunct Professor with Fudan University, Shanghai, China, and Shanghai Jiaotong University, Shanghai, China. He has authored or coauthored over 100 papers in referred journals and academic conferences. He is an Editorial Board member for the International Journal of Information Acquisition. He holds approximately 30 patents. His research interests have been in the fields of MEMS/nanoelectromechanical systems (NEMS), microelectromechanical/nanoelectromechanical sensors, and microelectromechanical/nanoelectromechanical integration technologies. Dr. Li is a Technical Program Committee (TPC) member for the IEEE International Conference on MEMS.

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Fringe Management for a T-Shaped Millimeter-Wave Imaging System Yue Li, Senior Member, IEEE, John W. Archer, Fellow, IEEE, Grahame Rosolen, Stuart G. Hay, Greg P. Timms, Member, IEEE, and Y. Jay Guo, Senior Member, IEEE

Abstract—Two methods, the modulated scene method and modulated beam method, are proposed in this paper to manage the fringe in a T-shaped correlating millimeter-wave imaging system. The modulated scene method incorporates the fringe into the scene to form a fringe-modulated scene. The pencil beam that corresponds to the beam of the system with a zero baseline scans the modulated scene to form an image. To recover the image of the original scene, an algorithm that involves demodulation and spectrum patching is used to process the original image after deconvolution. The resulting image is a super-resolution image of the scene. The advantage of the modulated scene method is that a phase shifter is not required. The modulated beam method incorporates the fringe into the beam. By dynamically adjusting the phase of a local oscillator, the fringe scans together with the beam. The advantages of this method are that demodulation is unnecessary and only a single output (real or imaginary) from the complex correlator is necessary to generate a super-resolution image. A disadvantage is that a rapidly adjustable phase shifter is needed. The performance of these methods is theoretically analyzed and tested with simulated data. Index Terms—Image reconstruction, millimeter-wave imaging, super resolution.

I. INTRODUCTION

M

ILLIMETER-WAVE imaging systems are able to see through clothing, fog, dust, and smoke [1]–[8]. Therefore, they have an advantage over optical and infrared sensors, even though they have less spatial resolution as a consequence of the much longer wavelength. The potential applications of millimeter-wave imaging include perimeter or border security (surveillance), bushfire hot-spot mapping, aircraft landing in fog, traffic control on highways, environmental monitoring, and medical imaging. Commercial systems (e.g., Brijot Imaging, Millivision, and QinetiQ) operating at frequencies from 35 to 100 GHz recently became available. Three main approaches have been used to acquire an image at millimeter-wave frequencies. The first is to place an array of millimeter-wave receivers in the focal plane of a lens [6], [7]. This allows high frame rates to be achieved because the entire image is acquired at once. The second approach is to build up an image over a 2-D plane by Manuscript received July 14, 2006; revised February 20, 2007. The authors are with the Wireless Technologies Laboratory, Information and Communication Technologies Center, Commonwealth Scientific and Industrial Research Organisation, Marsfield, N.S.W. 2122, Australia (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [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.2007.897694

Fig. 1. Cross-correlating millimeter-wave imaging system.

mechanically scanning an antenna with a single receiver [3], greatly reducing the cost of the system. Both of these approaches have shortcomings; the high cost of millimeter-wave receivers makes the cost of focal-plane-array systems prohibitively high for most applications and the long scanning time in the scannedantenna approach results in very low frame rates, ruling out potential applications such as aircraft landing in fog or security screening of airline passengers. The third approach offers a compromise with a sub-array being mechanically scanned to build up the image [4]. The sub-array typically contains 10–20 receivers, meaning that cost remains an issue in these systems. A cross-correlating millimeter-wave imaging system has been proposed [9]–[11] (Fig. 1). This system uses higher frequencies (around 190 GHz) to take advantage of the larger bandwidth and smaller aperture size possible at these higher frequencies. The ability to achieve the required resolution with a smaller aperture size is a major advantage for some applications such as aircraft landing where there is little space available to mount the imaging system. This system requires the use of only two receivers, thus reducing cost. It employs two orthogonally oriented co-polarized scanning fan-beam antennas (Fig. 1) arranged in a T-shaped configuration. A prototype of the antenna is shown in Fig. 2 and details about the antenna can be found in [9]. A rotating mirror makes it possible to perform fast scanning and each antenna scans in one direction. An image is built up by measuring the cross-correlation of the received signals from the two antennas at each beam intersection point as it scans a 2-D plane. We are developing a demonstrator system using this approach. The designed performance specifications of the demonm, depth of field 10 m, strator are focal distance 0.3 , corresponding to 20-cm resohalf-power beamwidth lution at 40-m range, and scan range 7 , corresponding to 5 5 m field of view at 40-m range.

0018-9480/$25.00 © 2007 IEEE

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Fig. 2. Prototype of the fan beam antenna. Fig. 5. Geometry for the theoretical analysis.

Fig. 3. Schematic of the millimeter-wave imaging demonstrator.

fringe. A spectrum-patching method is proposed to fill the part of the spatial spectrum of the image that is lost due to the fringe. Together with processes of demodulation (for the modulated scene method) and deconvolution with the Wiener filter (or another regulated filter), a super-resolution image of the scene can be generated. The performances of these methods are theoretically analyzed and then tested using simulated data. This paper is organized as follows. Section II theoretically analyses the performance of the cross-correlating millimeter-wave imaging system; modulated beam and modulated scene methods are proposed. These two methods are compared in Section III and tested using simulated data in Section IV, and a summary is given in Section V. II. THEORETICAL ANALYSIS

Fig. 4. Mills Cross imaging system with antenna arrays.

A simplified schematic of the signal processing system is shown in Fig. 3. The key components are the 190-GHz receivers and the broadband complex correlator [10], [11]. The complete imaging system will be described in a future publication. The proposed system is similar to the Mills Cross (or T-shaped) system used in astronomical imaging [12] (Fig. 4). The difference is that two fan beam antennas with mechanically scanning mirrors are used in the millimeter-wave imaging system compared with the two linear antenna arrays with electronic scanning used in the Mills Cross imaging system. It is difficult to spatially arrange the two fan beam antennas so that the phase centers of the two antennas coincide in the proposed system, thus a T configuration is used and, therefore, unlike the Mill’s Cross system, the baseline of the proposed system (the displacement between the two phase centers) is not zero (Fig. 1). The nonzero baseline generates a fringe that influences the performance of the imaging system. In this paper, two methods, i.e., the modulated scene method and modulated beam method, are proposed to deal with the

The geometry for the theoretical analysis is shown in Fig. 5. The baseline of the millimeter-wave imaging system shown in Fig. 1 is on the -axis with a length of . The center of the field-of-view is on the -axis. A point on the scene, which is in the far field, can be expressed with direc, where and tional cosine . Note that only two of the three directional cosines are independent. Therefore, a point on the . scene can be expressed as Assume that a single frequency signal is radiated from the scene. The signal that arrives at the phase center of antenna I of the (Fig. 1) from a unit solid angle in the direction scene is (1) where is the magnitude of the wave (assumed to be the same for both antennas due to the far-field assumption), is the wave is the frequency, is the propagation speed of the wave, and the phase center of distance between the scene at antenna I, and is the phase of the signal at the scene. It is assumed that depends on the position on the scene so that spatially incoherent scenes can be included. The received signal when the beam of of antenna I from a unit solid angle at is the antenna is steered to

(2)

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is the complex beam pattern of antenna I,

and the pseudoscene brightness distribution is defined as (3)

is the beam phase. and The received signal from the whole scene of (2) over all solid angles

is the integration

(11) then (9) becomes

(12) (4) The signal from the local oscillator (LO) for antenna I, i.e., can be expressed as

,

(5) is the LO frequency and is the LO phase. The where received signal is multiplied with the LO signal and filtered with an appropriate bandpass filter to become an IF signal

This is the output from the complex correlator (Fig. 3) if there is incorporated into the scene is no noise. The factor of in (11) instead of into the point spread function because this term is not shift invariant and it is an advantage to make the point spread function shift invariant. The exponential term in (12) is the fringe term due to the nonzero baseline and the phase difference between the two LO signals. Let the baseline’s projections on the -, -, and -axes be and , respectively, the phase of the fringe then becomes

(13) (6)

where

is the wavelength of the signal and the relationship of

Similarly, for antenna II, (14) is used. Since the baseline has only a

component in Fig. 5, (15)

(7) As shown in Fig. 3, the same LO is used for both antennas, however, the phase shifter is different for the two channels. The correlation function at zero lag between signals and is (8) where * denotes complex conjugation. It is assumed that the scanning is stepwise, i.e., the two beams scan to a pixel and integrate for a period and then they scan to the next pixel. When is sufficiently large and the scene is spatially incoherent, from (6) and (7), (8) becomes

(9)

For the Mills Cross imaging system shown in Fig. 4, the phase centers coincide. Therefore, the baseline length is zero and is zero if the two LO signals have identical phase. In this case, the imaging process described by (12) is that the brightness of the scene is smeared by a real point spread function. For the millimeter-wave imaging system shown in Fig. 1, generally is not zero and the fringe term needs to be considered. Two methods are proposed here to deal with the fringe term in (12). A. Modulated Scene Method Equation (15) shows that the phase of the fringe is indepenif the baseline dent of the steering direction of the beam is stable during scanning, i.e., when the two fan beams scan, the fringe does not move in space. It is equivalent to a Mills Cross system imaging a scene that is modulated (multiplied) by the and define the complex modulated scene as fringe. Let (16)

If the point spread function of the imaging system is defined as

Equation (12) becomes (10)

(17)

LI et al.: FRINGE MANAGEMENT FOR T-SHAPED MILLIMETER-WAVE IMAGING SYSTEM

Fig. 6. Spatial spectrum of the scene is shifted upward due to fringe modulation.

If the beam of each antenna is shift invariant, (17) becomes a 2-D complex convolution (18) The magnitude of the image is the same as that without the fringe only when the point spread function is a delta function or the scene only contains point targets that are from the image far apart. To recover the original scene , an algorithm is proposed in this paper, which includes steps of demodulation from the fringe, deconvolution from the . point spread function, and compensation for the factor of Deconvolution is a common problem for all imaging systems and demodulation is a special problem for the millimeter-wave imaging system with the modulated scene method. Performing a Fourier transform on both sides of (18) results in

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Fig. 7. Demodulation shifts the image spectrum downward and spectrum patching fills the spectrum so that it corresponds to the spectrum of a real scene.

is the noise power spectrum. Compared with where (20), (21) has an extra weighting factor. The second term in the denominator of the weighting factor is the noise-to-signal ratio at that frequency. This filter handles zero points in by setting the weighting to zero. It also increases the weightings on low noise-to-signal ratio frequencies and reduces the weightings on frequencies with a high noise-to-signal ratio. This method requires knowledge of the ratio between the noise power and the power spectrum of the scene. Since spectrum is usually unknown, the noise-to-signal ratio is often replaced with a constant . In this case, choosing a proper value becomes a problem and a trial-and-error method can be used. The demodulation process can be accomplished by shifting the spectrum in (21) by

(19) where is the spatial spectrum of the point spread function, is the spatial spectrum of the pseudoscene is the spais the spatial fretial spectrum of the image, and quency of the fringe. As shown in Fig. 6, the spectrum of the pseudoscene is shifted upward by the fringe. The beam’s spectrum is a square centered at the zero-frequency point if the aperture of the antenna is uniformly weighted. The spectrum of the is the spectrum of the modulated pseudoscene that image passes through the bandwidth of the point spread function. Ignoring noise, the obvious way to implement the deconvolution is to divide both sides of (19) by the spectrum of the point spread function (20) However, this deconvolution method has difficulties at points with a small spectrum magnitude when there is noise. In this case, the Wiener filter (or another regulated filter) can be used for deconvolution

(21)

(22) as shown in Fig. 7. For the example shown in Fig. 7, the downshifted spectrum is only nonzero in the negative region, and it is the spectrum of a complex valued scene. As the pseudoscene is real, the magnitude (phase) of its spatial spectrum is diagonally symmetrical (asymmetrical) with respect to the zero-frequency point, i.e., (23) Therefore, many unknown points in the shifted spectrum can be found according to (23) and the patched spectrum has a bigger bandwidth than the bandwidth of the point spread function (Fig. 7). After spectrum patching, a real image can be generated with an inverse Fourier transform of the patched spectrum. When the point spread function does not have a clear cutoff bandwidth (e.g., a Gaussian-shaped spectrum), frequency points in the negative part of the spectrum has a higher magnitude compared with their corresponding points in area after demodulation (for white noise). In the positive this case, the positive spectrum can be replaced with its corresponding point in the negative area according to (23). This replacement increases the signal-to-noise ratio because those

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points in the negative area have a higher signal-to-noise ratio area. than those in the positive In summary, the modulated scene method includes the following steps. Step 1) Deconvolve the image with the Wiener filter (or another deconvolution filter). Step 2) Demodulate the image by shifting the spectrum of the deconvolved images. Step 3) Patch the spectrum according to (23). The value with a higher signal-to-noise ratio is used to replace the value with a lower signal-to-noise ratio between two corresponding points. Step 4) Inverse Fourier transform the patched spectrum to generate the deconvolved, demodulated, and spectrum patched image. to generate the Step 5) Compensate for the factor of image of the scene from the image of the pseudoscene. B. Modulated Beam Method If the phases of the two LOs are dynamically adjusted during scanning so that

Fig. 8. Spatial spectrum of the beam is shifted upward due to fringe modulation and spectrum patching is used to fill the other half of the spectrum according to (23).

Step 3) Inverse Fourier transform the patched spectrum to generate the deconvolved and spectrum patched image. to generate the Step 4) Compensate for the factor of image of the scene from the image of the pseudoscene.

III. DISCUSSION (24) the fringe phase in (15) becomes (25) and the image in (12) can be expressed as a 2-D convolution (26) where it is assumed that the beam is shift invariant and (27) is defined as the modulated beam in this paper. In this case, the fringe is incorporated into the beam and the LOs’ phases are adjusted so that the fringe scans together with the beam. As a result, the modulated beam is also shift invariant. Performing a Fourier transform on both sides of (26) results in (28) The spectrum of the beam is shifted upward because of fringe modulation, as shown in Fig. 8. The modulated beam now acts as a filter that only allows part of the pseudoscene spectrum in area to pass. As a result, the image is the positive complex valued. With the a priori knowledge that the scene is real, it is again possible to patch the spectrum to a larger area according to (23) (Fig. 8). The modulated beam method includes the following steps. Step 1) Deconvolve the image with the Wiener filter (or another deconvolution filter). Step 2) Patch the spectrum according to (23).

Two methods for dealing with the fringe term in (12) have been discussed. In the modulated scene method, the fringe is incorporated into the scene. In the modulated beam method, the fringe is made to scan together with the beam by dynamically adjusting the phases of the two LO signals. Both methods require that the beam can be considered shift invariant and that the phase centers of the two fan beam antennas can be considered stable during scanning. These requirements can be satisfied with proper antenna design techniques. These two methods also require that the scene is spatially incoherent, and this makes it possible to patch the image spectrum. Note that, after spectrum patching, the bandwidth of the spectrum is doubled in the -direction. As a result, the resolution in the -direction is one-half of that in the -direction. This is made possible by the fringe. To form an image that has the same resolution in both the - and -direction, the aperture length of antenna II (Fig. 3) need to be twice as long as that of antenna I. For the modulated scene method, the outputs from both the real and imaginary channels of the complex correlator (Fig. 3) are required. If only one channel is available, the spectrum of the pseudoscene becomes mixed after modulation. For example, the real part of the modulated scene is

(29) and its spatial spectrum is

(30)

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Fig. 9. (a) Beam. (b) Real part of the fringe. (c) Imaginary part of the fringe. (d) Real part of the fringe-modulated beam. (e) Imaginary part of the fringe-modulated beam.

The spectrum of the pseudoscene is first shifted in two opposite directions and then added together in (30). The spectrum of the pseudoscene is mixed and can only be separated with both real and imaginary outputs from the complex correlator. For the modulated beam method, a single output from the complex correlator is sufficient to recover the spectrum of the pseudoscene within the passband. For example, the real part of the modulated beam is

where is the aperture length, is the wavelength, and are the vertical and horizontal scan angles of the antennas, respecand by tively (Fig. 5), and and are related to

(34) for a small field of view. Therefore, (35)

(31) and its spatial spectrum is

(32) The sum of the two shifted spectrums can be treated as the spectrum of a new beam and there is no need to separate them. The passband of the new beam already covers both positive and negareas. Therefore, the spectrum-patching step is unnecative essary. However, with two channels the signal-to-noise ratio is improved. The advantage of the modulated scene method is that it does not need phase shifters if the phase center is stable during scanning. The advantage of the modulated beam method is that only one channel of the complex correlator is needed. IV. SIMULATION RESULT Here, the performances of the modulated scene and modulated beam methods are tested with simulated data. A. Simulation Setup The two antennas are arranged in a T configuration (Fig. 1). The antennas have an aperture length of 450 mm and width of 5 mm with a uniform sensitivity weighting. The millimeter wave is assumed to contain a single frequency of 200 GHz (1.5-mm wavelength). The beam of each antenna can then be considered as a sinc function in the length direction and constant in the width direction due to the small size of the field of view 7 7 . The point spread function of the system is the multiplication of the two orthogonal fan beams (Fig. 9(a), all image sizes in this 2 or 349 349 pixels) and it can be expressed paper are 2 as

(33)

Note that the point spread function is real, but not positively defined. The resolution of this beam is approximately 0.3 in each direction. The phase center of the antenna is assumed to be at the middle of its aperture. Ignoring the boundaries of the antennas, the baseline length is one-half of the aperture length. The real (cosine) and imaginary (sine) fringes are shown in Fig. 9(b) and (c), respectively. The real and imaginary fringe-modulated beams are shown in Fig. 9(d) and (e), respectively. B. Modulated Scene Method A model scene was used, which contains many squares with various sizes and brightness values, as shown in Fig. 10(a). The brightness of the background is 20 dB below that of the brightest squares. There are also two point targets in the scene designed to demonstrate the point spread function of the image. The two point targets are too small to be seen in Fig. 10(a), but they can be seen in the formed images. The image of the scene formed with the beam in (35) [see Fig. 9(a)] without the fringe (Mills Cross image) is shown in Fig. 10(b). Zero mean Gaussian noise is added to the image. The signal-to-noise ratio (the brightness of the brightest square divided by the standard deviation of the noise) is 0 dB. The squares are not clearly distinguishable in the image. The image also has a speckle appearance due to the beam not been positively defined. The deconvolved Mills Cross image with the Wiener filter (22) shows improved resolution and signal-to-noise ratio [see Fig. 10(c)]. When the fringe is included, the real and imaginary parts of the modulated scene are shown in Fig. 10(d) and (e), re) spectively. The magnitude of the complex image ( of the complex modulated scene is shown in Fig. 10(f) with 0-dB Gaussian noise. The resolution of the image demonstrated by the two point targets is similar to that of the original Mills Cross image, but the speckle pattern changes due to the fringe. The “deconvolved,” “deconvolved and demodulated,” “deconvolved, demodulated and spectrum patched,” and the “demodulated and spectrum patched” images of the original

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Fig. 10. (a) Scene model I. (b) Mills Cross image of the scene with noise. (c) Deconvolved Mills Cross image. (d) Cosine modulated scene. (e) Sine modulated scene. (f) Magnitude image of the modulated scene. (g) Modulated scene image after deconvolution. (h) Modulated scene image after deconvolution and demodulation. (i) Modulated scene image after deconvolution, demodulation, and spectrum patching. (j) Modulated scene image after demodulation and spectrum patching. (k) Magnitude of the modulated beam image. (l) Image formed with the cosine modulated beam after deconvolution. (m) Image formed with the sine modulated beam after deconvolution. (n) Modulated beam image after deconvolution and spectrum patching. (o) Modulated beam image after spectrum patching.

complex image are shown in Fig. 10(g)–(j), respectively. The resolution in the “deconvolved” and the “deconvolved and demodulated” images are similar to that in the deconvolved Mills Cross image. The “deconvolved, demodulated and spectrum patched” image has much better resolution in the vertical direction due to the fringe and spectrum patching. The “demodulated and spectrum patched” image also shows better resolution in the vertical direction, but the resolution and signal-to-noise ratio are not as good as those in the “deconvolved, demodulated and spectrum patched” image due to the lack of deconvolution. The “demodulated and spectrum patched” image demonstrates the effect of demodulation and spectrum patching without deconvolution. Similar results are shown for a second scene model in Fig. 11(g)–(j). The second scene model contains a large target with two point targets. The background brightness is also 20 dB below the brightness of the large target. The Gaussian noise level added is also 0 dB.

The three images, i.e., Fig. 10(l)–(n), would be very similar if the deconvolution process were ideal. This is because their spectrums roughly have the same nonzero area (bandwidth). However, because the baseline length is exactly one-half of the aperture length, the spectrum of the sine modulated beam is zero due to the cancellation of the two shifted specwhen trums [similar to (32)] (36)

C. Modulated Beam Method

The spectrum values of the scene at cannot be recovered by the deconvolution process. Therefore, there is no zerofrequency component in Fig. 10(m), and this property causes the differences between the three images. The spectrum patched, but not deconvolved, original complex image [see Fig. 10(k)] is shown in Fig. 10(o), which is noisier than those deconvolved images. All of these processed images [see Fig. 10(l)–(o)] have an improved vertical resolution compared with the original image [see Fig. 10(k)]. This is because the modulated beams have a wider bandwidth in the vertical direction. Similar results are shown for scene model II in Fig. 11(k)–(o).

The magnitude image of scene model I formed with the complex modulated beam is shown in Fig. 10(k). It is similar to the original image of the modulated scene method [see Fig. 10(f)]. The images formed with cosine and sine modulated beams after deconvolution are shown in Fig. 10(l) and (m), respectively. The magnitude image formed with the complex modulated beam after deconvolution and spectrum patching is shown in Fig. 10(n).

V. SUMMARY In this paper, the modulated scene and modulated beam methods have been proposed to manage the fringe in a cross-correlating millimeter-wave imaging system with two fan beam antennas. These methods were successfully tested using simulated data. The modulated scene method keeps the fringe fixed with the scene during scanning. A complex image of the complex fringe-

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Fig. 11. Same images as those in Fig. 10, but for scene model II.

modulated scene is formed. The spatial spectrum of the complex image is a shifted version of the spectrum of the original scene that passes the beam filter. Therefore, it is a distorted image of the original scene. To mitigate this problem, the complex image is demodulated to shift its spectrum back to the correct position. A spectrum-patching method is then used to extend the spectrum bandwidth using the symmetric property of the spectrum of a real scene. The complex image is first deconvolved with the Wiener filter to improve its resolution and signal-to-noise ratio before demodulation and spectrum patching. The advantage of the modulated scene method is that a phase shifter is not needed. The disadvantage is that both outputs from the complex correlator are necessary. The modulated beam method makes the fringe scan together with the beam to form a modulated beam. This is accomplished by dynamically adjusting the phase of the LO in the receiver. Three images, formed with the cosine modulated beam, sine modulated beam, and complex modulated beam can be deconvolved using the beam that is used to form it and any one of the results can be used as the processed image. Note that the complex image also needs spectrum patching. The advantage of the modulated beam method is that only a single output from the complex correlator is needed and there is no need for demodulation. The disadvantage of this method is that a phase shifter is needed to scan the fringe together with the beam.

ACKNOWLEDGMENT The authors thank the anonymous reviewers for their constructive comments and G. Hislop, Wireless Technologies Laboratory, Information and Communication Technologies Center, Commonwealth Scientific and Industrial Research Organisation, Marsfield, Australia, for helpful discussions.

REFERENCES [1] K. Watabe, K. Shimizu, M. Yoneyama, and K. Mizuno, “Millimeter-wave active imaging using neural networks for signal processing,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1512–1516, May 2003. [2] L. Yujiri, H. Agravante, M. Biedenbender, G. S. Dow, M. Flannery, S. Fornaca, B. Hauss, R. Johnson, R. Kuroda, K. Jordan, P. Lee, D. Lo, B. Quon, A. Rowe, T. Samec, M. Shoucri, K. Yokoyama, and J. Yun, “Passive millimeter-wave camera,” in Proc. Int. Soc. Opt. Eng., 1997, vol. 3064 Passive Millimeter-Wave Imag. Technol., pp. 15–22. [3] R. Smith, B. Sundstrom, and B. Belcher, “Radiometric one second camera (ROSCAM) airborne evaluation,” in Proc. Int. Soc. Opt. Eng., 1999, vol. 3703 Passive Millimeter-Wave Imag. Technol. III, pp. 2–12. [4] R. Appleby, R. N. Anderton, S. Price, N. A. Salmon, G. N. Sinclair, J. R. Borrill, P. R. Coward, P. Papakosta, A. H. Lettington, and D. A. Robertson, “Compact real-time (video rate) passive millimetre-wave imager,” in Proc. Int. Soc. Opt. Eng., 1999, vol. 3703 Passive Millimeter-Wave Imag. Technol. III, pp. 13–19. [5] D. D. Ferris, Jr. and N. C. Currie, “Overview of current technology in MMW radiometric sensors for law enforcement applications,” in Proc. Int. Soc. Opt. Eng., 2000, vol. 4032 Passive Millimeter-Wave Imag. Technol. IV, pp. 61–71. [6] P. Moffa, L. Yujiri, K. Jordan, R. Chu, H. Agravante, and S. Fornaca, “Passive millimeter wave camera flight tests,” in Proc. Int. Soc. Opt. Eng., 2000, vol. 4032 Passive Millimeter-Wave Imag. Technol. IV, pp. 14–21. [7] C. Martin, J. Lovberg, S. Clark, and J. Galliano, “Real time passive millimeter-wave imaging from a helicopter platform,” in Proc. Int. Soc. Opt. Eng., 2000, vol. 4032 Passive Millimeter-Wave Imag. Technol. IV, pp. 22–28. [8] L. Yujiri, M. Shoucri, and P. Moffa, “Passive millimeter-wave imaging,” IEEE Micro., vol. 4, pp. 39–50, Sep. 2003. [9] S. G. Hay, J. Archer, G. Timms, and S. L. Smith, “A beam-scanning dual-polarized fan-beam antenna suitable for millimeter wavelengths,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2516–2524, Aug. 2005. [10] J. W. Archer, R. Lai, R. Grundbacher, M. Barsky, R. Tsai, and P. Reid, “An indium phosphide MMIC amplifier for 180–205 GHz,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 1, pp. 4–6, Jan. 2001. [11] J. W. Archer and M. G. Shen, “176–200 GHz receiver module using indium phosphide and gallium arsenide MMICs,” Microw. Opt. Technol. Lett., vol. 42, no. 6, pp. 458–462, Dec. 2004. [12] A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio Astronomy. New York: Wiley, 2001.

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Yue Li (M’90–SM’04) received the B.S. degree in physics from Peking University, Beijing, China, in 1982, the M.S. degree in physics from the Institute of Acoustics, Chinese Academy of Sciences, Beijing, China, in 1985, and the Ph.D. degree in ultrasonics from Drexel University, Philadelphia, PA, in 1990. From 1990 to 1992, he was a Research Post-Doctoral Fellow with Drexel University. Since 1992, he has been with the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Information and Communication Technologies Centre, Marsfield, N.S.W., Australia, where he is currently a Principal Research Scientist.

John W. Archer (F’90) was born in Sydney, Australia, in 1950. He received the B.Sc., B.E.(Hons. I), and Ph.D. degrees from the University of Sydney, Sydney, N.S.W., Australia, in 1970, 1972, and 1977, respectively. From 1977 to 1984, he was with the National Radio Astronomy Observatory, where his research focused on millimeter-wave systems for radio astronomy. In 1984, he joined the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Marsfield, N.S.W., Australia. He is a member of the Editorial Board of Microwave and Optical Technology Letters. He has achieved wide international recognition for his contributions to millimeter-wave receiver technology.

Grahame Rosolen received the Bachelor of Science and Bachelor of Engineering degrees from the University of Sydney, Sydney, N.S.W., Australia, in 1987 and 1989, respectively, and the Ph.D. degree from the University of Cambridge, Cambridge, U.K., in 1992. He was with Austek Microsystems, where he was involved with the design of custom very large scale integration (VLSI) integrated circuits. He was also with the University of New South Wales and Cambridge Instruments, where he developed advanced electron beam lithography instruments for fabricating and imaging nanostructures. He is currently with the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Marsfield, N.S.W., Australia. His research interests include developing specialized electron beam equipment, nanotechnology, and imaging instrumentation.

Stuart G. Hay received the B.E. and Ph.D. degrees in electrical engineering from the University of Queensland, Brisbane, Qld., Australia, in 1985 and 1994, respectively. From 1986 to 1989, and since 1994, he has been with the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Marsfield, N.S.W., Australia, where he is currently a Principal Research Scientist. He has various interests in electromagnetics and antennas including analysis and design techniques for reflectors and feeds, antenna arrays, and shaped-beam, multibeam, and beam-scanning antennas.

Greg P. Timms (M’05) received the B.Sc. (Hons.) and Ph.D. degrees in physics from the University of Sydney, Sydney, N.S.W., Australia, in 1993 and 1997, respectively. In 1997, he joined the Australian Nuclear Science and Technology Organization, where he investigated the environmental impacts of mining, focusing on the physical transport of reactants and pollutants within mine wastes. Since 2002, he has been with the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Marsfield, N.S.W., Australia, where he is engaged in research on millimeter-wave imaging and gigabit wireless networks.

Y. Jay Guo (SM’96) received the Ph.D. degree in antennas and scattering from Xian Jiaotong University, Xi’an City, China, in 1987, and the Ph.D. degree from the University of Bradford, Bradford, U.K., in 1997. From 1989 to 1994, he was a Research Fellow with the University of Bradford, Bradford, U.K. In 1994, he became a Senior Research Fellow and then Lecturer with the University of Bradford. From July 1997 to 2005, he was a Principal Engineer with the Fujitsu Europe Telecom Research and Development Centre. Since 2005, he has been the Director of the Wireless Technologies Laboratory, Information and Communication Technologies Centre (CSIRO), Marsfield, N.S.W., Australia.

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Letters Corrections to “CMOS Low-Noise Amplifier Design Optimization Techniques”

Accounting for those errors, the real and imaginary parts of the source impedance should be written as

Nam-Jin Oh

 Papers [1] and [2] corrected several equations in [3]–[5] on the analysis of low-noise amplifiers. However, there are more errors in [3]–[5] that require further correction. Based on [6], the sign of the drain and gate noise correlation factor c should be rewritten as

ig i3d i2g i2d

c=

 00:395j:

=1+

1

2 Rs gm

1 gd0 1

1+s

2

Cgs (Lg + Ls )

0(sCgsRs )

2

1

0 jcj 5

 1 0 j c j 5

2

2

0 5 1 0 jcj2 gm (sCgs)2 Rs2 0 s2 (Lg + Ls ) : The optimum noise impedance without an external capacitor is given by

0 0 Zopt = 1=Yopt =



0 j c j2 ) + j

1

!Cgs   (1 0 jcj2 ) + 5

1



5 2

(1

 !Cgs



5

(1

0 jcj 5

0 jcj 5

0jcj2)+ j CCgst 0 jcj 5

2  (1 0jcj2 )+ Ct 0 jcj 5 Cgs

0jcj2)

2 !Cgs   (1 0jcj2 ) + Ct 0 jcj 5 Cgs j C t 0  jc j  Cgs 5 2 !Cgs   (1 0jcj2 )+ Ct 0 jcj  5 Cgs 5

 5 2

2

= Re[Zs ]:

0 sLs = Im[Zs ]:

[1] J. Lu and F. Huang, “Comments on ‘CMOS low-noise amplifier design optimization techniques’,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, p. 3155, Jul. 2006. [2] T.-K. Nguyen, C.-H. Kim, G.-J. Ihm, M.-S. Yang, and S.-G. Lee, “Authors’ reply,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 3155–3156, Jul. 2006. [3] T.-K. Nguyen, C.-H. Kim, G.-J. Ihm, M.-S. Yang, and S.-G. Lee, “CMOS low-noise amplifier design optimization techniques,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1433–1442, May 2004. [4] T.-K. Nguyen, N.-J. Oh, C.-Y. Cha, Y.-H. Oh, G.-J. Ihm, and S.-G. Lee, “Image-rejection CMOS low-noise amplifier design optimization techniques,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 538–547, Feb. 2005. [5] T.-K. Nguyen, N.-J. Oh, V.-H. Le, and S.-G. Lee, “A low-power CMOS direct conversion receiver with 3-dB NF and 30-kHz flicker-noise corner for 915-MHz band IEEE 802.15.4 ZigBee standard,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 735–741, Feb. 2006. [6] D. K. Shaeffer and T. H. Lee, “Comment on corrections to ‘A 1.5-V, 1.5-GHz CMOS low noise amplifier’,” IEEE J. Solid-State Circuits, vol. 41, no. 10, pp. 2359–2359, Oct. 2006.

2

and the optimum noise impedance with external capacitor is given by

0 0 Zopt = 1=Yopt =

(1

REFERENCES

The noise factor F in [3]–[5] should be corrected as

F



5

 5

2

:

Manuscript received December 22, 2006; revised March 22, 2007. The author was with the Department of Engineering, Information and Communications University, Daejeon 305-732, Korea, and also with the Auto-ID Laboratory, Fudan University, Shanghai 201203, China. He is now with the Department of Electronics Engineering, Chungju National University, Chungju, Chungbuk 380-702, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.896818 0018-9480/$25.00 © 2007 IEEE

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Corrections to “A Low-Power CMOS Direct Conversion Receiver With 3-dB NF and 30-kHz Flicker-Noise Corner for 915-MHz Band IEEE 802.15.4 ZigBee Standard” Nam-Jin Oh Papers [1] and [2] describe the low-flicker noise direct conversion receiver for a 915-MHz ZigBee receiver. According to [3], the flicker noise corner frequency is defined as the intersection of the 1=f noise (inversely proportional to frequency) and the thermal noise (flat in frequency). Thus, the flicker noise corner frequency in Fig. 1 is 60 kHz, not 30 kHz, as described in the measured NF results in [1] and [2]. Thus, the title of [1] and [2] needs to be corrected to “A Low-Power CMOS Direct Conversion Receiver With 3-dB NF and 60-kHz Flicker-Noise Corner for 915-MHz Band IEEE 802.15.4 ZigBee Standard.” Also, [1, Table V] needs to be revised as shown here in Table I.

Fig. 1. Measured and simulated noise figure in [1] and [2]. TABLE I SIMULATED AND MEASURED RF RECEIVER PERFORMANCES

Manuscript received December 21, 2006; revised March 4, 2007. The author was with the Department of Engineering, Information and Communications University, Daejeon 305-732, Korea, and also with the Auto-ID Laboratory, Fudan University, Shanghai 201203, China. He is now with the Department of Electronics Engineering, Chungju National University, Chungju, Chungbuk 380-702, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

REFERENCES

Digital Object Identifier 10.1109/TMTT.2007.897837

[1] T.-K. Nguyen, N.-J. Oh, V.-H. Le, and S.-G. Lee, “A low-power CMOS direct conversion receiver with 3-dB NF and 30-kHz flicker-noise corner for 915-MHz band IEEE 802.15.4 ZigBee standard,” IEEE Tran. Microw. Theory Tech., vol. 54, no. 2, pp. 735–741, Feb. 2006. [2] T.-K. Nguyen, S.-G. Lee, and D.-K. Kang, “A 900 MHz CMOS RF direct conversion receiver front-end with 3-dB NF and 30-kHz 1=f noise corner,” in Asian Solid-State Circuits Conf., Nov. 2005, pp. 349–352. [3] B. Razavi, Design of Analog CMOS Integrated Circuits. New York: McGraw-Hill, 2001.

0018-9480/$25.00 © 2007 IEEE

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Digital Object Identifier 10.1109/TMTT.2007.901541

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Digital Object Identifier 10.1109/TMTT.2007.901538

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