[Journal] IEEE Transactions on Microwave Theory and Techniques. Vol. 64. No 9

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SEPTEMBER 2016

VOLUME 64

NUMBER 9

IETMAB

(ISSN 0018-9480)

PAPERS

EM Theory and Analysis Techniques Internal Network Boundary Condition Incorporated in TLM for Efficiently Modeling Thin Layer of Periodic Structures ............................................................... Y. Xiong, W. Che, D. Wang, Y. Han, and G. Shen Skin Effect Modeling in Conductors of Arbitrary Shape Through a Surface Admittance Operator and the Contour Integral Method ................................................................................... U. R. Patel and P. Triverio A Fully Coupled Nonlinear Scheme for Time-Domain Modeling of High-Power Microwave Air Breakdown ......... ............................................................................................................... S. Yan and J.-M. Jin Devices and Modeling Adaptive Decoupling Using Tunable Metamaterials ....... L. Zhang, S. Zhang, Z. Song, Y. Liu, L. Ye, and Q. H. Liu Compact Cascaded-Spiral-Patch EBG Structure for Broadband SSN Mitigation in WLAN Applications ............... .............................................................................................. C.-K. Shen, S. Chen, and T.-L. Wu Rapid Microwave Design Optimization in Frequency Domain Using Adaptive Response Scaling ....................... ..................................................................................................... S. Koziel and A. Bekasiewicz A New GaN HEMT Equivalent Circuit Modeling Technique Based on X-Parameters ..................................... ............................................................. R. Essaadali, A. Jarndal, A. B. Kouki, and F. M. Ghannouchi Iterative Learning Control for RF Power Amplifier Linearization ............................................................. ................................................................ J. Chani-Cahuana, P. N. Landin, C. Fager, and T. Eriksson Passive Circuits Integration of Interresonator Coupling Structures With Applications to Filter Systems With Signal Route Selectivity ................................................................................ B. Koh, B. Lee, S. Nam, T.-H. Lee, and J. Lee Real-Time Feedback Control System for Tuning Evanescent-Mode Cavity Filters .... M. Abu Khater and D. Peroulis Chirping Techniques to Maximize the Power-Handling Capability of Harmonic Waveguide Low-Pass Filters ......... ........................................................................................................................... F. Teberio, I. Arregui, A. Gomez-Torrent, I. Arnedo, M. Chudzik, M. Zedler, F.-J. Görtz, R. Jost, T. Lopetegi, and M. A. G. Laso Study of a New Planar-Type Balun Topology for Application in the Design of Balun Bandpass Filters ................ ............................................................................... J. Wang, F. Huang, L. Zhu, C. Cai, and W. Wu Double Dielectric Slab-Loaded Air-Filled SIW Phase Shifters for High-Performance Millimeter-Wave Integration .... .......................................................... F. Parment, A. Ghiotto, T.-P. Vuong, J.-M. Duchamp, and K. Wu

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Hybrid and Monolithic RF Integrated Circuits Codesign of Ka-Band Integrated Limiter and Low Noise Amplifier ................. P. Mahmoudidaryan and A. Medi Fundamental Limitations on the Output Power and the Third-Order Distortion of Balanced Mixers and Even Harmonic Mixers ............................................................. J. Hashimoto, K. Itoh, M. Shimozawa, and K. Mizuno InGaAs MMIC SPST Switch Based on HPF/LPF Switching Concept With Periodic Structure .......................... ....................................................................................... H. Mizutani, R. Ishikawa, and K. Honjo Frequency-Selective Limiters Utilizing Contiguous-Channel Double Multiplexer Topology ............................... ................................................................................................. E. J. Naglich and A. C. Guyette Instrumentation and Measurement Techniques Noncontact Measurement of Complex Permittivity of Electrically Small Samples at Microwave Frequencies ......... J. Dong, F. Shen, Y. Dong, Y. Wang, W. Fu, H. Li, D. Ye, B. Zhang, J. Huangfu, S. Qiao, Y. Sun, C. Li, and L. Ran Design and Characterization of Microwave Cavity Resonators for Noninvasive Monitoring of Plant Water Distribution ...................................................................... V. A. Sydoruk, F. Fiorani, S. Jahnke, and H.-J. Krause On-Wafer Measurement Errors Due to Unwanted Radiations on High-Q Inductors ........................................ ......................................................................... O. Bushueva, C. Viallon, A. Ghannam, and T. Parra Characterization for Multiharmonic Intermodulation Nonlinearity of RF Power Amplifiers Using a Calibrated Nonlinear Vector Network Analyzer ......................................................................................................... .................................. Y. Zhang, X. Guo, Z. He, H. Huang, J. Huang, M. Nie, L. Wang, A. Yang, and Z. Lu RF Systems and Applications A Radio Channel Sounder for Mobile Millimeter-Wave Communications: System Implementation and Measurement Assessment ......................................... P. B. Papazian, C. Gentile, K. A. Remley, J. Senic, and N. Golmie Highly Integrated Switched Beamformer Module for 2.4-GHz Wireless Transceiver Application ........................ .......................................................................................................... W.-T. Fang and Y.-S. Lin Differential Rectifier Using Resistance Compression Network for Improving Efficiency Over Extended Input Power Range ............................................................................................... Q. W. Lin and X. Y. Zhang MIMO FMCW Reader Concept for Locating Backscatter Transponders ..................................................... ................................................................. S. Appel, D. Berges, D. Mueller, A. Ziroff, and M. Vossiek Potential and Practical Limits of Time-Domain Reflectometry Chipless RFID .............................................. ................................................................ M. Pöpperl, A. Parr, C. Mandel, R. Jakoby, and M. Vossiek Toward a Reliable Chipless RFID Humidity Sensor Tag Based on Silicon Nanowires ..................................... ................................................................................. A. Vena, E. Perret, D. Kaddour, and T. Baron Reliable Orientation Estimation of Vehicles in High-Resolution Radar Images ............................................. ...................................................................... F. Roos, D. Kellner, J. Dickmann, and C. Waldschmidt Self-Calibration of a 3-D-Digital Beamforming Radar System for Automotive Applications With Installation Behind Automotive Covers ................................................... M. Harter, J. Hildebrandt, A. Ziroff, and T. Zwick System for Bulk Dielectric Permittivity Estimation of Breast Tissues at Microwave Frequencies ........................ ........................................................................................................ J. Bourqui and E. C. Fear Simultaneous Realization of Frequency Multiplication and Single Sideband Modulation by Exploiting Nonlinear Birefringent Effect .............................................................................. D. Feng, J. Sun, and H. Xie A Reconfigurable Photonic Microwave Mixer Using a 90° Optical Hybrid ......................... Z. Tang and S. Pan

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Digital Object Identifier 10.1109/TMTT.2016.2594858

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

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Internal Network Boundary Condition Incorporated in TLM for Efficiently Modeling Thin Layer of Periodic Structures Ying Xiong, Wenquan Che, Senior Member, IEEE, Desong Wang, Ye Han, and Guangxu Shen

Abstract— It is a problem when modeling the thin layer of periodic structures with numerical method by directly discretizing the inner part, because the fine mesh applied to simulate the thin layer would result in great computational data. The internal network boundary condition (INBC) incorporated in the transmission line matrix (TLM) scheme is proposed in this paper, to avoid the directly discretization of the thin layer and achieve a high accuracy. The thin layer of periodic structure is regarded as an easily analyzed two-port network. Vector fitting approach is used to approximate the network parameters into a series of rational expressions with either real term or complex conjugate pole–residue pairs. The INBC equation with discrete-time form is derived and then incorporated in TLM scheme by introducing the approximate pole–residue pairs to the TLM update equation. Compared with the extremely fine-discretized TLM unit cell used to deal with the thin layer by the conventional TLM scheme, a much larger coarse TLM unit cell is accurate enough to be used to discretize the structure by the proposed method. Compared with the conventional TLM scheme, substantial savings in computational storage are achieved. A common frequency selective surface (FSS) structure is first analyzed for the validation of the method. Good agreement in scattering characteristics is observed in frequency domain between the proposed method and simulation by HFSS software. To further demonstrate the validity of the proposed method, an aperture-coupled resonatorbased FSS is designed and fabricated. Good agreement among the S-parameters from the proposed method, simulation, and measurement is observed, verifying the accuracy of the proposed method. Index Terms— Frequency selective surface (FSS), internal network boundary condition (INBC), network theory, transmission line matrix (TLM) scheme, vector fitting (VF). Manuscript received February 25, 2016; revised June 25, 2016 and July 4, 2016; accepted July 4, 2016. Date of publication August 2, 2016; date of current version September 1, 2016. This work was supported by the 2012 Distinguished Young Scientist Program awarded by the National Natural Science Foundation Committee of China (61225001). Y. Xiong was with the Department of Communication Engineering, Nanjing University of Sciecne and Technology, Nanjing 210094, China. She is now with the China North Vehicle Research Institute, Beijing 100072, China (email: [email protected]). W. Che, Y. Han, and G. Shen are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; [email protected]; [email protected]). D. Wang is with the State Key Laboratory of Millimeter Waves and the Partner Laboratory, City University of Hong Kong, Hong Kong (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.2016.2592528

Fig. 1. TLM elementary unit cell. (a) Two pairs of tangential electric and magnetic fields sampled in each cube surface. (b) Wave amplitudes denoted by ai and bi . (c) Condensed symmetric TLM node in [6].

I. I NTRODUCTION

T

HE transmission line matrix (TLM) method has been usually used to numerically analyze the electromagnetic (EM) wave propagation in any kind of computational space [1]–[10]. It can be conceptually considered as an equivalent circuit approach to analyze EM field problems through replacing the spatial simulation space of a problem by a grid of transmission lines (TLs) [1]–[10]. The elementary TLM unit cell is shown in Fig. 1(a). There are two pairs of perpendicular tangential electric and magnetic field components sampled in the center of each cube surface. Fig. 1(b) gives the tangential field samples which denoted by incident waves ai and scattered waves bi . Therefore, in the TLM unit cell, the total 12 electric field components and 12 magnetic field components can be represented by incident waves with amplitudes a1 , . . . , a12 and scattered waves with amplitudes b1 , . . . , b12 . Fig. 1(c) shows the circuit representation of a 12-port TLM cell. However, most commonly, extra stubs are required to deal with modeling inhomogeneous materials, resulting in as many as 18 ports [1], [2]. In many engineering applications, such as reflectors or radar system, frequency selective surfaces (FSSs) have been widely used to achieve good frequency selective characteristics [11]–[14]. A limitation in the wider numerical study of such materials is the difficulty in constructing suitable efficient numerical models for such large-scale periodic structures, especially when modeling the fine part of the materials such as the thin layer with the layer thickness much smaller than the wavelength. If a regular TLM mesh with uniform size is used to construct the whole problem space, a very fine subwavelength resolution should be applied to satisfy the accuracy

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Fig. 2. Interfaces used to join the meshes of different spatial resolutions. (a) Triangular interface in 2-D problem space [15]. (b) Interface in 3-D problem space [16], [17].

requirement for modeling the fine part. However, large-scale simulation in EM wave propagation is normally based on a tenth of a wavelength resolution. Therefore, although a finer resolution is theoretically possible, increased computation data require huge memory up to several orders of magnitude. Some approaches can be used to solve such problems [7], [13]–[21]. 1) Constructing a 2-D interface arranged with triangular elements, TLM meshes with different spatial resolutions along each side of the interface could be connected [15], as shown in Fig. 2(a). However, this approach is proposed only in 2-D problem space. 2) Introducing nonuniform meshes to discretize the computational spaces by multigrid-subgridding TLM formulations [13], [14]. The idea is to introduce an interface boundary for adequately modeling the regions with TLM meshes of different spatial resolutions, as shown in Fig. 2(b). Although the uniform fine TLM mesh for modeling the whole problem space is avoided, the overall number of cells is still large, due to the fact that fine resolution is required for the discretization of the fine parts and thus leads to a substantial number of discretized cells by using fine mesh to deal with those fine parts. 3) A digital filtering (DF) network is used as a boundary to incorporate in TLM scheme to analyze the fine parts [18]–[21]. The method allows one to use a regular coarse mesh to simulate the whole problem space through introducing a boundary that includes the EM scattering characteristics of the fine parts. The aim is to convert the frequency-dependent properties of the parts into time domain by the DF technique, whose signal flow diagram was illustrated in [18]–[21]. Circuit network theory has been widely used also for EM wave propagation analysis [6], [21]–[24]. In this paper, the fine part of the periodic structure, such as the thin layer, is regarded as a short TL connected with the TLM TL to form a two-port circuit network, while the network parameters with whole information of the fine part are then regarded as an internal network boundary condition (INBC), which could provide detailed descriptions to the fine part and be incorporated in TLM scheme for accurate simulation. An accurate and efficient approach to deal with the thin layer of periodic structures is proposed in this paper. The frequency-dependent network parameter function is approx-

Fig. 3.

Derivation procedure of the proposed method.

imately matched as a model with pole–residue pairs by vector fitting (VF) method, and thus generates an internal network boundary including the scattering characteristics of the thin layer. Laplace transform and time differential methods can be used to obtain a discrete-time equation of INBC, which can then be directly incorporated in TLM scheme to update the electric fields on the network boundary. Compared with earlier work, this approach has the following advantages: 1) the complexity of the recursive convolution equations is avoided; 2) the computation time can be greatly reduced, since the fine resolution of layer thickness will not be a limitation to the unit of the discretization grid; 3) the approach is a considerably general tool to easily deal with thin layers of periodic structures or low-profile structures; and 4) the thickness of the thin layer is no longer limited to any restricted requirements. Moreover, the material consisting of the thin layer is also no longer restricted, and it could be either metallic, dielectric, lossy, or lossless material. This paper is organized as follows. A derivation procedure of the proposed method is first shown in Fig. 3. Next, the frequency-dependent scattering parameters matched as pole– residue pairs by VF method will be introduced in Section II. The discrete-time INBC formulations are derived and incorporated in TLM, which will be discussed in Section III. A common FSS structure is set as an example to demonstrate the validity of the proposed method, and the scattering characteristics calculated by INBC-TLM are presented and will be discussed in Section IV. Furthermore, an aperture-coupled resonator (ACR) FSS is designed and fabricated, and the S-parameters obtained from the proposed method, simulation, and measurement are compared and discussed in Section IV. II. A NALYSIS OF TL N ETWORK As we know, TLM scheme is based on the TL and circuit network theory [1]–[7], and allows the TL network to be easily incorporated in the TLM mesh system. Therefore, considering the fine part of a structure such as a thin layer, whose thickness

XIONG et al.: INBC INCORPORATED IN TLM FOR EFFICIENTLY MODELING THIN LAYER OF PERIODIC STRUCTURES

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Fig. 5. Interface of two adjacent TLM nodes. (a) Compact nodes. (b) Separated nodes.

Fig. 4. TLM cells connected with each other to discretize the problem space.

is very small, huge computation data might be generated by using a fine mesh to discretize the whole problem space. A two-port TL network of the thin layer is thus introduced and can be incorporated in TLM scheme to replace the fine spatial discretization of the fine part. In this section, the analysis of TL network will be introduced from two aspects. At first, the scattering matrix of two-port TL network will be derived from the chain (ABCD) matrix by circuit theory. Next, rational approximations of the frequency responses of a structure with a thin copper film are completed by using the VF method, and the accuracy of the rational approximation method can thus be verified and will be discussed. A. Scattering Matrix of Two-Port TL Network In the typical TLM scheme, the cube cells are connected with each other to spatially discretize the problem space, as shown in Fig. 4. The 12 voltage pulses scatter along the node arm of a cell and then pass into adjacent cells at each time step. Considering a boundary of two adjacent nodes along the y-axis shown in Fig. 4(a), with no other different mediums inside the mesh, the incident voltages of the node segment I at the next time step k + 1 are obtained by swapping with the reflected voltages at time step k in node segment II, and vice versa. Notice that the discussion is based on the boundary that is free to allow the reflected wave pass thoroughly to the adjacent node. If, for example, the boundary consists of a perfect electric conductor, the relationship between a and b on the boundary of two adjacent nodes should be presented as k+1 a ypx (I)

= −k b ypx (I) a (I) = −k b ypz (I) k+1 ypz

k+1 a ynx (II)

= −k b ynx (II) a (II) = −k b ynz (II) k+1 ynz

(1a) (1b)

where a and b are the incident and reflected pulses, and the three subscripts in the bottom-right of a or b indicate the direction of propagation (x, y, or z), the segment of the node along the negative or positive coordinate axis (n or p), and the polarization of the pulse (x, y, or z), respectively. Similarly, for different boundaries, such as PMC (RPMC = 1) and matched boundary (similar to absorbing boundary in FDTD, R M = 0), the expressions of the relationship between a and b on the boundary could be derived directly from the reflection coefficients.

Fig. 6.

TL-network for thin layer.

To give a clearer description and analysis of the boundary condition, two adjacent nodes are separated apart from each other for a better look of the impulses on the adjacent nodes, as shown in Fig. 5(b). Considering a more general boundary consisting of common materials, the expressions of the relationship between the incident and reflected pulses can be expressed by the scattering matrix S f (ω)   R11 (ω) T12 (ω) S f (ω) = (2) T21 (ω) R22 (ω) where the elements R11 (ω), T12 (ω), T21 (ω), and R22 (ω) represent the reflection coefficient of node I, transmission coefficient from node II to node I, transmission coefficient from node I to node II, and reflection coefficient of node II, respectively. Consider a case of a thin conductive layer placed between two dielectric materials, a coarse TLM cell (unit scale larger than thin layer thickness) could not be applied to accurately model the thin layer. To address this problem, the thin layer is treated as a short TL connected with the TL segments of the dielectric materials to construct a TL with a uniform TLM cell scale x. Fig. 6 gives a look of the TL-network, which consists of a conductive TL of intrinsic impedance Z 1 and thickness d1 placed in the middle of two TLs with intrinsic impedances Z 0 and Z 2 and thicknesses d0 and d2 , respectively. The chain matrix of a TL segment of length d can be found in [25, pp. 497–499]. Thus, the chain matrix of the TL segment of characteristic impedances Z 0 , Z 1 , and Z 2 can be represented as   cosh(γ0 d0 ) Z 0 sinh(γ0 d0 ) ld0 A = (3a) Z −1 sinh(γ0 d0 ) cosh(γ0 d0 )   0 cosh(γ1 d1 ) Z 1 sinh(γ1 d1 ) Ald1 = (3b) −1 Z sinh(γ1 d1 ) cosh(γ1 d1 )   1 cosh(γ2 d2 ) Z 2 sinh(γ2 d2 ) Ald2 = (3c) Z 2−1 sinh(γ2 d2 ) cosh(γ2 d2 )

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TABLE I E XTRACTED VALUES OF P OLES ( pn ) AND R ESIDUES (rn ) FOR THE S CATTERING C OEFFICIENTS OF TL-N ETWORK

TABLE II E XTRACTED VALUES OF d FOR THE S CATTERING C OEFFICIENTS OF TL-N ETWORK

where γ 0, γ 1, and γ 2 are the propagation coefficients of TL segments with the characteristic impedances Z 0, Z 1, and Z 2, respectively. The complete TL-network segment according to Fig. 6 is represented by the chain matrix AT L = Ald0 Ald1 Ald2 .

(4)

The scattering coefficients can be simply transferred from the ABCD matrix by two-port network theory in [25]. B. Rational Approximations of the Frequency Response Curves For the implementation of the INBC incorporated in the TLM scheme, the reflection and transmission coefficients must be expressed in discrete-time form. First, a numerical approximation (VF) method is used to approximate all the curves of scattering coefficients [26]–[33]. The approximation functions are in the form of Fs (s) =

N  n=1

rn +d s − pn

(5)

where s = j ω, N is the number of poles, rn and pn are the residues and the poles, in the form of either real quantities or complex conjugate pairs, and d is real. Consider a thin conductive layer of conductivity σ = 5.8 × 107 s/m and thickness d1 = 1 μm which is placed between free space and a material of relative permittivity εr = 2.2. The scattering coefficients are calculated and shown in Fig. 7(a)–(d), in which the curves approximated as the rational model of (5) with four poles are also given and compared with the analytical results. The corresponding parameters are presented in Tables I and II. It can be seen from Fig. 7(a)–(d) that all the curves agree well (actually, the approximate results will be more accurate if fitted with more pairs of pole–residues), thus verifying that the approximated rational model could accurately describe the scattering characteristics of the TL-network.

Fig. 7. Comparisons of scattering coefficients of TL-network between analytic result and approximation function. (a) Magnitudes of R11 and R22 . (b) Magnitude of T12 (T21 ). (c) Phases of R11 and R22 . (d) Phase of T12 (T21 ).

Notice that the approximate rational scattering functions are frequency-dependent, which should be necessarily converted into time-dependent form and applied in the TLM iteration equations. The detailed derivation procedures will be given in Section III.

XIONG et al.: INBC INCORPORATED IN TLM FOR EFFICIENTLY MODELING THIN LAYER OF PERIODIC STRUCTURES

III. INBC OF T HIN L AYER IN THE D ISCRETE T IME D OMAIN First of all, let us go back to the scattering matrix (2) and the complete relationships of the incidence and reflection impulses on the boundary [see Fig. 5(a) and (b)], which are expressed as      R11 (ω) T12 (ω) k+1 a ypx (I) k b ypx (I) = (6a) T21 (ω) R22 (ω) k+1 a ynx (II) k a ynx (II)      R11 (ω) T12 (ω) k+1 a ypz (I) k b ypz (I) = . (6b) T21 (ω) R22 (ω) k+1 a ynz (II) k a ynz (II) Write (6a) and (6b) as a series of equations as k+1 a ypx (I)

= R11 (ω)k b ypx (I) + T12 (ω)k a ynx (II)

(7a)

= T21 (ω)k b ypx (I) + R22 (ω)k a ynx (II) k+1 a ypz (I) = R11 (ω)k b ypz (I) + T12 (ω)k a ynz (II)

(7b) (8a)

k+1 a ynx (II) k+1 a ynz (II)

= T21 (ω)k b ypz (I) + R22 (ω)k a ynz (II).

(8b)

The reflection and transmission coefficients R11 , T12 , T21 , and R22 of (6a) and (6b) can be approximated as rational functions with a form of (5), which can be expressed as R11 (s) = T12 (s) =

N  n=1 N  n=1

T21 (s) = R22 (s) =

N  n=1 N  n=1

rn,R11 + d R11 (s − pn,R11 )

(9a)

rn,T12 + dT12 (s − pn,T12 )

(9b)

rn,T21 + dT21 (s − pn,T21 )

(9c)

rn,R22 + d R22 (s − pn,R22 )

(9d)

where s = j ω. By introducing the state variables rn,R11 rn,T12 X n,R11 = X n,T12 = (s − p R11 ) (s − pT12 ) rn,T21 rn,R22 X n,R22 = X n,T21 = (s − pT21 ) (s − p R22 )

R11 (s) =

X n,R11 + d R11 T12 (s) =

N 

n=1

n=1

N 

N 

Equations (12a) and (12b) could be converted into equations in time domain by using s → ∂/∂t, and the equations are expressed as ∂ X n,R11 /∂t − pn,R11 X n,R11 = rn,R11

(13a)

∂ X n,T12 /∂t − pn,T12 X n,T12 = rn,T12 ∂ X n,T21 /∂t − pn,T21 X n,T21 = rn,T21

(13b) (13c)

∂ X n,R22 /∂t − pn,R22 X n,R22 = rn,R22 .

(13d)

Equations (13a)–(13d) can then be converted into discretetime equations, and by enforcing all the equations at k − 1, thus yielding k−2 k − X n,R X n,R 11 11

2t k−2 k X n,T 12 − X n,T 12 2t k−2 k X n,T − X n,T 21 21 2t k−2 k X n,R − X n,R 22 22

n=1

X n,T21 + dT21 R22 (s) =

k X n,T 12

2 k−2 + X n,T 12

k X n,T 21

2 k−2 + X n,T 21

k X n,R 22

2 k−2 + X n,R 22

− pn,T12 − pn,T21 − pn,R22

2t where t is the sampling time.  k X n,R 11

= 

k X n,T = 12

 k X n,T = 21

 (10a)

k X n,R = 22

1 + pn,R11 t 1 − pn,R11 t 1 + pn,T12 t 1 − pn,T12 t 1 + pn,T21 t 1 − pn,T21 t 1 + pn,R22 t 1 − pn,R22 t

2

= rn,R11

(14a)

= rn,T12

(14b)

= rn,T21

(14c)

= rn,R22

(14d)



 k−2 X n,R 11

+ 

 k−2 + X n,T 12



 k−2 X n,T + 21



2t 1 − pn,T12 t 2t 1 − pn,T21 t

 k−2 X n,R + 22

2t 1 − pn,R11 t

 rn,R11

X n,T12 + dT12

X n,R22 + d R22

rn,T12 (15b)



2t 1 − pn,R22 t

rn,T21 

(15c) rn,R22 . (15d)

After the expressions of state variables are obtained, the INBC in the discrete time domain can be expressed by inserting (11a) and (11b) into (7a) and (7b), which yields k+1 a ypx (I)

= k b ypx (I)

N 

k X n,R + d R11 k b ypx (I) 11

n=1

n=1

(11b)

(15a)



(10b)

(11a) T21 (s) =

k−2 k X n,R + X n,R 11 11

− pn,R11

By rearranging, we obtain

(9a)–(9d) can be written as N 

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+ k a ynx (II)

N 

k X n,T + dT12 k a ynx (II) 12

(16a)

n=1

and we have s X n,R11 − pn,R11 X n,R11 = rn,R11 s X n,T12 − pn,T12 X n,T12 = rn,T12 s X n,T21 − pn,T21 X n,T21 = rn,T21 X n,R22 − pn,R22 X n,R22 = rn,R22 .

k+1 a ynx (II) = k b ypx (I)

(12a)

N 

21

n=1

+ k a ynx (II) (12b)

k X n,T + dT21 k b ypx (I)

N  n=1

k X n,R + d R22 k a ynx (II). (16b) 22

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TABLE III E XTRACTED VALUES OF P OLES ( pn ) AND R ESIDUES (rn ) FOR THE S CATTERING C OEFFICIENTS OF L OOP -S HAPED FSS

Fig. 8. Typical FSS structure’s unit cell: dual-rectangle loop-shaped FSS structure unit cell.

TABLE IV E XTRACTED VALUES OF d FOR THE S CATTERING C OEFFICIENTS OF L OOP -S HAPED FSS Fig. 9. Geometric setup of the computational space. d is the distance from excitation plane to matched boundary. Dz is the distance from INBC plane to observation (reflection/transmission) plane.

By inserting (15a)–(15d) into (16a) and (16b), we can get the new time step expressions of incident waves a ypx and a ynx , which relates to the state variables X n,R11 , X n,T 12 , X n,T 21 , and X n,R22 at the previous two time steps k+1 a ypx (I)

 1 + pn,R11 t k−2 = k b ypx (I) X n,R 11 1 − pn,R11 t    2t + rn,R11 1 − pn,R11 t   1 + pn,T12 t k−2 + k a ynx (II) X n,T 12 1 − pn,T12 t    2t + rn,T12 1 − pn,T12 t + d R11 k b ypx (I) + dT12 k a ynx (II) (17a) k+1 a ynx (II)   1 + pn,T21 t k−2 = k b ypx (I) X n,T 21 1 − pn,T21 t    2t rn,T21 + 1 − pn,T21 t   1 + pn,R22 t k−2 + k a ynx (II) X n,R 22 1 − pn,R22 t    2t + rn,R22 1 − pn,R22 t + dT21 k b ypx (I) + dT21 k b ypx (I) + d R22 k a ynx (II). (17b) 

The discrete-time equations of (8a) and (8b) for the impulses with z-polarization can be derived similarly. IV. A PPLICATION OF THE INBC FOR THE A NALYSIS OF P ERIODIC S TRUCTURES As it is known, the thickness of a thin layer is usually much smaller than the working wavelength, and a spatial

discretization should be fine enough to ensure the stability. The INBC of the thin layer can be applied to describe the characteristic of the thin layer and incorporate it into TLM to iterate the impulses on the TL-network boundary for avoiding direct discretization of the thin layer. A. Dual-Rectangle Loop-Shaped FSS Structure A typical FSS structure is used as an example to verify the feasibility of the INBC. The structure parameters of the FSS structure are given in Fig. 8. INBC-TLM is applied to calculate the scattering coefficients of the FSS structure, with the geometry setup of computational space shown in Fig. 9. An incident pulse is incited with the form of Vi (t) = [cos(ωt) + j sin(ωt)]e−((t −t0)/τ ) . 2

(18)

A start-up delay of t0 = 0.8τ is set to confirm that the excitation waveform starts near zero. The pulse behaviors in time and frequency domains will be illustrated, respectively. Considering the loop-shaped FSS structure shown in Fig. 8, Tables III and IV give the sets of data for approximate scattering coefficients, which are applied in the INBC for iteration. Then, the computational space is constructed and the wave is excited by the incident pulse of (18). The reflected and transmitted impulses in the observation planes will be observed and recorded at each time step, and thus, the curves of reflection and transmission in time domain can be drawn, as shown in Fig. 10(a) and (b). The waves in time domain are then converted into frequency domain by using fast Fourier transfer, and the waveforms in frequency domain are shown in Fig. 11(a). Moreover, the reflection and transmission coefficients can be obtained by R=

ftran (s) f ref (s) , T = f inc (s) finc (s)

(19)

XIONG et al.: INBC INCORPORATED IN TLM FOR EFFICIENTLY MODELING THIN LAYER OF PERIODIC STRUCTURES

Fig. 10. Diagrams of waveforms in time domain for dual-rectangle loopshaped FSS structure. (a) Reflected wave. (b) Transmitted wave.

where s = j ω, f ref (s), f tran (s), and f inc (s) are the functions of reflection, transmission, and incident impulses in frequency domain, respectively. The curves of the coefficients are shown in Fig. 11(b). Two resonant frequencies occurring at 8.9 and 14.5 GHz can be observed in Fig. 11(b), respectively. The simulated scattering coefficients by HFSS (a commercial FEM-based tool) are also given and compared with the computation results in Fig. 11(b). It can be observed that good agreement is achieved, indicating the validity of the sets of data in Tables III and IV. From the curves of time-dependent and frequencydependent scattering waveforms, it can be seen that high stability and accuracy are achieved for the INBC-TLM method. Moreover, the computation efforts of the proposed method are given in Table V and compared with the effort cost by the conventional uniform-mesh TLM and software simulation. The mesh size for spatial discretization is set as 0.5 mm in INBC-TLM method, which is chosen to satisfy the stability criterion related to the wavelength at the highest frequency of interest λmin [1, p. 11]. However, the mesh size is set much smaller for the uniform-mesh TLM method, owing to size limitation for the thin layer with fine resolution. The numbers of computational data for the computation space by using INBC-TLM method are given also in Table V, which is several orders of magnitude less than that generated by uniform-mesh TLM method and implies less time steps and a higher computation speed. Table V also gives the computation time for the full-wave simulation; obviously, a short computation time of 10 min is spent by INBC-TLM for a

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Fig. 11. Diagrams of wave behaviors in frequency domain for dual-rectangle loop-shaped FSS structure. (a) Waveforms in frequency domain. (b) Scattering coefficients.

full-wave simulation; in the meantime, the estimated time for the uniform-mesh TLM method is up to about 116 days, which is unbearable for the EM computation. Otherwise, in fact, a single common computer could not directly deal with such big computation data arisen by the uniform-mesh TLM method unless some parallel-computing techniques are used. The computational effort of the HFSS software simulation is also given in Table V, in which the tetrahedral mesh is used, and the time step depends on the frequency sweep type. Here, the total tetrahedron number and the time steps are 49 403 units and 2100 steps, respectively. The total computation time is 36 min, which is longer than the proposed method. B. 60-GHz ACR FSS Structure To give a further validation of the accuracy of the proposed method, a 60-GHz ACR FSS structure with high-frequency selectivity was fabricated. Fig. 12 shows the configuration of the FSS unit cell. The structure consists of three metallic layers separated by two dielectric layers; meanwhile, a crossed aperture is etched on each of the first and third conducting planes, and a cubic aperture was etched on the second conducting plane. The detailed structure parameters are given in Table VI. The structure was made on two 0.127-mm-thickness Rogers Duroid 5880 substrates with 17-μm-thickness electrodeposited copper foils, as shown in Fig. 13(a). Fig. 13(a) and (b) shows the fabricated 60-GHz ACR FSS structure sample and the measurement setup, respectively. There are

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TABLE V C OMPARISON OF C OMPUTATIONAL E FFORT OF INBC-TLM, U NIFORM -M ESH TLM M ETHODS , AND S IMULATION

Fig. 12. 3-D view of 60-GHz ACR FSS structure unit cell and front view of the ACR FSS structure. TABLE VI S TRUCTURE PARAMETERS OF ACR FSS

40 × 40 (1600) elements in the sample, and the total dimensions are 100 mm × 100 mm. Eight plastic screws with 0.2 mm diameter are used to align and keep good contact for the different layers. As shown in Fig. 13(b), the free-space measurement setup is applied to measure the reflection and transmission characteristics of the 60-GHz ACR FSS. Standard V -band horn antennas are used as the transmitting and receiving antennas and connected to Agilent E5052B Signal Source Analyzer to observe the corresponding frequency-dependent curves. The measured S-parameters are compared with the S-parameters from calculation by INBC-TLM and simulation by HFSS, as shown in Fig. 15(b). The approximate coefficients of INBC are listed in Tables VII and VIII. Fig. 14(a) and (b) gives the time-dependent reflected and transmitted waves, and Fig. 15(a) gives the frequency-dependent counterpart. The S-parameters from calculation and simulation have good agreement, as shown in Fig. 15(b). There are two transmission zeros lower than −60 dB at two sides of the narrow passband, and thereby, the frequency selectivity is

Fig. 13.

(a) Fabricated ACR FSS structure. (b) Measurement setup.

obviously improved. It can be observed that the return losses of measurement and simulation are 20 and 24 dB, respectively. The measured insertion loss is 2 dB, which is slightly larger than the simulated value of 1.4 dB. The measured and simulated center frequencies are 61.6 and 62.3 GHz, respectively. The 3-dB bandwidth of 3.34 GHz can also be observed from the measurement curve, while 3.71 GHz for the simulation.

XIONG et al.: INBC INCORPORATED IN TLM FOR EFFICIENTLY MODELING THIN LAYER OF PERIODIC STRUCTURES

Fig. 14. Diagrams of waveforms in time domain for ACR FSS structure. (a) Reflected wave. (b) Transmitted wave. TABLE VII E XTRACTED VALUES OF P OLES ( pn ) AND R ESIDUES (rn ) FOR THE S CATTERING C OEFFICIENTS OF TL-N ETWORK OF ACR FSS

Compared with the simulated results, the main reasons for the above larger insertion loss and lower operating frequency in the measurement are the measurement and fabrication errors. Another reason is because the two dielectric layers were not tightly bonded together like by using spray adhesive, so there must be some air existing between the dielectric layers, which, however, was not included in the simulation. V. C ONCLUSION An efficient method incorporated in TLM to deal with a thin layer of periodic structure is proposed and investigated, which allows the approximate scattering coefficients of the

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Fig. 15. Diagrams of wave behaviors in frequency domain for ACR FSS. (a) Waveforms in frequency domain. (b) Comparisons of S-parameters from calculation, simulation, and measurement. TABLE VIII E XTRACTED VALUES OF d FOR THE S CATTERING C OEFFICIENTS OF TL-N ETWORK OF ACR FSS

electrically thin layer to be directly applied in the TLM iteration equations to avoid a fully spatial discretization of the thin layer. The method first starts from the rational approximations of the scattering coefficients in frequency domain, and the INBC with discrete-time form is then derived and finally is incorporated into the TLM scheme to describe the thin layer with fine resolution. The implementation of the method in TLM code shows good agreement with the simulation. The proposed method can also result in a significant reduction of CPU and memory requirements when compared with the regular TLM presented in the literature for the thinlayer simulation. As it is known, extremely fine TLM mesh is needed to satisfy the stability requirement of the TLM mesh [1]–[10] when modeling thin parts by using regular TLM scheme, which might increase the burden of huge computational data. Compared with the regular TLM method, the stability requirement is much easier to satisfy by using the proposed INBC-TLM method, since it is based on a coarse TLM mesh for the whole space modeling.

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R EFERENCES [1] C. Christopoulos, The Transmission-Line Modeling (TLM) Method in Electromagnetics. San Rafael, CA, USA: Morgan & Claypool, 2006. [2] W. J. R. Hoefer, “The transmission-line matrix method—Theory and applications,” IEEE Trans. Microw. Theory Techn., vol. MTT-33, no. 10, pp. 882–893, Oct. 1985. [3] M. Krumpholz and P. Russer, “A field theoretical derivation of TLM,” IEEE Trans. Microw. Theory Techn., vol. 42, no. 9, pp. 1660–1668, Sep. 1994. [4] V. Trenkic, C. Christopoulos, and T. M. Benson, “Development of a general symmetrical condensed node for the TLM method,” IEEE Trans. Microw. Theory Techn., vol. 44, no. 12, pp. 2129–2135, Dec. 1996. [5] J. N. Rebel, M. Aidam, and P. Russer, “On the convergence of the classical symmetrical condensed node-TLM scheme,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 5, pp. 954–963, May 2001. [6] Y. Xiong, J. A. Russer, W. Che, G. Shen, Y. Han, and P. Russer, “Dispersion analysis of a fishnet metamaterial based on the rotated transmissionline matrix method,” IET Microw., Antennas Propag., vol. 9, no. 12, pp. 1345–1353, Sep. 2015. [7] Y. Xiong, W. Che, and Y. Han, “SIBC incorporated in conformal FDTD to efficiently simulate the thin conductive layer of periodic structure,” IET Microw., Antennas Propag., vol. 10, no. 4, pp. 353–361, Mar. 2016. [8] J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM—Part I: Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1528–1534, Oct. 1999. [9] J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM—Part 2: Materials with anisotropic properties,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1535–1542, Oct. 1999. [10] J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM—Part 3: Materials with nonlinear properties,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 997–1004, Jul. 2002. [11] D. S. Wang, P. Zhao, and C. H. Chan, “Design and analysis of a highselectivity frequency-selective surface at 60 GHz,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 6, pp. 1694–1703, Jun. 2016. [12] T. K. Wu, Ed., Frequency Selective Surface and Grid Array. New York, NY, USA: Wiley, 1995. [13] J. C. Vardaxoglou, Frequency Selective Surface: Analysis and Design. Taunton, U.K.: Res. Studies Press, 1997. [14] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York, NY, USA: Wiley, 2000. [15] P. Sewell, J. Wykes, A. Vukovic, D. W. P. Thomas, T. M. Benson, and C. Christopoulos, “Multi-grid interface in computational electromagnetics,” Electron. Lett., vol. 40, no. 3, pp. 162–163, Feb. 2004. [16] J. Wlodarczyk, “New multigrid interface for the TLM method,” Electron. Lett., vol. 32, no. 12, pp. 1111–1112, Jun. 1996. [17] L. Pierantoni and T. Rozzi, “A general multigrid-subgridding formulation for the transmission line matrix method,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 8, pp. 1709–1716, Aug. 2007. [18] J. Paul, V. Podlozny, D. W. P. Thomas, and C. Christopoulos, “TimeDomain simulation of thin material boundaries and thin panels using digital filters in TLM,” Turkish J. Elect. Eng., Comput. Sci., vol. 10, no. 2, pp. 185–198, 2002. [19] J. Paul, V. Podlozny, and C. Christopoulos, “The use of digital filtering techniques for the simulation of fine features in EMC problems solved in the time domain,” IEEE Trans. Electromagn. Compat., vol. 45, no. 2, pp. 238–244, May 2003. [20] V. Trenkic, A. P. Duffy, T. M. Benson, and C. Christopoulos, “Numerical simulation of penetration and coupling using the TLM method,” in Proc. Int. Symp. Electromagn. Compat., Rome, Italy, Sep. 1994, pp. 321–326. [21] H. Wakatsuchi, S. Greedy, J. Paul, and C. Christopoulos, “Efficient modelling of band-gap and metamaterial structures in EMC,” in Proc. EMC, vol. 9. 2009, pp. 745–748. [22] T. Kim, X. Li, and D. J. Allstot, “Compact model generation for on-chip transmission lines,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 3, pp. 459–470, Mar. 2004. [23] M. Parashar, J. S. Thorp, and C. E. Seyler, “Continuum modeling of electromechanical dynamics in large-scale power systems,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 9, pp. 1848–1858, Sep. 2004. [24] W. Macher, “Inter-reciprocity principles for linear network-waveguides systems based on generalized scattering, admittance and impedance matrices,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 59, no. 4, pp. 721–734, Apr. 2012. [25] P. Russer, Electromagnetics, Microwave Circuit, and Antenna Design for Communications Engineering, 2nd ed. Boston, MA, USA: Artech House, 2006.

[26] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1052–1061, Jul. 1999. [27] V. Nayyeri, M. Soleimani, and O. M. Ramahi, “A method to model thin conductive layers in the finite-difference time-domain method,” IEEE Trans. Electromagn. Compat., vol. 56, no. 2, pp. 385–392, Apr. 2014. [28] S. Luo and Z. Chen, “Extraction of causal time-domain network parameters from their band-limited frequency-domain counterparts using rational functions,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 6, pp. 1205–1210, Jun. 2005. [29] D. De Jonghe and G. Gielen, “Characterization of analog circuits using transfer function trajectories,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 59, no. 8, pp. 1796–1804, Aug. 2012. [30] B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del., vol. 21, no. 3, pp. 1587–1592, Jul. 2006. [31] T. Dhaene, D. Deschrijver, and N. Stevens, “Efficient algorithm for passivity enforcement of S-parameter-based macromodels,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 2, pp. 415–420, Feb. 2009. [32] S. Lefteriu and A. C. Antoulas, “On the convergence of the vectorfitting algorithm,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 4, pp. 1435–1443, Apr. 2009. [33] A. Beygi and A. Dounavis, “An instrumental variable vector-fitting approach for noisy frequency responses,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 9, pp. 2702–2712, Sep. 2012.

Ying Xiong was born in Gao’an, China, in 1989. She received the M.Sc. and Ph.D. degrees from the Nanjing University of Science and Technology, Nanjing, China, in 2012 and 2016, respectively. She was a Visiting Student with the Department of Electronic and Information Engineering, Technische Universitaet München, Munich, Germany, from 2013 to 2014. She is currently with the China North Vehicle Research Institute, Beijing, China. Her current research interests include computational electromagnetics, 3-D transmission methods and frequency-selected surfaces, and the electromagnetic compatibility study on pulsewidth modulation drive motor systems of electric vehicles.

Wenquan Che (M’01–SM’11) received the B.Sc. degree from the East China University of Science and Technology, Shanghai, China, in 1990, the M.Sc. degree from the Nanjing University of Science and Technology, Nanjing, China, in 1995, and the Ph.D. degree from the City University of Hong Kong, Hong Kong, in 2003. She was a Research Assistant with the City University of Hong Kong in 1999. In 2002, she was a Visiting Scholar with the Polytechnique de Montréal, Montréal, QC, Canada. From 2007 to 2008, she conducted academic research with the Institute of High Frequency Technology, Technische Universität München, Munich, Germany. From 2005 to 2006 and 2009 to 2012, she was with the City University of Hong Kong as a Research Fellow and a Visiting Professor. She is currently a Professor with the Nanjing University of Science and Technology. She has authored or co-authored over 110 internationally refereed journal papers and over 90 international conference papers. Her current research interests include electromagnetic computation, planar/coplanar circuits and subsystems in RF/microwave frequency, microwave monolithic-integrated circuits, and medical application of microwave technology. Dr. Che has been a Reviewer for IET Microwaves, Antennas, and Propagation. She is a Reviewer for the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES , the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION, the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS , and the IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS . She was a recipient of the 2007 Humboldt Research Fellowship by the Alexander von Humboldt Foundation of Germany, the 5th China Young Female Scientists Award in 2008, and the Distinguished Young Scientist by the National Natural Science Foundation Committee of China in 2012.

XIONG et al.: INBC INCORPORATED IN TLM FOR EFFICIENTLY MODELING THIN LAYER OF PERIODIC STRUCTURES

Desong Wang was born in Anhui, China, in 1988. He received the B.Sc. degree from Anhui University, Hefei, China, in 2010, and the M.Sc. degree from the Nanjing University of Science and Technology, Nanjing, China, 2013. He was an Exchange Student with Chang Gung University, Taoyuan, Taiwan, from 2012 to 2013. He is currently a Research Assistant with the City University of Hong Kong, Hong Kong. His current research interests include microwave, millimeterwave, and terahertz frequency-selective surfaces and passive circuits.

Ye Han was born in Xi’an, China, in 1989. She received the M.Sc. degree from the Nanjing University of Science and Technology, Nanjing, China, in 2012, where she is currently pursuing the Ph.D. degree. She was a Visiting Student with the Faculty of Engineering, Department of Electrical and Electronic Engineering, University of Nottingham, Nottingham, U.K., from 2013 to 2014. Her current research interests include microwave circuit analog absorbers, frequency-selected surfaces, and absorber measurement. Ms. Han attended the Cross Strait Quad-Regional Radio Science and Wireless Technology Conference in 2013. She was the recipient of the Excellent Paper Award.

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Guangxu Shen was born in Jiangsu, China. He received the B.Eng. degree from the Nanjing University of Science and Technology, Nanjing, China, in 2014, where he is currently pursuing the Ph.D. degree with the Department of Communication Engineering. He was an Exchange Student with the Institute of Nanoelectronics, Technische Universität München, Munich, Germany, in 2014. His current research interests include metamaterials, microwave passive circuits, and microwave/millimeter-wave antennas.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Skin Effect Modeling in Conductors of Arbitrary Shape Through a Surface Admittance Operator and the Contour Integral Method Utkarsh R. Patel, Student Member, IEEE, and Piero Triverio, Senior Member, IEEE

Abstract— An accurate modeling of skin effect inside conductors is of capital importance to solve transmission line and scattering problems. This paper presents a surface-based formulation to model skin effect in conductors of arbitrary cross section, and compute the per-unit-length impedance of a multiconductor transmission line. The proposed formulation is based on the Dirichlet–Neumann operator that relates the longitudinal electric field to the tangential magnetic field on the boundary of a conductor. We demonstrate how the surface operator can be obtained through the contour integral method for the conductors of arbitrary shape. The proposed algorithm is simple to implement, efficient, and can handle arbitrary cross sections, which is the main advantage over the existing approach based on eigenfunctions, which is available only for canonical conductor’s shapes. The versatility of the method is illustrated through a diverse set of examples, which includes transmission lines with trapezoidal, curved, and V-shaped conductors. Numerical results demonstrate the accuracy, versatility, and efficiency of the proposed technique. Index Terms— Contour integral method, impedance calculation, surface admittance operator, surface methods, transmission line.

I. I NTRODUCTION

T

RANSMISSION line modeling is a crucial part of the computer aided design of a variety of systems, including high-speed electronic boards [1], [2], integrated circuits [3], microwave systems [4], metamaterials [5], and power grids [6]. In order to create a transmission line model, one must first obtain the per-unit-length (p.u.l.) impedance and the admittance of the line [7]. In several applications, such as the investigation of signal integrity issues in high-speed electronic systems, such parameters are required over a wideband, extending from dc up to tens of gigahertz [1]. Across such a band, the development of the skin effect significantly changes the line behavior [2], and must be accurately described. The skin effect modeling is also important in scattering problems [8].

Manuscript received September 27, 2015; revised December 16, 2015 and July 4, 2016; accepted July 6, 2016. Date of publication August 5, 2016; date of current version September 1, 2016. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grant Program, in part under the Canada Research Chairs Program, and in part by CMC Microsystems. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2593721

In order to obtain the p.u.l. impedance of a transmission line, one must describe the electromagnetic fields both inside the conductors (interior problem) and the outside (exterior problem). For the exterior problem, integral equations [9]–[11] are commonly utilized. By requiring only a discretization of the conductors’ surface, integral equations are typically more efficient that volumetric approaches, such as the finite difference or finite-element method (FEM) [12], [13], which require the discretization of a large area around the conductors. For the interior problem, which is responsible for capturing the skin effect, both volumetric and surface methods have been used. Volumetric approaches include the FEM [13], [14], conductor partitioning [15], and volumetric integral equations [16]. Unfortunately, as frequency increases, these methods become computationally inefficient, since a very fine mesh is needed to capture the pronounced skin effect. Surface methods solve the interior problem using only a discretization of the boundary of the conductors. This is achieved by describing the electromagnetic behavior of the conductor through a surface operator that relates the electric and magnetic fields on the boundary. This operator is responsible for modeling skin and proximity effects [2]. The surface operator can be obtained analytically or numerically. In [17] and [18], a surface operator is derived using the so-called surface-impedance boundary conditions [19], under the approximation of small curvature and well-developed skin effect [20]. While high-order boundary conditions have been proposed to increase accuracy [21], this approach is mostly suitable for conductors with smooth boundaries. Numerically, the surface operator can be obtained using finite differences [22], finite elements [23], the electric field integral equation [24], [25], or the magnetic field integral equation [20]. All these approaches require, to calculate the surface operator, a volumetric discretization of the interior problem and the calculation of several kernel matrices whose size depends on mesh size. These features increase their computational cost and complexity. An interesting technique to derive the surface operator analytically was introduced by De Zutter and Knockaert [26]. Using the eigenfunctions of the Helmholtz equation, the surface admittance operator is obtained analytically avoiding a discretization of the interior problem. This key idea leads to a simple and efficient formulation. However, since eigenfunctions can realistically be obtained only for canonical

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PATEL AND TRIVERIO: SKIN EFFECT MODELING IN CONDUCTORS OF ARBITRARY SHAPE

geometries, this approach has been restricted so far to circular [26], [27], tubular [28], rectangular [26], and triangular conductors [29]. In this paper, we show how the surface admittance operator for the conductors of arbitrary shape can be efficiently obtained from a simple contour integral, through the so-called contour integral method [30]. While the possibility to extract the surface admittance operator from the contour integral is already known [31], it has been applied only to simple scattering problems [31]. All works that followed adopted the eigenfunctions approach [26], [29], [32]–[34], which is feasible for canonical shapes only. While arbitrary shapes can be decomposed into triangles [29], this significantly adds to the algorithm complexity. This paper provides a simple, unified way to handle arbitrary cross sections, and extracts accurate and broadband p.u.l. parameters for a variety of transmission lines. Numerical tests show that the proposed method is robust and computationally efficient, even when compared against the analytical approach based on eigenfunctions [26]. In this paper, the proposed method is applied to calculate transmission line parameters, but it can also be used for scattering problems [35]. This paper is organized as follows. In Section II, we define the problem and discuss how the surface admittance operator is related to the contour integral method from a theoretical standpoint. In Section III, we discuss the numerical implementation of the proposed approach, and in Section IV, how it can be used to compute the p.u.l. impedance of arbitrary transmission lines. Finally, in Section V, we demonstrate the accuracy, robustness, and computational efficiency of the proposed method through a comprehensive set of examples. II. S URFACE A DMITTANCE O PERATOR T HROUGH THE C ONTOUR I NTEGRAL M ETHOD A. Problem Definition We consider a system of the P conductors of arbitrary shape having conductivity σ , permittivity ε, and permeability μ. For the sake of generality, the conductors are assumed to be inside a stratified medium, where each layer has permittivity εl and permeability μl . If a conductor extends into two layers, it is decomposed into two parts, as discussed in [36]. Our final goal is to calculate the partial p.u.l. resistance R(ω) and inductance L(ω) matrices in the Telegrapher’s equation [2] ∂V = −[R(ω) + j ωL(ω)]I. (1) ∂z  T In (1), V = V1 . . . V P collects the potential V p of each T  conductor, while I = I1 . . . I P contains the current I p flowing in each conductor. The impedance parameters will be calculated with the surface admittance approach of [26] using, however, a contour integral to obtain the surface admittance operator. B. Surface Admittance Formulation We discuss the surface admittance formulation used to solve the interior problem by considering the pth conductor of a multiconductor transmission line. The conductor has an

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Fig. 1. Left: sample geometry of a conductor with arbitrary cross section inside a stratified medium. Right: equivalent configuration obtained after replacing the conductor with the surrounding medium, and introducing an ( p) equivalent current density Js (r ) on contour γ ( p) .

arbitrary cross section, has a simply connected contour γ ( p) , and is shown in Fig. 1 (left). The position vector of an arbitrary point on γ ( p) is denoted by r. Finally, the unit vectors normal and tangential to contour γ ( p) are denoted by nˆ and tˆ, respectively, as shown in Fig. 1. In order to derive the surface admittance operator, we initially follow [26], and consider a boundary value problem. Suppose that the z-directed electric field on contour γ ( p) is ( p) given by E z (r ). Under the quasi-TM assumption [26], the ( p) electric field E z inside contour γ ( p) can be obtained from the scalar Helmholtz equation [37] ( p)

∇ 2 Ez

( p)

+ k 2 Ez

=0

(2)

r ∈ γ ( p) .

(3)

subject to the boundary condition ( p)

( p)

Ez (r ) = E z (r )

In (2), k = (ωμ(ωε − j σ ))1/2 is the wavenumber inside the ( p) conductor. Furthermore, the tangential magnetic field Ht (r ) ( p) along the contour γ follows directly from Maxwell’s equations under the quasi-TM assumption [37]  ( p)  ∂Ez (r ) 1 ( p) (4) Ht (r ) = j ωμ ∂n ( p) r∈γ

where the derivative is taken along the direction normal to contour γ ( p) , as shown in Fig. 1 (left). Next, we replace the conductor by the material of the surrounding layer [26], as shown in Fig. 1 (right). From here onwards, we will call this the equivalent configuration. Due to this modification, the electric field inside the conductor ( p) changes to Ez (r ), which satisfies the Helmholtz equation ( p)

∇ 2 Ez

2 ( p) Ez = 0 + kout

(5)

subject to the same Dirichlet boundary condition (3) ( p) ( p) Ez (r ) = E z (r ) r ∈ γ ( p) . (6) √ In (5), kout = ω μl εl is the wavenumber inside the layer surrounding the conductor. In this new configuration, the tangential magnetic field along γ ( p) is  ( p)  1 ∂ Ez (r ) ( p) t (r ) = H . (7) j ωμl ∂n ( p) r∈γ

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

By replacing the conductor with the surrounding medium, we have modified the fields both inside and outside γ ( p) . Hence, to restore the original fields outside γ ( p) , we invoke the equivalence principle [38] and introduce an equivalent ( p) surface current density Js (r ) on γ ( p) [26], as shown in Fig. 1 (right). The equivalent current is directed along z, and is given by ( p)

( p)

Js (r ) = Ht

t( p)(r ). (r ) − H

(8)

It is important to note that an equivalent magnetic current on γ ( p) is not required, since the electric field on γ ( p) in the original and equivalent configuration remains the same due to the Dirichlet conditions (3) and (6). By substituting (4) and (7) into (8), we obtain [26]   ( p) ( p) 1 ∂ Ez (r ) 1 1 ∂Ez (r ) ( p) − Js (r ) = . (9) j ω μ ∂n μl ∂n ( p) r∈γ

( p) Ys

that Equation (9) defines a surface admittance operator relates the longitudinal electric field and equivalent current on γ ( p) as ( p)

( p)

( p)

Js (r ) = Ys E z (r ).

 Equation (11) stems from the planar Green’s and d = |d|. theorem, and shows that the electric field along the contour γ ( p) can be interpreted as the superposition of cylindrical waves that originate from points along γ ( p) . Constant C0 can be any complex number. If C0 = 1, Green’s function is the Hankel function of the second kind, which represents outgoing cylindrical waves. In Section III-E, we will discuss how to choose C0 at low and high frequencies to achieve high numerical robustness. The contour integral equation (11) relates the electric field on the boundary to its normal derivative. It can thus be used, after numerical discretization, to derive an explicit expression for the surface admittance operator (10). III. N UMERICAL F ORMULATION A. Discretization of Electric Fields, Magnetic Fields, and Equivalent Current We discretize the contour integral equation with the method of moments (MoM) [10], using point matching to test the resulting equation. We divide contour γ ( p) into N p segments ( p) γn , and expand the longitudinal electric field in terms of pulse basis functions

(10) ( p)

( p) Ys

can be written in terms of An explicit expression for the eigenfunctions of the Helmhotz equations (2) and (5), as shown in [26]. However, this approach is viable only for canonical conductor shapes, for which eigenfunctions are known analytically [26]–[29]. For arbitrary shapes, eigenfunctions can only be computed numerically. Since many eigenfunctions are needed to accurately model the operator, this approach can be very time consuming, and is typically avoided. In Section II-C, we show that the contour integral method [30] provides an efficient and robust way to compute such an operator numerically for arbitrary shapes.

E z (r ) =

Np 

( p)

( p)

en n (r )

(14)

n=1 ( p)

where n (r ) is the nth pulse basis function, which is one if r belongs to the nth segment, and zero otherwise. We also define rn to be the position vector of the midpoint of the nth partition. Similarly, we discretize the tangential magnetic ( p) t( p)(r ) that appear in (8) as fields Ht (r ) and H ( p) Ht (r )

=

Np 

( p)

( p)

h n n (r )

(15)

( p) ( p) h˜ n n (r ).

(16)

n=1

and C. Contour Integral Method Equating (9) and (10), we see that, in order to derive an explicit expression for the surface admittance operator, we ( p) need a relation between the electric field E z on the boundary ( p) and its normal derivative ∂Ez /∂n. With this goal in mind, we reconsider the way we solve (2). The contour integral method [30] gives the solution of the Helmholtz equation in terms of the electric field and its normal derivative on γ ( p)  ‰ j ∂ G(r , r ) ( p)  ( p) Ez (r ) = Ez (r ) 2 γ ( p) ∂n   ( p)  r)  ∂Ez ( − G(r , r ) (11) dr  ∂n 

t( p)(r ) = H

n=1

For simplicity of notation, we cast all into column vectors  ( p) ( p) E( p) = e1 e2 ...  ( p) ( p) ( p) H = h1 h2 ...  ( p) = h˜ ( p) h˜ ( p) . . . H 1 2

(12)

where J0 (.) and Y0 (.) are the zeroth order Bessel and Neumann functions [39], respectively. As shown in Fig. 1 (left), the distance between points r and r is denoted as d = r − r

(13)

expansion coefficients ( p) T

eN p

( p) T h Np ( p) T h˜ N p .

(17) (18) (19)

( p)

Similarly, we discretize equivalent current Js (r ) along the contour γ ( p) using pulse basis functions as Js (r ) =

where r and r are both on γ ( p) and n  is the unit vector normal to the contour at the point r . Green’s function is G(r , r ) = C0 J0 (kd) − j Y0 (kd)

Np 

Np 

( p)

( p)

jn n (r )

(20)

n=1 ( p)

with coefficients jn J

( p)



stored into vector ( p)

= j1

( p)

j2

...

( p) T

jN p

.

(21)

From (8), we have the following relation between the coefficients of equivalent current and magnetic fields: ( p) . J( p) = H( p) − H

(22)

PATEL AND TRIVERIO: SKIN EFFECT MODELING IN CONDUCTORS OF ARBITRARY SHAPE

( p) to the electric field coefficients Next, we relate H( p) and H ( p) E via the contour integral equation (11) to obtain the surface admittance operator. B. Magnetic Field in the Original Configuration

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C. Magnetic Field in the Equivalent Configuration Next, we find the magnetic field in the equivalent configuration by repeating all the steps of Section III-B, but with the material parameters of the conductor replaced by the parameters of the surrounding medium. These steps lead to

After combining (4) and (11), we substitute the electric and magnetic field expansions (14) and (15) into the contour equation to obtain Np 

j = 2

⎡

‰ γ ( p)

⎣

(31)

which is analogous to (30). In (31), the entries of Pout and Uout are calculated with (26), (28), and (29), with wavenumber k replaced by kout and permeability μ replaced by μl .

( p) ( p) em m (r )

m=1

−1  ( p) = Pout Uout E( p) H

Np

∂ G(r , r )  ( p) ( p)  en n (r ) ∂n  n=1

− j ωμG(r , r )

Np 

( p)

D. Surface Admittance Operator ⎤

( p)

h n n (r  )⎦ dr  .

Finally, by substituting (30) and (31) into (22), we obtain (23)

J( p) = Y( p)E( p)

(32)

−1 Y( p) = P−1 U − Pout Uout

(33)

n=1

Using point matching [38], we test (23) at the midpoints rm of all N p segments, obtaining ( p)

em

Np

j  ( p) = en 2 n=1

Np

ˆ

∂ G(rm ∂n 

( p) γn

ωμ  ( p) + hn 2 n=1

, r )

ˆ ( p)

γn

dr 

G(rm , r )dr 

(24)

for m = 1, . . . , N p . Note that in (24), the integration is ( p) performed only over the nth segment γn . All relations (24) can be compactly written in matrix form as UE( p) = PH( p)

where dm = r − rm and dm = |dm |, and we used the fact that the normal derivative of (12) can be written as ∂ G(r , r ) dm · nˆ  = −k [C0 J1 (kdm ) − j Y1 (kdm )]. (27) ∂n  dm When n = m, the contribution of the first term on the right-hand side of (24) is zero, because dm and nˆ  are orthogonal. Hence, the diagonal entries of U are given by The (m, n)th entry of P in (25) is given by ˆ ωμ [C0 J0 (kdm ) − j Y0 (kdm )]dr  . [P]m,n = 2 γn( p)

(28)

(29)

In all the numerical examples of Section V, the integrals in (26) and (29) were evaluated using a five-point Gaussian quadrature routine. When n = m, the integral in (29) can be evaluated analytically, as shown in Appendix A. From (25), we can express the magnetic field H( p) in terms of the electric field on the same conductor as H( p) = P−1 UE( p) .

is the discretized surface admittance operator of the pth conductor. Therefore, we see that with the proposed contour integral approach, it is sufficient to evaluate matrices P, Pout, U, and Uout to easily obtain the surface admittance operator for a conductor of arbitrary shape. In order to compact the notation, we collect the coefficients of the electric field and the equivalent current of all conductors into two column vectors E and J. The surface admittance relations (32) can thus be compactly written as

(25)

where U and P are square matrices with dimension N p × N p . Element (m, n) of matrix U, if m = n, is given by ˆ jk dm · nˆ  [U]m,n = [C0 J1 (kdm ) − j Y1(kdm )]dr  (26) 2 γn( p) dm

[U]m,m = 1.

where

(30)

J = Ys E where

⎡ (1) Y ⎢ ⎢ Ys = ⎢ ⎣

(34) ⎤ ⎥ ⎥ ⎥. ⎦

Y(2) ..

.

(35)

Y( P)

E. Choice of C0 In this section, we discuss how to set the constant C0 at low and high frequencies in order to achieve a well-conditioned algorithm. 1) Low Frequency: At low frequencies, where skin effect has not yet developed, computing (29) requires the evaluation of Green’s function (12) for very small arguments. For small arguments, the Neumann function Y0 (.) dominates the Bessel function J0 (.) [39]. Hence, to maintain a good numerical contrast between the Bessel and Neumann functions, we must set C0 to a high value, as discussed in [40]. The value of 106 provided accurate results for all numerical tests we performed, including the examples of Section V. 2) High Frequency: At high frequencies, we set C0 = 1. This makes Green’s function (12) become the Hankel function of the second kind, which is well-behaved at high frequency.

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3) Switching Condition: As previously discussed, two different values of C0 are appropriate at low and high frequencies. Numerical tests demonstrated that there is a large range of intermediate frequencies where both values of C0 provide accurate results. Therefore, the choice of the frequency where one should switch from one C0 value to the other is not critical. In our implementation, we use the low-frequency value of C0 on the pth conductor when p ≤t δp

(36)

where  p and δ p are, respectively, the minimum transversal dimension and the skin depth in the pth conductor. In all numerical test we performed, any t value between 0.2 and 0.5 gave good results, and t = 0.5 has been used in all numerical examples presented in Section V. When frequency increases and inequality (36) no longer holds, the high-frequency value of C0 is used. 4) Discussion: The choice of C0 and the switching condition (36) can be explained physically by looking at the wave phenomena that develop inside the conductors at low and high frequencies. The contour integral equation (11) can be interpreted using the equivalence principle [41]. In (11), the electric field on the contour is expressed as the superposition of the field radiated by a magnetic current on the contour (first term on the right-hand side) and the field radiated by an electric current (second term on the right-hand side). The fields are obtained from the currents through the multiplication by Green’s function and its derivative, respectively. This interpretation reveals that (11) expresses the field in the conductor as a superposition of cylindrical waves originating from the points of the contour. At high frequency, where (36) is not satisfied and skin depth is very small, the cylindrical wave emanating from a point on the contour attenuates appreciably before reaching the other side of the conductor. As a result, (2) the Hankel function of second kind H0 (.), which describes an outgoing cylindrical wave, is the most appropriate Green’s function for this regime. On the other hand, when (36) is satisfied, the cylindrical wave reaches the other side of the conductor without a significant attenuation, giving rise to standing waves, that are best modeled using a combination of Bessel and Neumann functions J0 (.) and Y0 (.). IV. E XTERIOR P ROBLEM AND I MPEDANCE C OMPUTATION In this section, we use the electric field integral equation to express the relation between the electric field on the conductors’ boundary, and the equivalent currents dictated by the region outside the conductors [26]. Combined with the surface admittance operator (34), this will lead to the p.u.l. impedance of the transmission line. On the contour of the pth conductor, the electric field integral equation [37] reads P ‰  ∂ Vp ( p) (q) (37) Js (r  )G 0 (r , r )ds  − E z (r ) = j ωμl ∂z γ (q) q=1

where r ∈ γ ( p) and the integration is performed over the contour γ (q) of each conductor (q = 1, . . . , P). The integral kernel G 0 (r , r ) is Green’s function of the

surrounding medium. As discussed in [42], under the quasiTM approximation, currents inside dielectrics can be safely ignored when calculating impedance parameters and the 2-D quasi-static Green’s function of free space can be used G 0 (r , r ) =

1 ln|r − r |. 2π

(38)

Following [16] and [26], we substitute (1), (14) and (20) into (37), obtaining Np 

( p)

( p)

em m (r ) = j ωμl

m=1

+

P ‰ 

Nq 

(q) q=1 γ n=1

P 

(q)

(q)

jn n (r  )G 0 (r , r )dsq

[R pq (ω) + j ωL pq (ω)]Iq

(39)

q=1

for p = 1, . . . , P. Using point matching [38], this integral equation can be converted into the system of algebraic equations E = j ωμl G0 J + Q[R(ω) + j ωL(ω)]I

(40)

where G0 is Green’s matrix. In (40), Q is a block diagonal matrix ⎤ ⎡ 11 ⎥ ⎢ 12 ⎥ ⎢ (41) Q=⎢ ⎥ .. ⎦ ⎣ . 1P where 1 p is a vector of size N p × 1 whose entries are all ones. Furthermore, as shown in [26], the total current inside each conductor is equal to the contour integral of the equivalent current (20), and hence I = QT WJ where W is a diagonal matrix in the form ⎤ ⎡ (1) w ⎥ ⎢ w(2) ⎥ ⎢ W=⎢ ⎥ .. ⎦ ⎣ . w( P)

(42)

(43)

where w( p) is a diagonal matrix of the size of N p × N p , where entry (n, n) is the width of the nth segment of γ ( p) . As shown in [27], we can manipulate (40) using (34) and (42) to obtain the partial p.u.l. resistance   (44) R(ω) = Re [QT W(1 − j ωμl Ys G0 )−1 Ys Q]−1 and partial p.u.l. inductance   L(ω) = ω−1 Im [QT W(1 − j ωμl Ys G0 )−1 Ys Q]−1 .

(45)

The p.u.l. impedance can be obtained from the partial p.u.l. impedance by taking one of the conductors as reference.

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Fig. 2. Cross section of the two-conductor lines analyzed in Section V-A. All dimensions are in millimeters. TABLE I E XAMPLE OF S ECTION V-A: CPU T IME R EQUIRED TO C OMPUTE THE p.u.l. I MPEDANCE AT O NE F REQUENCY

V. N UMERICAL R ESULTS In this section, we demonstrate the robustness of the proposed method, and compare its accuracy and CPU times against an FEM solver (COMSOL Multiphysics) and other surface admittance-based approaches in the literature [26], [27]. We illustrate the versatility of the technique by considering a comprehensive set of transmission lines made by circular, rectangular, trapezoidal, V-shaped, and conformal conductors. All computations were performed on a computer with 16 GB of memory and a 3.4-GHz processor. All techniques based on a surface admittance operator were implemented in MATLAB.

Fig. 3. P.u.l. resistance and inductance of the transmission line with two round conductors considered in Section V-A1, obtained with the proposed method and MoM-SO [27].

A. Two-Conductor Lines 1) Round Conductors: We first consider a transmission line made by two round conductors with radii a1 = 1 mm, a2 = 2 mm, and spacing d = 3.5 mm. The line cross section is shown in Fig. 2 (left). The conductivity of both conductors is σ = 5.8 · 107 S/m. We calculated the p.u.l. impedance using the proposed technique and the MoM-SO algorithm [27], which uses the eigenfunctions method to derive the surface admittance operator. Fig. 3 shows the p.u.l. resistance and inductance obtained with the proposed approach and with MoM-SO. The two methods are in excellent agreement. For the proposed approach, the boundaries enclosing the small and large conductors were discretized with N1 = 28 and N2 = 60 pulse basis functions, respectively. As shown in Table I, for each frequency point, the proposed approach took only 0.04 s. In this case, the eigenfunction approach is faster since it can capture, with few Fourier basis functions, the field distribution inside the circular conductors. The advantage of the proposed approach is generality: it can be applied to arbitrary shapes, while MoM-SO is limited to round conductors, either solid [26], [27] or hollow [28]. 2) Rectangular Conductors: We now consider the case of two rectangular conductors presented in [26]. Each conductor has conductivity σ = 5.6 · 107 S/m and dimension 2 mm × 0.2 mm, as shown in Fig. 2 (right). Fig. 4 shows the p.u.l. resistance and inductance for this transmission line computed with the proposed method and with [26]. The latter method computes the surface admittance operator analytically using sinusoidal functions, which are the eigenfunctions of

Fig. 4. P.u.l. resistance and inductance of the transmission line with two rectangular conductors considered in Section V-A2. Results obtained with [26] are labeled as De Zutter et al.

the Helmholtz equation on a rectangular domain. In order to interface the surface admittance operator to the electric field integral equation (39), the sinusoidal functions have to

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Fig. 5. Valley microstrip line considered in Section V-B. All dimensions are in micrometers.

be mapped onto pulse basis functions [26]. From Fig. 4, we see that the resistance and inductance values obtained with both approaches match very well. For the proposed approach, each conductor was discretized with N1 = N2 = 106 pulse basis functions. For the method of [26], the number of sinusoidal harmonics was chosen to be M = 400 (see [26] for definition of M), and each conductor was discretized with N1 = N2 = 280 pulse basis functions. The approach of [26] required more basis functions than the proposed approach to get accurate results at very high frequency. The computational time taken by both techniques is given in Table I. We see that the CPU time with the proposed approach is not far from the CPU time with [26], although the proposed method is more general, as it can handle arbitrary shapes. B. Valley Microstrip Line We consider a valley microstrip line from [43], having the cross section shown in Fig. 5. This line type is used in lowloss microwave integrated circuits, and has been considered in several previous publications [44], [45]. All conductors are made of copper (σ = 5.8 · 107 S/m). The top conductor is the signal line, while the two lower conductors form the reference line. Fig. 6 shows the p.u.l. inductance and resistance of the system obtained with an FEM solver (COMSOL Multiphysics) and with the proposed approach. The results from the two methods agree well both at very low (1 MHz) and very high (1 THz) frequencies. This test further validates the proposed method and shows its numerical robustness. In the FEM simulation, we had to use a fine mesh with 3 102 boundary elements and 188 107 triangular elements, in order to properly resolve the pronounced skin effect at high frequency. In the proposed technique, the boundary of the signal conductor was discretized with N1 = 194 pulse basis functions, while each ground conductor was discretized with N2 = N3 = 92 pulse basis functions. The CPU time taken by both techniques is reported in Table II and shows the efficiency of the proposed method, which requires only 0.52 s to extract the p.u.l. impedance at one frequency for this nontrivial line. While in such cases one can resort to a general 2-D FEM approach, this significantly increases the CPU time, as shown in Table II. C. On-Chip Transmission Line With Trapezoidal Conductors Next, we consider the four-conductor transmission line [46] shown in Fig. 7. All conductors are made of aluminum (σ = 35.7 MS/m). As in [46], conductivity is assumed to be constant up to the maximum frequency of interest (1 THz), since at this frequency the effect of relaxation time is still

Fig. 6. P.u.l. resistance and inductance of the valley microstrip line considered in Section V-B, computed with the proposed method and the FEM (COMSOL Multiphysics).

TABLE II E XAMPLES OF S ECTIONS V-B, V-C, AND V-D: CPU T IME R EQUIRED TO C OMPUTE THE p.u.l. I MPEDANCE AT O NE F REQUENCY W ITH THE P ROPOSED T ECHNIQUE AND AN FEM S OLVER (COMSOL M ULTIPHYSICS )

Fig. 7. On-chip interconnect of Section V-C. All dimensions are in micrometers.

negligible [47]. This type of transmission line is typical for interconnects in integrated circuits. The trapezoidal shape of the signal lines arises, which is a cause , for example, by under etching or electrolytical growth. As shown in [46], approximating signal lines with perfect rectangles results in a nonnegligible error on the p.u.l. resistance and inductance. Fig. 8 shows various entries of the p.u.l. resistance and inductance matrices of the transmission line, computed with the proposed technique and an FEM solver (COMSOL Multiphysics). The proposed technique correctly captures the nontrivial impedance behavior over frequency. The results shown in Fig. 8 also demonstrate

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TABLE III E XAMPLE IN S ECTION V-C: CPU T IME FOR THE FEM AND THE P ROPOSED M ETHOD FOR D IFFERENT M ESH S IZES

TABLE IV E XAMPLE OF S ECTION V-C: N ORMALIZED E RROR (%) IN S ELECTED p.u.l R ESISTANCE AND I NDUCTANCE C OEFFICIENTS , FOR THE FEM AND THE P ROPOSED M ETHOD , U SING D IFFERENT M ESH S IZES

Fig. 9. Cross section of the curved microstrips of Section V-D. Geometrical dimensions are: α1 = 10°, α2 = 20°, α3 = 30°, α4 = 35°, R1 = 80 μm, R2 = 100 μm, and h = 2 μm.

at high frequency, and fail to accurately resolve the narrow region where current is confined by skin effect. The proposed method, on the other hand, with only 236 pulse basis functions gives a maximum error of 2.10%, which drops below 1% if the number of basis functions is increased to 860. CPU times, reported in Table III, show the superior computational efficiency of the proposed method with respect to FEM. D. Curved Microstrip Lines

Fig. 8. P.u.l. resistance and inductance of the on-chip transmission line with trapezoidal conductors considered in Section V-C.

the robustness of the proposed method, at both low and high frequencies. In order to analyze the accuracy and CPU time scale with mesh size, we repeated the analysis for different levels of discretization, summarized in Table III. The normalized error in p.u.l. resistance and inductance is shown for selected selfand mutual-terms in Table IV, using the finest FEM mesh as a reference. For the FEM method, only the mesh FEM 4 gives reasonably accurate results. Meshes 1–3 are inaccurate

Flexible dielectrics allow for the creation of curved interconnects conformal to a cylindrical surface [48], [49]. We analyze the configuration shown in Fig. 9, which features four copper conductors (σ = 5.8 · 107 S/m). Conductors were chosen to be very thin and wide to demonstrate the robustness of the proposed approach in handling conductors with large aspect ratio. Fig. 10 shows the selected entries of the resistance and inductance matrices, computed with the proposed method and an FEM solver (COMSOL Multiphysics). A close match can be observed, which validates the proposed technique. For the FEM simulation, we discretized the geometry with 110 416 triangular and 28 360 boundary elements. In the proposed method, a total of 373 pulse basis functions were used to discretize the conductors boundary. Table II shows that,

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Future work will include the extension of this method to the calculation of p.u.l. capacitance and conductance, especially in transmission lines with semiconductors, where a proper modeling of charge dynamics inside semiconductors is required [42]. A PPENDIX A. Evaluation of Diagonal Entries of P Matrix For the diagonal entries of P, the integral in (29) reads ˆ wm 2 [P]m,m = ωμ [C0 J0 (kr  ) − j Y0 (kr  )]dr  (46) 0

where wm is the width of mth pulse basis function. The Neumann function Y0 (kr  ) has a singularity when its argument is equal to zero. Therefore, we must evaluate the above integral using the low argument approximation of the Bessel and the Neumann functions [39], as shown in [30]. For low and medium frequencies, the small-argument approximation holds for all the values of r  in the integral of (46), which can be evaluated analytically yielding     ωμwm ωμwm kwm − [P]m,m = C0 ln −1+γ (47) 2 π 4 where γ is Euler’s constant [39]. At high frequency, the integral in (46) is broken down into two integrals, corresponding to the small and large values of k r . For kr  < 0.1, the smallargument approximation of J0 (kr  ) and Y0 (kr  ) still holds, and a formula analogous to (47) can be used. For kr  > 0.1, the integral has to be evaluated numerically using, for example, Gaussian quadrature formulas, as done in the numerical tests in this paper. R EFERENCES

Fig. 10. Selected entries of the p.u.l. resistance and inductance matrices of the curved microstrip lines of Section V-D.

with the proposed method, the p.u.l. impedance of the conformal interconnect can be obtained in 0.51 s per frequency. VI. C ONCLUSION We presented an efficient approach to model the skin effect in the conductors of arbitrary cross section. The proposed approach is based on a surface admittance operator. We show how the operator can be efficiently computed from a contour integral. The proposed approach can handle the conductors of arbitrary shape, unlike the popular eigenfunctions method which is viable only for canonical shapes, where eigenfunctions are available analytically. This advantage comes at the expense of a moderately higher CPU time to calculate the surface operator. The novel method has been applied to the computation of the resistance and inductance of transmission lines with rectangular, circular, trapezoidal, V-shaped, and curved conductors. Numerical results demonstrate the efficiency, robustness, and accuracy of the proposed technique.

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Utkarsh R. Patel (S’13) received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 2012 and 2014, respectively, where he is currently pursuing the Ph.D. degree in electrical engineering. His current research interests include applied electromagnetics and signal processing.

Piero Triverio (S’06–M’09–SM’16) received the M.Sc. and Ph.D. degrees in electronic engineering from the Politecnico di Torino, Turin, Italy, in 2005 and 2009, respectively. He is an Assistant Professor with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada, where he is the Canada Research Chair in Modeling of Electrical Interconnects. His current research interests include signal integrity, electromagnetic compatibility, and model order reduction. Dr. Triverio was the recipient of several international awards, including the Best Paper Award of the IEEE T RANSACTIONS ON A DVANCED PACKAGING in 2007, the EuMIC Young Engineer Prize of the 13th European Microwave Week, and the Best Paper Award of the IEEE 17th Topical Meeting on Electrical Performance of Electronic Packaging.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

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A Fully Coupled Nonlinear Scheme for Time-Domain Modeling of High-Power Microwave Air Breakdown Su Yan, Member, IEEE, and Jian-Ming Jin, Fellow, IEEE

Abstract— In this paper, the air breakdown problem encountered with high-power microwave operation is modeled using a fully coupled nonlinear Newton scheme in the time domain. As a highly nonlinear process, the air breakdown is resulted from the complicated electromagnetic–plasma interactions, which can be described by a coupled system where Maxwell’s equations govern the electromagnetic fields, and a simplified plasma fluid equation governs the plasma current. The resulting nonlinear Maxwell’s equations are solved by the time-domain finite-element method with a proposed Newton’s method, while the simplified plasma fluid equation is solved with another point-wise Newton’s method. These two sets of equations are coupled together using a proposed inner–outer iterative scheme to guarantee the convergence and accuracy of the numerical solution. Numerical examples are presented to characterize the nonlinear phenomenon of the breakdown process and the self-sustaining property of the plasma current. Index Terms— Air breakdown, fluid model, fully coupled scheme, high-power microwave (HPM), Newton’s method, nonlinear modeling, plasma physics, self-sustaining, time-domain finiteelement method (TDFEM).

I. I NTRODUCTION IGH-POWER microwave (HPM) devices and systems have very important military and civilian applications. To design better HPM devices that generate higher electromagnetic power and longer pulsewidth, extensive research effort has been devoted to the development of microwave sources [1], [2], the design of output windows [3], [4], and the optimization of advanced cathodes. However, as the power density increases, the air breakdown can be triggered in the HPM devices and systems, which is usually hazardous and can become a limiting factor for the high-power radiation to be generated and transmitted. During the transmission of the HPM pulses in air, the neutral gas is ionized into free electrons and ions, which are accelerated by the high-intensity electromagnetic fields. The motions of these charged particles produce impact on other gas particles through collision, which can release more free

H

Manuscript received February 29, 2016; revised June 24, 2016 and July 5, 2016; accepted July 6, 2016. This work was supported by AFRL/RDMP (Kirtland AFB) under the program BAARDK-2012-0001 with Contract FA9451-14-1-0349. The authors are with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801-2991 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2593724

electrons and ions into the device. These newly generated charged particles can also be accelerated by the electromagnetic fields and impact other particles. Meanwhile, they also radiate the secondary electromagnetic fields, which can either enhance or weaken the original incident fields. If the power density of the HPM pulses is sufficiently high, such a process continues and becomes highly nonlinear. This will result in an exponential increase of the total number of the charged particles, which is commonly known as breakdown. This may lead to the malfunction or even damage of the HPM devices and systems. To have a better understanding of the breakdown mechanism and process, research investigation has been conducted experimentally [3]–[6], theoretically [7], and computationally [8]–[11]. By comparing the numerical results obtained from numerical simulations with experimental observations, the breakdown theories can be validated, and the underlying physics can be better understood. However, in order to simulate the breakdown process accurately, two major issues must be addressed properly. One is related to a good physical model that describes the breakdown process with a sufficient accuracy. The other is related to a numerical method that is capable to solve the resulting nonlinear system with a high accuracy and good robustness. In [11], a nonlinear conductivity model that describes the conduction current as a nonlinear function of the electric field has been employed to simulate the dielectric breakdown process in solid materials, which has been solved with the time-domain finite-element method (TDFEM) [12] and Newton’s method. More complicated physical model with higher fidelity can also be employed. For example, by taking the first two moments of the Boltzmann equation, an ionization–diffusion model is obtained [10]. Such a model has been coupled with Maxwell’s equations and solved using a coupled discontinuous Galerkin time-domain method [13] to simulate the plasma formation and shielding effect during HPM breakdown in [14]. In Section II of this paper, a simplified plasma fluid model [15] that is based on the momentum transfer equation with particle collision taken into consideration is employed to describe the air breakdown phenomenon. Due to the nonlinearity of the physical process and the physical model itself, the resulting Maxwell’s equations also become nonlinear. To model such a nonlinear phenomenon, the numerical methods should be able to solve nonlinear Maxwell’s equations accurately. In [15], the coupled electromagnetic–plasma

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system is solved with a leap-frog coupling scheme, where Maxwell’s equations are treated as linear equations and solved at integer time steps, and the plasma fluid equation is solved with a nonlinear Newton’s method at half-integer time steps. Such a scheme is usually referred to as the one-way coupling scheme, and cannot obtain accurate solutions in a long time integration if a strong nonlinearity is experienced. To obtain accurate solutions, the time-step size required in this scheme must be extremely small. In Section IV of this paper, we solve the nonlinear Maxwell’s equations using a proposed Newton’s method based on the TDFEM, and solve the nonlinear plasma fluid equation using another point-wise Newton’s method at each Gaussian quadrature point throughout the simulation domain. The nonlinear solutions of these two sets of equations are coupled at the same time step using a proposed fully coupled scheme to achieve a converged solution before marching onto the next time step. Since the fully coupled scheme updates both solutions at the same time step, it is a two-way coupling scheme, which is able to obtain accurate solutions during the breakdown process. It should be pointed out that although the breakdown model used in this paper is relatively simple, other more sophisticated plasma models can be incorporated in a similar manner and solved by the proposed method. II. P LASMA M ODEL In this paper, a simplified plasma fluid model [15] is adopted to describe the air breakdown phenomenon. The model is based on the unmagnetized momentum transfer equation with particle collision taken into account, which relates the plasma current density J p with the electric field intensity E as 1 ∂ Jp + ρ p Jp = E ε0 ω2p ∂t

(1)

where ε0 denotes the permittivity in air, ρ p = νm /ε0 ω2p denotes the plasma resistivity, which is a function of the electron collision frequency νm , and the plasma frequency ω p = (ne2 /mε0 )1/2 , where n stands for the electron density, e and m stand for the electron charge and mass at rest, respectively. The plasma frequency, as one of the most important concepts in plasma physics, is the intrinsic frequency of plasma oscillation under the restoring Coulomb force when displaced by a small distance from its rest position. The resistivity transition during breakdown can be described by expressing ρ p as a simple nonlinear function [15]    ρmin − ρmax π  Jp  ρmin + ρmax + tanh ρp = −1 2 2 2 Jt (2) which is an empirical formulation used to mimic the I –V characteristic of glow discharge fluorescent lamps. In (2), Jt stands for the transition point of breakdown, which can also be used to determine the dielectric strength E max ≈ (1/2)ρmax Jt , since the breakdown occurs when the plasma current exceeds the transition current such that  J p  ≥ Jt . Clearly, with the increase of the plasma current, the plasma resistivity decreases from its maximum value ρmax to its minimum value ρmin . Fig. 1 shows the relation between the plasma resistivity and the plasma current density. Typical values of the parameters are

Fig. 1.

Plasma resistivity as a function of the plasma current density.

chosen as those suggested in [15], where ρmax = 1012 -m, ρmin = 7 × 10−3 -m, and Jt = 6 × 10−6 A/m2 . In the figure, the value of Jt is indicated by the vertical dashed gray line. Apparently, when  J p  < Jt , the plasma resistivity ρ p stays as a constant, which makes (1) a linear equation that relates J p and E. However, when breakdown happens, the magnitude of J p starts to exceed the transition current Jt , and the plasma resistivity starts to drop drastically, which can change over 14 orders of magnitude when the plasma current increases for only one order of magnitude. This makes the plasma equation (1) highly stiff and nonlinear. After breakdown happens, ρ p approaches ρmin , which is almost zero, and the plasma equation becomes 1 ∂ J p ≈ E. ε0 ω2p ∂t

(3)

As a result, there is a 90◦ phase difference between J p and E (in terms of a time delay in the time domain), which dissipates no electric energy, since in the frequency domain, J p · E is purely imaginary, and is responsible for the self-sustaining effect of the oscillating plasma, as will be shown and elaborated in Section VI. To understand the physical significance of the simplified plasma fluid model, one can substitute the definitions of the plasma frequency ω p , the plasma resistivity ρ p , and the plasma current J p = enu (with u being the electron velocity) back into (1) to obtain m ∂enu νm m + enu = E ne2 ∂t ne2 which becomes the momentum transfer equation

(4)

e ∂u + νm u = E (5) ∂t m if the electron density n is assumed to be time invariant. As a result, the simplified plasma fluid equation can be interpreted as the momentum transfer equation with a time-invariant electron density n (and therefore, a fixed plasma frequency ω p ), and a time-dependent collision frequency νm , which is varying as a function of the electric field. III. C OUPLED E LECTROMAGNETIC –P LASMA S YSTEM With the plasma modeled by the simplified fluid model, the air breakdown problem is governed by the coupled

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Fig. 2.

Illustration of the coupled time-stepping scheme.

electromagnetic–plasma system ∂ H =0 ∂t ∂ ∇ × H − ε0 E = J p (E) + Ji ∂t 1 ∂ Jp + ρ p ( Jp ) Jp = E ε0 ω2p ∂t ∇ × E + μ0

(6) (7) (8)

where μ0 stands for the permeability in air and Ji stands for the impressed current source that provides excitation to the system. During breakdown, since ρ p is a nonlinear function of J p , which makes J p a nonlinear function of E, the coupled system (6)–(8) is essentially nonlinear. In order to simulate the breakdown process accurately, these three coupled equations have to be solved simultaneously to preserve the self-consistency of the physical system. In this paper, a fully coupled self-consistent Newton’s method is proposed to solve the coupled electromagnetic–plasma system in the time domain. To this end, the two Maxwell’s equations are first reformulated to obtain a second-order wave equation to reduce the number of unknown variables. Newton’s method is then applied to both the wave equation and the plasma fluid equation in a coupled manner. A traditional approach to obtain the second-order wave equation is to eliminate the magnetic field H from Maxwell’s equations, which results in the so-called E-formulation as ε0

∂2 ∂ 1 ∂ E + J p (E) + ∇ × ∇ × E = − Ji . 2 ∂t ∂t μ0 ∂t

(9)

However, as pointed out in [16], the E-formulation suffers from the later-time linear drift problem, which, in a lossless and source-free case, permits a spurious solution E = −(at + b)∇φ that is linearly increasing with respect to time. Moreover, to apply Newton’s method based on the E-formulation, the partial derivative of ∂ J p (E)/∂t with respect to E has to be performed in order to obtain the Jacobian matrix, which is quite problematic given the complicated relation between J p and E. To overcome these issues, we employ the A-formulation proposed in [16] by defining the time integral of the electric field E as an auxiliary vector  t E(r, τ )dτ (10) A(r, t) = −

Comparing the A-formulation (13) with the E-formulation (9), it is clear that the plasma current term is no longer inside the time derivative, which will facilitate the application of Newton’s method. Moreover, the spurious solution of the electric field can be effectively suppressed, since in a lossless and source-free problem solved by the A-formulation, E = −∂ A/∂t = a∇φ, which is no longer a linear function of time and can be eliminated by enforcing the initial conditions. The coupled electromagnetic–plasma system is now reformulated and can be described by (13) and (8), with the electric field E and the auxiliary vector A related by (11). IV. C OUPLED N EWTON S OLVER In this section, a coupled Newton solver is applied to solve the coupled nonlinear electromagnetic–plasma system accurately. Newton’s methods for the A-formulation and the plasma equation are derived, and their coupling scheme is then introduced. To derive Newton’s method for the nonlinear A-formulation, the FEM is first used to discretize (13) into a matrix equation by expressing the auxiliary vector as the linear combination of the vector basis functions [17]–[19] as A=

∂2 η0 η0 1 [M] {a} − {J } + [S]{a} = {b} 2 2 ∂t c0 c0 c0 where

∂ A(r, t) ∂t B(r, t) = ∇ × A(r, t).

(11) (12)

Substituting (11) and (12) back into (7) yields A − J p ( A) + ∇ ×

1 ∇ × A = Ji . μ0

(13)

(14)

(15)

 Mi j = Ji = Si j =

  V   V   V

bi =

Ni · N j d V

(16)

Ni · J p d V

(17)

∇ × Ni · ∇ × N j d V

(18)

Ni · Ji d V

(19)

V

where c0 is the speed of light, η0 = (μ0 /ε0 )1/2 is the wave impedance in vacuum, and V is the solution domain. The semi-discrete equation is then integrated in time using the well-known Newmark-β method [20]–[22] to obtain a unconditionally stable system with a second-order accuracy [K ]n+1 {a}n+1 − η0 c0 t 2 {J }

˜ = −[K ]n {a}n − [K ]n−1 {a}n−1 + {b}

E(r, t) = −

∂t 2

a j (t)N j (r)

and testing the A-formulation with the testing functions Ni (r) to yield the semi-discrete equation

such that

ε0

N  j =1

−∞

∂2

3

(20)

where 1 [K ]n±1 = [M] + c02 t 2 [S] 4 1 n [K ] = −2[M] + c02 t 2 [S] 2  1 1 2 1 ˜ {b}n+1 + {b}n + {b}n−1 {b} = η0 c0 t 4 2 4

(21) (22) (23)

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Fig. 3. TEM wave travels through a coaxial waveguide from z = −100 to 100 mm. The waveguide is divided into three segments in z-direction. The first and third segments are filled with vacuum, while the second (the middle) segment is filled with air. (a) Front view. (b) Side view.

which can be solved for the unknown coefficients {a}n+1 at the (n + 1)th time step from the known solutions at the (n − 1)th and the nth time steps. Once the auxiliary vector A is obtained at the (n + 1)th time step, the electric field E can be recovered and used as the source term of the plasma equation. Note that since J p is a nonlinear function of E, which is a function of {a}n+1 as ∂A E=− ∂t N  ∂a j (t) N j (r) =− ∂t j =1

N 1   n+1 a j − a n−1 Nj =− j 2 t

(24)

j =1

the {J } vector is a nonlinear function of the unknown coefficients {a}n+1 at time step n + 1. To solve the nonlinear equation (20) for the unknown coefficients {a}n+1 , Newton’s method can be applied to update iteratively as its estimation {a}n+1 k n+1 n+1 − [J ]−1 {a}n+1 k+1 = {a}k k {r }k

(25)

where {r }n+1 is the nonlinear residue evaluated using the k current estimation {a}n+1 k {r }n+1 k

= [K ]

{a}n+1 k n n

n+1

2

− η0 c0 t {J }

˜ + [K ] {a} + [K ]n−1 {a}n−1 − {b}

(26)

and [J ]k is the Jacobian matrix defined as the partial derivative of the nonlinear residue with respect to the unknown coefficient as Jk,i j =

n+1 ∂rk,i n+1 ∂ak, j

= K in+1 − η0 c0 t 2 j

∂ Ji n+1 ∂ak, j

.

(27)

The derivative on Ji can be further expressed as  ∂ Jp ∂ Ji = Ni · n+1 d V n+1 ∂ak, j ∂ak, j V  ∂ Jp ∂E · n+1 d V = Ni · ∂ E ∂a V k, j  1 Ni · σ¯ d · N j d V. (28) =− 2 t V Equation (24) is used in reaching the above expression, where σ¯ d = ∂ J p /∂ E is the differential conductivity tensor. Substituting (28) back into (27) yields the final explicit expression for the Jacobian matrix element 1 Jk,i j = K in+1 + η0 c0 t G k,i j (29) j 2

Fig. 4. (a) Electric field recorded in the nonlinear region at z = −79 mm, with the comparison made between results obtained from Newton’s method (NT), the fixed-point (FP) method, and the one-way coupling scheme (OW), with various time-step sizes. (b) Zoomed-in view from 2 to 4 ns.

with

 Ni · σ¯ d · N j d V.

G k,i j =

(30)

V

Since the relation between J p and E is very complicated, it is difficult to obtain an expression for the differential conductivity tensor in closed form. However, one can still use finite difference to obtain σ¯ d as a 3 × 3 matrix ⎡ ⎤

J px J px J px ⎢ E x

E y E z ⎥ ⎢ ⎥ ⎢

J

J

J py py J py ⎥ p d ⎢ ⎥ =⎢ (31) σ¯ =

E y E z ⎥

E ⎢ E x ⎥ ⎣ J pz

J pz J pz ⎦

E x

E y E z which needs to be evaluated at every Gaussian quadrature point of every tetrahedral element to complete the volume integral in the construction of the Jacobian matrix. Once the Jacobian matrix is obtained, the unknown coefficients can be updated using Newton’s iteration (25) until convergence is reached. The electric field can then be recovered by E = −∂ A/∂t and used as the source term in (8) to solve for the plasma current J p at every Gaussian quadrature point. To achieve the same second-order accuracy, the trapezoidal rule   1 n+ 1 n− 1 J p 2 + Jp 2 (32) Jp = 2   ∂ 1 n+ 1 n− 1 Jp = J p 2 − Jp 2 (33) ∂t

t

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5

Fig. 5. Nonlinear convergence of the fixed-point method for (a) wave equation and (b) plasma fluid equation.

Fig. 6. Nonlinear convergence of Newton’s method for (a) wave equation and (b) plasma fluid equation.

is used to discretize (8) as     ρp 1 n+ 12 n− 12 n+ 12 n− 12 J J + = E n (34) − J + J p p p p ε0 ω2p t 2

Once J p 2 is obtained, the plasma current can be updated and fed back into (31) to calculate the differential conductivity tensor, which is then used for the next Newton iteration in (25). Because of the strong nonlinearity experienced during the breakdown process, the coupled electromagnetic–plasma system has to be solved concurrently to ensure the accuracy of the solution. In this paper, a fully coupled Newton scheme is designed to solve both (13) and (8) using Newton’s method. As described above, the wave equation is solved at integer time steps n − 1, n, and n + 1, and the plasma fluid equation is solved at half-integer time steps n − (1/2) and n + (1/2), as shown in Fig. 2. The coupling between these two equations is carried out through an inner–outer Newton iteration scheme, shown in Algorithm 1, to ensure that all the physical quantities are converged before marching onto the next time step. In the inner Newton iteration, the plasma fluid equation is solved at all the Gaussian quadrature points in parallel, since it is valid point wisely and cheap to solve. In the outer Newton iteration, the wave equation is solved globally, which is much more expensive than solving the plasma fluid equation, and therefore, is designed as the outer iteration in the coupling scheme.

where

   1 n+ 12 n− 12 ρp = ρp J p + Jp 2

(35)

calculated in accordance with (2). The plasma current at (n+(1/2))th time step can be obtained by another Newton’s iteration n+ 1

n+ 1

n+ 12

2 = J p,s 2 − [J]−1 J p,s+1 s { f }s

(36)

where the nonlinear residue is n+ 1 { f }s 2

n+ 1

n− 12

J p,s 2 − J p = ε0 ω2p t

+

ρ p,s 2



n+ 1

n− 12

J p,s 2 + J p

 − En (37)

and the Jacobian matrix is a 3 × 3 matrix, which can be expressed analytically as   1 1 + ρ p,s [I] + κ{J p }{J p }T [J]s = (38) ε0 ω2p t 2 with [I] being a 3 × 3 identity matrix, and {J p } = {J px , J py , J pz }     Jp  π ρmin − ρmax 2 π κ = sech −1 . 8  J p Jt 2 Jt T

n+ 1

V. N UMERICAL VALIDATION (39) (40)

In this section, the proposed coupled Newton’s method is validated, and the accuracy of the one-way coupling scheme is investigated. To this end, a simple fixed-point

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Fig. 7. Electric field and plasma current recorded near (a)–(c) first interface and (d)–(f) second interface that separate the linear and nonlinear regions, which demonstrate the behavior when the air breaks down and does not breakdown, respectively. (a) and (d) Electric field versus time. (b) and (e) Plasma current versus time. (c) and (f) J–E relation.

Algorithm 1 Fully Coupled Newton Scheme for the Coupled Electromagnetic–Plasma System n− 1 Input: An−1 , An , and Jp 2 n+ 1 Output: An+1 , and Jp 2 Set initial guess: An+1 = An , 0

n+ 1 ,0

Jp,0 2

n− 12

= Jp

;

Solve for auxiliary variable A: for each Newton step k = 1, 2, . . . do Construct the Jacobian matrix [J ]k ; Solve the Jacobian matrix equation for An+1 k ;  n+1 1 n Update the electric field Ek = − 2 t Ak − An−1 ; n+ 12 ,k

Solve for plasma current at every point Jp for each Newton step s = 1, 2, . . . do Construct the Jacobian matrix [J]s ;

:

n+ 1 ,k

Solve the Jacobian matrix equation for Jp,s 2 ; end  n+ 1 ,k n− 1 Update the plasma current Jpn,k = 12 Jp 2 + Jp 2 ; Calculate the differential conductivity tensor σ¯ d ; end n+ 12

Output An+1 , and Jp

;

method [11], [23] is used as a reference solution. In the fixed-point method, the plasma current J p is not treated as an unknown term, but as a known current source term. This results in a linear system that can be updated iteratively until the convergence of the unknown vector {a} is reached.

Such a procedure can be depicted as (k = 1, 2, . . .) n n n−1 ˜ {a}n−1 + {b} [K ]n+1 {a}n+1 k+1 = −[K ] {a} − [K ]   1 1 n+ 2 2 n− 21 + η0 c0 t {J }k + {J } . (41) 2

At the kth step of the fixed-point iteration, the plasma current n+(1/2) {J }k is obtained by solving the plasma fluid equation using Newton’s method presented in (36). Since in the fixedpoint method, the construction and solution of the Jacobian matrix equation are avoided, it is much cheaper to conduct the fixed-point iteration compared with the Newton iteration. However, the iterative convergence of the fixed-point method is only linear, as opposed to the quadratic convergence of Newton’s method. To guarantee the convergence, a much smaller time-step size is usually required, so that the iteration always starts from a good initial guess. Moreover, it is recognized that the one-way coupling scheme is the same as the fixed-point method, if the number of the fixed-point iteration is limited to one at each time step. The geometric model used in this investigation is a coaxial waveguide shown in Fig. 3. Laying along the z direction, the waveguide has a total length of 200 mm (z ∈ [−100, 100] mm), an inner and outer radius of 2 and 4.6 mm, respectively, and is divided into three segments at z = −80 mm and z = 80 mm. The first and third segments are linear regions filled with vacuum, and the middle segment is a nonlinear region filled with air, which has a plasma frequency of ω p = 30 × 109 rad/s. The plasma resistivity and transition current in the air region are set

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Fig. 8. Port voltage recorded at (a) input port and (b) output port as a function of time. The results from a linear case, a nonlinear case with a high plasma frequency, and a nonlinear case with a low plasma frequency are shown.

as those given in Section II, which results in a dielectric strength of E max ≈ 3 MV/m. The coaxial waveguide is operating at the TEM mode and excited by a modulated Gaussian pulse with a center frequency of 5 GHz, a 100% inc = 25 kV, bandwidth, and a maximum pulse magnitude of Vmax which is high enough to trigger the breakdown in the coaxial waveguide. This problem is solved using the proposed Newton’s method, the reference fixed-point method, and the one-way coupling scheme, respectively. A time-step size of t = 1 ps is used in Newton’s method, while a much smaller time-step size of t = 0.05 ps is employed in the fixed-point method to guarantee its convergence. For the one-way coupling scheme, both the larger and the smaller time-step sizes are considered. The electric fields recorded at z = −79 mm in all the three methods are presented and compared in Fig. 4. From Fig. 4, it is clear that in early times from 0 to about 1.8 ns, the results from all the methods agree well with each other. However, from 1.8 ns on, the result obtained from the one-way coupling scheme starts to show a larger and larger deviation from the solution of Newton’s method. It is very interesting to see that by reducing the time-step size of the one-way coupling scheme twenty times from t = 1 ps to t = 0.05 ps, its result converges toward Newton’s solution, but still exhibits observable difference from the latter one. Finally, if the fixedpoint method with t = 0.05 ps is used, its solution becomes identical to Newton’s solution. This example validates the proposed Newton’s method, and demonstrates its accuracy

7

even when a large time-step size is used. From this investigation, it is also very clear that since it does not converge to an accurate result at each time step, the one-way coupling scheme will eventually generate erroneous results in a long time integration. For both the fixed-point and Newton’s methods, a relative residue of 10−7 on the nonlinear function is set as the convergence criterion. Figs. 5(a) and 6(a) show the fixed-point and Newton iteration counts in solving the wave equation, respectively. It can be seen from Fig. 6(a) that the global Newton iteration (25) for the wave equation usually converges within one to three steps, which confirms the quadratic nonlinear convergence of the proposed Newton’s method. From Fig. 5(a), it usually takes the fixed-point iteration (41) seven to eight steps to converge, which confirms the linear convergence rate of the fixed-point method. Fig. 5(a) also indicates that the erroneous results of the one-way coupling scheme is due to the fact that the fixed-point method cannot produce converged results if only one iteration is allowed per time step. Figs. 5(b) and 6(b) show the respective Newton iteration counts in solving the plasma equation in these two methods. The point-wise Newton iteration (36) for the plasma equation takes many more steps to converge to the same tolerance [Figs. 5(b) and 6(b)], which explains why the Newton iteration for the wave equation needs to be the outer iteration in the proposed coupling scheme. In Figs. 5(b) and 6(b), the maximum, minimum, and average iteration counts at the nth time step are defined as   (42) Cmax (n) = max Cin i  n Cmin (n) = min Ci (43) i

Cave (n) =

 1 Cin M×P

(44)

i

respectively, where Cin stands for the Newton iteration count at the i th quadrature point and the nth time step, i = 1, 2, . . . , M × P, with M being the total number of discretization elements in the nonlinear region and P being the number of Gaussian quadrature points in each element. VI. N UMERICAL E XAMPLES In this section, two sets of numerical examples are presented to demonstrate the capability of the proposed method and to show the nonlinear phenomena during the breakdown process. A. Investigation of Air Breakdown Behavior For the first set of examples, the nonlinear behavior of air breakdown under high-power operation is investigated, which includes a demonstration of the electric field and plasma current behaviors when breakdown is and is not encountered, and an investigation of different breakdown behaviors at different plasma frequencies. The geometric model used in this section is the same as that used in Section V.

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9 L 9 Fig. 9. Electric field and plasma current recorded near the first interface in the case where (a)–(c) ω H p = 65 × 10 rad/s and (d)–(f) ω p = 7 × 10 rad/s for the air region, which demonstrate the different behaviors for the breakdown plasma at different plasma frequencies. (a) and (d) Electric field versus time. (b) and (e) Plasma current versus time. (c) and (f) Power spectrum of the fields.

1) Breakdown Versus No Breakdown: This investigation intends to demonstrate different behaviors when the air breakdown does and does not take place. With the plasma frequency set as ω p = 30 × 109 rad/s, the electric fields and plasma currents near the first and second interfaces that separate the linear and nonlinear regions are recorded and shown in Fig. 7. Specifically, Fig. 7(a) shows the electric fields recorded at two neighboring points close to the first interface in the linear and nonlinear regions with z = −81 mm and −79 mm, respectively, as functions of time. From Fig. 7(a), it can be seen that the electric field is enhanced due to the air breakdown in the nonlinear region. It is very interesting that even after the incident pulse has traveled through, the electric field and plasma current in the nonlinear region continue to oscillate, as can be seen from Fig. 7(b). This is due to the self-sustaining electron plasma and the plasma current it generates. The nonlinear relation between the electric field and the plasma current is shown in Fig. 7(c) (plotted from 2 ns), from which a 90◦ phase difference can be observed clearly. Such a 90◦ phase difference between the field and the current is critical to the self-sustaining of the plasma, since it results in a purely imaginary dissipative power. Considering J p = enu with e and n being constants, the plasma energy is merely converting back and forth between the kinetic energy that is stored in the electron motion and the electric energy that is stored in the electric field oscillation. Due to the conservation of energy, however, since all the energy in the system is supplied by the incident pulse, if a portion of the energy is trapped in the nonlinear region close to the first interface at z = −80 mm, the remainder of the energy that propagates to the second

Fig. 10. Illustration of the geometry induced air breakdown. (a) Standard rectangular waveguide with two needle-shaped metal probes. (b) Zoomed-in view of the probes and the mesh of the geometry.

interface at z = 80 mm is not high enough to cause the air breakdown. As can be seen from Fig. 7(a)–(f), without breakdown there is no significant difference between the electric fields in the linear and the nonlinear regions, where z = 81 and 79 mm, respectively, except for a very small time delay [Fig. 7(d)], and the plasma current and the electric field are in phase [Fig. 7(e)] and have a linear relation [Fig. 7(f)]. It should also be noticed from the comparison between Fig. 7(b) and (e) that the magnitudes of the plasma currents between the breakdown and nonbreakdown cases differ by twelve orders of magnitude, while those of the electric fields are only about three times difference at their maxima. 2) Breakdown at Different Plasma Frequencies: In this example, the effect of plasma frequency in the breakdown process is investigated, where two different plasma frequencies 9 are considered. One is ω H p = 65 × 10 rad/s that is higher than the center frequency of the input modulated Gaussian pulse ω0 = 31.4 × 109 rad/s ( f 0 = 5 GHz). The other

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9

Fig. 11. Snapshot of the electric field distribution inside the rectangular waveguide recorded at t = 12 ns. The waveguide is filled with (a) vacuum, (b) air 9 with a plasma frequency ω Lp = 2.4 × 109 rad/s, and (c) air with a plasma frequency ω H p = 4.8 × 10 rad/s, respectively.

is ω Lp = 7 × 109 rad/s that is lower than ω0 . The port voltage recorded at the input and output ports are shown in Fig. 8, with the result from a linear case (σ = 0) presented as a reference. It is obvious that at both ports, the voltage recorded in the low-plasma-frequency case is almost identical to that in the linear case, indicating that the signal can propagate though the coaxial waveguide without much disturbance even with the air breakdown taking place, as will be seen in a moment. In the high-plasma-frequency case, however, because of the air breakdown, the incident pulse could not travel through the waveguide, causing a significant reflection of the incident pulse back to the input port, and an incomplete transmission of the pulse to the output port. Such phenomena observed in these two figures agree well with the plasma physics. Because when the plasma frequency is higher than the incident frequency, the plasma can oscillate fast enough to cancel the incident wave, and therefore, acts like an electromagnetic shield that blocks the incident field. When the plasma frequency is lower than the incident frequency, the plasma cannot respond fast enough to cancel the incident field, and as a result, the incident wave is able to pass through the plasma region without much disturbance. When looked closer to what happened inside the waveguide, the physical process becomes more clear. As shown in Fig. 9(a) and (b), in the high-plasma-frequency case, a significant amount of energy is trapped in the nonlinear region near the first interface in terms of the plasma oscillation. A portion of the electric field travels back across the interface between the linear and nonlinear regions and results in the oscillatory tail of the electric field at z = −81 mm shown in Fig. 9(a). When the Fourier transform is performed on the electric field at z = −79 mm, two major frequency components can be observed in Fig. 9(c). One is a wide frequency band around 5 GHz, which is due to the forced oscillation of the plasma when the incident pulse travels through. The other is a narrow frequency band centered at ω H p , which is the intrinsic oscillation of the plasma after the incident pulse is gone. Different from the high-plasma-frequency case, in the lowplasma-frequency case, very little electric energy is trapped in the nonlinear region [Fig. 9(d)], indicating that the majority of energy travels through the observing point. From Fig. 9(e),

Fig. 12. Snapshot of the plasma current distribution inside the rectangular waveguide recorded at t = 12 ns. The waveguide is filled with air with a 9 plasma frequency (a) ω Lp = 2.4 × 109 rad/s and (b) ω H p = 4.8 × 10 rad/s, respectively.

it can be seen that air breaks down when the magnitude of the electric field exceeds the dielectric strength of air, which induces a strong plasma current. When the Fourier transform is performed, two major frequency components can also be observed in Fig. 9(f), which correspond to the physics discussed earlier. B. Geometry Induced Air Breakdown Geometry induced air breakdown is investigated in this section. To induce the air breakdown, two tiny needle-shaped metal probes are placed tip-to-tip inside a standard rectangular waveguide WR-2300 with a cross-sectional dimension of 584.2 × 292.1 mm2 , as shown in Fig. 10. A modulated Gaussian pulse with a center frequency of f 0 = 400 MHz (corresponding to ω0 = 2.5 × 109 rad/s), a 100% bandwidth, inc = 0.5 MV in and a maximum pulse magnitude of Vmax TE10 mode is transmitted through the waveguide. Three different cases are considered. In the first case, the waveguide is filled with vacuum, which will not encounter any breakdown and will serve as a reference result. In the second and third cases, the waveguide is filled with air, with the plasma fre9 quency being ω Lp = 2.4 × 109 rad/s and ω H p = 4.8 × 10 rad/s, respectively. With all other plasma parameters set the same as those in the preceding examples, the air in the waveguide has a dielectric strength of E max ≈ 3 MV/m. In all the three cases, the relative permittivity and permeability are εr = 1 and μr = 1, respectively.

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Fig. 13. Electric field recorded at the middle point between two needle tips as a function of time. The waveguide is filled with (a) vacuum and (b) air 9 with a plasma frequency ω H p = 4.8 × 10 rad/s, respectively. The input signal is shown as a reference.

The electric field distribution inside the rectangular waveguide recorded at t = 12 ns is shown in Fig. 11, from which clear differences can be observed. In the vacuum case shown in Fig. 11(a), the needles act as the metal scatterers, which enhance the field around them due to the singularity of the electromagnetic fields near sharp tips. In both air cases shown in Fig. 11(b) and (c), the enhanced electric fields exceed the dielectric strength of the air and result in air breakdown around the needles. Nevertheless, the breakdown patterns in these two cases are different. In Fig. 11(b), the breakdown fields (shown in red) are very close to the needles. In Fig. 11(c), however, the breakdown fields distribute not only close to the needles, but also some distance away from the needles. This is because in the third case, the breakdown air with a higher plasma frequency has a better capability of trapping energy, which causes more breakdown around the needles. Similar phenomena can also be observed in Fig. 12, which shows the strong plasma current induced by the air breakdown. Fig. 13 compares the electric fields recorded at the middle of the two needle tips between the vacuum and 9 the air (ω H p = 4.8 × 10 rad/s) cases. From these figures, it can be seen that in the vacuum case, the electric field is much stronger than that in the air case, but dies out quickly as the incident wave passes by, because the needles in the vacuum case only act as scatterers and cannot sustain energy. In the air case, although the maximum value of the electric field is not as high as that in the vacuum case due to the lossy nature of the plasma current during the breakdown process, it can oscillate for a very long time after the air breaks down

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Fig. 14. Plasma current recorded at the middle point between two needle tips. The waveguide is filled with air with a plasma frequency ω H p = 4.8 × 109 rad/s. (a) Plasma current versus time. The input signal in time domain is shown as a reference. (b) Power spectrum of the plasma current. The input signal in frequency domain is shown as a reference.

(resulting in the 90◦ phase shift between current and field), and the incident field is gone. From the plasma current recorded at 9 the same point in the air (ω H p = 4.8 × 10 rad/s) case, shown in Fig. 14, it is clear that the plasma current oscillates in the incident frequency band when the incident signal pass through, and at the plasma frequency after the incident field is gone. More interestingly, the induced plasma current has a very strong dc component [Fig. 14(b)], which is highly undesirable in the design of HPM devices. VII. D ISCUSSION From the numerical examples presented in Section VI, it is clear that the advantages of the proposed method are its fast nonlinear convergence and its high numerical accuracy. Thanks to the proposed Newton’s method, the nonlinear convergence is quadratic, which is significantly faster than the fixed-point method, which converges only linearly, while the one-way coupling scheme cannot generate accurate results in a long time integration. Since the inner–outer iteration scheme guarantees the convergence and accuracy at each time step, it can endure the long time integration without sacrificing the numerical accuracy. In simulations with increased complexities, the proposed method will also converge, as long as the Jacobian matrix is constructed correctly and the initial guess of Newton’s iteration is chosen close enough to the true solution. The first requirement can be satisfied using the formulations developed in Section IV, especially (29), (38), and Algorithm 1. The second requirement

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is met with the time-domain method for the simulation of a nonlinear problem, since the time-step size can be set small enough such that the solution from the previous time step can be utilized as a very good initial guess of Newton’s method for the next time step. The limitation of the proposed numerical method mainly comes from the construction and solution of the Jacobian matrix equation, which can be computationally intensive. Fortunately, this limitation can be alleviated by using more efficient nonlinear update methods [24]. We would like to point out that to obtain an accurate and meaningful simulation result for a physical process, a physical model with sufficient fidelity is needed. Although the plasma model adopted in this paper is relatively simple, more sophisticated plasma models can be incorporated in a similar manner and solved by the proposed method to account for more physics of interest in the numerical simulation. For example, since the ionization and the diffusion of plasma are two dominant physical mechanisms in an HPM air breakdown, the ionization–diffusion model [10], [14], [25], [26] can be employed to describe the physical process with a higher fidelity. VIII. C ONCLUSION In this paper, the air breakdown phenomena modeled with a nonlinear plasma fluid equation are simulated numerically. The nonlinearity of the plasma current results in nonlinear Maxwell’s equations that govern the electric and magnetic fields. To solve the nonlinear Maxwell’s equations without encountering the later-time linear drift problem, the A-formulation is employed, which can also facilitate the derivation of Newton’s method in solving the nonlinear wave equation. A fully coupled Newton’s method is proposed, where a nonlinear TDFEM is applied to solve the nonlinear A-formulation for the electromagnetic fields, and a point-wise nonlinear solver is applied to solve the nonlinear plasma fluid equation for the plasma currents. The fully coupled nonlinear scheme solves the two coupled nonlinear equations simultaneously to achieve an accurate solution of the physical system. Numerical validations and examples are presented in this paper to demonstrate the capability of the proposed numerical scheme, and the nonlinear phenomena taking place in the air breakdown process. R EFERENCES [1] S. H. Gold and G. S. Nusinovich, “Review of high-power microwave source research,” Rev. Sci. Instrum., vol. 68, no. 11, pp. 3945–3974, Nov. 1997. [2] J. Benford and G. Benford, “Survey of pulse shortening in highpower microwave sources,” IEEE Trans. Plasma Sci., vol. 25, no. 2, pp. 311–317, Jun. 1997. [3] A. Neuber, J. Dickens, D. Hemmert, H. Krompholz, L. L. Hatfield, and M. Kristiansen, “Window breakdown caused by high-power microwaves,” IEEE Trans. Plasma Sci., vol. 26, no. 3, pp. 296–303, Jun. 1998. [4] D. Hemmert, A. A. Neuber, J. Dickens, H. Krompholz, L. L. Hatfield, and M. Kristiansen, “Microwave magnetic field effects on high-power microwave window breakdown,” IEEE Trans. Plasma Sci., vol. 28, no. 3, pp. 472–477, Jun. 2000.

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[5] Y. Hidaka, E. M. Choi, I. Mastovsky, M. A. Shapiro, J. R. Sirigiri, and R. J. Temkin, “Observation of large arrays of plasma filaments in air breakdown by 1.5-MW 110-GHz gyrotron pulses,” Phys. Rev. Lett., vol. 100, pp. 035003-1–035003-4, Jan. 2008. [6] Y. Hidaka et al., “Plasma structures observed in gas breakdown using a 1.5 MW, 110 GHz pulsed gyrotron,” Phys. Plasmas, vol. 16, no. 5, pp. 055702-1–055702-7, 2009. [7] S. K. Nam and J. P. Verboncoeur, “Theory of filamentary plasma array formation in microwave breakdown at near-atmospheric pressure,” Phys. Rev. Lett., vol. 103, pp. 055004-1–055004-4, Jul. 2009. [8] J. T. Krile, A. A. Neuber, H. G. Krompholz, and T. L. Gibson, “Monte Carlo simulation of high power microwave window breakdown at atmospheric conditions,” Appl. Phys. Lett., vol. 89, no. 20, pp. 201501-1–201501-3, 2006. [9] P. Zhao, C. Liao, W. Lin, L. Chang, and H. Fu, “Numerical studies of the high power microwave breakdown in gas using the fluid model with a modified electron energy distribution function,” Phys. Plasmas, vol. 18, no. 10, p. 102111, 2011. [10] J.-P. Boeuf, B. Chaudhury, and G. Q. Zhu, “Theory and modeling of selforganization and propagation of filamentary plasma arrays in microwave breakdown at atmospheric pressure,” Phys. Rev. Lett., vol. 104, no. 1, pp. 015002-1–015002-4, 2010. [11] S. Yan and J.-M. Jin, “Three-dimensional time-domain finite-element simulation of dielectric breakdown based on nonlinear conductivity model,” IEEE Trans. Antennas Propag., vol. 64, no. 7, pp. 3018–3026, Jul. 2016. [12] J.-M. Jin, The Finite Element Method in Electromagnetics, 3rd ed. Hoboken, NJ, USA: Wiley, 2014. [13] J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. New York, NY, USA: Springer, 2008. [14] S. Yan, A. D. Greenwood, and J.-M. Jin, “Modeling of plasma formation during high-power microwave breakdown in air using the discontinuous Galerkin time-domain method,” IEEE J. Multiscale Multiphys. Comput. Techn., vol. 1, pp. 2–13, Apr. 2016. [15] A. A. Costa, W. A. Artuzi, and M. J. do Couto Bonfim, “Finite-element time-domain simulation of electric discharges,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 6, pp. 1435–1439, Jun. 2008. [16] W. A. Artuzi, Jr., “Improving the Newmark time integration scheme in finite element time domain methods,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 898–900, Dec. 2005. [17] H. Whitney, Geometric Integration Theory. Princeton, NJ, USA: Princeton Univ. Press, 1957. [18] J. C. Nédélec, “Mixed finite elements in R3 ,” Numer. Math., vol. 35, no. 3, pp. 315–341, 1980. [19] J. P. Webb, “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, Aug. 1999. [20] N. M. Newmark, “A method of computation for structural dynamics,” J. Eng. Mech. Division, vol. 85, no. 3, pp. 67–94, Jul. 1959. [21] O. C. Zienkiewicz, “A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach,” Earthquake Eng. Struct. Dyn., vol. 5, no. 4, pp. 413–418, 1977. [22] S. D. Gedney and U. Navsariwala, “An unconditionally stable finite element time-domain solution of the vector wave equation,” IEEE Microw. Guided Wave Lett., vol. 5, no. 10, pp. 332–334, Oct. 1995. [23] S. Yan and J.-M. Jin, “Theoretical formulation of a time-domain finite element method for nonlinear magnetic problems in three dimensions (invited paper),” Prog. Electromagn. Res., vol. 153, no. 11, pp. 33–55, 2015. [24] S. Yan, J.-M. Jin, C.-F. Wang, and J. Kotulski, “Numerical study of a time-domain finite element method for nonlinear magnetic problems in dimensions,” Prog. Electromagn. Res., vol. 153, no. 11, pp. 69–91, 2015. [25] B. Chaudhury and J.-P. Boeuf, “Computational studies of filamentary pattern formation in a high power microwave breakdown generated air plasma,” IEEE Trans. Plasma Sci., vol. 38, no. 9, pp. 2281–2288, Sep. 2010. [26] G. Q. Zhu, J.-P. Boeuf, and B. Chaudhury, “Ionization–diffusion plasma front propagation in a microwave field,” Plasma Sour. Sci. Technol., vol. 20, no. 3, pp. 035007-1–035007-9, 2011.

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Su Yan (S’08–M’12) received the B.S. degree in electromagnetics and microwave technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2005, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana–Champaign, Urbana, IL, USA, in 2012 and 2016, respectively. He became an Instructor with the University of Illinois at Urbana–Champaign in 2012, where he is currently a Post-Doctoral Research Associate with the Department of Electrical and Computer Engineering. He has authored or co-authored over 70 papers in refereed journals and conferences. His current research interests include nonlinear electromagnetic and multiphysics problems, electromagnetic scattering and radiation, numerical methods in computational electromagnetics, especially finite element methods, integral equation based methods, fast algorithms, and preconditioning techniques. Dr. Yan is a Member of the Applied Computational Electromagnetics Society (ACES). He serves as a Reviewer for multiple journals, including the P ROCEEDINGS OF THE IEEE, the IEEE Antennas and Propagation Magazine, the IEEE T RANSACTIONS ON A NTEN NAS AND P ROPAGATION , the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES , and the IEEE T RANSACTIONS ON M AG NETICS . He was a recipient of the Best Student Paper Award by the IEEE Chengdu Section in 2010, the Best Student Paper Award (the First Place Winner) at the 27th International Review of Progress in ACES, Williamsburg, VA, USA, in 2011, the USNC/URSI Travel Fellowship Grant Award by the National Academies in 2015, and the Best Student Paper Award (the First Place Winner) at the IEEE ICWITS/ACES 2016 Conference, Honolulu, HI, USA, in 2016. He was also a recipient of the Yuen T. Lo Outstanding Research Award and the P. D. Coleman Outstanding Research Award by the Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, in 2014 and 2015, respectively. His name appeared in the University of Illinois at Urbana–Champaign’s List of Teachers Ranked as Excellent by Their Students in 2012 with an outstanding rating (top 10%).

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Jian-Ming Jin (S’87–M’89–SM’94–F’01) received the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 1989. He joined the University of Illinois at Urbana– Champaign, Urbana, IL, USA, in 1993, where he is currently the Y. T. Lo Chair Professor of Electrical and Computer Engineering and the Director of the Electromagnetics Laboratory and the Center for Computational Electromagnetics. He was appointed as the First Henry Magnuski Outstanding Young Scholar with the Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, in 1998, and later as a Sony Scholar in 2005. He was also appointed as a Distinguished Visiting Professor with the Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA, in 1999. He has been an Adjunct, Visiting, Guest, or Chair Professor for 11 institutions around the world. He has authored or co-authored over 250 papers in refereed journals and 22 book chapters. He has also authored The Finite Element Method in Electromagnetics—First Edition (Wiley, 1993), The Finite Element Method in Electromagnetics—Second Edition (Wiley, 2002), and The Finite Element Method in Electromagnetics—Third Edition (Wiley, 2014); Electromagnetic Analysis and Design in Magnetic Resonance Imaging (CRC Press, 1998); and Theory and Computation of Electromagnetic Fields— First Edition (Wiley, 2010) and Theory and Computation of Electromagnetic Fields—Second Edition (Wiley, 2015), and co-authored Computation of Special Functions (Wiley, 1996); Fast and Efficient Algorithms in Computational Electromagnetics (Artech, 2001); and Finite Element Analysis of Antennas and Arrays (Wiley, 2008). His current research interests include computational electromagnetics, scattering and antenna analysis, electromagnetic compatibility, high-frequency circuit modeling and analysis, bioelectromagnetics, and magnetic resonance imaging. Dr. Jin is a Fellow of the Electromagnetics Academy and the Applied Computational Electromagnetics Society (ACES), and a Member of the URSI Commission B. He served as an Associate Editor and a Guest Editor for the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION, Radio Science, Electromagnetics, Microwave and Optical Technology Letters, and Medical Physics. He was a recipient of the 1994 National Science Foundation Young Investigator Award, the 1995 Office of Naval Research Young Investigator Award, the 1999 ACES Valued Service Award, the 2014 ACES Technical Achievement Award, the 2016 ACES Computational Electromagnetics Award, the 2015 IEEE Antennas and Propagation Society Chen-To Tai Distinguished Educator Award, and the 2015 IEEE Antennas and Propagation Edward E. Altschuler IEEE Antennas and Propagation Magazine Prize Paper Award. He was also the recipient of the 1997 Xerox Junior Research Award and the 2000 Xerox Senior Research Award by the College of Engineering, University of Illinois at Urbana–Champaign. His name appeared 22 times in the University of Illinois at Urbana–Champaign’s List of Excellent Instructors. His students have been the recipient of Best Paper Awards of the IEEE 16th Topical Meeting on Electrical Performance of Electronic Packaging and the 25th, 27th, 31st, and 32nd Annual Review of Progress in Applied Computational Electromagnetics. He was elected by ISI as one of the world’s most cited authors in 2002.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Adaptive Decoupling Using Tunable Metamaterials Liang Zhang, Member, IEEE, Shaoqing Zhang, Zhengyong Song, Yanhui Liu, Member, IEEE, Longfang Ye, and Qing Huo Liu, Fellow, IEEE

Abstract— A novel method of decoupling two closely placed antennas is proposed based on tunable metamaterials. The experimental results show that this method has more than 110-dB isolation ratio, frequency and bandwidth reconfigurable ability, and environmental adaptivity. These features are realized by placing multiple varactor-based tunable resonant structures between two antennas within the near-field region and by using artificial intelligent algorithm to optimize the transmission coefficient between the antennas. A case of suppressing mutual coupling between two orthogonally polarized arrays is given to demonstrate the versatility of this method. Index Terms— Antenna decoupling, metamaterial, near-field interference.

I. I NTRODUCTION

I

T IS well known that mutual coupling between elements has long been a serious problem for multiple-antenna systems [1]–[6]. For a multiple-input multiple-output (MIMO) system, the antennas are usually supposed to be isolated from each other and considered as independent radiating elements [1]–[4]. For arrays, mutual coupling may have adverse influence on the input impedance, gain, sidelobe level, and radiation pattern shape [4], [5]. Mutual coupling is also known as a key factor, which degrades the polarization purity of dual-polarized base-station antenna systems. Basically, physical/analog decoupling can be realized at two stages: propagation path and analog circuit. Placing metallic partitions between antennas is a classical shie lding method mentioned in electromagnetic compatibility text books [7], [8]. Following this idea, the artificial magnetic conductor and the electromagnetic bandgap are used to reduce the mutual coupling by setting up obstacle in the propagation path [9], [10]. Unfortunately, a very large Manuscript received August 17, 2015; revised November 30, 2015, June 25, 2016 and July 4, 2016; accepted July 4, 2016. Date of publication July 29, 2016; date of current version September 1, 2016. (Corresponding author: Zhengyong Song.) L. Zhang is with the Institution of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen 361005, China, and also with the Information Engineering College, Jimei University, Xiamen 361021, China (e-mail: [email protected]). S. Zhang is with the Aviation Key Laboratory of Science and Technology on Electromagnetic Environmental Effects, Shenyang Aircraft Design and Research Institute, Shenyang 110035, China (e-mail: [email protected]). Z. Song, Y. Liu, and L. Ye are with the Institution of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen 361005, China (e-mail: [email protected]; [email protected]; [email protected]). Q. H. Liu is with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 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.2016.2590385

metallic screen is required to achieve a high isolation ratio. Unlike the shielding idea, decoupling networks/circuits use feedback signal to cancel the coupling signal [11]–[14]. In this method, a higher isolation ratio is realized and space occupation is reduced. Although fairly good isolation is achieved, the methods reported are almost static, which means in a varying environment, they cannot maintain the same performance. Recently, metamaterials-based insulator and absorber are applied to improve the isolation between closely spaced antennas [15]–[23]. If the signal reaches another antenna through different channels and the amplitude and phase responses of the channels meet certain requirements, it is possible to make the signals cancel each other at the port of the other antenna. A specially designed device that controls the amplitude and phase responses of the channels is also required. Recently, using tunable frequency selective surfaces (FSSes) [24]–[26] and tunable metamaterials [27]–[30] in the near-field region to manipulate the far-field radiation pattern of the antennas has been demonstrated. Although these studies focus on the performance of the far-field radiation, the possibility of using tunable FSSes and metamaterials in the near-field region to tailor the electromagnetic field is well demonstrated. Thus, it is possible to use such tunable metamaterials to create signal channels and manipulate the amplitude and phase responses of the channels for the purpose of isolation. Previously, we finished an experimental investigation and proved that narrow band high-level isolation can be achieved by using multiple tunable metamaterial structures [31], [32]. In this paper, we extended this research. Basic theory is introduced in Section II. In Section III, we experimentally investigate and prove the possibility of manipulating near-field signal distribution using tunable metamaterials for the purpose of high-level isolation. A screen made of multiple tunable metamaterial units is placed in the middle of two closely placed (0.8λ) copolarized monopoles. The tunable metamaterial screen is composed of eight rows of horizontally extended unit cells. The tunable ability is achieved by varying the bias voltage of the varactors mounted on the patch structure. The bias voltages of the eight rows of unit cells are individually controllable. To achieve high-level isolation without analytical solution, a genetic algorithm-based system is developed. With this system, the isolation ratio can exceed 110 dB (S21 < −110 dB) and the isolation frequency can be tuned from 2.0 to 2.6 GHz. After these fundamental experiments with the dual-monopole system, we designed a real case of a pair of cross-polarized arrays and 15 rows of metamaterials in Section IV. Not only is 104.3-dB isolation achieved at a

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ZHANG et al.: ADAPTIVE DECOUPLING USING TUNABLE METAMATERIALS

Fig. 1. Theoretical signal model for near-field transmission between two antenna elements with resonators in the middle.

single frequency, with bandwidth constrictions, 95.2–76-dB isolation ratio is achieved within 4–22 MHz, but also this method can adapt to a dramatically varying environment with a high-level isolation. In addition, when the metamaterial screen is installed, the influence on the match performance and the radiation characteristics of the original arrays is slight. II. BASIC T HEORETICAL M ODEL A theoretical model of the propagation procedure when multiple metamaterial resonators are placed between two antenna elements is demonstrated in Fig. 1. The signal emitted from antenna A reaches the resonator rows and is reemitted to antenna B. Here R1 is the distance from antenna A to row1. Diffraction occurs at the edges of the metamaterial screen and the gaps between the unit cells. Under the assumption of near field, the signal received at the port of antenna B can be expressed as n  E(ϕ)e− j k2Ri Ti e− j θi + E(ϕ)De− j d EB = 2Ri

(1)

i=1

where k is the wavenumber; Ti and θi are the amplitude and phase coefficient of resonator i ; D and d are the amplitude and phase coefficients of diffraction signal; n is the number of resonators. By tuning the bias voltages, the statuses of the resonators are controlled, and thus, Ti and θi are manipulated accordingly. If the parameters are properly adjusted, it is possible to cancel the signal received at antenna B. It is noted that this method is frequency-dependent, which means that the cancellation happens only in a narrow band. Although the above physical model is simple, it is very difficult to find out the proper parameters to control the metamaterial. When multiple antennas are placed closely (assuming within one wavelength) in complex electromagnetic environments, the coupling between them is complicated. Theoretical analysis based on S-parameters is well described in [33]. However, as a part of the space net (defined in [33]), the coupling between the metamaterial structures is unknown. Since the metamaterial rows are individually controlled and have different statuses, the coupling becomes more complicated and it seems difficult to measure the coupling strength (no feed port) or to simulate (infinity combinations).

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Fig. 2.

Photograph of the prototyped dual-monopole system.

So, we have to consider this system as a black box. The input of this black box is multiple-channel bias-voltage signals and the output is the transmission coefficient. By tuning the bias voltages the black box provides different responses. With the help of artificial intelligence algorithm, the high-level isolation can be realized. III. E XPERIMENTAL S UBSTANTIATION An experimental system is built to examine the method in Section II. As shown in Fig. 2, two vertically positioned monopoles are installed symmetrically with respect to a metamaterial screen. The centers of the monopoles and the screen are at the same height. The length of the monopoles is 30.6 mm, and they resonate at 2.4 GHz. The distance between these monopoles is 10 cm, which is 0.8λ. A circular ground plate with a diameter of 40 mm is added for getting a stable matching performance. This metamaterial screen is constructed with eight rows of resonators. There are three uniform resonators within each row. A bias voltage is applied individually at the end of each row, and thus, the resonators in the same row share the same bias voltage. The voltages of different rows are independent. The other parts of this prototype are made of FR4 board and nylon sticks to support the structures in position. By altering the bias voltages, the center frequency of the metamaterial can be controlled. Thus, the field distribution is manipulated. The fabricated metamaterial screen and the structure are shown in Fig. 3. The original design of this patch can be traced back to [34], and it is also known as the electric-LC resonator. It was later successfully modified into an active FSS [35]. This structure is sensitive to vertically polarized incident wave. In this paper, it is deposited on a single layer FR4 board. The varactors BB857 are produced by Infineon and can be tunable from 0.54 to 6.6 pF. When the varactor is reversely biased, the maximum current is 200 nA (V R = 30 V), implying that the minimum resistance is 150 M, which is much larger than the 10-k resistors. Thus, the dc bias-voltage drop down caused by the choke resistors is negligible. The dimensions are as follows: d = 44 mm, a = 30 mm, l1 = 4 mm, l2 = 4.25 mm, l3 = 6 mm, W1 = W2 = W3 = 1.5 mm, s = 0.75 mm, and g1 = 1 mm.

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Fig. 5. Transmission coefficients when the entire screen is biased to stop 2.0, 2.2, 2.4, and 2.6 GHz.

Fig. 3. system.

Structure of the tunable metamaterial used by the dual-monopole

Fig. 6.

Fig. 4. Simulation results of transmission rate of the metamaterial when biased at 0, 10, 20, and 30 V.

The simulation results of the metamaterial under Floquet boundary condition are shown in Fig. 4. When the varactors are biased from 0 V (6.6 pF) to 30 V (0.54 pF), the stopband of this metamaterial is tuned from 1.89 to 2.61 GHz. The transmission ratio of the peak is from −12.1 to −17.4 dB. Similar to a notch filter, the phase response shifts dramatically. Furthermore, the amplitude and phase responses are not individually tunable. They are both determined by the bias voltages. The relation of amplitude and phase is decided by the patch structure and varactors’ parameters. Although the metamaterial has the ability to alter the amplitude and phase responses, (1) cannot be solved analytically. When simulated numerically, all the units are working at the same bias voltage. In this paper, the resonator rows are supposed to have different bias voltages. When the rows have individual bias voltages, at present, the mutual coupling cannot be measured (no feed port), simulated (infinity combinations and slow), or calculated (no analytical model). Thus, the complicated mutual coupling is another main reason why we use artificial intelligence algorithm for optimization.

Genetic algorithm-based experiment system.

Before optimization, we measured the cases when the entire screen is biased at the same certain voltages to make the screen’s stopband at 2.0, 2.2, 2.4, and 2.6 GHz. As shown in Fig. 5, the related voltages are 3.7, 4.7, 18.5, and 19.3 V. This is the traditional method that a metamaterial is used as a space filter to stop the signal of a certain frequency from passing through. It is functional, however, far from high level. As shown in Fig. 6, the experiments are taken in a microwave anechoic chamber. The varactors can be biased from 0 to 30 V. Thus, there are eight float numbers within 0–30 to be decided. A self-made multichannel voltage controller connects the metamaterial screen to provide eight channels of bias voltages. A GA-based program that run on a computer controls the bias voltages and reads the corresponding S21 feedback from the vector network analyzer (VNA). In this paper, an Agilent (now known as Keysight) PNA N5227A is employed. A higher score is given to a lower S21 . With the evolution of the program, a lower level is achieved. Since the metamaterial is tunable from 1.89 to 2.61 GHz, we optimized the isolation at 2.0, 2.2, 2.4, and 2.6 GHz. The results compared with the direct transmission (nothing between the monopoles) and metal plate isolation (a metal plate with the same dimensions of the metamaterial screen) are shown in Figs. 7–10. At the targeting frequencies,

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Fig. 7.

Measured optimization results at 2 GHz.

Fig. 10.

Measured optimization results at 2.6 GHz.

Fig. 8.

Measured optimization results at 2.2 GHz.

Fig. 11. Measured optimization results at 2.4 GHz when the monopole space from 0.5 to 1.7λ.

the high-level isolation cannot be obtained. We believe that when the monopoles are very close to the screen, some unit cells cannot work properly as under plane wave incidence. IV. R EALISTIC C ASE OF D ECOUPLING

Fig. 9.

Measured optimization results at 2.4 GHz.

the transmission coefficients are −112.0, −112.0, −107.1, and −111.2 dB, respectively. It almost reaches the sensitivity limit of the employed instrument. These experiments have been repeated several times to ensure that the results are correct. All the data are acquired when interfrequency bandwidth of the VNA is set to 1 Hz. During the experiments, we are wondering if this method is only functional when these two monopoles have particular distance. To experimentally find out the answer, we keep antenna A and the isolation screen at their position but move antenna B from 0.5 to 1.7λ from antenna A stepping each 0.3λ. Then, we tried the optimization at each spot, and the results are shown in Fig. 11. Only when the distance is 0.5λ,

In Section III, the isolation method is validated by the experiment, which is highly efficient and frequency adaptive. This method is supposed to be a general purpose isolation solution for various applications. The cost of such a system is low. The tunable metamaterial screen is made of FR4 PCB and some varactors and resistors. The voltage controller can be built with entry level digital to analog converters and microcontroller units. Another advantage of this solution is that the weight of the whole system is affordable by many applications, such as base stations, repeaters, vehicle-based devices, and so on. In addition, the power consumption is very low. That is, because the varactors, which are the only active component, are reversely biased, the maximum current is 200 nA (30 V biased, at 85 °C temperature), which means that the maximum power consumption of a varactor is only 6 μW. To prove the versatility, we applied this method to enhance the isolation ratio between two cross-polarized arrays. A. Design of the Array System The cross-polarized arrays are often employed by base stations and MIMO systems. The isolation ratio between these two arrays is a key performance characteristic for

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Fig. 12.

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Fig. 13.

Schematic of the arrays.

Fig. 14.

Layout of the isolation screen.

Design of the array system.

these applications. However, achieving high isolation, high axial ratio, broad bandwidth, and some other characters all at the same time is challenging. In this paper, we employ the proposed adaptive isolation method to solve the isolation problem stand-alone, which means the isolation between these two arrays is not considered when designing the antennas and feed networks. An isolation screen is placed after the arrays are prototyped. The design is shown in Fig. 12. The system is constructed with an isolation screen with respect to two coplanar fourelement arrays. The method is to isolate the arrays by interfering the coupling field between them using the isolation screen. The structure of the arrays is shown in Fig. 13. Arrays A and B are symmetrical with all details. Each array is constructed with four patch elements and an optimized power divider. Array A is +45° polarized, while Array B is −45° polarized. Some important dimensions are given as follows: L 1 = 32 mm, L 2 = 9 mm, W1 = 32 mm, W2 = 9 mm, and D = 104 mm. They are made of dual-layer FR4 board. Since these two arrays are symmetrical, they have the same return loss and symmetrical radiation characters. They resonate at 2.44 GHz. The array is +45° polarized with low crosspolarization level. As shown in Fig. 14, the isolation screen is composed of multiple rows of metamaterial unit cells and certain controlling circuits. There are 15 rows and two unit cells within each row. The structure of the unit cells is the same as those used in Fig. 3 in Section III. The voltage of each row is individually controlled by a multiple-channel voltage controller through connectors A and B. The dimensions of the metamaterial part are 70 mm by 165 mm. As shown in Fig. 12, this screen is vertically placed between the arrays. The material of this screen is the same as FR4 PCB as shown in Fig. 3. The prototype is fabricated with LPKF S103 machine. Dielectric supporting materials, including FR4 board, nylon sticks, and skews, are added to support the arrays and isolation screen as shown in Fig. 15. The isolation screen is inserted

through a vertical slot in the middle of these two arrays. Thus, the circuit part of the screen and bias voltage cables, SMA connectors, and coaxial cable are all hidden behind the array plane. In this way, the influence on the high-frequency signal is reduced. B. Preliminary Optimization The prototyped system is installed in a microwave anechoic chamber, and S-parameters of the arrays measured without the isolation screen compared with the simulated results are shown in Fig. 16. The difference between the simulated and measured results is mainly caused by fabrication and FR4 material parameter errors, which are not considered in simulations. Similar to the experiments shown in Section III, we performed an isolation optimization at 2.41 GHz, which is the resonant center of the arrays. As shown in Fig. 17, the isolation ratio reaches 104.3 dB (S21 = −104.3 dB), which is a pretty high level. Compared with the original transmission coefficient at this frequency, which is −33 dB, and the case when a

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Fig. 15.

Prototyped arrays with isolation screen.

Fig. 18.

Fig. 16.

Measured S-parameters of the arrays.

Fig. 19. Measured transmission coefficient with bandwidth constraint (zoomed-in-view).

Fig. 17.

Preliminary optimized transmission coefficient.

metal screen having the same shape with the isolation screen, the isolation efficiency (SE) is 71.3 dB. This result shows that this theory is applicable for isolating mutual coupled arrays. C. Bandwidth Constrained Optimization So far, the single-frequency isolation performance is proved to be highly efficient. However, in the real world, most

Measured transmission coefficient with bandwidth constraint.

noise sources have certain bandwidth, which means a singlefrequency isolation method is insufficient. For this system, there are 15 individual bias voltages, in other words there are 15 variables. According to (1), only one equation needs to be satisfied. Thus, this problem is very likely to be underdetermined. If so, it is possible to find out multiple solutions around the target frequency, which means the isolation bandwidth can be broadened. So, in this paper, we experimentally investigated the isolation performance with bandwidth constraints. To do so, we modified the program to fetch back all the S21 data from the VNA over a given bandwidth (4, 8, and 22 MHz); then, the program finds the maximum one and uses this value as the score of the corresponding 15-channel voltage combination. Thus, the goal of this optimization is to find out a solution to lower the transmission coefficient within the target spectrum range. The results for 4 MHz (2.408–2.412 GHz), 8 MHz (2.406–2.414 GHz), and 22 MHz (2.4–2.422 GHz) are shown in Fig. 18 and more clearly in Fig. 19. The results show that the transmission coefficient lower than −95.2, −92.1, and −76.0 dB can be realized within 4 MHz (2.408–2.412 GHz), 8 MHz (2.406–2.414 GHz), and 22 MHz (2.4–2.422 GHz). These experiments show that this method can provide not only single frequency but also reconfigurable bandwidth high-level isolation. Since we have already demonstrated the

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Fig. 20. Measured transmission coefficient when a bottle of water is placed at 45° in front of the system. Fig. 22. Reoptimized isolation results when a bottle of water is placed at 45° in front of the system.

Fig. 21. Measured transmission coefficient when a bottle of water is placed at 90° in front of the system.

frequency reconfigurable ability of this method in Section III, this method can be used to deal with frequency and bandwidth variable noise sources. Although we use bandwidth constraint method to extend the isolation bandwidth and its works, it is still not broad and the SE is influenced. The essential reason of this problem is the phenomenon of dispersion. For a given frequency, decoupling happens when the reradiated signals through different paths cancel each other when they add up at the port of Antenna B. When the frequency varies, the electrical lengths of these propagation paths vary positively related. Thus, the decoupling cannot remain. The decoupling bandwidth can be extended when some antidispersive method is employed. When tunable left-hand materials and right-hand materials are mixed, this purpose may be achieved. However, we are still not confident with this point. D. Environmental Adaption Since the bandwidth constrained optimizations show that the high-level isolation can be realized within a certain bandwidth, the system should be underdetermined. In that case, different boundary conditions are also possible to be satisfied. If so, this system will be capable of achieving the high-level isolation in different environments. To verify this feature, we put a bottle of water (600 ml) in front of the isolated arrays at 90° and 45° directions, both with the distance of 25 cm. For these cases, the coupling between these arrays is dramatically influenced. As a certain consequence, the high-level isolation is totally gone for both the cases, as shown in Figs. 20 and 21.

Fig. 23. Reoptimized isolation results when a bottle of water is placed at 45° in front of the system (zoomed-in-view).

Obviously, when the bottle of water is placed at 90° in front of the system, the influence to the transmission coefficient is greater. That is because both the main lobes of these two arrays point to this direction, so the reflection is stronger and easier to be collected by another array. Then, we run the optimization program again to verify if the isolation screen can achieve high isolation again for these cases. The results are shown in Figs. 22–25. The robustness of this system is well illustrated. For the cases when the bottle of water is placed at 45° direction in front of the system, the reoptimized results are −109.7 dB (single frequency), −83 dB (4 MHz), −81.1 dB (8 MHz), and −75.0 dB (22 MHz). For the cases when the bottle of water is placed at 90° direction in front of the system, the reoptimized results are −103.4 dB (single frequency), −73.5 dB (4 MHz), −85.5 dB (8 MHz), and −70.7 dB (22 MHz). These experiments show that the proposed method has the ability of environmental adaption, which means if such a system was installed in the real world, it would be physically possible for such a system to maintain high-level isolation when the environment changes. E. Influences on the Original Arrays The isolation performance, frequency reconfigurable, bandwidth reconfigurable, and environmental adaptive abilities are

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Fig. 24. Reoptimized isolation results when a bottle of water is placed at 90° in front of the system.

Fig. 27.

Measured radiation pattern comparison in the azimuth plane.

Fig. 25. Measured transmission coefficient when a bottle of water is placed at 90° in front of the system (zoomed-in-view).

Fig. 28.

Measured radiation pattern comparison in the elevation plane.

Fig. 26. Influence on the return loss of the arrays when the isolation screen is optimized with different methods.

verified in Sections III and IV. However, the influence on the arrays’ performance should also be evaluated. First, we measured the S11 parameters when the high-level isolations are achieved and the results are shown in Fig. 26. The results show that the influence is very limited.

Then, we measured the radiation pattern at 2.41 GHz when the arrays are isolated and compared with the original radiation pattern. The instrument we use is an MVG (SATIMO) Starlab near-field measurment system. The results in Figs. 27 and 28 show that in the azimuth plane, the influence on the radiation pattern is very slight within ±30° region of the main lobe pointing direction. Outside this region, the influence is more significant, because the isolation screen is installed within this region. There is also some influence on the cross polarization radiation. In the elevation plane, the influence on the main polarization is small for the main lobe and side lobes. The cross polarization is also influenced. Furthermore, the radiation patterns for both the single frequency and 22 MHz constrained are almost the same. Basically, although there is some influence on the radiation patterns, the radiation characters are still good for application. We also measured the radiation efficiency from 2.35 to 2.45 GHz to see if the isolation screen absorbs much energy. As a matter of fact, the absolute efficiency measured by a near-field system is not very accurate. To make this comparison more straightforward, we normalized the data to the original radiation efficiency of the array. The data in Fig. 29 shows that this screen absorbs a little more energy than the case of a metal screen.

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Fig. 29.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Measured radiation efficiency (normalized) comparison.

V. C ONCLUSION In this paper, a novel method of isolating two closely placed antennas is presented and experimentally verified. The experiments demonstrated that this method is highly efficient, frequency and bandwidth reconfigurable, and environmental adaptive. When it is applied to solve the isolation problem of two arrays, the influence on the return loss and radiation pattern is small. Furthermore, this method is supposed to be a general purpose solution to solve the isolation problems for different applications. ACKNOWLEDGMENT The authors would like to thank F. Yu for helping programming, X. Xu for helping building the experiment environment, and T. Ding for proof reading. R EFERENCES [1] K. Fujimoto and J. R. James, Mobile Antenna Systems Handbook. Norwood, MA, USA: Artech House, 2001. [2] K.-C. Chim, K. C. L. Chan, and R. D. Murch, “Investigating the impact of smart antennas on SAR,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1370–1374, May 2004. [3] A. Derneryd and G. Kristensson, “Signal correlation including antenna coupling,” Electron. Lett., vol. 40, no. 3, pp. 157–159, Feb. 2004. [4] R. Addaci et al., “Dual-band WLAN multiantenna system and diversity/MIMO performance evaluation,” IEEE Trans. Antennas Propag., vol. 62, no. 3, pp. 1409–1415, Mar. 2014. [5] I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag., vol. AP-31, no. 5, pp. 785–791, Sep. 1983. [6] T. Macnamara, Introduction to Antenna Placement and Installation. New York, NY, USA: Wiley, 2010. [7] H. W. Ott, Electromagnetic Compatibility Engineering. New York, NY, USA: Wiley, 2011. [8] W. P. Kodali, Engineering Electromagnetic Compatibility. New York, NY, USA: Wiley, 2001. [9] M. M. B. Suwailam, M. S. Boybay, and O. M. Ramahi, “Mutual coupling reduction in MIMO antennas using artificial magnetic materials,” in Proc. 13th Int. Symp. Antenna Technol. Appl. Electromagn. Can. Radio Sci. Meeting (ANTEM/URSI), Feb. 2009, pp. 1–4. [10] L. Inclan-Sanchez, J.-L. Vaquez-Roy, and E. Rajo-Iglesias, “High isolation proximity coupled multilayer patch antenna for dualfrequency operation,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1180–1183, Apr. 2008. [11] X. Tang, K. Mouthaan, and J. C. Coetzee, “Flexible design of decoupling and matching networks for two strongly coupled antennas,” Electron. Lett., vol. 49, no. 8, pp. 521–522, Apr. 2013.

[12] J. C. Coetzee and Y. Yu, “New modal feed network for a compact monopole array with isolated ports,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3872–3875, Dec. 2008. [13] C. H. Niow, Y. T. Yu, and H. T. Hui, “Compensate for the coupled radiation patterns of compact transmitting antenna arrays,” IET Microw., Antennas Propag., vol. 5, no. 6, pp. 699–704, Apr. 2011. [14] S. C. Chen, Y. S. Wang, and S. J. Chung, “A decoupling technique for increasing the port isolation between two strongly coupled antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3650–3658, Dec. 2008. [15] H.-X. Xu, G.-M. Wang, and M.-Q. Qi, “Hilbert-shaped magnetic waveguided metamaterials for electromagnetic coupling reduction of microstrip antenna array,” IEEE Trans. Magn., vol. 49, no. 4, pp. 1526–1529, Apr. 2013. [16] A. B. Numan, O. Hammi, M. S. Sharawi, and D. N. Aloi, “A printed MIMO antenna system with CLLs for isolation enhancement,” in Proc. 7th Eur. Conf. Antennas Propag. (EuCAP), Apr. 2013, pp. 2012–2015. [17] G. Zhai, Z. N. Chen, and X. Qing, “Enhanced isolation of a closely spaced four-element MIMO antenna system using metamaterial mushroom,” IEEE Trans. Antennas Propag., vol. 63, no. 8, pp. 3362–3370, Aug. 2015. [18] X. Jiang, Y.-J. Qiu, and L. Peng, “Novel metamaterial insulator for compact array isolation,” in Proc. IEEE Int. Conf. Signal Process., Commun. Comput. (ICSPCC), Aug. 2014, pp. 649–652. [19] N. Yoon, H. Kim, and C. Seo, “Design of absorber based on metamaterial structure to improve the isolation of WCDMA indoor repeater,” in Proc. Asia–Pacific Microw. Conf., Dec. 2012, pp. 568–570. [20] H. Qi, X. Yin, H. Zhao, and L. Liu, “Enhancing isolation between two closely spaced patch antennas using parasitic elements,” in Proc. IEEE Int. Symp. Antennas Propag. USNC/URSI Nat. Radio Sci. Meeting, Jul. 2015, pp. 386–387. [21] S. K. Dhar and M. S. Sharawi, “An isolation enhanced ultra-wideband semi-ring monopole MIMO antenna,” in Proc. IEEE Int. Symp. Antennas Propag. USNC/URSI Nat. Radio Sci. Meeting, Jul. 2015, pp. 2309–2310. [22] K. Okuda, H. Sato, and M. Takahashi, “Decoupling method for twoelement MIMO antenna using meander branch shape,” in Proc. Int. Symp. Antennas Propag. (ISAP), Nov. 2015, pp. 1–2. [23] G. Zhai, Z. N. Chen, X. Qing, and M. Jiang, “Isolation enhancement between two closely spaced substrate integrated cavity-backed slot antennas using mushroom,” in Proc. Asia–Pacific Microw. Conf. (APMC), vol. 3. Dec. 2015, pp. 1–3. [24] M. N. Jazi and T. A. Denidni, “Agile radiation-pattern antenna based on active cylindrical frequency selective surfaces,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 387–388, 2010. [25] A. Edalati and T. A. Denidni, “High-gain reconfigurable sectoral antenna using an active cylindrical FSS structure,” IEEE Trans. Antennas Propag., vol. 59, no. 7, pp. 2464–2472, Jul. 2011. [26] L. Zhang, Q. Wu, and T. A. Denidni, “Electronically radiation pattern steerable antennas using active frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 61, no. 12, pp. 6000–6007, Dec. 2013. [27] T. Jiang et al., “Low-DC voltage-controlled steering-antenna radome utilizing tunable active metamaterial,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 1, pp. 170–178, Jan. 2012. [28] J. C. Myers, P. Chahal, E. Rothwell, and L. Kempel, “A multilayered metamaterial-inspired miniaturized dynamically tunable antenna,” IEEE Trans. Antennas Propag., vol. 63, no. 4, pp. 1546–1553, Apr. 2015. [29] C. Jouvaud, J. de Rosny, and A. Ourir, “Adaptive metamaterial antenna using coupled tunable split-ring resonators,” Electron. Lett., vol. 49, no. 8, pp. 518–519, Apr. 2013. [30] J. Liang and H. Y. D. Yang, “Microstrip patch antennas on tunable electromagnetic band-gap substrates,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1612–1617, Jun. 2009. [31] L. Zhang, Z. Song, L. Ye, Y. Liu, and Q. H. Liu, “Experimental investigation on high efficiency decoupling using tunable metamaterials,” in Proc. Int. Symp. Antennas Propag. (ISAP), Nov. 2015, pp. 1–4. [32] L. Zhang, Y. Liu, L. Ye, H. Liu, and Q. H. Liu, “Dynamic copolarization decoupling method using tunable resonators,” in Proc. IEEE MTT-S Int. Microw. Workshop Ser. Adv. Mater. Process. RF THz Appl. (IMWS-AMP), Jul. 2015, pp. 1–3. [33] C.-H. Liang, X.-J. Dang, N. Wang, and H.-B. Yuan, “Generalized isolation between antennas for EMC problems in complex EM environments,” IEEE Trans. Electromagn. Compat., vol. 53, no. 3, pp. 645–652, Aug. 2011.

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[34] D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett., vol. 88, no. 4, pp. 041109-1–041109-3, Jan. 2006. [35] W. Withayachumnankul, C. Fumeaux, and D. Abbott, “Planar array of electric-LC resonators with broadband tunability,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 577–580, 2011.

Liang Zhang (S’12–M’14) received the B.S. degree in communication engineering and the M.Eng. and Ph.D. degrees in microelectronics and solid state electronics from the Harbin Institute of Technology, Harbin, China, in 2004, 2007, and 2013, respectively. He served as a Post-Doctoral Fellow from 2013 to 2015. He joined the College of Information Engineering, Jimei University, Xiamen, China, in 2016, where he is currently an Associate Professor. His current research interests include frequency selective surfaces, smart antennas, and high-level isolation. Dr. Zhang served as a Session Chair for APS/URSI 2012 in Chicago, IL, USA. He was a recipient of the Student Paper Contest and the Young Scientist Travel Grant of ISAP 2012, Nagoya, Japan.

Shaoqing Zhang received the B.S., M.S., and Ph.D. degrees from the Harbin Institute of Technology, Harbin, China, in 2004, 2007, and 2013, respectively. He joined the Shenyang Institute of Aircraft Design and Research, Shenyang, China, in 2013. He is currently the Associate Director of the Aviation Key Laboratory of Science and Technology on Electromagnetic Environmental Effects. His current research interests include antenna layout, the simulation of electromagnetic transients, and the electromagnetic compatibility of airplanes.

Zhengyong Song, photograph and biography not available at the time of publication.

Yanhui Liu (M’2015) received the B.S. and Ph.D. degrees in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2004 and 2009, respectively. He was a Visiting Scholar with the Department of Electrical Engineering, Duke University, Durham, NC, USA, from 2007 to 2009. He has been an Associate Professor with the Department of Electronic Science, Xiamen University, Xiamen, China, since 2011. He has authored or co-authored over 70 peerreviewed journal and conference papers. He holds several granted Chinese patents. His current research interests include antenna array design, array signal processing, and microwave imaging methods. Dr. Liu was the recipient of the UESTC Outstanding Graduate Award in 2004 and the Excellent Doctoral Dissertation Award of the Sichuan Province of China in 2011.

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Longfang Ye received the Ph.D. degree in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China, Chengdu, China, in 2013. He was a Visiting Student with the Massachusetts Institute of Technology, Cambridge, MA, USA, from 2011 to 2013. He is currently an Assistant Professor with Xiamen University, Xiamen, China. His current research interests include microwave circuits, antennas, terahertz waveguides, and graphene-based devices.

Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, Xiamen, China, in 1983 and 1986, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana– Champaign, Champaign, IL, USA, in 1989. He was with the Electromagnetics Laboratory, University of Illinois at Urbana–Champaign as a Research Assistant from 1986 to 1988, and as a PostDoctoral Research Associate from 1989 to 1990. He was a Research Scientist and a Program Leader with Schlumberger-Doll Research, Ridgefield, CT, USA, from 1990 to 1995. He was an Associate Professor with New Mexico State University, Las Cruces, NM, USA, from 1996 to 1999. He has been with Duke University, Durham, NC, USA, where he is now a Professor of electrical and computer engineering, since 1999. He has authored over 300 papers in refereed journals and 450 papers in conference proceedings. His current research interests include computational electromagnetics and acoustics, inverse problems, and their application in nanophotonics, geophysics, biomedical imaging, and electronic packaging. Dr. Liu is a Fellow of the Acoustical Society of America, Electromagnetics Academy, and the Optical Society of America. He is a Member of Phi Kappa Phi, Tau Beta Pi. He is full Member of the U.S. National Committee of URSI Commissions B and F. He currently serves as the founding Editorin-Chief of the new IEEE J OURNAL ON M ULTISCALE AND M ULTIPHYSICS C OMPUTATIONAL T ECHNIQUES , the Deputy Editor-in-Chief of Progress in Electromagnetics Research, an Associate Editor for the IEEE T RANSACTIONS ON G EOSCIENCE AND R EMOTE S ENSING , and an Editor of the Journal of Computational Acoustics. He also served as Guest Editor for the P RO CEEDINGS OF THE IEEE. He serves as an IEEE Antennas and Propagation Society Distinguished Lecturer from 2014 to 2016. He was the recipient of the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Compact Cascaded-Spiral-Patch EBG Structure for Broadband SSN Mitigation in WLAN Applications Chi-Kai Shen, Student Member, IEEE, Siang Chen, and Tzong-Lin Wu, Fellow, IEEE

Abstract— A compact cascaded-spiral-patch electromagnetic bandgap (CSP-EBG) structure for wideband simultaneously switching noise suppression is proposed. Instead of using double open stubs to provide dual-band noise suppression for a wireless local area network (WLAN), the proposed CSP-EBG structure offers an even wider bandgap, covering multiple adjacent WLAN bands from 2.4 to 6 GHz. A 1-D equivalent circuit model is proposed and validated by full-wave simulation and measurement. The circuit model could also be adopted to investigate design concepts, optimizing the EBG design by the theory of stepped-impedance resonator. The stopband of the proposed EBG structure covers a frequency range from 2.4 to 6.4 GHz with 91% fractional bandwidth (FBW). It not only provides three times wider FBW than single and double open stubs EBG but also occupies 95% smaller unit-cell size than double open stubs EBG and four times smaller than the mushroom EBG. Index Terms— Electromagnetic bandgap (EBG), mixed signal system, radio-frequency interference (RFI), simultaneously switching noise (SSN), stepped-impedance resonator (SIR).

I. I NTRODUCTION

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IGAHERTZ wireless communication grows extraordinarily fast in recent years. In order to shrink the size of the mixed-signal equipment, an antenna circuitry is usually located close to digital one. However, transient current offered by switching digital devices flows through the parasitic inductance of a power distribution network, inducing power noise or simultaneously switching noise (SSN). Such noise can cover a very wide frequency range with several gigahertzs [1], [2]. Besides spreading on power/ground (P/G) plane and leading to the problems of power integrity and digital signal integrity, SSN shown in Fig. 1 also induces noise coupling between adjacent metal planes and radiation at discontinuities, such as slots and edge of the P/G plane. Such a coupling and radiation degrade the signal-to-noise ratio (SNR) of nearby antenna components, causing radio-frequency interference (RFI). Traditional solutions to prohibit SSN propagation, such as decoupling capacitors [3], [4] and isolating slots [5], [6], hardly provide effective noise mitigation for the frequency of Manuscript received August 8, 2015; revised January 18, 2016 and April 16, 2016; accepted June 18, 2016. Date of publication August 11, 2016; date of current version September 1, 2016. This work was supported by the Ministry of Science and Technology (MOST), Taiwan, under Grant MOST 103-2221-E-002-049-MY3. The authors are with the Department of Electrical of Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]; b00901103@ ntu.edu.tw; [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.2016.2594073

Fig. 1. Propagation of SSN noise by conductive, coupling, and radiative paths in mixed signal systems.

gigahertz range, because of parasitic inductance and mutual coupling cross slots. An electromagnetic bandgap (EBG) structure could stop the propagation of gigahertz noise in a wide frequency range due to its periodicity. However, because the EBG structure periodically cascades several unit-cells to form the bandgap, it often takes up large space in the whole layout. In order to suppress the broadband SSN, the bandgap of the EBG structure needs to be enhanced as much as possible. Therefore, miniaturization and bandwidth enhancement [7]–[9] are the two main goals for the EBG structure design. For RFI applications, SSN propagation could simply be suppressed within some specific narrow bands [10]–[12]. The open stub EBG structure uses the transmission line behavior of an open stub to reduce its size [10]. An EBG structure with double open stubs provides two individual stopbands at 2.4- and 5.2-GHz wireless local area network (WLAN) band [11], [12]. However, currently used WLAN band between 2.4 and 6 GHz contains four channels, and different channels would be adopted in different regions. The double open stub EBG design would not support the requirements of WLAN channels in different regions. In this paper, a proposed cascaded-spiral-patch EBG (CSPEBG) structure provides wideband SSN mitigation for WLAN applications, as shown in Section II. An equivalent circuit model will be adopted to predict the bandgap behavior in Section II. In Section III, the divide-and-conquer method is adopted to separate the circuit model into the transmission line part and the shunt susceptance part, and cutoff frequencies could be correlated with the resonant frequencies of an open stub. Then, the stepped-impedance analysis on an open stub is used to illustrate design concepts and optimized design. In Section IV, EBG test boards for applications in suppressing noise propagation in conductive, coupling, and radiative ways

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Fig. 2. Structure of CSP-EBG. (a) Oblique view of cascaded 2 ×2 unit-cells. (b) Top view. (c) Side view.

are built, and the measurement results validate the circuit model and design concepts, showing effective noise reduction in the WLAN band. The conclusions are drawn in Section V. II. CSP-EBG A. Description of EBG Structure The investigated CSP-EBG structure with three metal layers is shown in Fig. 2. The oblique view of 2 × 2 cascaded unit-cells in a real layout is shown in Fig. 2(a), where a continuous power and a ground plane is on the top and bottom layers. A through hole via of unit-cell connects the bottom layer to the middle metal layer composed of a cascadedspiral open stub. The cascaded-spiral open stub in the proposed structure, unlike the previous open stub EBG structure, adopts two sections with different widths, which correspond to two different characteristic impedances of asymmetric stripline. Views from the top and side of a unit-cell with structural and material parameters are shown in Fig. 2(b) and (c). The related structural parameters are: the side length of a square unit-cell a, the width of the inner and outer stubs wi and wo , the minimum spacing s, the inner spiral stub length li from the center of unit-cell to discontinuity of impedance, the outer stub length lo from the discontinuity to open end of spiral, the gap width between outmost round of stub to periphery of unit-cell g, the thickness of dielectric substrates tup and tdown , and the diameter of via and pad Dvia and Dpad , respectively. The dielectric constant of the substrate is εr . B. 1-D Equivalent Circuit Model Fig. 3(a) shows a full-wave simulated 2-D dispersion diagram of the proposed CSP-EBG structure with geometric

Fig. 3. (a) 2-D dispersion diagram. (b) 1-D circuit model. (c) Validation of circuit model by full-wave simulation.

parameters (a, tup , tlow , g, wi , wo , s, Dvia , Dpad , li , and lo ) = (2.18, 0.05, 0.47, 0.05, 0.1, 0.5, 0.1, 0.2, 0.46, 3, and 7.8 mm) and relative permittivity of substrate εr = 4.2. It is worth to note that the bandgap determined by 2-D dispersion diagram is the same as that given by 1-D dispersion diagram from  to X in reciprocal lattice. Therefore, 1-D circuit model might be capable to analyze the proposed EBG structure more efficiently. Fig. 3(b) shows 1-D equivalent circuit model of a unit-cell. The circuit model contains parallel-plate transmission line whose characteristic impedance is Z t and propagation constant is βt , a via inductance L Via and input impedance of cascadedspiral asymmetric stripline Z stub. The characteristic impedance of the inner and outer asymmetric striplines is Z Si and Z So , whose propagation constants are βSi and βSo , respectively. Dispersion relation for predicting bandgap could be derived as follows. First, Z stub of a cascaded-spiral open stub is Z So − Z Si tan βSili tan βSolo Z stub = − j Z Si (1) Z So tan βSili + Z Si tan βSolo √ where βSi = βSo = ω εr /c with angular frequency ω and light speed c [13]. Next, the susceptance b composed of Z stub and L Via could be obtained as 1 Y ≡ jb = j (2) Z So −Z Si tan βSi li tan βSo lo Z S Z Si tan βSili +Z So tan βSo lo − ω · L Via where L Via is estimated from [14] as     μ0 tdown 1  L Via = +α −1 ln 4π α

(3)

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Fig. 5. Divide-and-conquer method. (a) Periodic structure. (b) Frequency dependence of suspectance part Z e and transmission line parts Z L and Z H for f L and f H .

Fig. 4. Accuracy of 1-D equivalent circuit model with different (a) wi /wo and (b) li /lo values.

and α  = π D 2Via /4a 2 . Finally, the dispersion relation derived by periodic boundary and ABCD-matrix of a unit-cell [15] is b · Zt sin(βt a) (4) cos(phase) = cos(βt a) − 2 where b is susceptance mentioned in (2). Fig. 3(c) compares the dispersion relation predicted by the adopted 1-D circuit model in Fig. 3(b) with full-wave simulation. Good agreement shows that the 1-D circuit model is feasible to model and analyze the CSP-EBG structure more efficiently. C. Variation Test of Adopted Circuit Model Since the proposed 1-D equivalent circuit model will be adopted to discuss the design concepts and find the optimized ratio of wi /wo and li /lo for CSP-EBG design, the applicability of the circuit model with different ratios of wi /wo and li /lo needs to be verified. Fig. 4(a) shows the variation of the first lower bound frequency f L and upper bound cutoff frequency f H under a different ratio of wi /wo with (a, wi , li , and lo ) = (2.29, 0.13, 2.92, and 6.35 mm) and fixed all the other parameters as previous usage. In addition, Fig. 4(b) shows the variation of these two cutoff frequencies under a different ratio of li /(lo + li ) with (a, wi , wo , and li + lo ) = (3.05, 0.13, 0.65, and 9.14 mm) and fixed all the other parameters. The results from the circuit model approximate those from full-wave simulation for both variation, which shows the applicability of the proposed circuit model for later discussion.

Moreover, Fig. 4 implies the design concepts that lower f L and wider bandgap could be achieved if the inner stripline of cascaded-spiral is narrower and shorter than the outer stripline. This phenomenon will be analyzed and optimized in Section-III. III. D ESIGN C ONCEPTS OF CSP-EBG The 1-D equivalent circuit model mentioned earlier is adopted and simplified here to illustrate the design concepts. In order to differentiate and analyze passband and stopband, the behavior of dispersion relation (4) at cutoff frequencies f L and f H is critical. As shown in Fig. 3(b), f L and f H occur at phase = 180° and 0°, which would be substituted into (4), respectively. Next, the divide-and-conquer method is adopted to separate a unit-cell in Fig. 5(a) into two parts: transmission line part with characteristic impedance Z t and propagation constant βt , and susceptance part denoted by jb. By rearranging (4) and separating the susceptance part from the transmission line part at cutoff frequencies f L and f H , the relationship between these two parts could be derived as Z t sin(βt a) −1 = ≡ Ze( f ) (5) 2 · (±1 − cos(βt a)) b where Z e ( f ) represents the equivalent impedance behavior of the susceptance part jb in Fig. 5(a). Table I shows formula of Z e ( f ) for different EBG structures. In addition, since two different functions of transmission part for f L and f H exist in (5), here, they are defined as Z t sin(βt a) (6) ZL ≡ 2 · (−1 − cos(βt a)) Z t sin(βt a) ZH ≡ . (7) 2 · (1 − cos(βt a))

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TABLE I F ORMULAS OF D IFFERENT S USCEPTANCE S TRUCTURES

Fig. 6. Separation of open stub from inductance via Z e ( f ) for observation.

Such definition is valid for first several bandgaps with βt a  1, where Z L is always negative, and Z H is always positive. Fig. 5(b) shows three functions appearing from (5) to (7), Z L , Z H , and Z e . The actual f L is given by the intersection point A of Z e and Z L , and f H is given by the intersection point B of Z e and Z H . Since Z e is monotonic inside a bandgap defined between curves Z L and Z H , A portion of curve Z e inside gray region in Fig. 5(b) represents the behavior of the susceptance part in bandgap. In Sections III-A–III-C, in order to simplify the analysis, the actual f L and f H would be approximately related to the behavior of susceptance part, and even attributed only to open stub inside the susceptance part. A. f L Versus λ/4 Resonance of Open Stub For the first bandgap, since guided wavelength is much larger than periodicity a, condition βt a  1 is satisfied, (6) could be simplified as Z t βt a Z t sin(βt a) ZL ≡ ≈− ≈ 0. (8) 2 · (−1 − cos(βt a)) 4 As shown in Fig. 5(b), f L is determined by the intersection of Z e and Z L , and (8) implies that f L is very close to the frequencies at Z e ( f ) = 0, which corresponds to the short resonant frequency of the susceptance part. Moreover, Z e ( f ) in Table I could be further separated into part of inductance via, ω · L Via , and part of open stub with tangential behavior or patch for mushroom. Since L Via in most printed circuit boards (PCBs) is around few nanohenry, it is relatively negligible as compared with abruptly changing tangent function of open stub, as shown in Fig. 6. Therefore, Z e ( f ) = 0 could be further simplified by Z e = ω · L Via − |Z stub| ≈ −|Z stub| ≈ 0.

(9)

|Z stub| = 0 in (9) occurs at frequency of λ/4 open stub resonance, which could be interpreted as open-to-short resonant frequency f O2S . Earlier discussion not only infers that the resonance frequency of unit-cell does not represent the center frequency of bandgap, but also gives a rule of thumb that f L could be approximated by fO2S of open stub for most cases on PCB with negligible overestimation.

B. f H Versus λ/2 Resonance of Open Stub As discussed in Section III-A, (7) could also be simplified by Taylor expansion as   Zt 1 βt a Z t sin(βt a) (10) ≈ − ZH ≡ 2 · (1 − cos(βt a)) 2 βt a 6 where both the first and second nonzero terms are kept due to higher frequency than previous discussion of f L . For further simplification, as the lowest bandgap of the EBG structure with small size a is concerned, the term 1/(βt a) in (10) would be quite large, and Z H might be approximated to infinity. If Z H is infinite, the intersection point of Z e and Z H is at Z e ( f ) → ∞, which corresponds to the frequency of λ/2 open stub resonance, or open-to-open resonant frequency f O2O . Comparing with the approximation for f L , the simplification method here is much rougher, which means that the actual f H is not that close to f O2O . However, as shown in Fig. 7, if only the lowest bandgap is considered, such simplification seems fairly applicable for wider width of open stub or smaller size of unit-cell. Therefore, f O2O would still be adopted to discuss the influence of open stub on f H in Section III-C. C. Stepped-Impedance Resonator As mentioned earlier, f L and f H could be simply approximated by fO2S and f O2O , respectively, for the first bandgap. Hence, the stepped-impedance resonator (SIR) [16] could be adopted to analyze the characteristic of cascaded-spiral open stub. Fig. 8(a) shows the structure of cascaded-spiral open stub and defines the electrical length of the inner and outer stubs as θi and θo . For convenience, the total electrical length of open stub resonator, θi + θo , and the impedance ratio of the outer and inner striplines, Z So /Z Si , are defined as θT and K , respectively. The input impedance Z stub of cascaded-spiral open stub in Fig. 8(a) is given as tan θi tan θo − K Z stub = j Z Si . (11) K · tan θi + tan θo The conditions of open-to-short and open-to-open resonance occur at Z stub = 0 and Z stub → ∞ in (11), which is expressed as tan θi tan θo − K = 0

(12)

K · tan θi + tan θo = 0.

(13)

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Fig. 7. Influence of different (a) open stub widths and (b) unit-cell size on Z e ( f ), showing overestimation of f H with f O2O .

The relationship between θT and θi is derived from (12) and (13) as   K 1 (open-to-short) (14) tan θT = + tan θi 1 − K tan θi 1 tan θT = (1 − K ) 1 (open-to-open). (15) + K tan θi tan θi Based on (14) and (15), Fig. 8(b) shows the resonance condition of cascaded open stubs at f O2S and f O2O for different K values. In Fig. 8(b), the variation of total electrical length at these two resonance frequencies is plotted with respect to different ratios of inner to total electrical length, θi /θT , which is defined as α. For opento-short resonance, θT gets shorter than 90° if K < 1, which implies that such resonance occurs at lower f O2S than uniform open stub (K = 1) with the same physical length. That is, shorter physical length could be achieved for fixed f O2S . For open-to-open resonance, θT becomes longer than 180° instead if K < 1 and α < 0.5, which implies that this resonance occurs at higher fO2O than uniform open stub with the same physical length. Therefore, bandwidth enhancement and miniaturization could be accomplished at the same time as K < 1 and α < 0.5, which corresponds to narrower and shorter inner stripline. This phenomenon is more obvious if impedance ratio K gets even smaller than one. Fig. 8(c) shows possible characteristic impedance that can be realized in an asymmetric stripline. Because characteristic impedance for small W/ h region drops fast as W/ h increases,

Fig. 8. (a) SIR. (b) Resonance condition of cascaded open stubs at f O2S and f O2O for different K values. (c) Achievable impedance of asymmetry stripline.

choosing smaller W/ h for ZSi and larger W /h for ZSo in this region could achieve K < 1 more efficiently. As shown in Fig. 9, design concept based on SIR with K < 1 and α < 0.5 could be explained simply by Smith chart. For open-to-short resonance, input impedance moves from the right end of Smith chart to the left end. Cascadedspiral open stub offers a discontinuity between the striplines with characteristic impedance Z So and Z Si , so input impedance would be renormalized as Z in,after = K · Z in,before

(16)

where Z in,before and Z in,after represent input impedance on open stub before and after discontinuity, respectively. If K < 1 is satisfied, Z in,after would be smaller than Z in,before , equivalently offering additional phase to the circling input impedance for open-to-short resonance, as shown by thick solid line in

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Fig. 9.

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Explanation of SIR effect by Smith chart.

Fig. 9. In other words, the discontinuity plays as a role of phase accelerator of open-to-short resonance for K < 1, shortening θT of stripline for resonance. For open-to-open resonance, the discontinuity would instead acts like a phase decelerator as K < 1 if outer stripline is longer than quarter-wavelength of f H , which corresponds to α < 0.5, as shown by thick dashed line in Fig. 9. Such phase decelerator lengthens θT of stripline for resonance, raises f H and enhances the bandwidth of EBG structure. Finally, the case in Section II-B is used to validate the design concepts and describe design procedure as follows. 1) Choose a possible K with practical tup , wi , and wo . K < 1 is suggested for bandwidth enhancement. For example, K is chosen to be 0.35 in Section II. 2) Use (14) and (15) to plot the relationship between θT and α like Fig. 8(b), and determine optimized α for desired fractional bandwidth (FBW). If the optimized α still cannot achieve desired FBW, go back to step 1 and choose another K . In Section II, the optimized α is 0.28. 3) Determine length of spiral stub for desired f L . For instance, because θT at f O2S decreases from 90° to 65° for K = 0.35 and α = 0.28, the spiral length for fO2S = 2.4 GHz in substrate εr = 4.2 could be shrunk from 15.25 to 11.01 mm, which is close to li + lo = 10.7 mm in Section II. 4) Calculate the corresponding f H . In Section II, the stub length of 11.01 mm with K = 0.35 and α = 0.28 makes f O2O = 7.7 GHz, since θT at f O2O increases from 180° to 209° under the same physical length, which shortens the corresponding wavelength from 22.02 to 18.96 mm. As mentioned previously, f O2O offers larger overestimation of f H , which is 7 GHz in Fig. 3(c). IV. M EASUREMENT VALIDATION The equivalent circuit model and the proposed design concepts are validated by measurement in this section with three kinds of noise propagation mechanism: conductive path, coupling path, and radiative path.

Fig. 10. (a) Unit-cell and (b) test boards with 9×9 unit-cells of different EBG structures with mushroom, single stub, double stubs, and CSP-EBG from left to right. (c) Measurement setup. (d) Measurement validation of CSP-EBG. (e) Comparison of measured results between different EBG structures.

A. Suppression on Conductive Noise Fig. 10(a) shows all four kinds of EBG structures for comparison. Structural parameters of CSP-EBG keep the same

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TABLE II C OMPARISON B ETWEEN D IFFERENT EBG S TRUCTURES

as those in Section II-B, which provides K = 0.36 and α = 0.28. The other three EBG structures follow the same vertical stacking and optimized to f L of 2.4 GHz under minimum linewidth and spacing in PCB. Fig. 10(b) demonstrates the fabricated test boards of these four kinds of EBG structure with 9 × 9 unit-cells. Each test board has four pairs of measuring pads to show EBG effects with different number of cells. The measurements for conductive noise and coupling noise are carried out by an Agilent four-port vector network analyzer (VNA; Agilent N5230A) with two 250-μm ground–signal–ground probes, which are calibrated by the short-open-load-thru method. The measurement setup is shown in Fig. 10(c). Fig. 10(d) shows the comparison of |S11 | and |S21 | between the simulated and measured results of CSP-EBG test board. Good agreement between them justifies the applicability of design concepts. The proposed EBG structure offers stopband with |S21 | under −40 dB from 2.4 to 6.4 GHz. A slight discrepancy of f H compared with the prediction of the circuit model in Section II is observed due to lower selectivity around f H . Fig. 10(e) compares the measured results of all four kinds of EBG structures, and Table II lists the cutoff frequencies, FBW, and normalized size with respect to the guided wavelength at f L , which is named λ f L . The proposed CSP-EBG offers two and three times wider FBW than single stub and double stubs EBG structure. Although the mushroom EBG structure could give 30% larger FBW than the CSP-EBG structure, it takes almost four times larger layout area than CSP-EBG structure. Therefore, proposed CSP-EBG provides fairly wide bandgap in a compact area. B. Suppression on Coupling Noise As previously shown in Fig. 1, if two metal planes are very closer to each other, noise would couple from one plane to another by capacitive or inductive coupling. Such coupling mechanism would become significant if resonance of either plane occurs. Fig. 11(a) shows coupling test boards with two 30 mm × 30 mm isolated metal planes with a gap of 0.1 mm between them, which provide half-wavelength resonance near 2.4 GHz. Excitation port (Port 1) is placed at the side of one plane for exciting half-wavelength resonance well. Ports 2–4 are located at the other plane and are denoted in Fig. 11(a). Comparing the measured and simulated results of |S31 |, Fig. 11(b) shows a good consistency for both reference board

Fig. 11. (a) Test boards for validation on coupling noise suppression. (b) Measurement validation of |S31 |. (c) Comparison of the measured results of |S21 |, |S31 |, and |S41 |.

and EBG board. To study the influence of ports at different positions, Fig. 11(c) shows the measurement results of |S21 |, |S31 |, and |S41 | in Fig. 11(a). Although different responses would be observed due to relation between port location and parallel-plate resonant modes, the proposed CSP-EBG offers superior suppression on noise coupling in designed bandgap from 2.4 to 6.4 GHz. Average suppression levels of EBG test board compared with those of reference boards are from 11.7 to 15.4 dB in designed bandgap for different port locations.

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rectangle metal planes, which are resonant at 2.4 GHz, are adopted to verify EBG effect on radiative noise suppression. Anechoic chamber shown in Fig. 12(b) is adopted to measure antenna efficiency, which is defined as the ratio of radiated power Prad to input power Pin . However, since broadband measurements are conducted here, such antenna efficiency would be affected seriously by input matching. To remove the matching problem, VNA is used to obtain |S11 |, and alternative radiation efficiency er could be defined by the ratio of radiated power Prad to accepted power Pacc of radiative structure, which is given as Prad 1 Prad er ≡ = · (17) Pacc 1 − |S11 |2 Pin where er is the radiation efficiency independent to input matching. Fig. 12(c) shows the measurement validation of simulated er . Although some difference exists between the simulated and measured results due to the noise from feeding cable, the effect of EBG structure on radiation suppression could be observed in designed bandgap from 2.4 to 6.4 GHz. Drop of er offered by the EBG structure is about 7% on average and 11% maximally. From point of view of suppression ratio, measured radiative noise of EBG test board is 14 times smaller than that of reference board on average by surrounding metal plane with two rows of proposed CSP-EBG structure. V. C ONCLUSION

Fig. 12. (a) Test boards for validation on radiative noise suppression. (b) Anechoic chamber for measuring antenna efficiency. (c) Measurement validation of er .

In addition, maximum enhancements of noise suppression level provided by CSP-EBG are 28, 31, and 56.5 dB for |S21 |, |S31 |, and |S41 |, respectively. C. Suppression on Radiative Noise Noise propagating to discontinuities, such as edge of planes and slots, would cause radiation, especially at resonant frequencies of parallel planes. This radiation would degrade SNR of nearby antenna and receiver circuits, causing serious RFI issues. One way to mitigate noise radiation is surrounding radiative portion with EBG structure, preventing power noise from propagating to edge of plane. Fig. 12(a) shows the test board of radiative noise suppression. Two 30 mm × 30 mm

A compact CSP-EBG structure, which offers large enhancement of bandwidth for SSN mitigation in multiple WLAN channels, has been proposed. A 1-D equivalent circuit model is adopted with full-wave validation, providing a quick prediction on dispersion diagrams. Divide-and-conquer method is also proposed to further simplify the circuit model, correlating cutoff frequencies with resonant frequencies of cascaded open stubs. Therefore, the theory of SIR could then be adopted to illustrate design concepts physically and optimize design efficiently. Finally, test boards are built to demonstrate the effects of proposed EBG on suppressing noise propagation in conductive, coupling, and radiative ways. The measured results agree well with the simulated ones, validating the proposed circuit model and claimed design concepts. The CSP-EBG structure has been proven to provide a wide band from 2.4 to 6.4 GHz with size 2.18 mm × 2.18 mm only, which offers two to three times wider FBW with smaller space occupation than single and double open stubs EBG, and takes only one fourth area of traditional mushroom. R EFERENCES [1] M. Swaminathan, J. Kim, I. Novak, and J. P. Libous, “Power distribution networks for system-on-package: Status and challenges,” IEEE Trans. Adv. Packag., vol. 27, no. 2, pp. 286–300, May 2004. [2] T.-L. Wu, H.-H. Chuang, and T.-K. Wang, “Overview of power integrity solutions on package and PCB: Decoupling and EBG isolation,” IEEE Trans. Electromagn. Compat., vol. 52, no. 2, pp. 346–356, May 2010. [3] J. Fan et al., “Quantifying SMT decoupling capacitor placement in DC power-bus design for multilayer PCBs,” IEEE Trans. Electromagn. Compat., vol. 43, no. 4, pp. 588–599, Nov. 2001. [4] J. Fan, W. Cui, J. L. Drewniak, T. P. Van Doren, and J. L. Knighten, “Estimating the noise mitigation effect of local decoupling in printed circuit boards,” IEEE Trans. Adv. Packag., vol. 25, no. 2, pp. 154–165, May 2002.

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[5] T. Hubing et al., “Power bus noise reduction using power islands in printed circuit board designs,” in Proc. IEEE Int. Symp. Electromagn. Compat., Seattle, WA, USA, Aug. 1999, pp. 1–4. [6] T. H. Kim, J. Lee, H. Kim, and J. Kim, “3 GHz wide frequency model of ferrite bead for power/ground noise simulation of high-speed PCB,” in Proc. IEEE Elect. Perform. Electron. Packag., Oct. 2002, pp. 217–220. [7] T.-L. Wu, Y.-H. Lin, T.-K. Wang, C.-C. Wang, and S.-T. Chen, “Electromagnetic bandgap power/ground planes for wideband suppression of ground bounce noise and radiated emission in high-speed circuits,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 9, pp. 2935–2942, Sep. 2005. [8] T.-K. Wang, T.-W. Han, and T.-L. Wu, “A novel power/ground layer using artificial substrate EBG for simultaneously switching noise suppression,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 5, pp. 1164–1171, May 2008. [9] C.-K. Shen, C.-H. Chen, D.-H. Han, and T.-L. Wu, “Modeling and analysis of bandwidth-enhanced multilayer 1-D EBG with bandgap aggregation for power noise suppression,” IEEE Trans. Electromagn. Compat., vol. 57, no. 4, pp. 858–867, Aug. 2015. [10] H. Toyao, N. Ando, and T. Harada, “Electromagnetic bandgap (EBG) structures using open stubs to suppress power plane noise,” IEICE Trans. Commun., vol. E93-B, no. 7, pp. 1754–1759, Jul. 2010. [11] Y. Kasahara, H. Toyao, and T. Harada, “Open stub electromagnetic bandgap structure for 2.4/5.2 GHz dual-band suppression of power plane noise,” in Proc. IEEE Elect. Design Adv. Packag. Syst. Symp., Dec. 2011, pp. 1–4. [12] Y. Kasahara and H. Toyao, “‘Open-stub electromagnetic bandgap’ structures loaded with capacitive transmission line segments for bandgap frequency control,” in Proc. IEEE Int. Symp. Electromagn. Compat., Raleigh, NC, USA, Aug. 2014, pp. 351–356. [13] Y.-Z. Wang and M.-L. Her, “Compact microstrip bandstop filters using stepped-impedance resonator (SIR) and spur-line sections,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 153, no. 5, pp. 435–440, Oct. 2006. [14] S. Clavijo, R. E. Diaz, and W. E. McKinzie, “Design methodology for Sievenpiper high-impedance surfaces: An artificial magnetic conductor for positive gain electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2678–2690, Oct. 2003. [15] D. M. Pozar, Microwave Engineering, 4th ed. New York, NY, USA: Wiley, 2012. [16] M. Makimoto and S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Techn., vol. MTT-28, no. 12, pp. 1413–1417, Dec. 1980. Chi-Kai Shen (S’14) received the B.S. degree in physics and the M.S. degree in electro-optical engineering from National Taiwan University, Taipei, Taiwan, in 2008 and 2010, respectively, where he is currently pursuing the Ph.D. degree at the Graduate Institute of Communication Engineering. His current research interests include electromagnetic bandgap structure design and power integrity design in advanced packages and PCBs. Mr. Shen is a Member of the Phi Tau Phi Scholastic Society. He was a recipient of the Best Student Paper Award of 2015 APEMC and the President Memorial Award of the 2015 Joint ISEMC and EMC Europe.

Siang Chen was born in Taipei, Taiwan, in 1993. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 2015, where he is currently pursuing the M.S. degree at the Graduate Institute of Communication Engineering. His current research interests include electromagnetic bandgap structure design.

Tzong-Lin Wu (S’93–M’98–SM’04–F’13) received the B.S.E.E. and Ph.D. degrees from National Taiwan University (NTU), Taipei, Taiwan, in 1991 and 1995, respectively. He was a Senior Engineer with Microelectronics Technology Inc., Hsinchu, Taiwan, from 1995 to 1996. From 1996 to 1998, he was with the Central Research Institute, Tatung Company, Taipei, Taiwan, where he was involved in the analysis and measurement of electromagnetic compatibility (EMC)/electromagnetic interference (EMI) problems of high-speed digital systems. From 1998 to 2005, he was with the Electrical Engineering Department, National Sun Yat-sen University, Kaohsiung, Taiwan. In 2008, he was a Visiting Professor with the Electrical Engineering Department, University of California at Los Angeles, Los Angeles, CA, USA. He is currently a Distinguished Professor with the Department of Electrical Engineering, NTU, where he serves as a Director of the Graduate Institute of Communication Engineering. His current research interests include EMC/EMI, signal/power integrity design for high-speed digital/optical systems, and microwave circuits. Prof. Wu was the Chair of the Taipei Section of IEICE from 2007 to 2011, the Treasurer of the IEEE Taipei Section from 2007 to 2008, and a Member of the Directors of the IEEE Taipei Section from 2009 to 2010 and 2013 to 2014. He was the recipient of the Excellent Research Award and Excellent Advisor Award of NSYSU in 2000 and 2003, the Wu Ta-You Memorial Award of the National Science Council (NSC) in 2005, the Technical Achievement Award of the IEEE EMC Society in 2009, the Best Paper Award of the IEEE T RANSACTIONS ON A DVANCED PACKAGING in 2010, and the Outstanding Research Award from NSC both in 2010 and 2013. He was a Distinguished Lecturer of the IEEE EMC Society from 2008 to 2009. He was the Co-Chair of the IEEE EDAPS Workshop in 2007, the TPC Chair of the IEEE EDAPS Symposium in 2010 and 2012, and the General Chair of the Asia–Pacific EMC Symposium in 2015. He is serving as an Associate Editor of the IEEE T RANSACTIONS ON E LECTROMAGNETIC C OMPATIBILITY and the IEEE T RANSACTIONS ON C OMPONENTS , PACKAGING , AND M ANUFACTURING T ECHNOLOGY.

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Rapid Microwave Design Optimization in Frequency Domain Using Adaptive Response Scaling Slawomir Koziel, Senior Member, IEEE, and Adrian Bekasiewicz

Abstract— In this paper, a novel methodology for cost-efficient microwave design optimization in the frequency domain is proposed. Our technique, referred to as adaptive response scaling (ARS), has been developed for constructing a fast replacement model (surrogate) of the high-fidelity electromagneticsimulated model of the microwave structure under design using its equivalent circuit (low-fidelity model). The basic principle of ARS is a nonlinear frequency and amplitude response scaling aimed at accommodating the discrepancies between the low- and high-fidelity models at the reference design and, subsequently, at tracking the low-fidelity model changes that occur during the optimization run. The surrogate model prediction is obtained by applying appropriately composed scaling functions to the highfidelity model at the reference design. ARS is a parameterless and simple-to-implement method that can be applied to a wide range of microwave structures. The ARS surrogate features excellent generalization capability that translates into improved reliability and reduced design cost. It is demonstrated using an eighth-order microstrip bandpass filter and a miniaturized rat-race coupler. Comparison with several space mapping algorithms is provided. The numerical results are supplemented by measurements of the fabricated optimum designs of the considered structures. Index Terms— Adaptive response scaling (ARS), computeraided design, electromagnetic (EM)-driven design, microwave design, surrogate-based optimization (SBO).

I. I NTRODUCTION ULL-WAVE electromagnetic (EM) simulation has become one of the fundamental tools of contemporary microwave engineering. In particular, one of the important steps of the microwave design process is EM-driven optimization and design closure. It aims at adjustment of geometry and/or material parameters of the structure at hand (e.g., a filter) to ensure that the prescribed design specifications (concerning, e.g., return loss and transmission) are fulfilled. Deviations from the ideal/required characteristics are normally the effect of inaccuracies of the simplified representations such as equivalent circuit models that utilized to obtain the initial (or pretuning) design.

F

Manuscript received November 18, 2015; revised January 18, 2016 and July 5, 2016; accepted July 8, 2016. Date of publication July 26, 2016; date of current version September 1, 2016. This work was supported in part by the Icelandic Centre for Research (RANNIS) under Grant 163299051 and in part by the National Science Centre of Poland under Grant 2014/15/B/ST7/04683. The authors are with the School of Science and Engineering, Reykjavík University, Reykjavik 101, Iceland, and also with the Faculty of Electronics, Telecommunications and Informatics, Gda´nsk University of Technology, Gda´nsk 80-233, Poland (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2590551

For the sake of automation, it is generally desirable that EM-based design closure is conducted through numerical optimization. This, however, may be computationally expensive or even prohibitive when conventional off-the-shelf algorithms such as gradient-based search with numerical derivatives [1] or global methods (e.g., population-based metaheuristics [2]) are utilized. Over the last years, there have been numerous techniques proposed to speed up EM-driven design, most of which rely on the surrogate-based optimization (SBO) paradigm [3]. Popular methods include many variations of space mapping (SM), e.g., [4]–[6], response correction techniques [7]–[9], feature-based optimization [10]–[13], artificial neural networks [14], as well as simulation-based tuning and tuning SM [15], [16]. A comprehensive review of SBO methods with the emphasis on response correction techniques can be found in [17]. Furthermore, the availability of cheap adjoint sensitivities [18] revived, to some extent, the interest in gradient-based optimization [19], also in connection with SBO [20]. On the other hand, utilization of adjoints is not yet widespread in the microwave community. Despite the aforementioned developments, SBO methods have not yet been widely accepted by the designers and industry. Due to certain issues of SBO techniques, particularly those exploiting physics-based surrogates [21], [22], their successful application is rather difficult (especially by users who are not experts in optimization). In particular, they require a rather careful implementation, as well as certain engineering insight into the problem. One of the issues pertinent to SM is the necessity of appropriate selection of the type and parameters of the surrogate model (such as preassigned parameters for implicit SM (ISM) [23]). Straightforward application of such methods may lead to unpromising results, including convergence issues [24]. Methods such as simulation-based tuning are much more robust yet limited in terms of application scope and software requirements. Other techniques, such as shapepreserving response prediction (SPRP) [8], require satisfaction of specific assumptions concerning the response shape of the structure of interest and relatively complex implementation. In this paper, we propose a simple adaptive response scaling (ARS) technique for fast design optimization of microwave devices. ARS is based on tracking the changes (both frequency- and amplitude-wise) of the underlying low-fidelity model (e.g., an equivalent circuit) that occur during the surrogate model optimization stage. As explained in Section III, the tracking is implemented as nonlinear scaling

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involving spline functions. This ensures good generalization capability of the surrogate model. As ARS does not make any special assumptions concerning the type/shape of the response being handled, it is generic and can be applied to a wide class of microwave devices. In this paper, it is demonstrated in Section V using an eighth-order microstrip bandpass filter and a compact rat-race coupler. For the sake of additional verification, ARS-optimized designs have been fabricated and measured. Furthermore, comprehensive numerical comparisons indicate that ARS outperforms, both in terms of the computational cost and reliability, benchmark methods based on various combinations of SM algorithm. II. SBO BASICS In this section, we formulate EM-driven design problem and briefly recall the fundamentals of SBO. Surrogate model construction scheme using the proposed ARS is provided in Section III. A. Microwave Design Problem as Optimization Task Let R f (x) be a response of the EM-simulated (high-fidelity) model of a device (e.g., filter) under design, where x is a vector of adjustable (geometry) parameters and R f represents relevant responses such as S-parameters versus frequency. The problem to be solved is x∗ = arg min U (R f (x)) x

(1)

where U encodes the given design requirements (e.g., minimax with lower/upper specifications) and x∗ is the optimum design to be found. B. Physics-Based Surrogate-Assisted Optimization In many situations, direct solving of (1) is computationally too expensive due to high cost of evaluating the high-fidelity model. SBO [3], [4] allows for reducing the design cost through replacing (1) by an iterative procedure (i+1)

x

=

arg min U (R(i) s (x)) x

(2)

where x(i) , i = 0, 1, . . ., is a sequence of approximations to (i) x∗ and Rs is a surrogate model at iteration i . The surrogate may be a data-driven model (obtained by approximating sampled high-fidelity model data) or a physics-based one, i.e., obtained by suitable correction/enhancement of the underlying low-fidelity model Rc (x). In microwave engineering, the application of approximation models is rather limited to low-dimensional cases (with notable exception of artificial neural networks [25]) and for construction of multiple-use library models. Physics-based SBO algorithms are more efficient and therefore popular, and the low-fidelity model of choice is typically an equivalent circuit representation of the structure under design. C. Properties of SBO The efficiency of physics-based SBO methods comes from exploitation of the knowledge about the system under design

embedded in the low-fidelity model. Typically, a few iterations of (2) are sufficient to find a satisfactory design. The differences between various realizations of SBO algorithms are in the strategies for constructing the surrogate model. The most popular SBO approach in microwave engineering is SM [4], where the surrogate is obtained by usually simple correction of the low-fidelity model of the form Rs (x) = Rc (x; p), where p denotes the SM parameters extracted or calculated to reduce the misalignment between the surrogate and the low-fidelity model at the most recent design or at several previous designs encountered at the optimization path [26]. Examples include input SM with Rs (x) = Rc (Bx + c), ISM [Rs (x) = Rc (x; p), where p denotes the socalled preassigned parameters, e.g., the substrate height and permittivity [27]], or output SM [Rs (x) = Rc (x) + , where  is typically a constant vector, e.g.,  = R f (x(i) )−Rc (x(i) )]. Response correction methods belong to another group of SBO techniques that include, among others, manifold mapping [7], adaptive response correction [9], or SPRP [8]. Techniques such as SPRP offer improved efficiency by exploiting the problem-specific knowledge to the fuller extent, however, at the expense of certain limitations [8]. From the point of view of SBO performance, it is important that at least zero-order consistency between the surrogate and (i) the high-fidelity model [i.e., R f (x(i) ) = R(i) s (x )] is satisfied. Because majority of physics-based SBO algorithms do not rely on high-fidelity model derivatives, the first-order consistency [agreement of the model Jacobians at x(i) ] is rarely satisfied, which may lead to convergence problems [24]. Ensuring it by output SM as in the previous paragraph may not work well for highly nonlinear (e.g., filter) responses [25]. On the other hand, even within a specific method such as SM, algorithm performance heavily depends on a particular choice of the SM surrogate (input/implicit/output and combinations thereof). An optimum choice of the SM model is not straightforward and problem dependent [21], [24]. III. S URROGATE M ODEL C ONSTRUCTION BY M EANS OF ARS In this section, we describe an ARS methodology developed to construct a surrogate model for the SBO scheme (2). The important prerequisite of ARS is to fully exploit the knowledge about the microwave structure (e.g., a filter) of interest embedded in its low-fidelity model yet be as generic as possible (as opposed to, e.g., SPRP [8] whose operation depends on satisfaction of rather strict requirements concerning the shape of the model responses). A. ARS: Reference Design Scaling The ARS technique aims at constructing the surrogate model that: 1) preserves zero-order consistency (i) Rs (x(i) ) = R f (x(i) ) and 2) exhibits good generalization capability by accounting for both frequency and amplitude changes of the low-fidelity model responses during the optimization process. Fig. 1 shows the high- and low-fidelity model responses (here, |S11 |) of an exemplary sixth-order microstrip filter at the

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Fig. 1. Responses of the sixth-order microstrip bandpass filter. |S11 | at a reference design x(i) (top) and at another design x (bottom). High- and lowfidelity models are shown using the solid and dashed lines, respectively.

reference design x(i) and another design x. It can be observed that the models are significantly misaligned yet relatively well correlated. The ARS attempts to explore these correlations using the procedure described below. The response scaling is realized at the level of complex S-parameter responses, separately for the respective real and imaginary parts. In the first step, the frequency relationships between the low- and high-fidelity models at the reference design x(i) [i.e., the design being the starting point for the subsequent iteration of (2)] are retrieved by solving the following nonlinear regression problem:  ωmax (i) |r f (x(i) , ω) − rc (x(i) , F(ω))|dω F (ω) = arg min F

ωmin

(3) where F is a nonlinear frequency scaling function, here, implemented using cubic splines with 20 control points within the frequency range of interest; r f /rc are the high-/low-fidelity model responses of interest [e.g., Re(S11 )]. Problem (3) is solved for all relevant responses (such as S11 , S21 , S31 , and S41 in the case of coupler structures, see Section V-B), separately for the real and imaginary parts. That is why in (3) and the following equations, the symbols r f /rc are used rather than R f /Rc , i.e., to account for the fact that the operations are performed independently for each response. Cubic splines used here are essentially compositions of second-order polynomials connected at the control points in a smooth way (i.e., so that they agree with respect to the function value and first-order derivatives [26]). The aforementioned number of control points is sufficient to ensure good flexibility of the scaling curve. Problem (3) is solved with the control point locations along the frequency axis as unknown. Upon extraction, F (i) minimizes the discrepancy between the responses within the frequency range of interest ωmin -ωmax . As mentioned before, the main purpose here is identification of the frequency-wise model misalignment. As indicated in Fig. 2, the frequency-scaled low-fidelity model at the reference design is well aligned with the high-fidelity one in terms of the frequency location of the response minima/ maxima.

Fig. 2. ARS, Stage I (reference scaling). (a) Frequency scaling function F (i) (ω) (solid line) extracted using (3) for the sixth-order filter at the design shown in Fig. 1. The plot restricted to the range of 4–7 GHz. For comparison, the identity function is shown as the dotted line. (b) Re(S11 ) of the high- (solid line) and low-fidelity (dotted line) models at the reference design x(i) as well as the frequency-scaled low-fidelity model response (circles). Good frequency alignment of the response minima/maxima can be observed.

B. ARS: Model Prediction At the prediction stage, the objective is to account for the changes of the low-fidelity model between the reference design x(i) and the current design x [encountered in the course of the surrogate model optimization run (2)]. In order to do this, first, the scaling function similar to (3) is computed to identify the frequency changes of the low-fidelity model response. In particular, we have  ωmax |rc (x, ω) − rc (x(i) , F(ω))|dω. (4) F(x, ω) = arg min F

ωmin

In other words, (4) allows for determination of the nonlinear frequency scaling function between rc responses evaluated at the designs x(i) and x. It should be emphasized that both (3) and (4) are solved with respect to the scaling coefficients as unknowns. The response of the frequency scaled low-fidelity model in (3) and (4) is obtained by interpolating the known response at the original frequency sweep. Consequently, the computational cost of solving both (3) and (4) is negligible (in practice, a fraction of a second). Furthermore, the amplitude changes of the low-fidelity model at the design x and design x(i) scaled by (4) are calculated as   A(x, ω) = [rc (x, ω) + 1] ÷ rc (x(i) , F(x, ω)) + 1 . (5) Here, ÷ denotes component-wise division with respect to frequency. The shift by +1 is introduced in order to avoid division by zero [for frequencies for which rc (x(i) , F(x, ω)) = 0] and to avoid too large values of |A| (for rc close to zero). The choice of this particular value is not critical although the shift should be sufficiently large to ensure that rc (x(i) , F(x, ω)) + 1 > 0 for the entire range of frequencies of interest.

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At this point, it should be explained why frequency scaling (3) and then (4) are separated from amplitude correction (5). The reason is that mixing these two corrections together would not allow for finding the frequency relationship between the low- and high-fidelity models [as in (3)] and between the low-fidelity model at the reference design and the current design [as in (4)]. Arranging the sequence of corrections as described, i.e., when the amplitude correction is applied for the frequency-scaled responses, leads to better generalization of the surrogate because the amplitude adjustment is made on the responses that have well-aligned minima/maxima and slopes (in particular, this reduces a possibility of creating irregularities such as spikes in the surrogate model response during its optimization process, see [9]). The prediction stage of the surrogate model is then realized as follows. First, the reference high-fidelity model response r f (x(i) , ω) is scaled in frequency using F(x, ω) in order to account for the changes of the low-fidelity model while moving from x(i) to x (the low- and high-fidelity models are assumed to be well correlated although perhaps misaligned in the absolute terms). Then, the amplitude scaling function A(x, ω) is scaled in frequency using F (i) (x, ω) in order to accommodate the frequency relationships between the low- and high-fidelity models at the reference design. Finally, it is applied to correct the surrogate response in amplitude. Formally, the surrogate model is defined as   (6) rs (x, ω) = A(x, F (i) (ω)) ◦ r f (x(i) , F(x, ω)) + 1 − 1 where ◦ denotes component-wise multiplication. In other words, in (6), the amplitude changes determined by (5) are rescaled by (3) to account for frequency misalignment between the high- and low-fidelity models. Note that the scaling function F (i) is calculated only once per iteration (2). At the same time, (4) and (5) are computed for each evaluation of the surrogate model that allows for better utilization of the knowledge embedded in the low-fidelity model through tracking its changes on the optimization path. Fig. 3 shows the plots of the scaling function F(x, ω) corresponding to the model responses of Figs. 1 and 2, the effect of the low-fidelity model scaling with F(x, ω), and the amplitude scaling function A(x, ω). Fig. 4 shows Re(S11 ) of the responses presented in Fig. 1, as well as the response of the surrogate model constructed according to ARS, i.e., using (6). It can be observed that that the prediction power of the model is very good, especially given considerable misalignment between the low- and high-fidelity models as well as considerable change of the responses between the designs x(0) and x. Fig. 5 shows how this translates to |S11 | prediction. At the same time, the quality of the conventional output SM prediction is poor as it does not account for the model response changes in frequency. It should be mentioned here that decent correlation between the low- and high-fidelity models is important for the success of all physics-based surrogate-assisted optimization methods. Unfortunately, rigorous assessment of whether the quality of the low-fidelity model will be sufficient for the optimization algorithm to succeed is difficult. There have been several theoretical and algorithmic attempts to address this issue

Fig. 3. ARS, Stage II (prediction). (a) Frequency scaling function F(x, ω) (solid line) extracted using (4) and the identity function (dotted line). (b) Re(S11 ) of the low-fidelity model responses at the reference design x(i) (dotted line), at the design x (solid line), and the low-fidelity model response at x(i) scaled using F(x, ω) (circles). (c) Amplitude scaling function A(x, ω) (solid line) computed with (5).

Fig. 4. Responses of the sixth-order filter at x(i) (top) and at x (bottom) (see Fig. 1): R f (solid line) and Rc (dashed-dotted line). Surrogate model response determined by (6) is shown using circles.

(see [28] and [29]); however, in general, visual inspection of the model responses and their correlation is still the most reliable way of making such an assessment. As demonstrated in Section V, the proposed ARS technique is much more immune to possible deficiencies of the low-fidelity model compared with state-of-the-art methods such as SM. IV. D ESIGN O PTIMIZATION U SING ARS Figs. 6 and 7 show the flowcharts of the two stages of the ARS surrogate modeling process. As mentioned before, Stage I is executed only once for each iteration of the

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Fig. 5. High-fidelity model response at x (solid line), and surrogate model responses obtained using ARS (circles) and conventional output SM (stars) for the example sixth-order microstrip filter.

Fig. 8. Design optimization using ARS: within the optimization loop, the ARS surrogate model is prepared (Stage I) as shown in Fig. 6, and then optimized [see (2)] to obtain the next design x(i+1) . For each candidate design suggested by the optimizer, Stage II of the ARS modeling process is executed (see Fig. 7).

Fig. 6. ARS, Stage I (reference design scaling). The frequency relationships between the high- and low-fidelity model responses at the reference design x(i) , represented by the scaling function F (i) , are obtained as in (3), separately for the real and imaginary parts of each response of interest (S11 , S21 , etc.). Stage I is executed only once for each iteration of the SBO algorithm (2).

Fig. 9.

Geometry of the eighth-order microstrip bandpass filter.

is convergence in argument, i.e., ||x(i+1) − x(i) || < ε for a user-defined threshold ε. ARS can be used as a stand-alone methodology or it can be combined with other surrogate modeling techniques such as SM. In particular, the low-fidelity model used by ARS can be preconditioned by the initial alignment with the highfidelity model using, e.g., implicit and/or frequency SM. V. V ERIFICATION E XAMPLES In this section, we demonstrate ARS using two examples: an eighth-order microstrip bandpass filter and a miniaturized rat-race coupler. In both cases, a comprehensive comparison between ARS and several benchmark techniques is provided and discussed. A. Eighth-Order Microstrip Bandpass Filter Fig. 7. ARS, Stage II (model prediction). The input arguments are the current design x, as well as the high- and low-fidelity model responses at the reference design and the reference scaling function F (i) . The frequency and amplitude relations between the low-fidelity models at x(i) and x are obtained using (4) and (5), respectively. Finally, the scaling functions F(x, ω) and A(x, ω) are applied to compute the surrogate model response as in (6).

SBO algorithm (2), whereas Stage II is carried out for each candidate design x. Fig. 8 shows the flowchart of the entire ARS-based optimization process. The termination condition

Our first example is the eighth-order bandpass filter shown in Fig. 9. The structure is implemented on a 0.762-mmthick Taconic RF-35 dielectric substrate (εr = 3.5 and tanδ = 0.0018). The design variables are x = [w1 w2 w3 w4 d1 d2 d3 d4 l1 l2 l3 l4 ]T . The EM model is implemented in CST Microwave Studio (∼160 000 mesh cells and simulation time 6 min). The low-fidelity model is an equivalent circuit model implemented in Agilent ADS. The design specifications are |S11 | ≤ −20 dB for 4 GHz ≤ ω7 GHz, and |S11 | ≥ −3 dB for ω ≤ 3.92 GHz and ω ≥ 7.08 GHz. The initial design is

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Fig. 10. Eighth-order microstrip bandpass filter: initial (dashed-dotted line) and final (solid line) filter responses obtained using the ARS technique.

Fig. 11. Photograph of the fabricated eighth-order bandpass filter at the optimized design found by the ARS technique.

TABLE I E IGHTH -O RDER F ILTER : ARS V ERSUS B ENCHMARK M ETHODS

Fig. 12. Simulated (dashed-dotted line) versus measured (solid line) characteristics of the optimized eighth-order microstrip bandpass filter.

x(0) = [1.0 1.0 1.0 1.0 5.0 5.0 5.0 5.0 17.0 17.0 17.0 17.0]T . The final design is x∗ = [0.61 1.16 1.49 1.97 6.90 3.14 3.03 2.38 15.51 15.53 16.87 16.87]T . The filter responses at the initial design and the design optimized using ARS are shown in Fig. 10. The ARS algorithm has been compared with various SM-based algorithms in terms of reliability and computational cost. The benchmark routines include ISM, a combination of frequency SM and output SM, and input SM. For the sake of fair comparison, each SBO process has been carried out in an unattended manner. The results are gathered in Table I. The only techniques that converged are ARS and ISM with 16 preassigned parameters. Nonetheless, the latter required more iterations and was over two times more expensive than ARS. The reason is that ARS does not require a parameter extraction step, which is mandatory for ISM. It should be noted that the ISM with eight preassigned parameters and the combination of frequency and output SM have been both prematurely terminated due to divergence. The input SM converged, but its final response violates design specifications. Apart from limited reliability and higher computational cost, there are other issues related to the benchmark techniques reported here. In particular, performance of SM-based algorithms is problem specific and heavily depends on selected SM transformations (or their combinations) [21]. Due to this ambiguity, any particular SM model selection may or may not be successful. In particular, inappropriate setup being a result

of utilization of inadequate model or selection of insufficient (or too large) number of degrees of freedom for parameter extraction cannot be identified until the algorithm is executed. In practice, several variations have to be tried out in order to find the one that is suitable for a given problem. As indicated by the data gathered in Table I, this results in wasting of computational resources for performing optimizations that end with divergence of the algorithm or produce designs with unsatisfactory characteristics. On the contrary, ARS does not exhibit this sort of problem because it is parameterless. For the sake of additional verification, the optimized filter design has been fabricated (see Fig. 11) and measured. As indicated in Fig. 12, the agreement between the simulated and the measured filter characteristics is satisfactory. The discrepancies between the obtained responses result from using a simplified EM model of the filter that lacks the SMA connectors. Other (minor) sources of misalignment are fabrication tolerances [30]. Simulated and measured characteristics are shifted in frequency by 20 MHz, whereas their maximum vertical discrepancy within the frequency of interest band is 5 dB. B. Compact Microstrip Rat-Race Coupler Our second example is a miniaturized equal-split rat-race coupler shown in Fig. 13 [31]. The structure is composed of six compact microstrip resonant cells with shunt stubs. The cells are meandered to ensure small footprint. The coupler is implemented on a Taconic RF-35 dielectric substrate (εr = 3.5, tanδ = 0.0018, and h = 0.762). The design variables are x = [w d1 d2 l2 l3r l4r l5r ]T . Relative parameters

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Fig. 13. Geometry of the compact microstrip rat-race coupler. Digits in circles represent port numbers. Fig. 15. Compact rat-race coupler: initial (thin lines) and final (thick lines) coupler responses obtained using the ARS technique. TABLE II C OMPACT C OUPLER : ARS V ERSUS B ENCHMARK M ETHODS

Fig. 14. Generalization capability of the ARS technique applied to the compact rat-race coupler: high-fidelity model at the reference design (dasheddotted line), high-fidelity model at the test design (solid line), and ARS surrogate model at the same test design (circles). It can be observed that the prediction power of the ARS model is excellent despite of the considerable change of the model response between the reference and the test design.

are l1 = 2d1 + d2 + 2.5w, l3 = 0.1l2 · l3r , l4 = 0.1l3 · l4r , and l5 = 0.1l2 · l5r , whereas the dimension w0 = 1.7 remains fixed to ensure 50- input impedance. The unit of geometrical parameters is mm. The structure is designed to operate at 1 GHz. The EM model is implemented in CST Microwave Studio (∼225 000 mesh cells and average simulation time 22 min). The low-fidelity model is an equivalent circuit model implemented in Agilent ADS. The design specifications are as follows: 1) obtain equal power split, i.e., |S21 | = |S31 | at the operating frequency of 1 GHz; 2) increase −20-dB bandwidth for matching |S11 | and isolation |S41 |, symmetrically around the operating frequency; and 3) bring the minima of |S11 | and |S41 | possibly close to 1 GHz. The initial design is x(0) = [0.4 0.2 2.0 12.0 12.0 4.0 2.0]T mm. Fig. 14 shows excellent generalization capability of the ARS method applied to the coupler structure of Fig. 13. The final design x∗ = [0.494 0.200 2.497 14.469 9.912 4.773 2.276]T has been obtained after three iterations of ARS. The filter responses at the initial design and the optimized designs are shown in Fig. 15. The results of comparison with various SM-based algorithms are gathered in Table II. Similarly as for the previous example, the optimization runs have been performed in an unattended manner. The ARS required only three iterations to converge (with the total cost corresponding to ∼4.8 R f simulations). Among the benchmark methods, only ISM with

Fig. 16. Measurement of the compact rat-race coupler found by the ARS technique. (a) Photograph of fabricated design. (b) Utilized measurement setup.

six preassigned parameters was successful, whereas the other algorithms have been diverged. This is consistent with the previous example, i.e., demonstrating that determination of the appropriate setup for SM-based algorithms is challenging. It should be noted that the computational cost of ARS is over two times lower than that of the successful version of ISM. The optimized coupler structure has been fabricated and measured (see Fig. 16). As indicated in Fig. 17, the agreement between the simulated and the measured characteristics is very good. The only notable discrepancies include a small (around 20 MHz) frequency shift between the responses, as well as slightly larger losses of the measured prototype (below 0.1 dB

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Fig. 17. Simulated (thin lines) versus measured (thick lines) characteristics of the optimized microstrip rat-race coupler filter.

for the operating frequency). The observed discrepancies are mostly due to using a simplified EM model that lacks SMA connectors. VI. C ONCLUSION In this paper, a novel ARS technique has been proposed for cost-efficient design optimization of microwave structures. ARS is a simple and generic method that is based on identifying frequency and amplitude relationships between the lowand high-fidelity models of the structure of interest, as well as similar relationships between the low-fidelity model responses across the design space. As opposed to simpler response correction methods, ARS exhibits very good generalization capability that translates into low cost of the ARS-based design optimization process. Operation and performance of ARS have been demonstrated using an eighth-order microstrip filter and a compact rat-race coupler, including experimental verification of the fabricated prototypes of the optimized structures. Comprehensive numerical comparison indicates that ARS outperforms competitive surrogate-assisted methods. ACKNOWLEDGMENT The authors would like to thank Computer Simulation Technology AG, Darmstadt, Germany, for making CST Microwave Studio available. R EFERENCES [1] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. New York, NY, USA: Springer, 2006. [2] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms. New York, NY, USA: Wiley, 2001. [3] S. Koziel, X.-S. Yang, and Q.-J. Zhang, Eds., Simulation-Driven Design Optimization and Modeling for Microwave Engineering. London, U.K.: Imperial College Press, 2013. [4] J. W. Bandler et al., “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 337–361, Jan. 2004. [5] S. Koziel, Q. S. Cheng, and J. W. Bandler, “Space mapping,” IEEE Microw. Mag., vol. 9, no. 6, pp. 105–122, Dec. 2008. [6] M. Sans, A. Rodríguez, J. Bonache, V. E. Boria, and F. Martín, “Design of planar wideband bandpass filters from specifications using a two-step aggressive space mapping (ASM) optimization algorithm,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 3341–3350, Dec. 2014. [7] D. Echeverría and P. W. Hemker, “Space mapping and defect correction,” Int. Comput. Methods Appl. Math., vol. 5, no. 2, pp. 107–136, 2005. [8] S. Koziel, “Shape-preserving response prediction for microwave design optimization,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 11, pp. 2829–2837, Nov. 2010.

[9] S. Koziel, J. W. Bandler, and K. Madsen, “Space mapping with adaptive response correction for microwave design optimization,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 2, pp. 478–486, Feb. 2009. [10] S. Koziel and A. Bekasiewicz, “Fast simulation-driven feature-based design optimization of compact dual-band microstrip branch-line coupler,” Int. J. RF Microw. Comput.-Aided Eng., vol. 26, no. 1, pp. 13–20, 2016. [11] N. Leszczynska, L. Szydlowski, and M. Mrozowski, “Zero-pole space mapping for CAD of filters,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 9, pp. 581–583, Sep. 2014. [12] C. Zhang, F. Feng, V.-M.-R. Gongal-Reddy, Q. J. Zhang, and J. W. Bandler, “Cognition-driven formulation of space mapping for equal-ripple optimization of microwave filters,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 7, pp. 2154–2165, Jul. 2015. [13] S. Koziel and J. W. Bandler, “Reliable microwave modeling by means of variable-fidelity response features,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 12, pp. 4247–4254, Dec. 2015. [14] D. Gorissen, L. Zhang, Q.-J. Zhang, and T. Dhaene, “Evolutionary neuro-space mapping technique for modeling of nonlinear microwave devices,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 2, pp. 213–229, Feb. 2011. [15] Q. S. Cheng, J. W. Bandler, and S. Koziel, “Tuning space mapping: The state of the art,” Int. J. RF Microw. Comput.-Aided Eng., vol. 22, no. 6, pp. 639–651, 2012. [16] D. G. Swanson and R. J. Wenzel, “Fast analysis and optimization of combline filters using FEM,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, USA, May 2001, pp. 1159–1162. [17] S. Koziel and L. Leifsson, Simulation-Driven Design by KnowledgeBased Response Correction Techniques. New York, NY, USA: Springer, 2016. [18] P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Theory of self-adjoint S-parameter sensitivities for lossless non-homogenous transmission-line modelling problems,” IET Microw. Antennas Propag., vol. 2, no. 3, pp. 211–220, Apr. 2008. [19] J. I. Toivanen, J. Rahola, R. A. E. Makinen, S. Jarvenpaa, and P. Yla-Oijala, “Gradient-based antenna shape optimization using spline curves,” Annu. Rev. Prog. Appl. Comput. Electromagn., Tampere, Finland, 2010, pp. 908–913. [20] S. Koziel, S. Ogurtsov, J. W. Bandler, and Q. S. Cheng, “Reliable spacemapping optimization integrated with EM-based adjoint sensitivities,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 10, pp. 3493–3502, Oct. 2013. [21] S. Koziel and J. W. Bandler, “Space-mapping optimization with adaptive surrogate model,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 3, pp. 541–547, Mar. 2007. [22] S. Koziel, J. W. Bandler, and Q. S. Cheng, “Constrained parameter extraction for microwave design optimisation using implicit space mapping,” IET Microw., Antennas Propag., vol. 5, no. 10, pp. 1156–1163, Jul. 2011. [23] J. W. Bandler, Q. S. Cheng, N. K. Nikolova, and M. A. Ismail, “Implicit space mapping optimization exploiting preassigned parameters,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 378–385, Jan. 2004. [24] S. Koziel, J. W. Bandler, and K. Madsen, “Quality assessment of coarse models and surrogates for space mapping optimization,” Optim. Eng., vol. 9, no. 4, pp. 375–391, 2008. [25] J. E. Rayas-Sanchez, “EM-based optimization of microwave circuits using artificial neural networks: The state-of-the-art,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 1, pp. 420–435, Jan. 2004. [26] S. Koziel, J. W. Bandler, and K. Madsen, “Towards a rigorous formulation of the space mapping technique for engineering design,” in Proc. Int. Symp. Circuits Syst. (ISCAS), vol. 1. May 2005, pp. 5605–5608. [27] C. de Boor, A Practical Guide to Splines. New York, NY, USA: Springer-Verlag, 2001. [28] S. Koziel and S. Ogurtsov, “Model management for cost-efficient surrogate-based optimisation of antennas using variable-fidelity electromagnetic simulations,” IET Microw. Antennas Propag., vol. 6, no. 15, pp. 1643–1650, Dec. 2012. [29] S. Koziel and S. Ogurtsov, Antenna Design by Simulation-Driven Optimization. New York, NY, USA: Springer, 2014. [30] S. Koziel and A. Bekasiewicz, “Expedited geometry scaling of compact microwave passives by means of inverse surrogate modeling,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 12, pp. 4019–4026, Dec. 2015. [31] C.-H. Tseng and H.-J. Chen, “Compact rat-race coupler using shuntstub-based artificial transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 11, pp. 734–736, Nov. 2008.

KOZIEL AND BEKASIEWICZ: RAPID MICROWAVE DESIGN OPTIMIZATION IN FREQUENCY DOMAIN DOMAIN USING ARS

Slawomir Koziel (M’03–SM’07) received the M.Sc. and Ph.D. degrees in electronics engineering from the Gda´nsk University of Technology, Gda´nsk, Poland, in 1995 and 2000, respectively, and the M.Sc. degrees in theoretical physics and mathematics and the Ph.D. degree in mathematics from the University of Gda´nsk, Gda´nsk, Poland, in 2000, 2002, and 2003, respectively. He is currently a Professor with the School of Science and Engineering, Reykjavík University, Reykjavik, Iceland. He is also a Visiting Professor with the Gda´nsk University of Technology. His current research interests include CAD and modeling of microwave circuits, simulation-driven design, surrogate-based optimization, space mapping, circuit theory, analog signal processing, evolutionary computation, and numerical analysis.

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Adrian Bekasiewicz received the M.Sc. degree in electronics engineering from the Gda´nsk University of Technology, Gda´nsk, Poland, in 2011, where he is currently pursuing the Ph.D. degree in wireless communication engineering. He is also a Research Associate with the School of Science and Engineering, Reykjavík University, Reykjavik, Iceland. He has authored or co-authored over 150 peer-reviewed papers. His current research interests include multiobjective optimization, metaheuristic algorithms, design of compact microwave antennas, and miniaturization of microwave/RF components.

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A New GaN HEMT Equivalent Circuit Modeling Technique Based on X-Parameters Riadh Essaadali, Student Member, IEEE, Anwar Jarndal, Member, IEEE, Ammar B. Kouki, Senior Member, IEEE, and Fadhel M. Ghannouchi, Fellow, IEEE Abstract— In this paper, a new accurate small- and largesignal equivalent-circuit-based modeling technique for gallium nitride (GaN) HEMT transistors grown on silicon substrate is presented. Despite X-parameters are developed as tools for the development of black-box modeling, they are used for equivalentcircuit-based model extraction. Unlike traditional modeling that uses the small-signal data to build with an indirect manner a nonlinear model, the proposed model is extracted from X-parameter measurements directly. However, similar to the equivalent-circuitbased models discussed in the literature, the new model is subdivided into extrinsic and intrinsic parts. The extrinsic part consists of linear elements and is related to the physical layout of the transistor. The intrinsic part can be extracted with the proposed analytical de-embedding technique. The nonlinear intrinsic elements are represented by new nonlinear lumped impedances and admittances whose extraction is carried out using a newly proposed technique. This new technique uses nonlinear network parameters, various X-parameter conversion rules, and basic analysis techniques of interconnected nonlinear networks. It is accurate and more advantageous than traditional transistor modeling techniques. The modeling procedure was applied to a 10 µm × 200 µm GaN HEMT with a gate length of 0.25 µm. A very good accordance between model simulations and measurements was obtained, validating the modeling approach. Index Terms— Equivalent circuit based model, large-signal model, nonlinear lumped element admittance, nonlinear lumped element impedance, parameter extraction, X-parameter de-embedding, X-parameters.

I. I NTRODUCTION

A

CCURATE large-signal models are used to describe the nonlinear behavior of microwave devices and are mandatory for nonlinear circuit design. In conventional approaches, compact models are defined by nonlinear equivalent circuits or by a system of nonlinear ordinary differential equations describing current–voltage and charge–voltage relations for nonlinear lumped elements [1]–[4]. Nonlinear models for microwave- and millimeter-wave devices are commonly based

Manuscript received September 9, 2015; revised December 16, 2015 and July 10, 2016; accepted July 15, 2016. Date of publication August 9, 2016; date of current version September 1, 2016. R. Essaadali and A. B. Kouki are with the Laboratoire de Communications et d’Intégration et de la Microélectronique, École de Technologie Supérieure, Montreal, QC H3C 1K3, Canada (e-mail: [email protected]; [email protected]). A. Jarndal is with the Electrical and Computer Engineering Department, University of Sharjah, Sharjah 27272, United Arab Emirates (e-mail: [email protected]). F. M. Ghannouchi is with the Intelligent RF Radio Laboratory, Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (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.2016.2594234

on dc and small-signal S-parameter measurements [3]–[5] and their nonlinear elements are described in terms of state functions [6]. These nonlinear elements are typically determined by integrating the intrinsic capacitances and conductances, which are extracted from multibias S-parameter measurements [1], [7]. This technique is based on smallsignal approximations of the nonlinear state functions at different bias points and requires a large set of S-parameter measurements [1], [2], [4], [5], [7]. Nonlinear RF measurements, such as Pin –Pout (AM/AM and AM/PM) are not part of the model construction and parameter extraction but are used for the model parameter value optimization and final model validation [1]. The accuracy limitations of this approach arise from several sources. The main source is that S-parameters are not suitable for the extraction of nonlinear elements. “Conventional S-parameters are defined only for passive systems or active systems behaving linearly with respect to a small signal applied around a static operating point” [8]. Furthermore, dc and S-parameter measurements cannot be made at the same time. Consequently, devices may not be characterized under the exact same conditions leading to possible errors due to temperature differences and memory effects. In 2008, Agilent introduced a new precision network analyzer, PNA-X, which is a mixer-based nonlinear vector network analyzer (NVNA) [9], [10] capable of measuring the newly developed X-parameters [10]. X-parameters represent new nonlinear scattering parameters, applicable to passive and active circuits under small and large signal excitations [9], [11], and are considered as a mathematical superset of small-signal S-parameters and large-signal S-parameters (LSSPs) [8], [9]. They capture the device behavior at the fundamental and harmonic frequencies in a single measurement for a given bias point. As such, they offer great potential for device modeling that will: 1) do away with dc measurements and 2) incorporate nonlinear measurements in the model construction and parameter extraction process. Some work has already begun to explore the application of X-parameters to device modeling [1], [7], [10], [12]–[16]. In [1], [7], and [10], the advantages of using NVNA measurement systems and X-parameters for transistor characterization and modeling are highlighted. It was shown that parameter extraction, model tuning, and validation using NVNA waveform data combined with the artificial neural network (ANN) modeling technique is more advantageous than the conventional technique. It was also mentioned that X-parameters are becoming useful and increasingly important in the

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future. X-parameters are explored for transistor modeling applications [12]–[14], [16]. However, they are still used as a model only at the external terminals of the device, which means that the internal structure of the equivalent circuit topology is not defined. An equivalent-circuit-based or compact model is preferred since it is able to indicate which of the material and structural properties of a given transistor affect its electrical performance: “it is worth pointing out that it is common to try and place physical meanings and origins on the circuit elements in the model” [17]. Equivalent circuit models can be measurement or physically based. The measurement-based equivalent circuit parameter values can be carried out over a range of bias voltages, loads, and input power, and the values can be stored in a table. Interpolation can be used to calculate a nonstored case. Mathematical function fittings can also be applied to obtain a nonlinear function that describes the behavior of the extracted element. In this paper, a new small- and large-signal equivalent circuit modeling technique using X-parameter measurements is proposed. The overall equivalent circuit topology is derived from the commonly used physics-based models and incorporates extrinsic (linear) and intrinsic (nonlinear) elements. The intrinsic elements are represented by new nonlinear lumped impedances and admittances whose extraction is carried out using a newly proposed technique. This new technique uses nonlinear network parameters, various X-parameter conversion rules, and basic analysis techniques of interconnected nonlinear networks. The main contribution of this paper is the development of an X-parameter-based equivalent circuit nonlinear modeling methodology instead of the S-parameterbased equivalent circuit modeling as reported in [4]. The remainder of this paper is organized as follows. Section II outlines the concept of two-port X-parameters and nonlinear network parameters. The new expressions of nonlinear lumped element impedance and admittance, derived from X-parameters, are also presented in this section. Section III defines the equivalent circuit topology and presents the extrinsic parameter extraction technique. In Section IV, a new technique for de-embedding extrinsic parasitic elements from the measured X-parameters is presented. Section V provides the details of the intrinsic parameter extraction from de-embedded X-parameter measurements. Section VII presents several validation results.

of indices i and j is from one to the number of system ports; indices kand l range from one to the highest harmonic index; and Fik , Sik, j l , and Tik, j l are complex functions of input power, dc biasing, and source and load terminations, respectively,

II. X-PARAMETERS AND N ONLINEAR C HARACTERISTIC PARAMETERS

where the voltage and current vectors at port p are [V p ] = [ v p1 v∗p1 . . . v pn v∗pn ]T and [I p ] = [ i p1 i ∗p1 . . . i pn i ∗pn ]T . The expressions of [Z nonlin−pq ] β submatrices in terms of Z αpmq j and Z pmq j are ⎤ ⎡ β β Z αp1q1 Z p1q1 . . . Z αp1qn Z p1qn ∗ ⎥ ⎢ β∗ ∗ ∗ ⎢ Z p1q1 Z αp1q1 . . . Z βp1qn Z αp1qn ⎥ ⎥ ⎢ ⎢ .. .. .. ⎥. .. [Z nonlin− pq ] = ⎢ ... . . . . ⎥ ⎥ ⎢ β β ⎥ ⎢ Zα α ⎣ pnq1 Z pnq1 . . . Z pnqn Z pnqn ⎦ ∗ ∗ ∗ ∗ β β Z pnq1 Z αpnq1 . . . Z pnqn Z αpnqn (4)

A. X-Parameter Formalism According to [8] and [10], “X-parameters are mathematically rigorous supersets of S-parameters and are applicable to linear and nonlinear components under both small-signal and large-signal conditions”. For a given device under test (DUT) driven by a large signal, the function describing the relation between reflected and incident waves is linearized and leads to an X-parameter expression in (1) [8]. In (1), the terms bik (B-waves) and a j l (A-waves) denote the scattered and incident traveling voltage waves, respectively. The range

bik = Fik (|a11|, Vgs , Vds , s ,  L )P k  + Sik, j l (|a11 |, Vgs , Vds , s ,  L )P k−l a j l ( j,l) =(1,1)

+



Tik, j l (|a11|, Vgs , Vds , s ,  L )P k+l a ∗j l .

( j,l) =(1,1)

(1) B. Nonlinear Network Parameters Thanks to the conversion rules of X-parameters to other nonlinear parameters, such as nonlinear network impedance, admittance, and ABCD parameters, and the analysis of cascade, series, and parallel configurations of purely nonlinear networks, as well as hybrid configurations of linear and nonlinear components, it is possible to use X-parameter measurements or data to manipulate different nonlinear circuit topologies. These tools allow the modeling of series topologies, Z-parameters, parallel topologies, Y-parameters and cascaded topologies, ABCD parameters, and any combination thereof. Just as X-parameters of a two-port relate harmonic components of B-waves and A-waves at both ports, nonlinear impedance parameters relate harmonic current to harmonic voltage components at both ports. Furthermore, in the same way that X-parameters include Sik, j l and Tik, j l terms, which are associated with a j l and a ∗j l [see (1)], it can be shown that β nonlinear Z-parameters include the Z αpmq j and Z pmq j terms ∗ associated with i q j and its conjugate i q j as follows:  β Z αpmq j (|a11|) i q j + Z pmq j (|a11|) i q∗j (2) v pm = (q, j )

where p and q refer to port numbers, m is the harmonic index at port p, and j is the harmonic index at port q. Equation (2) can be put in matrix form, which, for a two-port network, gives     [I1 ] [V1 ] = [Z nonlin] [V2 ] [I2 ]    [I1 ] [Z nonlin−11 ] [Z nonlin−12 ] (3) = [Z nonlin−21 ] [Z nonlin−22 ] [I2 ]

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The equivalent nonlinear impedance of a series configuration of N purely nonlinear, purely linear, or a mix of linear and nonlinear elements is the sum of the impedances of each component [Z nonlin] Eq =

N 

[Z nonlin]i .

(5)

i=1

Similarly, for nonlinear admittances, we use two terms β α ∗ Y pmq j and Y pmq j associated with vq j and to its conjugate vq j such that  β α ∗ i pm = Y pmq (6) j (|a11 |)vq j + Y pmq j (|a11 |)vq j . (q, j )

In matrix form for a two-port network, (6) can be written as     [V1 ] [I1 ] = [Ynonlin] [I2 ] [V2 ]    [Ynonlin−11 ] [Ynonlin−12 ] [V1 ] = . (7) [Ynonlin−21 ] [Ynonlin−22 ] [V2 ] The expression of [Ynonlin− pq ] submatrices β α Y pmq j and Y pmq j is ⎡ β α α Y p1q1 Y p1q1 . . . Y p1qn ⎢ β∗ ∗ β∗ α ⎢Y p1q1 Y p1q1 . . . Y p1qn ⎢ ⎢ .. .. .. [Ynonlin− pq ] = ⎢ ... . . . ⎢ β ⎢Y α α Y . . . Y ⎣ pnq1 pnqn pnq1 β∗

Y pnq1

∗

α Y pnq1

...

β∗

Y pnqn

in terms of β

Y p1qn

⎤

α∗ ⎥ ⎥ Y p1qn ⎥ .. ⎥. . ⎥ ⎥ β Y pnqn ⎥ ⎦ α∗ Y pnqn

(8)

The equivalent nonlinear admittance parameters of a parallel configuration of N purely nonlinear, purely linear, or a mix of linear and nonlinear elements are the sum of the admittances of each nonlinear component [Ynonlin ] Eq =

N 

[Ynonlin]i .

Nonlinear ABCD matrix includes four parameter categories: A, B, C, and D terms. A parameters relate input harmonic voltages to output harmonic voltages. B parameters relate input harmonic voltages to output harmonic currents. C parameters relate input harmonic currents to output harmonic voltages. D parameters relate input harmonic currents to output harmonic currents. Each ABCD parameter includes two-term categories associated, respectively, with a harmonic component and its conjugate of voltage and current present at the second port. The formulation of the nonlinear model based on nonlinear ABCD parameters is  β Aα1m2 j (|a11|)v2 j + A1m2 j (|a11|)v2∗ j v1m = (q, j )



β

α ∗ B1m2 j (|a11 |)i 2 j + B1m2 j (|a11 |)i 2 j

(10)

(q, j )

i 1m =



β

α ∗ C1m2 j (|a11 |)v2 j + C1m2 j (|a11 |)v2 j

(q, j )

+



(q, j )

β

Large-signal model for GaN HEMT intrinsic part [5].

In matrix form, (10) and (11) can be written as     [V1 ] [V2 ] = [ ABC Dnonlin] [I1 ] [I2 ]    [V2 ] [ Anonlin] [Bnonlin] . = [Cnonlin] [Dnonlin ] [I2 ]

α ∗ D1m2 j (|a11 |)i 2 j + D1m2 j (|a11 |)i 2 j . (11)

(12)

The expressions of [ ABC D nonlin] submatrices: [ Anonlin], [Bnonlin], [Cnonlin ], and [Dnonlin ] in terms of Aαpmq j , B αpmq j , β β β β C αpmq j , D αpmq j , A pmq j , B pmq j , C pmq j , and D pmq j are in (13). For simplification purpose, the letter R symbolizes A, B, C, and D symbols ⎤ ⎡ α β β α R1121 R1121 . . . R112n R112n β∗ ⎢ β∗ α∗ α∗ ⎥ . . . R112n R112n ⎥ ⎢ R1121 R1121 ⎥ ⎢ . . . . . ⎢ .. .. .. .. ⎥ [Rnonlin] = ⎢ .. ⎥. (13) ⎥ ⎢ α β β α ⎣ R1n21 R1n2n . . . R1n2n R1n2n ⎦ β∗

R1n21

∗

α R1n21

...

β∗

R1n2n

∗

α R1n2n

The equivalent nonlinear ABCD parameters of a cascade configuration of N purely nonlinear, purely linear, or a mix of linear and nonlinear elements is the product of the ABCD parameters of each nonlinear component

(9)

i=1

+

Fig. 1.

[ ABC Dnonlin] Eq =

N 

[ ABC Dnonlin_i ].

(14)

i=1

C. Nonlinear Lumped Element Impedance and Admittance Typically, the intrinsic transistor can be modeled and represented by the circuit shown in Fig. 1 [5]. The intrinsic part is composed of nonlinear current and charge sources that are obtained by path integration of the intrinsic conductances and capacitances [5]. The intrinsic nonlinear model can be modeled as a π network of nonlinear lumped element impedances. These parameters can be extracted directly from X-parameter measurement. Traditionally, a nonlinear circuit can be described by a Taylor series expansion of its nonlinear current/voltage, charge/voltage, or flux/current characteristic [18]–[20]. Power series and Volterra series can be used to describe or implement the behavior of a nonlinear component in a CAD software program. Unfortunately, none of these descriptions is suitable to the X-parameter concept: it is difficult to extract their expression from nonlinear network parameters. In this paper, new expressions of nonlinear lumped element impedance and admittance

ESSAADALI et al.: NEW GaN HEMT EQUIVALENT CIRCUIT MODELING TECHNIQUE BASED ON X-PARAMETERS

Fig. 2. Two-port network consisting of (a) series nonlinear impedance and (b) parallel nonlinear admittance between the input and output ports.

that are accurate and easy to construct from X-parameters are introduced. The expressions of the new nonlinear lumped element impedance and admittance and their implementation in a CAD software program are validated in Section IV-D. Linear impedance is employed when the current is proportional to the applied voltage [21]. In a nonlinear case, the current feeding nonlinear impedance is not linearly proportional to the voltage. The impedance of a nonlinear component can be defined from X-parameters. The harmonic voltage components of the nonlinear component are expressed in terms of nonlinear impedance and current [V ] = [ Z¯¯ NL ][I ]

(15)

where the harmonic voltage and current vectors are [V ] = [ v1 v1∗ . . . vn vn∗ ]T and [I ] = [ i 1 i 1∗ . . . i n i n∗ ]T , respectively; the terms vk and i k represent the voltage and current frequency components, respectively; and n is the highest harmonic order. To determine the expression of the nonlinear impedance [Z NL ], the expression of the nonlinear ABCD matrix can be used for the two-port network consisting of a series nonlinear impedance between the input and output ports, as illustrated in Fig. 2(a). The expression of harmonic voltage and current components at the first port can be expressed as     [V2 ] [V1 ] = [ ABC Dnonlin] [I1 ] [I2 ]    [Id ] [Bnonlin] [V2 ] = (16) [0] [Id ] [I2 ] where

⎡

α B1121

⎢ β∗ ⎢B ⎢ 1121 ⎢ ¯ ¯ [Bnonlin] = [ Z NL ] = ⎢ ... ⎢ ⎢Bα ⎣ 1n21 β∗

B1n21

β

B1121

...

α B112n

α B1121 .. .

B112n .. .

B1n21

... .. . ...

α B1n2n

α B1n21

...

B1n2n

∗

β

∗

β∗

β∗

n  j =1

Z iαj i j +

n  j =1

β

Z i j i ∗j .

where i is the frequency component index and n is the harmonic order. Equation (17) can be rewritten in a matrix form as follows: ⎤ ⎡ β β α α ⎤ ⎡ ⎤ ⎡ Z . . . Z Z Z 11 1n 11 1n i1 v1 ⎥ ⎢ ∗ ∗ ∗ ∗ β β α α ⎥ ⎢ i∗ ⎥ ⎢ v∗ ⎥ ⎢ Z Z . . . Z Z 11 11 1n 1n 1 ⎥ ⎢ ⎥⎢ 1 ⎥ ⎢ ⎢ .. ⎥ ⎢ .. .. .. .. ⎥ ⎢ .. ⎥. (18) . . = ⎢ ⎥ ⎢ . ⎥ ⎢ . . . . . ⎥ ⎥ ⎢ ⎥⎢ . ⎥ ⎢ β β ⎥⎣ i ⎦ ⎣ vn ⎦ ⎢ Z α α n ⎣ n1 Z n1 . . . Z nn Z nn ⎦ ∗ vn∗ i n∗ β∗ β∗ α∗ α Z n1 Z n1 . . . Z nn Z nn A nonlinear ABCD matrix can be used for the two-port network consisting of a parallel nonlinear admittance between the input and output ports, as illustrated in Fig. 2(b). The expressions of nonlinear submatrices [ Anonlin], [Bnonlin], [Cnonlin], and [Dnonlin] of nonlinear admittance are [ Anonlin] = [Dnonlin] = [Id ], [Bnonlin] = [0], and ⎤ ⎡ β β α α C1121 . . . C112n C112n C1121 ⎥ ⎢ β∗ β∗ α∗ α∗ ⎥ ⎢C ⎢ 1121 C1121 . . . C112n C112n ⎥ ⎢ .. .. ⎥. .. .. [Cnonlin] = [Y¯¯NL ] = ⎢ ... . . . ⎥ . ⎥ ⎢ β β ⎥ ⎢ Cα α ⎣ 1n21 C1n2n . . . C1n2n C1n2n ⎦ ∗ ∗ β β α∗ α∗ C1n21 C1n21 . . . C1n2n C1n2n The expression of each harmonic current component as a function of nonlinear impedances and harmonic current components is as follows: ii =

n 

Yiαj v j +

j =1

n 

β

Yi j v∗j .

(19)

j =1

The admittance can easily be deduced from nonlinear impedance ⎡ [Y¯¯NL ] = [ Z¯¯ NL ]−1

α Y11

β

β

⎢ β∗ ⎢ Y11 ⎢ ⎢ = ⎢ ... ⎢ ⎢ α ⎣ Yn1

Y11

...

∗

α Y1n

Y1n

α Y11 .. . β Yn1

... .. . ...

α Y1n .. . β Ynn

Yn1

∗

Y1n .. . α Ynn

α Yn1

...

Ynn

α Ynn

β∗

β∗

β∗

∗

∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

⎤

(20)

⎥ α∗ ⎥ B112n ⎥ .. ⎥. . ⎥ ⎥ β ⎥ B1n2n ⎦ α∗ B1n2n

The electrical nonlinear π network (Fig. 3) has three nonlinear impedance branches connected in series to form a closed circuit, with the three junction points forming an output terminal, an input terminal, and a common output and input terminal. A nonlinear π network model can be extracted directly from nonlinear two-port admittance parameters. The expression of nonlinear harmonic current components as a function of nonlinear harmonic voltage components is

  ¯   −[Y¯¯NL3 ] [Y¯NL1 ] + [Y¯¯NL3 ] [V1 ] [I1 ] = . [I2 ] [V2 ] −[Y¯¯NL3] [Y¯¯NL2 ] + [Y¯¯NL3 ] (21)

β

B112n

Consequently, the expression of each harmonic voltage component vi can be expressed as a function of nonlinear β impedances (Z iαj , Z i j ) and harmonic current components vi =

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(17)

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Fig. 3.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Nonlinear π network model.

Fig. 4. (a) Photograph of 200 μm × 10 μm GaN HEMT on Si substrate and (b) its equivalent circuit model [26].

III. GaN HEMT X-PARAMETERS ’ E XTRINSIC PARAMETER E XTRACTION Throughout the remainder of this paper, we will consider an on-silicon (Si) gallium nitride (GaN) HEMT (high-electron mobility transistor) device for which the modeling technique is applied. A. Model Topology Definition The Nitronex GaN HEMT structure to be modeled was grown on a Si substrate. A small-signal equivalent circuit for GaN HEMTs on a Si substrate was presented in [22]. The topology of the developed model is illustrated in Fig. 4. In the extrinsic part of this model, Cgp and Cdp represent parasitic capacitances due to pad, fingers, and crossover connections [22]. Rd , Rs , and Rg include contact and semiconductor resistances, whereas L d , L s , and L g include model metallization inductances. Rgg and Cgg (Rdd and Cdd ) are added to take into account extra parasitic substrate/buffer loading effects of GaN on Si substrate. B. Extrinsic Parameter Extraction Using Open Structures and Cold Measurements In order to accurately characterize the intrinsic of the device, the impact of the parasitic elements must be removed. The parasitic elements of the pad contacts can be extracted from open de-embedding structure fabricated on the same wafer of the DUT, while cold forward measurements for the same device can be used to extract the parasitic inductances and resistances. Fig. 5 shows the implemented on-wafer open de-embedding structure and its equivalent circuit model [23]–[25]. For passive components, X-parameters reduce to S-parameters [8], and all nonlinear Z β terms and all cross frequency Z αpmq j,m= j terms vanish. The expressions of the nonlinear Z αpmqm terms reduce to linear impedance in

Fig. 5. (a) On-wafer GaN open de-embedding structure for a 2-mm GaN HEMT and (b) its equivalent circuit model.

linear operation mode and are equal to Z pq . The extrinsic elements Rgg and Rdd can be extracted from the real part of parasitic gate and drain impedances. The Z-parameters of the open structure equivalent circuit can be expressed as follows:

2 C ω Rgg Rgg gg 1 α (22) Z 1111 = −j + 2 R2 2 R2 1 + ω2 Cgg ωCgp 1+ω2Cgg gg gg

2 C ω Rdd Rdd 1 dd α Z 2121 = . (23) − j + 2 R2 2 R2 ωCd p 1+ω2Cdd 1 + ω2 Cdd dd dd Extrinsic resistances Rgg and Rdd can be extracted by α extrapolating the curves of the measured real part of Z 1111 α and Z 2121 at ω = 0. However, the reliability of extraction depends on the accuracy of extrapolation and measurement uncertainties. Another more accurate method is to extract Rgg and Rdd from the slope of the curves α ])(1/(ω2 real[Z α ])) versus 1/ω2 . The of 1/(ω2 real[Z 1111 2121 expressions of these curves are given by   1 1 1 2  α  = (24) + Rgg Cgg Rgg ω2 ω2 real Z 1111   1 1 1 2  α  = . (25) + Rdd Cdd Rdd ω2 ω2 real Z 2121 Parasitic capacities Cgg and Cdd are derived from the slope α ](1/real[Z α ]) versus ω2 . The of the curves of 1/real[Z 1111 2121 expressions of these curves are 1 1 2 2  α  = + Rgg Cgg ω Rgg real Z 1111 1 1 2 2  α  = + Rdd Cdd ω . Rdd real Z 2121

(26) (27)

α ] At higher frequencies, the equation imag[Z 1111 α ω(imag[Z 2121]ω) tends toward −1/Ceqg = −1/Cgp − 1/Cgg(−1/Ceqd = − 1/Cdp − 1/Cdd )  α  lim imag Z 1111 ω ω→∞   2 ω2 Cgg Rgg 1 1 =− + (28) = lim − 2 C 2 ω2 ω→∞ Cgp 1 + Rgg C eqg gg  α  ω lim imag Z 2121 ω→∞   2 ω2 Cdd Rdd 1 1 =− + . (29) = lim − 2 C 2 ω2 ω→∞ Cd p C 1 + Rdd eqd dd

ESSAADALI et al.: NEW GaN HEMT EQUIVALENT CIRCUIT MODELING TECHNIQUE BASED ON X-PARAMETERS

Fig. 6.

GaN HEMT equivalent circuit model.

Parasitic capacities Cgp (Cd p ) are deduced from Ceqp (Ceqd ) as follows: Ceqg Cgg Cgp = (30) Cgg − Ceqg Ceqd Cdd Cd p = . (31) Cdd − Ceqd The measured Y-parameters converted from X-parameters of the open structure are then de-embedded from cold Y-parameter measurements at Vds = 0 V and Vgs ≥ 0 V. Extrinsic inductances L g , L d , and L s can then be extracted from the curves of the imaginary part of the stripped Z-parameter measurements by linear regression [27]. The extrinsic resistances Rg , Rd , and Rs can also be extracted from the curves of the real part of the stripped Z-parameter measurements. The whole extracted values of the extrinsic elements are then optimized to find the best fit for the previously stated cold measurements. The same procedure is used to find the optimal value for each model element [27]. IV. X-PARAMETER -BASED D E -E MBEDDING T ECHNIQUE There are many papers published on the topic of de-embedding techniques for S-parameters [28]–[30] and for large-signal measurements, but very little has been published on X-parameters [15], [31]. A typical or generalized equivalent circuit model of a GaN transistor is shown in Fig. 6. In order to avoid overloading the demonstrations, admittances G 1 and G 2 are designated to represent the parasitic gate and drain elements, respectively. Impedances Z 1 , Z 2 , and Z 3 represent the gate, source, and drain cascaded configurations of inductance with resistance, respectively. The values of admittances (G 1 and G 2 ) and impedances (Z 1 , Z 2 , and Z 3 ) ⎡

Z 1 (ω) + Z 3 (ω) ⎢ 0 ⎢ ⎢ .. [Z 11 ] = ⎢ . ⎢ ⎣ 0 0 ⎡ Z 2 (ω) + Z 3 (ω) ⎢ 0 ⎢ ⎢ .. [Z 22 ] = ⎢ . ⎢ ⎣ 0 0

0 Z 1∗ (ω) + Z 3∗ (ω) .. .

2763

are determined by the extrinsic elements’ extraction procedure presented in the previous section. X-parameters of the intrinsic part can be easily determined by applying the X-parameters de-embedding procedure explained below. The measurements of X-parameter can be performed using a PNA-X network analyzer with an NVNA option. X-parameters of the intrinsic part are calculated or deduced by means of transformations between X-parameters, nonlinear Y-parameters, and nonlinear Z-parameters. Admittances G 1 and G 2 are in parallel. To subtract their effects, the measured X-parameters [X] of the transistor should be converted into nonlinear parameters [Ynonlin] using the following conversion rule: [Ynonlin] =

1 [[I d] + [X]]−1 [[I d] − [X]] Zc

where [Id ] and Z c are the identity matrix and the characteristic impedance, respectively. The current vector [Ii ] = ∗ . . . i i ∗ ]T can be expressed as a function of the [ i i1 i i1 in in ∗ ∗ T . . . vin vin ] through voltage vector [Vi ] = [ vi1 vi1     [I1 ] [V1 ] = [Ynonlin] [I2 ] [V2 ]    [Ynonlin−11 ] [Ynonlin−12 ] [V1 ] = . (33) [Ynonlin−21 ] [Ynonlin−22 ] [V2 ] The harmonic current components [I  1 ] and [I  2 ] that flow through impedances [Z 1 ] and [Z 2 ] can be calculated as follows:         [I1 ] [I1 ] [G 1 ] [0] [V1 ] = − [I2 ] [I2 ] [0] [G 2 ] [V2 ]   [V1 ] = [Ynonlin_A ] (34) [V2 ] where

[G i ]i=1,2

⎡

G i (ω) ⎢ 0 ⎢ ⎢ = ⎢ ... ⎢ ⎣ 0 0

0 G ∗i (ω) .. . 0 0

... ... .. . ... ...

0 0 .. .

G i (nω) 0

0 0 .. . 0 G ∗i (nω)

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

The resulting nonlinear Y-matrix after de-embedding the admittances G 1 and G 2 is   [G 1 ] [0] [Ynonlin_ A ] = [Ynonlin] − . (35) [0] [G 2 ] ... ... .. .

0 0 .. .

0 0 .. .

0 0

... ...

Z 1 (nω) + Z 3 (nω) 0

0 Z 1∗ (nω) + Z 3∗ (nω)

0 Z 2∗ (ω) + Z 3∗ (ω) .. .

... ... .. .

0 0 .. .

0 0 .. .

0 0

(32)

... ...

Z 2 (nω) + Z 3 (nω) 0

0 Z 2∗ (nω) + Z 3∗ (nω)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2764

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

The harmonic voltage components at the input and the output of the intrinsic part can be expressed as           I [V1DUT] [V1 ] [Z 11 ] [Z 12]  1  (36) = − [V2DUT] [V2 ] [Z 21 ] [Z 22] I2 and matrices Z 11 and Z 22 , shown at the bottom of the previous page, and ⎡ ⎤ Z 3 (ω) 0 ... 0 0 ⎢ 0 Z 3∗ (ω) . . . 0 0 ⎥ ⎢ ⎥ ⎢ .. . . .. ⎥. . .. .. .. [Z 12 ] = [Z 21] = ⎢ . . ⎥ ⎢ ⎥ ⎣ 0 0 ⎦ 0 . . . Z 3 (nω) 0 0 ... 0 Z 3∗ (nω)

Fig. 7.

Large-signal intrinsic model to be extracted from X-parameters.

Fig. 8.

Transistor intrinsic part modeling with FDD components.

The resulting nonlinear Y-matrix [Ynonlin_ A ] is transformed into a nonlinear Z-matrix. The effects of series impedances Z 1 , Z 2 , and Z 3 are subtracted as follows:   [Z 11] [Z 12 ] −1 [Z nonlin_B ] = [Ynonlin_ A ] − (37) [Z 21] [Z 22 ] where [Z nonlin_B ] is the nonlinear impedance matrix relating the nonlinear voltage harmonic components at the input and output of the intrinsic parts [V1DUT] and [V2DUT], respectively, to nonlinear current harmonic components [I1DUT ] and [I2DUT ]     [I1DUT ] [V1DUT ] = [Z nonlin_B ] . (38) [V2DUT ] [I2DUT ] The X-parameters of the intrinsic part are obtained through the conversion rule from nonlinear Z-parameters to X-parameters. The expression of the X-parameter matrix is  −1   1 1 [X DUT ] = [Id ] + [Z nonlin_B ] [Z nonlin_B ] − [Id ] . Zc Zc (39) V. I NTRINSIC PARAMETER E XTRACTION A. X-Parameter-Based Large-Signal Intrinsic Model Topology Definition and Extraction Once the extraction of the extrinsic elements is complete, the de-embedding of the parasitic parameters enables the X-parameters of the intrinsic part to be determined. Typically, the intrinsic transistor can be modeled by the equivalent circuit in Fig. 1. The intrinsic part is composed of nonlinear gate and drain current and charge sources that are obtained by path integration of the intrinsic gate and drain conductances and capacitances. The intrinsic nonlinear model can be modeled as a π network of nonlinear impedances, as shown in Fig. 7. Nonlinear impedances [Z GS ], [Z GD ], [Z DG ], and [Z DS ] model the gate– source, gate–drain, drain–gate, and drain–source nonlinear junctions, respectively. The nonlinear impedances can be extracted directly from the de-embedded X-parameters. The intrinsic elements are extracted as a function of the extrinsic gate–source VGS and drain–source VDS voltages. The nonlinear admittance matrix of the intrinsic part of an active device is

  ¯   −[Y¯¯GD ] [V1 ] [Y¯GS ] + [Y¯¯GD ] [I1 ] = . (40) [I2 ] [V2 ] −[Y¯¯ ] [Y¯¯ ] + [Y¯¯ ] DG

DS

DG

Nonlinear impedances of intrinsic transistor junctions—gate source [Z GS ], drain source [Z DS ], gate drain [Z GD ], and drain gate [Z DG ]—can be easily determined from (40). Their expressions are as follows: [ Z¯¯ GD (VGS , VDS )] = −[Y12 ]−1 [ Z¯¯ DG (VGS , VDS )] = −[Y21 ]−1 [ Z¯¯ GS (VGS , VDS )] = [[Y11 ] + [Y12 ]]¨−1 [ Z¯¯ DS (VGS , VDS )] = [[Y22 ] + [Y21 ]]¨−1 .

(41) (42) (43) (44)

B. Transistor Intrinsic Part Modeling With Frequency-Domain Defined Devices In the Advanced Design System (ADS), a frequency-domain defined (FDD) component enables the spectral values of current and voltage to be expressed in terms of other harmonic components of voltages and currents through algebraic relationships. This component simplifies the development of nonlinear models. An FDD is ideal for modeling the transistor intrinsic part. As illustrated in Fig. 8, the intrinsic nonlinear part of the transistor is modeled by a cascade of FDD components. The nonlinear impedances shown in Fig. 8 are intrinsic. However, FDD components can describe input and output voltage components or current components; therefore, the relation between the input or output spectral component voltages and currents is required. There are two types of configurations in Fig. 8: series and parallel configurations. The model of series nonlinear impedance can be described by an FDD component, as shown in Fig. 9(a). The equations of the FDD model that describe the

ESSAADALI et al.: NEW GaN HEMT EQUIVALENT CIRCUIT MODELING TECHNIQUE BASED ON X-PARAMETERS

Fig. 9.

(a) Series and (b) parallel nonlinear impedance FDD modeling.

behavior of a nonlinear component in a series configuration are [I1 ] = [ Z¯¯ NL ]−1 [[V1 ] − [V2 ]] [I2 ] = [ Z¯¯ NL ]−1 {[V2 ] − [V1 ]}.

(45) (46)

The model of parallel nonlinear impedance can be described by the FDD component, as illustrated in Fig. 9(b). The equations of the FDD model that describe the behavior of a nonlinear component in a parallel configuration are [V1 ] = [Y¯¯NL ]−1 {[I1 ] + [I2 ]} [V2 ] = [Y¯¯NL ]−1 {[I1 ] + [I2 ]}.

(47) (48)

C. Trapping and Self-Heating Effects in Model Embedding Trapping effects are related to buffer and surface traps in the active region of the transistor [5]. The surface trapping is attributed mainly to polarization-induced surface states. The buffer traps refers to the hollows in the nucleon layer between the GaN buffer layer and the silicon substrate. The lattice mismatch between the GaN layer and the Si substrate results in higher concentration of free ions that are responsible for electron trapping. In normal operation, the electrons move in the two-dimensional electron gas (2-DEG) channel, but they could deviate into the buffer where they can be trapped. Consequently, they will not be conducted and will not follow the RF and/or dc signals. The trapped electrons induce current dispersion because their negative charges deplete the 2-DEG channel. In [32], “it has been shown that modeling the trapping effects improves the large-signal simulation results, particularly when the output loads deviate from the optimum matching conditions”. The effects of the traps are taken into account by modifying the command voltage of the current source by adding transients to gate-to-source voltage. These delay times are related to the capture or the emission of charges by traps. Typically, the parameters of the lag circuits are extracted from pulsed IV and/or S-parameters measurements. In [33], a nonlinear automated measurement system, which is based on low-frequency multiharmonic signal sources, is presented. The system is used for in-depth investigation of low-frequency dispersion. Another concept of the low-frequency dispersion modeling is presented in [34] using an integral transform

2765

approach for the description of the drain current. In [35], an accurate table-based large-signal model for AlGaN/GaN HEMTs accounting for trapping- and self-heating-induced current dispersion is presented. To include these effects, the RF drain current is modeled as a linear combination of the isothermal dc current and the deviation in the drain current due to the surface-trapping, buffer-trapping, and self-heating effects. The RF drain current is derived from pulsed I –V measurement. The amount of trapping-induced current dispersion is controlled by the averaged values of the intrinsic voltages that are extracted using RC high-pass circuits at gate and drain sides. For the self-heating, the amount of the induced current dispersion is controlled by a low-pass circuit that determines the value of the normalized channel temperature rise. In [36], a general analytical formulation for the description of low-frequency dispersion is presented. The model avoids the simplifying approximations of the description of the complex phenomena related to low-frequency dispersion. The RF current depends not only on the instantaneous values of the voltage at the device ports but also on other variables, such as average values of the voltage. The trapping and self-heating effects modeling approach presented in this paper is based on X-parameters and consistent with the small-signal Z-parameter model. Z-parameters in the model are functions of the input power, dc voltages, fundamental frequency, and source and load terminations. The question is how trapping and thermal effects can be embedded in the model. These two effects result in extremely different values for the drain at low and high RF frequencies. The gate charge could be neglected for frequency less than the intrinsic transient frequency f T . Thus, [Z GS ] and [Z GD ] could be kept without any modification [ Z¯¯ GS (|a11 |, Vgs , Vds , s ,  L , fo )] = [ Z¯¯ GS,RF (|a11|, Vgs , Vds , s ,  L , f o )] (49) ¯ ¯ [ Z (|a |, V , V ,  ,  , f )] GD

11

gs

ds

s

L

o

= [ Z¯¯ GD,RF (|a11 |, Vgs , Vds , s ,  L , fo )].

(50)

Thermal effect influences the electron mobility and saturation velocity of electrons and thus both channel conductance gds and transconductance gm will be affected under smallsignal regime. This will also be reflected on the values of [Z DS ] and [Z DG ] under large-signal condition. Thus, these two elements should be a function of channel or junction temperature T in addition to Vgs and Vds [ Z¯¯ DS (|a11 |, Vgs , Vds , s ,  L , fo )] = [ Z¯¯ (|a |, V , V ,  ,  , f )] + [α ]T (51) DS,RF

11

gs

ds

s

L

o

T1

[ Z¯¯ DG (|a11|, Vgs , Vds , s ,  L , f o )] = [ Z¯¯ DG,RF (|a11|, Vgs , Vds , s ,  L , f o )] + [αT 2 ]T. (52)

Surface and buffer trapping influence the channel electron concentration due to backgating. The negative charge due to the trapped electrons in the surface and in the buffer modifies the depletion region and thus drains current. However, these two effects are observable only under RF and thus the quiescent (Vgso, Vdso )with fitting matrices [α D ] and [αG ] can be

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

The measured Z-parameters (converted from X-parameters) at quiescent voltages (bias voltages) Vds = 0 V and Vgs < VPinch−off and RF frequency (>100 MHz) can be described by (58) and (59). In this condition, the static (average) power Vds Ids is negligible and thus T = 0 [ Z¯¯ DS (Vgs , Vds )] = [ Z¯¯ DS,RF (Vgs , Vds )] [ Z¯¯ DG (Vgs , Vds )] = [ Z¯¯ DG,RF (Vgs , Vds )] + [αG ]Vgso .

(58) (59)

The measured Z-parameters at quiescent voltages (bias voltages) Vds = 0 V and Vgs = 0 V and RF frequency (>100 MHz) can be described by (60) and (61). The dissipated power is also negligible in this case [ Z¯¯ DS (Vgs , Vds )] = [ Z¯¯ DS,RF (Vgs , Vds )] [ Z¯¯ DG (Vgs , Vds )] = [ Z¯¯ DG,RF (Vgs , Vds )]. Fig. 10. Equivalent circuit models for an intrinsic GaN HEMT model coupled with thermal and trapping effects.

+ [αT 1 ]T + [α D ]Vdso

[ Z¯¯ DG (|a11|, Vgs , Vds , s ,  L , f o )] = [ Z¯¯ DG,RF (|a11 |, Vgs , Vds , s ,  L , fo )] + [αT 2 ]T + [αG ]Vgso.

(53)

(54)

The expressions of the matrices [αT 1 ], [αT 2 ], [α D ], and [αG ] are in (55). For simplification purpose, the letter F represents αT 1 , αT 2 , α D , and αG symbols ⎡ α β β ⎤ α F1n F11 F11 . . . F1n ∗ ∗ ⎢ β β α∗ α∗ ⎥ ⎥ ⎢ F11 F11 . . . F1n F1n ⎥ ⎢ ⎢ .. .. .. .. ⎥. .. (55) [F] = ⎢ . . . . . ⎥ ⎥ ⎢ ⎥ ⎢ α β β α ⎣ Fn1 Fn1 . . . Fnn Fnn ⎦ β∗

Fn1

∗

α Fn1

...

β∗

Fnn

∗

α Fnn

The equivalent circuit topology of the nonlinear intrinsic model including thermal and trapping effects circuit model is shown in Fig. 10. The circuit in the top models the intrinsic part of the model based on X-parameters. The circuit in the middle predicts the power dissipation induced temperature. The other two subcircuits are for simulating the dynamic trapping effects. At single operating frequency and single input power (with 50 source and load terminations), the measured X-parameters and Z-parameters will be a function of just Vgs and Vds (the intrinsic voltages including the RF and dc components). Thus, [Z DS ] and [Z DG ] can be formulated as [ Z¯¯ (V , V )] DS

gs

ds

= [ Z¯¯ DS,RF (Vgs , Vds )] + [αT 1 ]T + [α D ]Vdso

[ Z¯¯ DG (Vgs , Vds )] = [ Z¯¯ DG,RF (Vgs , Vds )] + [αT 2 ]T + [αG ]Vgso.

(56) (57)

(61)

The measured Z-parameters at quiescent voltages (bias voltages) Vds = Vds,h >30 V and Vgs < VPinch−off and RF frequency (>100 MHz) can be described by [ Z¯¯ DS (Vgs , Vds )] = [ Z¯¯ DS,RF (Vgs , Vds )] + [α D ]Vdso [ Z¯¯ DG (Vgs , Vds )] = [ Z¯¯ DG,RF (Vgs , Vds )] + [αG ]Vgso .

used to simulate this effect as follows: [ Z¯¯ DS (|a11 |, Vgs , Vds , s ,  L , fo )] = [ Z¯¯ DS,RF (|a11|, Vgs , Vds , s ,  L , f o )]

(60)

(62) (63)

For static operating condition and at low frequency ( 1 vanish as well as all cross frequency Sikj l,k=l terms and all Tikj l,k=l terms. S-parameters are compared with the following terms: F X 11 = S1111 a11 XF = 21 = S2111 a11 = S1121 = S2121 .

S11 =

(76)

S21

(77)

S12 S22

(78) (79)

Fig. 26. Comparison between the logarithmic amplitudes of measured and simulated X-parameters of a 2-mm GaN HEMT on Si substrate measured at (a) 750 MHz for Vds = 3 V, Id = 113 mA, Vgs = −1.28 V, and Ig = 11 mA, (b) 1500 MHz for Vds = 1 V, Id = 30 mA, Vgs = −1.48 V, and Ig = 13 mA, and (c) 6 GHz for Vds = 25 V, Id = 125 mA, Vgs = −1 V, Meas and Ig = 8 mA. S Meas pmq j and T pmq j are the terms of measured X-parameters Model and S Model pmq j and T pmq j are the terms of signal model X-parameters. The set of X-parameter terms are chosen randomly.

The model is simulated under small-signal S-parameters and LSSPs and compared with the S-parameter measurements. The small-signal S-parameter simulation runs, but it gives an

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Fig. 27. Comparison between the phase of measured and simulated X-parameters of a 2-mm GaN HEMT on Si substrate measured at (a) 750 MHz for Vds = 3 V, Id = 113 mA, Vgs = −1.28 V, and Ig = 11 mA, (b) 1500 MHz for Vds = 1 V, Id = 30 mA, Vgs = −1.48 V, and Ig = 13 mA, and (c) 6 GHz for Vds = 25 V, Id = 125 mA, Vgs = −1 V, and Ig = 8 mA. Meas Model S Meas pmq j and T pmq j are the terms of measured X-parameters and S pmq j and Model are the terms of signal model X-parameters. The set of X-parameter T pmq j terms are chosen randomly.

unexpected result. This is due to the fact that the FDD is not fully compatible with all the different circuit analysis modes of ADS. However, the LSSP simulation gives the expected results. LSSP simulation of the xnp file that includes

Fig. 28. Extrinsic input and output voltage and current waveform simulations (solid lines) compared with measurements (symbols) of a 2-mm GaN HEMT on Si substrate at (a) 750 MHz for Vds = 3 V, Id = 113 mA, Vgs = −1.28 V, and Ig = 11 mA and for an input power sweep between 1 and 15 dBm with a step of 2 dBm, (b) 1500 MHz for Vds = 1 V, Id = 30 mA, Vgs = −1 V, and Ig = 13 mA and for an input power sweep between 1 and 15 dBm with a step of 2 dBm, and (c) 6 GHz for Vds = 25 V, Id = 125 mA, Vgs = −1 V, Ig = 8 mA and for an input power sweep between 1 and 17 dBm with a step of 2 dBm.

X-parameter measurements is compared with the simulation of the equivalent circuit model. Then, they are compared with S-parameters measured by the NVNA: the NVNA is able

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Fig. 30. Extrinsic and intrinsic (de-embedded) dynamic loadline simulations (solid lines) compared with measurements (symbols) of a 2-mm GaN HEMT on Si substrate at (a) 750 MHz for Vds = 3 V and Vgs = −1.28 V and for an input power of 15 dBm, (b) 1500 MHz for Vds = 1 V and Vgs = −1 V and for an input power of 15 dBm, and (c) 6 GHz for Vds = 25 V and Vgs = −1 V and for an input power of 17 dBm.

The model is validated within 50- terminations for a set of frequencies between 500 MHz and 6 GHz, for Vds = 10 V, Ids = 189 mA, Vgs = −1 V, and Igs = 14 mA, and for an input power equal to −20 dBm where the device operates linearly. Good accordance between the measured S-parameters and the predicted ones is seen in Fig. 33. The S-parameter measurements are also compared with the X-parameters mentioned in (76)–(79). VII. D ISCUSSION

Fig. 29. Intrinsic (de-embedded) input and output voltage and current waveform simulations (solid lines) compared with de-embedded measurements (symbols) of a 2-mm GaN HEMT on Si substrate at (a) 750 MHz for Vds = 3 V, Id = 113 mA, Vgs = −1.28 V, and Ig = 11 mA and for an input power sweep between 1 and 15 dBm with a step of 2 dBm, (b) 1500 MHz for Vds = 1 V, Id = 30 mA, Vgs = −1 V, and Ig = 13 mA and for an input power sweep between 1 and 15 dBm with a step of 2 dBm, and (c) 6 GHz for Vds = 25 V, Id = 125 mA, Vgs = −1 V, and Ig = 8 mA and for an input power sweep between 1 and 17 dBm with a step of 2 dBm.

to run S-parameter measurement, which allows keeping the same setup and the same calibration, and therefore the same conditions.

The advantage of using this new modeling technique is that it significantly reduces the number of required measurements by extracting the nonlinear impedances of the intrinsic part of the transistor directly from X-parameter measurements. For an amplifier working in a 50- environment and for a given bias voltage, input power, and load, there is no need to run a high set of measurements compared with the conventional modeling technique. Another advantage is that the model is easy and fast to construct and mimic the physical operation of the transistor. It is consistent with the physical structure of the transistor and is more suitable for design improvement. For this purpose, after removing the effect (de-embedding) of parasitic elements, the intrinsic part, which is related to the core of the transistor, is obtained by a direct extraction of the intrinsic elements from X-parameter measurements. In order to keep connection to the basic device physics, the intrinsic part is represented by four nonlinear impedances describing the gate–source, drain–gate, gate–drain, and drain–source impedances. Thus, it has a good link to the physical operation, which is advantageous for circuit level simulation: it allows a seamless transition from the transistor level to circuit level. In addition to these benefits, the model is accurate in either small- or large-signal operation mode, which is due to the use of X-parameters to characterize the device and its capability to capture the nonlinear behavior accurately, better than dc and S-parameters. Thus, the developed model is able to operate

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Fig. 31. Output power simulations (solid lines) compared with measurements (symbols) of a 2-mm GaN HEMT on Si substrate at (a) 750 MHz for Vds = 3 V, Id = 113 mA, Vgs = −1.28 V, and Ig = 11 mA, (b) 1500 MHz for Vds = 1 V, Id = 30 mA, Vgs = −1.48 V, and Ig = 13 mA, and (c) 6 GHz for Vds = 25 V, Id = 125 mA, Vgs = −1 V, and Ig = 8 mA.

Fig. 32. AM–AM and AM–PM simulations (solid lines) compared with measurements (symbols) of a 2-mm GaN HEMT on Si substrate at (a) 750 MHz for Vds = 3 V, Id = 113 mA, Vgs = −1.28 V, and Ig = 11 mA, (b) 1500 MHz for Vds = 1 V, Id = 30 mA, Vgs = −1.48 V, and Ig = 13 mA, and (c) 6 GHz for Vds = 25 V, Id = 125 mA, Vgs = −1 V, and Ig = 8 mA.

under large signal stimulus and accurately predict the harmonics and mismatch affects. Its accuracy and generality guarantee its exploitation with confidence in active RF circuit design for modern communication systems. Thanks to this modeling technique, the development of die model is quick, as well as the development of package model, just by adding the bonding and package models. As any equivalent-circuit model, this model takes into account electrothermal and trapping effects. It is easier to

embed such a complex phenomenon in an equivalent-circuitbased model. Then, it is possible to identify the sources of memory effects and their extent, which can be advantageous for memory effects removing or reducing trials. The different types of simulations performed in ADS with the proposed model such as harmonic balance, LSSP, and X-parameters are fast and no convergence issues were observed. The proposed approach is better than the conventional procedure that uses the small-signal data to build with an

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be inaccurate under large-signal operation conditions since they are extracted under small-signal conditions and then used to simulate the behavior of the device under largesignal excitations [38], [40]. Moreover, they have additional drawbacks: 1) discrepancy between dc and small-signal characteristics due to time-dependent trapping effects; 2) error in load-pull measurements; and 3) nonphysical model due to numerical optimization techniques [41]. On the other side, the new approach directly uses the nonlinear measured response (X-paramaters) of the device under realistic stimulus: in the model extraction process, the DUT is in a similar state as the usual operating conditions. Thus, this new approach can guarantee the accuracy and the reliability of the large-signal model. Moreover, it benefits from X-parameters that accurately measure the different harmonics. That is why no optimization algorithms are used to extract the nonlinear intrinsic parameters in this procedure.

Fig. 33. Measured S-parameters [(a) S11 , (b) S12 , (c) S21 , and (d) S22 ] compared with the simulated ones of the equivalent-circuit based model with ADS LSSP and with the simulated X parameters (S1111 , S1121 , S2111 , and S2121 ) of the model with ADS X-parameters. The transistor is a 2-mm GaN HEMT on Si substrate biased with Vds = 10 V, Ids = 189 mA, Vgs = −1 V, and Igs = 14 mA and has an input power of −20 dBm. The frequency range is between 500 MHz and 6 GHz.

indirect manner a nonlinear model. The models obtained by the conventional method may be accurate only under dc and small-signal operating states [38], [39]. However, they can

VIII. C ONCLUSION The GaN HEMT is applied in diverse high-efficient RF power amplifiers because of its good performance. To design a high-efficient RF power amplifier, a large-signal model is necessary. A transistor can be a physics model, a black box model, or a lumped element based model. Equivalent-circuitbased or compact models are preferred because they have good performance in terms of convergence, operating range, extrapolation accuracy, physical transistor insight, modeling process easiness, and usability for circuit design. Conventional large-signal compact transistor models are extracted from small-signal data (multibias S-parameters) indirectly. This method limits its accuracy to predict nonlinear distortion. To overcome this limitation, the development and the extraction of compact models based on X-parameters instead of multibias S-parameters are proposed in this paper. However, in its form, X-parameters are not suitable to develop an equivalent-circuit-based model. Thus, new modeling tools are needed. That is why a set of new nonlinear two-port network parameters that are extracted from X-parameters, conversion rules, and equations for different nonlinear circuit topologies was developed. These modeling tools are helpful for the development of the new equivalent-circuit-based modeling technique, especially for the de-embedding and intrinsic part extraction. Similar to the equivalent-circuit-based models discussed in the literature, the new model is subdivided into passive and active parts. The extrinsic part consists of passive elements and is related to the physical layout of the transistor. After determining the values of extrinsic elements, the X-parameters of the intrinsic part can be determined with the proposed analytical de-embedding technique. The embedding technique is also developed in this paper that allows translating the inner reference plane at the device intrinsic boundary to the measurement reference plane at the external and accessible ports of the device. The intrinsic part is modeled as a modified π network of nonlinear lumped element impedances of drain-to-source, gate-to-source, drain-to-gate, and gate-to-drain junctions. The concept of nonlinear lumped element impedances or

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admittances derived from X-parameters is presented. It is shown that nonlinear lumped element impedances or admittances can be easily extracted from X-parameters in series, parallel, or π network configuration. The topology of the transistor intrinsic part is modified to describe the nonunilaterality. The extraction procedure and the implementation of the model in CAD software are presented and are validated successfully. In the future, the model will be extended to include thermal and memory effect extraction results. Mathematical function fitting techniques will be applied to the intrinsic part element parameters. ANNs would be potentially used because they can fit data in any dimension [7]. Combining the ANN-based model with the table-based model can be effective. ANN allows overcoming the limitations of tables and interpolation schemes used in the proposed model. ANN is considered as a powerful tool for function approximation. Thus, it can be used to model any nonlinear behavior having even a high number of independent variables [1]. The equivalent-circuit based model is extracted for a 10 μm × 200 μm GaN HEMT with a gate length of 0.25 μm. This model can be scaled to other device geometries. Scaling study that will be carried out in the future will allow simulating other devices with different geometrical dimensions manufactured with the same foundry process. Moreover, the model has already a lower number of parameters, but regridding will certainly reduce the set of dependent parameters of the nonlinear intrinsic elements. R EFERENCES [1] D. E. Root, “Future device modeling trends,” IEEE Microw. Mag., vol. 13, no. 7, pp. 45–59, Nov./Dec. 2012. [2] D. M. M.-P. Schreurs, J. Verspecht, B. Nauwelaers, A. Van de Capelle, and M. Van Rossum, “Direct extraction of the non-linear model for twoport devices from vectorial non-linear network analyzer measurements,” in Proc. 27th Eur. Microw. Conf., Israel, Palestine, 1997, pp. 921–926. [3] A. H. Jarndal, “Large-signal modeling of GaN device for high power amplifier design,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Kassel, Kassel, Germany, 2006. [4] A. Jarndal, R. Essaadali, and A. B. Kouki, “A reliable model parameter extraction method applied to AlGaN/GaN HEMTs,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 35, no. 2, pp. 211–219, Feb. 2016. [5] A. Jarndal, B. Bunz, and G. Kompa, “Accurate large-signal modeling of AlGaN-GaN HEMT including trapping and self-heating induced dispersion,” in Proc. 18th IEEE Int. Symp. Power Semiconductor Devices IC’s (ISPSD), Naples, Italy, Jun. 2006, pp. 1–4. [6] D. M. M.-P. Schreurs, J. Verspecht, S. Vandenberghe, and E. Vandamme, “Straightforward and accurate nonlinear device model parameterestimation method based on vectorial large-signal measurements,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 10, pp. 2315–2319, Oct. 2002. [7] D. E. Root et al., “Compact and behavioral modeling of transistors from NVNA measurements: New flows and future trends,” in Proc. IEEE Custom Integr. Circuits Conf. (CICC), Sep. 2012, pp. 1–6. [8] J. Verspecht and D. E. Root, “Polyharmonic distortion modeling,” IEEE Microw. Mag., vol. 7, no. 3, pp. 44–57, Jun. 2006. [9] J. Horn, D. Gunyan, L. Betts, C. Gillease, J. Verspecht, and D. E. Root, “Measurement-based large-signal simulation of active components from automated nonlinear vector network analyzer data via X-parameters,” in Proc. IEEE Int. Conf. Microw., Commun., Antennas Electron. Syst. (COMCAS), May 2008, pp. 1–6. [10] D. E. Root, J. Verspecht, J. Horn, and M. Marcu, X-parameters: Characterization, Modeling, and Design of Nonlinear RF and Microwave Components. Cambridge U.K.: Cambridge Univ. Press, 2013. [11] J. Verspecht, “Large-signal network analysis,” IEEE Microw. Mag., vol. 6, no. 4, pp. 82–92, Dec. 2005. [12] C.-S. Chiu et al., “Characterization of annular-structure RF LDMOS transistors using polyharmonic distortion model,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 977–980.

[13] J. Horn, D. E. Root, and G. Simpson, “GaN device modeling with X-parameters,” in Proc. IEEE Compound Semiconductor Integr. Circuit Symp. (CSICS), Oct. 2010, pp. 1–4. [14] D. E. Root, J. Xu, J. Horn, M. Iwamoto, and G. Simpson, “Device modeling with NVNAs and X-parameters,” in Proc. IEEE Workshop Integr. Nonlinear Microw. Millim.-Wave Circuits (INMMIC), Apr. 2010, pp. 12–15. [15] D. E. Root, M. Marcu, J. Horn, J. Xu, R. M. Biernacki, and M. Iwamoto, “Scaling of X-parameters for device modeling,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2012, pp. 1–3. [16] G. Simpson, J. Horn, D. Gunyan, and D. E. Root, “Load-pull + NVNA = enhanced X-parameters for PA designs with high mismatch and technology-independent large-signal device models,” in Proc. 72nd ARFTG Microw. Meas. Symp., 2008, pp. 88–91. [17] P. H. Aaen, J. A. Plá, and J. Wood, Modeling and Characterization of RF and Microwave Power FETs. Cambridge, U.K.: Cambridge Univ. Press, 2007. [18] S. A. Maas, Nonlinear Microwave and RF Circuits. London, U.K.: Artech House, 2003. [19] S. M. Homayouni, D. M. M.-P. Schreurs, G. Crupi, and B. K. J. C. Nauwelaers, “Technology-independent non-quasi-static table-based nonlinear model generation,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 2845–2852, Dec. 2009. [20] M. C. Currás-Francos, “Comparison of HEMT non-linear model extraction approaches based on small signal and on large signal measurements,” Int. J. Numer. Model., Electron. Netw., Devices Fields, vol. 16, no. 1, pp. 41–51, Jan. 2003. [21] D. M. Pozar, Microwave Engineering. Don Mills, ON, Canada: Addison-Wesley, 1990. [22] A. Jarndal, “AlGaN/GaN HEMTs on SiC and Si substrates: A review from the small-signal-modeling’s perspective,” Int. J. RF Microw. Comput.-Aided Eng., vol. 24, no. 3, pp. 389–400, May 2014. [23] G. Crupi et al., “Accurate multibias equivalent-circuit extraction for GaN HEMTs,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 10, pp. 3616–3622, Oct. 2006. [24] P. M. Cabral, J. C. Pedro, and N. B. Carvalho, “Nonlinear device model of microwave power GaN HEMTs for high power-amplifier design,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 11, pp. 2585–2592, Nov. 2004. [25] R. Essaadali, A. Kouki, A. Jarndal, and F. M. Ghannouchi, “Modeling of extrinsic parasitic elements of Si based GaN HEMTs using two step de-embedding structures,” in Proc. WAMICON, 2015, pp. 1–4. [26] A. Jarndal, A. Z. Markos, and G. Kompa, “Improved modeling of GaN HEMTs on Si substrate for design of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 3, pp. 644–651, Mar. 2011. [27] A. Jarndal and G. Kompa, “A new small-signal modeling approach applied to GaN devices,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 11, pp. 3440–3448, Nov. 2005. [28] E. P. Vandamme, D. M. M.-P. Schreurs, and G. van Dinther, “Improved three-step de-embedding method to accurately account for the influence of pad parasitics in silicon on-wafer RF test-structures,” IEEE Trans. Electron Devices, vol. 48, no. 4, pp. 737–742, Apr. 2001. [29] C. Cha, Z. Huang, N. M. Jokerst, and M. A. Brooke, “Test-structure free modeling method for de-embedding the effects of pads on device modeling,” in Proc. Electron. Compon. Technol. Conf., 2003, pp. 1694–1700. [30] H. Cho and D. E. Burk, “A three-step method for the de-embedding of high-frequency S-parameter measurements,” IEEE Trans. Electron Devices, vol. 38, no. 6, pp. 1371–1375, Jun. 1991. [31] J. G. Leckey, “A scalable X-parameter model for GaAs and GaN FETs,” in Proc. Eur. Microw. Integr. Circuits Conf. (EuMIC), Oct. 2011, pp. 13–16. [32] O. Jardel et al., “An electrothermal model for AlGaN/GaN power HEMTs including trapping effects to improve large-signal simulation results on high VSWR,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 12, pp. 2660–2669, Dec. 2007. [33] A. Raffo, S. Di Falco, V. Vadala, and G. Vannini, “Characterization of GaN HEMT low-frequency dispersion through a multiharmonic measurement system,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 9, pp. 2490–2496, Sep. 2010. [34] F. van Raay et al., “New low-frequency dispersion model for AlGaN/GaN HEMTs using integral transform and state description,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 154–167, Jan. 2013.

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[35] A. Jarndal and G. Kompa, “Large-signal model for AlGaN/GaN HEMTs accurately predicts trapping- and self-heating-induced dispersion and intermodulation distortion,” IEEE Trans. Electron Devices, vol. 54, no. 11, pp. 2830–2836, Nov. 2007. [36] A. Raffo et al., “Nonlinear dispersive modeling of electron devices oriented to GaN power amplifier design,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 4, pp. 710–718, Apr. 2010. [37] W. Van Moer and Y. Rolain, “A large-signal network analyzer: Why is it needed?” IEEE Microw. Mag., vol. 7, no. 6, pp. 46–62, Nov. 2006. [38] J. W. Bandler, Q.-J. Zhang, S. Ye, and S. H. Chen, “Efficient largesignal FET parameter extraction using harmonics,” IEEE Trans. Microw. Theory Techn., vol. 37, no. 12, pp. 2099–2108, Dec. 1989. [39] A. Werthorf, F. van Raay, and G. Kompa, “Direct nonlinear FET parameter extraction using large-signal waveform measurements,” IEEE Microw. Guided Wave Lett., vol. 3, no. 5, pp. 130–132, May 1993. [40] B. Kopp and D. D. Heston, “High-efficiency 5-watt power amplifier with harmonic tuning,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 2. May 1988, pp. 839–842. [41] C.-J. Wei, Y. E. Lan, J. C. M. Hwang, W. J. Ho, and J. A. Higgins, “Waveform-based modeling and characterization of microwave power heterojunction bipolar transistors,” IEEE Trans. Microw. Theory Techn., vol. 43, no. 12, pp. 2899–2903, Dec. 1995. Riadh Essaadali (S’12) was born in Thala, Tunisia. He received the B.Eng. degree in wireless communications from the École Supérieure des Communications de Tunis, Ariana, Tunisia, in 2006, and the M.Sc. and Ph.D. degrees in electrical engineering from the École de Technologie Supérieure (ÉTS), Montreal, QC, Canada, in 2010 and 2015, respectively. He was a Radio Frequency Engineer with Divona Telecom, London, U.K., in 2006, and then with Ultra Electronics TCS, Montreal, QC, Canada, from 2011 to 2012. Since 2007, he has been with the LACIME Laboratory, ÉTS. He was a Teaching Assistant with ÉTS from 2009 to 2013, and the Université du Québec à Montréal, Montreal, QC, Canada, from 2008 to 2011. He has been a Lecturer with ÉTS since 2010, and the Université de Sherbrooke, Sherbrooke, QC, Canada, since 2015. From 2012 to 2014, he was a Test and Validation Engineer with Ultra Electronics TCS. He was a Research Associate with the NSERC-ULTRA Electronics Chair from 2013 to 2014. He was a Consultant Radio Frequency Engineer with Vigilant Global, Montreal, QC, Canada, in 2015. In 2016, he was a Research Engineer with Comprod Inc., Boucherville, QC, Canada. He is currently a Senior Radio Frequency Processing Engineer with Aviat Networks, Calgary, AB, Canada. His current research interests include embedded systems, smart antennas, multistandard transceivers, transistor modeling, power amplifier, digital communication, analog and digital signal processing, and channel modeling. Anwar Jarndal (M’04) was born in Zabid, Yemen, in 1973. He received the B.Sc. degree (Hons.) in electronics and communication engineering from Applied Sciences University, Amman, Jordan, in 1998, the M.Sc. degree in communication and electronics engineering from the Jordan University of Science and Technology, Irbid, Jordan, in 2001, and the Ph.D. degree in electrical engineering from the University of Kassel, Kassel, Germany, in 2006. He was a Teaching Assistant with the Jordan University of Science and Technology from 1999 to 2001. From 2001 to 2003, he was a Lecturer with Hodeidah University, Hodeidah, Yemen, where he became an Assistant Professor with the Department of Computer Engineering in 2007. In 2008, he was a Post-Doctoral Fellow with the École de Technologie Supérieure, Montreal, QC, Canada. From 2011 to 2013, he was an Assistant Professor with the Department of Electrical and Computer Engineering, University of Nizwa, Nizwa, Oman. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, University of Sharjah, Sharjah, United Arab Emirates. He has authored over 40 internationally peer-reviewed publications, including, journal papers, book chapters, and conference papers. His current research interests include active devices modeling, measurements and characterization techniques, power amplifiers design, local and global optimizations, artificial neural networks modeling, fuzzy logic modeling, and radio channel modeling. Dr. Jarndal is a Member of the Editorial Boards of a number of international journals and reviewing boards of many IEEE journals and conferences.

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Ammar B. Kouki (S’88–M’92–SM’01) received the B.S. (Hons.) and M.S. degrees in engineering science from Pennsylvania State University, State College, PA, USA, in 1985 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Champaign, IL, USA, in 1991. He was a Consultant with the National Center for Supercomputing Applications, Urbana, IL, USA, from 1988 to 1991. From 1991 to 1993, he was a Post-Doctoral Fellow with the Microwave Research Laboratory, École Polytechnique de Montréal, Montreal, QC, Canada. From 1994 to 1998, he was a Senior Microwave Engineer with the Microwave Research Laboratory, École Polytechnique de Montréal, where he was involved in power amplifier linearization techniques. In 1998, he co-founded AmpliX Inc., Montreal, QC, Canada, a company that specialized in RF linearizers for wireless and SatCom applications. In 1998, he joined the faculty of the École de Technologie Supérieure, Montréal, QC, Canada, where he is currently a Full Professor of Electrical Engineering, the Director of the LACIME Laboratory, one of the co-founders of ISR Technologies which is a software defined radio-company, and the Founding Director of the school’s LTCC and Microsystems Laboratory. He has authored or coauthored over 170 peer-reviewed publications and holds 8 patents. His current research interests include radio communication and navigation with a focus on devices, intelligent and efficient RF front-ends/transceiver architectures, antenna and propagation, active device modeling and characterization, poweramplifier design, linearization and efficiency enhancement techniques, and computational electromagnetic techniques for the modeling and design of passive microwave structures, multiple antenna systems, and intelligent antenna research.

Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’07) is currently a Professor and the iCORE/CRC Chair with the Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, and the Director of the Intelligent RF Radio Laboratory. His research activities have led to more than 600 publications, 15 U.S. patents (five pending), and 3 books. His current research interests include microwave instrumentation and measurements, nonlinear modeling of microwave devices and communications systems, design of power and spectrum efficient microwave amplification systems, and design of intelligent RF transceivers for wireless and satellite communications.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

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Iterative Learning Control for RF Power Amplifier Linearization Jessica Chani-Cahuana, Per Niklas Landin, Christian Fager, Senior Member, IEEE, and Thomas Eriksson, Member, IEEE

Abstract— This paper proposes a new technique to identify the parameters of a digital predistorter based on iterative learning control (ILC). ILC is a well-established control theory technique that can obtain the inverse of a system. Instead of focusing on identifying the predistorter parameters, the technique proposed here first uses an iterative learning algorithm to identify the optimal power amplifier (PA) input signal that drives the PA to the desired linear output response. Once the optimal PA input signal is identified, the parameters of the predistorter are estimated using standard modeling approaches, e.g., least squares. To this end, in this paper, we present a complete derivation of an ILC scheme suitable for the linearization of PAs, which includes convergence conditions and the derivation of two learning algorithms. The proposed ILC scheme and parameter identification technique were demonstrated experimentally and compared with the indirect learning architecture (ILA) and direct learning architecture (DLA). The experimental results show that, even for the most difficult cases, the proposed ILC scheme can successfully linearize the PA. The experimental results also indicate that the proposed parameter identification technique is more robust to measurement noise than ILA and can provide better linearity performance when the PA nonlinearities are strong. In addition, the proposed parameter identification technique can achieve similar or better linearity performance than DLA but with a simpler identification process. Index Terms— Digital predistortion (DPD), iterative algorithm, linearization, nonlinear distortion, nonlinear systems, nonlinearity, power amplifier (PA).

I. I NTRODUCTION

H

IGH efficiency and linearity are two important requirements for radio frequency (RF) power amplifiers (PAs) used in modern wireless communication systems. Unfortunately, due to the inherent nonlinear behavior of PAs, these requirements are not easy to fulfill simultaneously. In order to maximize the efficiency, PAs have to be operated close Manuscript received February 20, 2015; revised May 18, 2015, August 14, 2015, January 9, 2016, and January 28, 2016; accepted June 5, 2016. This work was carried out at the Gigahertz Centre under a joint project supported by VINNOVA, by the Chalmers University of Technology, by Ericsson, by Gotmic, by Infineon Technologies, by National Instruments, by Ampleon, and by Saab. J. Chani-Cahuana and T. Eriksson are with the Department of Signals and Systems, Chalmers University of Technology, Gothenburg 412 96, Sweden (e-mail: [email protected]; [email protected]). P. N. Landin is with Ericsson AB, Kumla 692 33, Sweden (e-mail: [email protected]). C. Fager is with the Department of Microtechnology and Nanoscience, Chalmers University of Technology, Gothenburg 412 96, Sweden (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.2016.2588483

Fig. 1.

Block diagram of the ILA.

to saturation, where they exhibit strong nonlinear behavior. The PA nonlinear behavior does not only produce in-band distortion but also generates spectral regrowth that causes interference to neighboring channels. The nonlinear behavior can be reduced by backing off the amplifier far from saturation. Unfortunately, due to the large peak-to-average power ratio (PAPR) that modern communication signals present, this results in very low average efficiencies. In order to satisfy the efficiency requirement without sacrificing the linearity, PA linearization is required. Among the linearization techniques reported in the literature, digital predistortion (DPD) is one of the most suitable for modern communication systems. DPD introduces a predistorter block before the PA whose main role is to provide a complementary nonlinearity to that of the PA so that the cascade of the predistorter and PA ideally behaves like a linear system. In reality, due to problems encountered in the identification process and limitations of the models used in the predistorter, residual distortions can be found at the output of the amplifier. Commonly known techniques used to identify the parameters of a digital predistorter are the indirect learning architecture (ILA) [1] and the direct learning architecture (DLA) [2], [3]. ILA is based on the inverse modeling approach, where a postinverse of the PA is identified using the output of the amplifier to model the input, as depicted in Fig. 1. After identifying the parameters of the postinverse (also known as postdistorter), ILA copies the parameters to an identical model that is used as predistorter [4]. Although ILA simplifies the predistorter identification process from a nonlinear optimization problem to an iterated linear optimization problem [5], it suffers from two fundamental drawbacks. First, due to the presence of noise in the measured output

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Fig. 2.

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Block diagram of the DLA.

signal y(n) in Fig. 1, the parameter estimates converge to a biased solution [4]. Second, ILA provides limited performance when the PA nonlinearity is severe [3]. Although in high-power applications, such as macrocell base stations, the measurement noise may be considered negligible, there are other applications where that assumption is not necessarily true. For instance, in low-power applications, such as small-cell base stations, in order to meet the power consumption and cost targets, the requirements imposed on the components used in the DPD feedback chain are often relaxed and higher noise levels must be accepted, e.g., high precision and power inefficient analog to digital (A/D) converters are replaced by low-precision A/D converters, which introduces higher noise levels [5], [6]. Higher noise level are also present in the linearization of microwave links and the remote predistortion of satellite systems, where the predistorter is calculated using the signal at the receiver [7]. To address the ILA noiseinduced bias problem, Morgan et al. [4] and Landin et al. [5] proposed a modified version of ILA where the inverse model is identified using the output signal of a forward model of the PA instead of the noisy PA output. While their approach yields better linearity performance than ILA, its performance depends on the accuracy of the PA forward model. DLA, on the other hand, does not present any of the ILA problems. In DLA, the predistorter parameters are estimated by comparing the input signal yd (n), in Fig. 2, and the error e(n) produced at the output of the PA [2], [3]. There are several DLA algorithms available in the literature [2], [8], [9]. A recent addition to those algorithms is the closed-loop estimator proposed in [10], [11], which uses a least mean squares algorithm to minimize the residual distortion in the output of the PA. Although DLA algorithms can provide unbiased parameter estimates, most of them are computationally expensive, complex in structure, and present slow convergence. For those reasons, ILA is the most used parameter identification technique for DPD for RF PAs. In recent years, different iterative techniques have been proposed for the linearization of PAs. In [12], an iterative block level methodology is introduced where a lookup table is used to compensate for the PA static nonlinearity, and an iterative algorithm is used to compensate for the deterministic memory effects of the PA. However, the convergence conditions of the algorithm were not reported. In [13], different iterative methods based on successive approximation and Newton’s method were investigated. However, those algorithms are computationally expensive and are based on the assumption

that the structure of the PA is known. Therefore, modeling errors consisting of mismatches between the PA and the model limit their performance [13]. Recently, Hotz and Vogel [14] proposed an iterative algorithm based on the Richardson iteration and an alternative view of the Volterra series. However, that algorithm is also based on the assumption that a model of the PA is known; in addition, all the results reported are based on simulations only. In this paper, we introduce the concept of iterative learning control (ILC) for the linearization of RF PAs and propose a new technique to identify the parameters of digital predistorters. ILC is an iterative technique used in control theory to invert the dynamics of linear and nonlinear dynamical systems. Instead of focusing on identifying the parameters of the predistorter, the technique proposed here uses an iterative learning algorithm to find the optimal PA input signal that produces the desired linear output response. Once the optimal PA input signal is found, the parameters of the predistorter are estimated using standard modeling approaches, e.g., linear least squares (LS). To the best of the authors’ knowledge, this is the first time that one has access to the desired output values of the predistorter (which is identical to the input to the PA) before estimating its parameters. To this end, in this paper, we present a complete derivation of an ILC scheme for the linearization of PAs, from establishing the conditions for convergence to the derivation of two iterative learning algorithms suitable for PAs. Unlike other iterative techniques, the algorithms proposed here do not require much prior information of the PA and are computationally less expensive. The experimental results presented in this paper indicate that even for the most difficult cases, the proposed ILC scheme can successfully identify the optimal PA input signal that linearizes the PA. The results also shown that in contrast to ILA, the proposed DPD identification technique is more robust to measurement noise and provides better linearity performance when the PA nonlinearities are strong. Compared with DLA, the proposed DPD identification technique can achieve better linearity performance at low signal-to-noise ratios (SNRs) and similar linearity performance when the PA is in deep compression, but with a simpler identification process. This paper is organized as follows. Section II describes the concept of ILC. In Section III, an ILC scheme for the linearization of PAs is presented. Section IV presents a new technique to identify the parameters of digital predistorters for RF PAs. Section V describes the experimental setup and performance evaluation criteria. The measurement results are presented and discussed in Section VI, and finally the conclusions are provided in Section VII. II. ILC ILC is a well-established control theory technique used to improve the transient response and tracking performance of systems that operate repetitively over a fixed time interval [15]. First introduced in [16] and later developed as a formal theory in [17], ILC has been applied to a wide range of applications including industrial robots, mechanical systems, manufacturing, chemical industry, and aerodynamic

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will learn the new most optimal input signal without having to reconfigure the learning algorithm [15]. ILC differs from other adaptive control schemes in that other strategies modify either the controller or the parameters of the controller. In an ILC scheme, it is the input signal that is modified after each iteration [18], [19]. Further information about ILC can be found in [15], [18], and [19]. III. ILC FOR PA L INEARIZATION Fig. 3.

ILC scheme for RF PAs.

systems, to name a few [18]. This section describes the idea behind ILC.

A. ILC Scheme for PA Linearization

A. ILC General Description Consider a system F defined by y(n) = Fs [u(n)]. For this system, we want to drive the output y(n) to a desired response defined by yd (n). The problem consists in finding an optimal input u ∗ (n), so that when this signal is applied to the system, the output is as close as possible to the desired response yd (n), where close is measured in the sense of some arbitrary norm. ILC is an iterative technique for finding such u ∗ (n). The basic idea of ILC is illustrated in Fig. 3, where the system F is considered to be a PA. All the signals shown are assumed to be defined over a finite discrete-time interval n ∈ [0, N − 1], where N is the number of samples of the signals. Also note that the subscript k is used to indicate the iteration number. The ILC scheme works as follows [15]: during the kth iteration, the system is driven by an input signal u k (n) that produces an output response yk (n). The learning controller then uses the error observed between the desired output and the actual output ek (n) = yd (n) − yk (n) to compute a new input signal u k+1 (n) that will be used during the next iteration. The learning controller is designed in such a way as to guarantee that the error produced by the new input signal u k+1 (n) is smaller than the error produced by the current input u k (n). This scheme is run iteratively until the desired performance is reached. The main task in the design of an ILC scheme is then to find a learning algorithm defined by FL [u, yd − y] that generates a sequence of inputs u k+1 (n) = FL [u k (n), yd (n) − yk (n)], n ∈ [0, N − 1] = FL [u k (n), yd (n) − Fs [u k (n)]]

Although ILC is generally associated with control theory, it is a powerful technique to obtain the inverse of a system [20]. Therefore, after some modifications, it can be made suitable for the linearization of PAs. In this section, we derive an ILC scheme for the linearization of PAs. A summary of the proposed ILC scheme can be found at the end of this section.

(1)

such that u k (n) converges to a fixed point u ∗ (n) that minimizes the error between the desired output yd (n) and the actual output yk (n). Note that in an ILC scheme, it is usually assumed that all initial conditions of the system are reset to the same value at the beginning of each iteration. Also, minimum information about the system F should be used in the design of the learning algorithm. The convergence of the learning algorithm should not depend on a given desired response yd (n). If a new desired response yd (n) is introduced, the learning controller

The proposed ILC scheme for the linearization of PAs is illustrated in Fig. 3. All the signals shown are defined over a finite discrete-time interval n ∈ [0, N − 1], where N is the number of samples of each signal. The signal yd (n) is the desired output signal of the PA and is defined within reasonable limits. The signals u k (n) and yk (n) are the input and output of the amplifier during the kth iteration, respectively. The output of the amplifier yk (n) can be described as yk (n) = Fs [u k (n), u k (n − 1), . . .]

(2)

where Fs represents the nonlinear dynamic transfer function of the PA. The error produced after the kth iteration ek (n) is defined by ek (n) = yd (n) − yk (n).

(3)

The learning algorithm proposed for this scheme is of the form [18] uk+1 = uk + ek

(4)

where uk = [u k (0), u k (1), . . . , u k (N − ek = [ek (0), ek (1), . . . , ek (N − 1)]T , and  is denoted by the learning matrix that controls the convergence speed of the algorithm. Before discussing how to select the learning matrix , let us derive the convergence conditions of the learning algorithm. 1)]T ,

B. Convergence Conditions An important aspect in the design of the learning algorithm is to guarantee that it converges in successive iterations. In the following, an approximate criterion for convergence of the algorithm given in (4) is determined. For this purpose, it is assumed that the transfer function Fs is continuous in the region of interest. The convergence of the algorithm is measured using the LS criterion, i.e., the L 2 -norm squared of the error ek . Let us denote the system description in (2) in vector form yk = Fs (uk )

(5)

where yk = [yk (0), yk (1), . . . , yk (N −1)]T and Fs is a vectorvalued function defined by    (N−1) T    Fs (uk ) = f 0 uk0 , . . . , f n ukn , . . . , f N−1 uk (6)

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where ukn = [u k (0), u k (1), . . . u k (n)] and f n is a causal function that represents the relation between ukn and the output yk at the time instant n   yk (n) = f n ukn = f n (u k (0), . . . , u k (n)). (7)

TABLE I S UMMARY OF ILC A LGORITHMS

In a similar way, the error signal obtained after the kth iteration in (3) can be rewritten in vector form as ek = yd − Fs (uk )

(8)

where yd = [yd (0), yd (1), . . . , yd (N − 1)]T . Referring to (4) and (8), the error produced by the input uk+1 is given by ek+1 = yd − Fs (uk+1 ) = yd − Fs (uk + ek ).

(9)

Since it is assumed that the function Fs is continuous in the region of interest, the expression Fs (uk + ek ) in (9) can be approximated using the first-order Taylor expansion Fs (uk + ek ) ≈ Fs (uk ) + JF (uk )ek

(10)

where JF (uk ) is the Jacobian matrix of Fs with respect to uk and is defined by     ⎤ ⎡ ∂ f 0 uk0 ∂ f 0 uk0 · · · ⎢ ∂u k (0) ∂u k (N − 1) ⎥ ⎥ ⎢ .. .. ⎥ ⎢ .. JF (uk ) = ⎢ ⎥. . . . ⎢  (N−1)  ⎥ ⎦ ⎣ ∂ f (N−1) u(N−1)  ∂ f (N−1) uk k ··· ∂u k (0) ∂u k (N − 1) (11) Inserting the approximation given in (10) into (9) and using the error definition given in (8), (9) becomes ek+1 = yd − Fs (uk ) − JF (uk )ek = ek − JF (uk )ek .

(12)

Factorizing and taking the L 2 -norm squared of both sides, it follows that: ek+1 22 = (I − JF (uk ))ek 22

(13)

where I denotes the identity matrix and .2 denotes the L 2 -norm. Using the submultiplicative property of norms [21], (13) becomes ek+1 22 ≤ I − JF (uk )2i2 ek 22

(14)

where .i2 denotes the matrix norm induced by the L 2 -norm, also known as the spectral norm [21]. The spectral norm of a matrix A is defined as 1 2

Ai2 = λmax {A H A}

(15)

where λmax and (.) H denote the maximum eigenvalue and conjugate transpose, respectively. Returning to (14), it can be said that the convergence of the error is guaranteed, i.e., limk→∞ ek 22 = 0, if I − JF (uk )2i2 < 1.

Note that although theoretically the error converges to zero when k → ∞, this does not represent a limitation in a practical scheme. The statement above only implies that if the condition given in (16) is satisfied, the error will be reduced after each iteration. In a practical system, the number of iterations can be chosen so that the final error satisfies the requirements.

(16)

C. Learning Algorithm Once the convergence condition has been established, the next step in the design of the learning algorithm is to select an appropriate  that satisfies the convergence condition, provides fast convergence speed, and requires minimal information about the PA. In this section, we propose two learning algorithms that satisfy those requirements. From (16), the most straightforward choice of  would be  = JF (uk )−1

(17)

which gives rise to the algorithm uk+1 = uk + JF (uk )−1 ek .

(18)

In the ILC framework, this learning algorithm is known as the Newton-type algorithm [22] and was also proposed in [13]. While this algorithm is desired for fast convergence speed, its implementation presents some practical difficulties. To calculate the Jacobian matrix, precise knowledge of the structure of the PA is required. If a model of the PA is known, the performance of the algorithm is limited by the accuracy of the model [13], [18]. Furthermore, the Jacobian matrix must be evaluated and inverted at each iteration, which makes this algorithm computationally expensive. Fortunately, there exist other choices of  that can satisfy the convergence condition established in (16). In this paper, we propose two learning algorithms that are computationally less expensive than the Newton-type algorithm, require less information about the PA, and also present good convergence speed. A summary of the ILC algorithms proposed in this section is given in Table I. For comparison reasons, the Newton-type algorithm is also included in Table I. 1) Instantaneous Gain-Based ILC Algorithm: The instantaneous gain-based algorithm is given by uk+1 = uk + G(uk )−1 ek

(19)

where G(uk ) is a diagonal matrix with entries G(uk ) = diag{G[u k (0)], . . . , G[u k (N − 1)]}

(20)

where G[u k (n)] is the instantaneous complex gain of the PA, which is defined by G[u k (n)] =

yk (n) . u k (n)

(21)

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The derivation of the instantaneous gain-based algorithm is shown in Appendix A. Note that since this learning algorithm uses the instantaneous complex gain of the amplifier G(uk ) instead of the Jacobian matrix JF (uk ), it does not require precise knowledge of the structure of the PA. Moreover, since G(uk ) requires only the knowledge of u k (n) and yk (n), this algorithm is less computationally expensive than the Newton-type algorithm. Based on extensive experiments on a real PA, it will be shown in Section VI that this learning algorithm presents a similar convergence speed to the Newton-type algorithm. 2) Linear ILC Algorithm: The computational complexity of the learning algorithm can be further reduced using a simplified version of the algorithm given in (4) uk+1 = uk + γ ek

(22)

where the learning matrix  is set to γ I, where γ is denoted by the learning gain. Note that this algorithm updates the input signal uk+1 using a constant gain for all iterations. This algorithm is also known as first-order linear-type ILC algorithm [19], [22]. Assuming that the PA is memoryless, it is shown in Appendix B that the convergence of this algorithm is guaranteed if γ satisfies the following condition: 2 (23) Jmax where Jmax denotes the supremum of the diagonal entries of JF (uk ) given a desired output response yd (n). For PAs, this value is approximately equal to the PA small-signal gain Jmax ≈ G ss that can be obtained from the PA datasheet or from measurements. Moreover, as it is shown in Appendix B, the convergence speed of this algorithm is optimized when γ is chosen to be 0 1) can be reduced by an increase in Pp . The calculated output power at 1 dB gain compression P1 dB is shown in Fig. 6. IP1 dB of both the mixers becomes higher with a higher value of Pp . OP1 dB of the balanced diode mixer increases with an increase in Pp . For the even harmonic mixer, OP1 dB has a saturated characteristic versus Pp , as shown in (25). The calculated conversion loss L c is shown in Fig. 7. The fundamental limitation on L c of both the mixers is 3.92 − 10 log K c1 (dB). L c of the balanced diode mixer has a saturated characteristic versus Pp . In the case of the even harmonic mixer, L c has a single peak without the saturation versus Pp . B. Two-Tone Case The calculated output power of the balanced diode mixer with two-tone input signals is shown in Fig. 8. The output

Fig. 16.

Measured conversion loss L c .

Fig. 17. Measured output power of the balanced diode mixer with two-tone input signals.

power Pout of the output signal becomes higher with a higher Pp . In the low Pin region, the output power Pim3 of IM3 can be reduced by an increase in Pp . The calculated output power of the even harmonic mixer with two-tone input signals is shown in Fig. 9. Pout of the output signals become

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GaAs SBD’s dc characteristics are shown in Fig. 12. The built-in voltage Vt (0.79 V) and the series resistance Rs (3.1 ) are extracted from the measurement results. From the SPICE model in Fig. 12, the calculated cutoff frequency of the SBD is determined to be 277 GHz and is sufficiently higher than the measurement frequency of 100 MHz. A. One-Tone Case

Fig. 18. Measured output power of the even harmonic mixer with two-tone input signals.

The measured output power of the balanced diode mixer and the even harmonic mixer with a one-tone input signal are shown in Figs. 13 and 14. The calculated and measured values are in agreement. In the measurements, the output power Pout of the balanced diode mixer, with a one-tone input signal is higher than that of the even harmonic mixer, as formulated. The measured output power at 1 dB gain compression P1 dB is shown in Fig. 15. In the higher Pp region, the calculated and measured values of both the mixers are in agreement. This is because of the approximation conditions | p|  1, |q|  1, and |s|  1 for the power series (4), (20). These conditions require an LO voltage V p that is sufficiently √ higher than Vm + 2Vt in (4) or Vm + Vt in (20). In the case of the balanced mixer that requires the higher LO voltage V p , the accuracies of IP1 dB and OP1 dB are rapidly degraded with a lower LO level less than 10 dBm. In addition, the saturation on OP1 dB of the even harmonic mixer is confirmed, as formulated. The measured conversion loss L c is shown in Fig. 16. In the region with the higher local power, the theoretical and measured values are in agreement within 1.1 dB. The measured results support the same dependence on the local power: the single peak of the even harmonic mixer and the saturation of the balanced diode mixer. B. Two-Tone Case

Fig. 19.

Measured (a) IIP 3 and (b) OIP3 .

higher with an increase in Pp . In the low Pin region, the output power Pim3 of IM3 can be reduced by an increase in Pp . The calculated third-order intercept points IP3 are shown in Fig. 10. IIP3 of both the mixers increases with an increase in Pp . OIP3 of the balanced diode mixer becomes higher with a higher Pp . In the case of the even harmonic mixer, OIP3 has a saturation versus Pp , as shown in (33). As a conclusion of the above discussions, the formulated characteristics are summarized in Table I. There is an output power limitation restricted by Vt in the even harmonic mixer case. VI. E XPERIMENTAL R ESULTS In this section, the experimental investigations are demonstrated for verification of the derived formulas. The experimental setup at 100 MHz is shown in Fig. 11. In the experimental setup, the output frequency is a 100 MHz band uninfluenced by the junction capacitance. In the following discussions, the measured values are corrected with the measured insertion loss of the measurement setups shown in Fig. 11.

The measured output power of the balanced diode mixer and the even harmonic mixer with two-tone input signals are shown in Figs. 17 and 18. The calculated and measured values are in agreement. In the measurements, the output signal of the balanced diode mixer is higher than that of the even harmonic mixer, as formulated. In addition, the IM3 of the balanced diode mixer is lower than that of the even harmonic mixer. The measured IP3 is shown in Fig. 19. In the higher Pp region, the theoretical and measured values of both the mixers are in agreement. As in the measured P1 dB , this is because of the same approximation | p|  1, |q|  1, |r |  1, |s|  1. In addition, saturation on OIP3 of the even harmonic mixer is confirmed, as formulated. As mentioned above, the analysis results on the output power and the third-order intermodulation of the diode mixers were confirmed with the experimental investigations. VII. C ONCLUSION In this paper, we theoretically describe the fundamental limitations on the output power and the third-order distortion of the balanced diode mixers and the even harmonic mixers. The output power characteristics of both the diode mixers

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are represented as universal charts. Further, the closed-form formulas of P1 dB and IP3 of both the diode mixers are presented. As a result of the analysis, the lower output power capability of the even harmonic mixer is theoretically clarified. This theory does not consider the influence of the junction capacitance of the SBD. Thus, the described formulas and universal charts are confirmed with experimental investigations at 100 MHz. R EFERENCES [1] H. C. Torrey and C. A. Whitmer, Crystal Rectifiers. New York, NY, USA: McGraw-Hill, 1948. [2] D. G. Tucker, Modulators and Frequency-Changers for AmplitudeModulated Line and Radio Systems. London, U.K.: Macdonald & Co., 1953. [3] D. G. Tucker, Circuits With Periodically-Varying Parameters. London, U.K.: Macdonald & Co., 1964. [4] S. A. Maas, Microwave Mixers, 2nd ed. Norwood, MA, USA: Artech House, 1993. [5] B. Henderson and E. Camargo, Microwave Mixer Technology and Applications. Norwood, MA, USA: Artech House, 2013. [6] M. Cohn, J. E. Degenford, and B. A. Newman, “Harmonic mixing with an antiparallel diode pair,” IEEE Trans. Microw. Theory Techn., vol. MTT-23, no. 8, pp. 667–673, Aug. 1975. [7] M. V. Schneider and W. W. Snell, “Harmonically pumped stripline downconverter,” IEEE Trans. Microw. Theory Techn., vol. MTT-23, no. 3, pp. 271–275, Mar. 1975. [8] T. Tokumitsu, “K-band and millimeter-wave MMICs for emerging commercial wireless applications,” IEEE Trans. Microw. Theory Techn., vol. MTT-49, no. 11, pp. 2066–2072, Nov. 2001. [9] S. A. Maas, “Two-tone intermodulation in diode mixers,” IEEE Trans. Microw. Theory Techn., vol. MTT-35, no. 3, pp. 307–314, Mar. 1987. [10] D. N. Held and A. R. Kerr, “Conversion loss and noise of microwave and millimeterwave mixers: Part 1—Theory,” IEEE Trans. Microw. Theory Techn., vol. MTT-26, no. 2, pp. 49–61, Feb. 1978. [11] J. C. Pedro and N. B. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Norwood, MA, USA: Artech House, 2003. [12] S. A. Maas and D. Neilson, “Modeling MESFETs for intermodulation analysis of mixers and amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 38, no. 12, pp. 1964–1971, Dec. 1990. [13] M. T. Terrovitis and R. G. Meyer, “Intermodulation distortion in currentcommutating CMOS mixers,” IEEE J. Solid-State Circuits, vol. 35, no. 10, pp. 1461–1473, Oct. 2000. [14] S. He and C. E. Saavedra, “An ultra-low-voltage and low-power ×2 subharmonic downconverter mixer,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 2, pp. 311–317, Feb. 2012. [15] J. Hashimoto, K. Itoh, M. Shimozawa, and K. Mizuno, “Fundamental limitations on the output power of balanced mixers and even harmonic mixers in modulator operation,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 3085–3094, Dec. 2014. [16] K. Itoh, K. Tajima, K. Kawakami, O. Ishida, and K. Mizuno, “Fundamental limitations on output power and conversion loss of an even harmonic mixer in an up-conversion operation,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1997, pp. 849–852. [17] A. J. Kelly, “Fundamental limits on conversion loss of double sideband resistive mixers,” IEEE Trans. Microw. Theory Techn., vol. MTT-25, no. 11, pp. 867–869, Nov. 1977. [18] K. Itoh and M. Shimozawa, “Fundamental limitations of conversion loss and output power on an even harmonic mixer with junction capacitance,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, pp. 1333–1336.

Jun Hashimoto (M’15) received the B.S. and M.S. degrees in electrical engineering from the Kanazawa Institute of Technology, Nonoichi, Japan, in 2012 and 2014, respectively. He joined Mitsubishi Electric Corporation, Nagoya, Japan, in 2014.

Kenji Itoh (M’91–SM’04) received the B.S. degree in electrical engineering from Doshisha University, Kyoto, Japan, in 1983, and the Ph.D. degree in electrical engineering from Tohoku University, Sendai, Japan, in 1997. He joined Mitsubishi Electric Corporation, Nagoya, Japan, in 1983, where he was involved in the research and development of microwave and millimeter-wave transmitters, receivers, and semiconductor circuits for satellite communication systems, land mobile communication systems, and radar systems. Since 2009, he has been a Professor with the Department of Electronics, Kanazawa Institute of Technology, Nonoichi, Japan. Prof. Itoh is a Member of the Institute of Electronics, Information, and Communication Engineers. He was a recipient of the OHM Technology Award of the Promotion Foundation for Electrical Science and Engineering of Japan in 2002, and the N. Walter Cox Award of the IEEE Microwave Theory and Techniques Society in 2014. He has served as a Technical Program Review Committee Member of the IEEE International Microwave Symposium since 2002 and the Chair of the IEEE MTT-S Nagoya Chapter since 2016, and served as an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES from 2004 to 2007, and an elected IEEE MTT-S Administrative Committee Member from 2006 to 2008, 2010, and from 2012 to 2014.

Mitsuhiro Shimozawa (M’08) received the B.S. and M.S. degrees in electrical engineering from the University of Electro-Communications, Tokyo, Japan, in 1989 and 1991, respectively, and the Ph.D. degree in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 2008. He joined the Mitsubishi Electric Corporation, Nagoya, Japan, in 1991, where he was involved in the research and development of microwave mixers, converters, and transceivers. Prof. Shimozawa was the recipient of the Electronics Society Award of the Institute of Electronics, Information and Communication Engineers, Japan, in 2011.

Koji Mizuno (M’72–SM’72–F’93–LF’07) received the B.E., M.E., and D.E. degrees from Tohoku University, Sendai, Japan, in 1963, 1965, and 1968, respectively, all in electronic engineering. He joined the Department of Electronic Engineering, Tohoku University, in 1968, as a Research Associate. He spent a sabbatical with Queen Mary University of London, London, U.K., under the sponsorship of the Science Research Council, U.K., from 1972 to 1973. He was an Associate Professor with the Research Institute of Electrical Communication, Tohoku University, in 1972, and a Professor of electron devices in 1984. In 1990, he spent a six-month sabbatical with the California Institute of Technology, Pasadena, CA, USA, and the Queen Mary University of London, under the sponsorship of Monbusho (Ministry of Education, Science and Culture, Japan). From 1990 to 1998, he was a Team Leader with the Photodynamics Research Center, Institute of Physical and Chemical Research, Sendai, Japan, where he ran a laboratory for submillimeter-wave research at the same time as he ran the laboratory with Tohoku University. In 2004, he retired from his appointment to become an Emeritus Professor. He has been interested in the millimeter-and submillimeter-wave (THz) region of the electromagnetic wave spectrum since 1965. He has been continuously developing technologies for detection, generation, and applications in this frequency regime. Mr. Mizuno was a recipient of the Kenneth J. Button Medal in 1998, the Minister Award of MEXT Japan, in 2003, the Distinguished Educator Award of the IEEE Microwave Theory and Techniques Society in 2005, and the Exceptional Service Award of the International Society of Infrared, Millimeter, and Terahertz Waves in 2015.

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InGaAs MMIC SPST Switch Based on HPF/LPF Switching Concept With Periodic Structure Hiroshi Mizutani, Member, IEEE, Ryo Ishikawa, Member, IEEE, and Kazuhiko Honjo, Fellow, IEEE

Abstract— This paper presents the analysis of a novel highpass filter/low-pass filter (HPF/LPF) switching concept. Since an HPF/LPF switching concept has a periodic structure, its equivalent circuit is almost the same as an LPF for the ON-state and is the same as an HPF for the OFF -state. The broadband isolation characteristics with low insertion loss can be achieved by designing its cutoff frequency. A three-stage single pole single throw InGaAs pseudomorphic high electron mobility transistor monolithic microwave integrated circuit switch based on the HPF/LPF switching concept is successfully demonstrated with an insertion loss of less than 1.6 dB and isolation of more than 82 dB below 6 GHz, with a size of 1.1 mm × 1.0 mm. The RF performances are in good agreement with the theoretical calculations. The measured input power of 1-dB insertion loss compression, P1dB, and the measured third-order intercept point, IIP3, are 19 and 27.7 dBm, respectively, at 1.95 GHz. The measured ON-time is 5.5 ns without cable delay. The measured rise time is as fast as 1.4 ns. Index Terms— High-pass filter (HPF), low-pass filter (LPF), monolithic microwave integrated circuit (MMIC), RF switch.

I. I NTRODUCTION N THE era of the Internet of Things (IoT), wireless communication systems play more important roles in connecting the IoT devices to the Cloud [1]. RF switches are essential components for wireless communication systems, because they can change the pathway and control the amplitude of the electromagnetic wave propagation. To obtain high-performance RF switches including low insertion loss, high isolation, a large power handling capability, good linearity, and high switching speed, two possible approaches have usually been selected. One approach is the device approach with inherent device switching characteristics, such as in a field-effect transistor (FET), diode, and microelectromechanical systems (MEMS). The other is the circuit approach in which the required pass resistance and isolation are realized with circuit configurations. For the device approach, the most suitable switching device should have low ON-resistance and small OFF-capacitance in the corresponding bias state. MEMS devices indicate low

I

Manuscript received August 26, 2015; revised February 9, 2016, April 22, 2016, and June 11, 2016; accepted June 16, 2016. Date of publication July 29, 2016; date of current version September 1, 2016. H. Mizutani is with the Electrical Engineering Department, Salesian Polytechnic, Tokyo 194-0215, Japan, and also with the ESICB Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected]). R. Ishikawa and K. Honjo are with the Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2591547

ON -resistance derived from the contact resistance between two metals of the same kind and small OFF-capacitance because of a low dielectric constant. From the viewpoint of power handling capability and linearity, the contact resistance and the capacitance between two metals are almost constant, because they are independent of the input power level. Therefore, a higher power handling capability and good linearity can be easily obtained. However, for an MEMS device, the switching speed, such as the ON-time, tON , turn-ON-time, tr , and turnOFF -time, t f , tends to be slower at most submicrosecond orders as compared with the speed of other devices, such as the nanosecond order of FETs [6]. This difference cannot be overlooked. The diode is one of the most attractive components for the switching device. But it needs a dc forward current operated in the forward-biased state. Hence, it dissipates the dc power during the switching operation, especially for largesignal operation. In this paper, a pseudomorphic high electron mobility transistor (pHEMT) was used for demonstrating the novel proposed switch circuits, which has a low ON-resistance due to its high-speed electron mobility and a small OFF -capacitance because of its short gate length. For the circuit approach, four types of switch circuit configurations have been presented, namely, the series–shunt configuration [2], the traveling wave configuration [3], the resonant circuit configuration, which has an FET with an inductor between source and drain [4], and the impedance-transformed circuit configuration in which the impedance of the sourcegrounded FET is transformed via a quarter-wavelength transmission line [5]. The former two types of circuit configurations indicate broadband characteristics, with low insertion loss and high isolation. The latter two types provide relatively narrow band characteristics around the resonant frequency. Recently, another type of switch circuit configuration has emerged as a third approach, where inherent device reactance and resistance in FETs and MEMS are appropriately used as parts of filtering circuits [7]–[12]. Originally, the series–shunt configuration switches the RC filter characteristics between a low-pass filter (LPF) and a high-pass filter (HPF). Such a configuration can be classified as belongings to the same group as a switching filter configuration with a conventional series–shunt configuration. To improve the switching frequency response of the conventional series–shunt configuration, a novel HPF/LPF switching concept was proposed in our previous work [13]. In our previous study, FETs were directly installed in the intrinsic element of the proposed unit circuit [13]. The unit cell was identical to the series–shunt FET configuration. The FET

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Fig. 1.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Schematic of the proposed HPF/LPF switching concept.

was connected with a high-value resistor to the gate terminal. For the RF signal, the gate-isolated FET approximately serves as the only resistor in the open-channel state and as the only capacitor in the pinched-off state [2]. The simplified equivalent circuits for the ON- and OFF-states were the same as the unit circuit of an LC LPF and HPF, respectively. This type of switch provides a low insertion loss for the passband of the LPF and a high isolation for the stopband of the HPF. For a more precise circuit expression, the unit cell of the HPF is identical to that of a composite right-/left-handed transmission line, which has a periodic structure [14]–[17] in the case of fabricating on the substrate for higher frequencies, such as the gigahertz band. The proposed novel switch circuit has a periodic structure to improve the conventional series–shunt configuration. In this paper, the theoretical analysis of the HPF/LPF switching circuit concept is presented in Section II. The single pole single throw (SPST) monolithic microwave integrated circuit (MMIC) switch is demonstrated in Section III. Finally, the advantage of the HPF/LPF compared with the conventional series–shunt configuration will be discussed in Section IV.

Fig. 2.

Unit circuit diagram of the proposed HPF/LPF switching concept.

Fig. 3. Unit equivalent circuit diagram, where FETs are expressed as variable impedance ZFET.

Fig. 4. Equivalent two-port component of the gate-isolated FET in each bias state.

II. HPF/LPF S WITCHING C ONCEPT A. Circuit Configuration In our previous work, a novel HPF/LPF switching concept was presented [13]. Fig. 1 shows a schematic of the proposed HPF/LPF switching concept. This circuit is composed of a plural cascaded connection of the unit cell, which is surrounded by a dashed rectangle. The unit cell is composed of a series transmission line, which connects the two halves of series gate-isolated FETs. In the center of the series transmission line, a gate-isolated shunt FET is connected to the ground via an inductance. The unit circuit diagram has been shown in Fig. 2. The series transmission line is expressed by the series inductance, L e , and the shunt capacitance, Ch . The unit equivalent circuit diagram is shown in Fig. 3, where gate-isolated FETs are expressed as variable impedances Z FET,e for series FETs or Z FET,h for shunt FETs. The impedance of a gate-isolated FET changes in accordance with its bias. Fig. 4 shows the equivalent two-port component of the gate-isolated FET in each bias state [2]. Figs. 5 and 6 are the unit equivalent circuit diagrams for the ON - and OFF -states, respectively. In Fig. 5, the series FET is in the open-channel state and the shunt FET is in the pinched-off

Fig. 5. Unit equivalent circuit diagram for the ON-state. The series FET is in the open-channel state and the shunt FET is in the pinched-off state.

Fig. 6. Unit equivalent circuit diagram for the OFF-state. The series FET is in the pinched-off state and the shunt FET is in the open-channel state.

state; the FET bias state in Fig. 6 is vice versa. Fig. 5 is the same unit circuit as that for the LPF. Fig. 6 shows the same unit circuit as that for the HPF.

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B. S-Parameters If ABCD parameters of the unit circuit fulfill the following equations:   A B FUnitCell = C D     Vn+1 Vn = FUnitCell In+1 In

(1) (2)

and the ABCD matrix of a T-type unit circuit such as that in Figs. 5 and 6 is derived as follows: 

A C

B D



 =

ZY + 1 Y

 Z (Z Y + 2) . ZY + 1

Fig. 7. Circuit diagram of the demonstrated three-stage SPST MMIC switch.

(3)

The periodic structure shown in Fig. 1 is composed of a series connection of n (positive integer) unit circuits. The ABCD matrix of an n-stage periodic structure is derived as follows:  n = FUnitCell

A C

B D



n =

ZY + 1 Y

Z (Z Y + 2) ZY + 1

n . (4)

From Cayley–Hamilton’s theorem, the following expression is obtained: n FUnitCell = an FUnitCell + bn E, n = 1, 2, 3 . . .

(5)

as follows:   1 1 (13) + j ωL e Z = 2 j ωCFET,e 1 Y = + j ωCh (14) RFET,h + j ωL via   ⎧ 1   L via ω2 L e − CFET,e 1⎨ 1 2 ZY = − ω L e Ch + 2 2 ⎩ CFET,e RFET,h + ω2 L 2via   ⎫ 1 RFET,h ⎬ ωL e − ωCFET,e +j . (15) 2 ⎭ RFET,h + ω2 L 2via

where an and bn are given by the following expressions: n t n − t− an = + t+ − t−  n−1 n−1  − t− t+ t− t+ bn = t− − t+  t± = Z Y + 1 ± Z Y (Z Y + 2).

III. E XPERIMENT (6) (7) (8)

Finally, the transmission scattering coefficient can be calculated by the following expression: S21 =

2 2{an (Z Y + 1) + bn } +

an Z (Z Y +2) Z0

+ Z 0 an Y

(9)

where Z 0 is the port impedance, 50  in this case. For the ON-state, the impedance Z and the admittance Y of the unit circuit shown in Fig. 5 can be described with the following equations: 1 (RFET,e + j ωL e ) (10) 2   CFET,h + Ch Y = jω (11) 1 − ω2 L via CFET,h    1 2 CFET,h −ω L e + j ω RFET,e ZY = + C h . 2 1 − ω2 L via CFET,h (12) Z =

For the OFF-state, the impedance Z and the admittance Y of the unit circuit shown in Fig. 6 can be described

A. Circuit Design and Fabrication Fig. 7 shows the circuit diagram of the demonstrated threestage SPST switch based on the HPF/LPF switching concept. In this case, three unit circuits are connected in cascade. The demonstrated circuit was designed using a PP25-21 process design kit, which was prepared for a WIN Semiconductor 0.25-μm InGaAs pHEMT. An MMIC was fabricated in the foundry process of WIN Semiconductor. The circuit and layout were designed by using Keysight Technologies ADS. The gate unit finger length of an FET was 150 μm. The gate widths of all the series FETs were 600 μm and those of shunt FETs were 300 μm. B. Measurement Fig. 8 is a photograph of the demonstrated three-stage SPST MMIC switch. The chip size was very small at 1.1 mm × 1.0 mm. The control biases were applied to the terminal of Vc1 for the gate of the series FETs and to the terminal of Vc2 for the gate of the shunt FETs. For the ON-state, the shunt FETs were biased by −8 V and the series FETs were biased by 0 V. The S-parameters were measured by using an Agilent E8364B network analyzer ranging from 10 MHz to 50 GHz. The measured insertion loss and isolation are shown in Fig. 9. The high isolation characteristics of more than 82-dB SPST MMIC switch with less than 1.6-dB insertion loss have been successfully demonstrated below 6 GHz. The return loss for

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Fig. 8.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Photograph of the demonstrated three-stage SPST MMIC switch.

Fig. 11. Measured insertion loss and isolation of the demonstrated three-stage SPST MMIC switch with respect to input power.

Fig. 9. Measured and modeled insertion loss and isolation with respect to frequency.

Fig. 12. Measured power linearity of the demonstrated three-stage SPST MMIC switch.

Fig. 10. Measured reflection loss of the demonstrated three-stage SPST MMIC switch for the ON-state.

the ON-state is shown in Fig. 10, which was more than 10 dB below 6.85 GHz. The measured insertion loss and isolation with respect to the input power are shown in Fig. 11 at 1.95 GHz. The input power of 1-dB insertion loss compression, P1dB, was 19 dBm. The isolation was not changed from the change

of input power from 20 to 25 dBm. Below 20 dBm, an increase in isolation was observed from around 80 dB at a small signal input of −30 dBm to 46 dB at an input power of 20 dBm. The variation in the measured isolation with respect to the input power is thought to be caused by the average OFF-capacitance changes. Generally, the average FET OFF -capacitance increases with respect to the input voltage swing in the case where the control bias voltage is set under the threshold voltages. In this case, the control voltage and threshold voltage were −8 and −3 V, respectively. The degradation of isolation usually occurs at the voltage clip channel breakdown in OFF-state FETs or at the current saturation in ON -state FETs. In this case, the constant isolation between 20 to 25 dBm may be obtained, because the input voltage swing is not clipped at any FETs. The power linearity was also measured with two signals, which were the main signal of 1.95 GHz ( f 0 ) and the adjacent signal of 1.76 GHz ( f 1 ). The third-order intermodulation distortion at 2.14 GHz (2 f 0 − f 1 )

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Fig. 15. Equivalent circuit diagram of the demonstrated SPST MMIC switch for the OFF-state. TABLE I PARAMETER VALUES OF E QUIVALENT C IRCUIT

Fig. 13. Measured switching speed of the demonstrated three-stage SPST MMIC switch.

been calculated by the parameter values of the equivalent circuits listed in Table I. The modeled curves were in good agreement with the measured ones. It was found that the previously mentioned theory could stand for the demonstrated SPST MMIC switch. Fig. 14. Equivalent circuit diagram of the demonstrated SPST MMIC switch for the ON-state.

is shown in Fig.12. The input power at the third-order intercept point, IIP3, was 27.7 dBm. The measured switching speed of the developed MMIC is shown in Fig. 13 with the input gate control signal waveform. The ON-time, tON , which was the time difference between the time at 50% gate input pulse height and the time at 90% output RF voltage height, was very fast at 8.8 ns, including a coaxial cable propagation delay of 3.3 ns. The rise time, t R , which was the time difference between the turn-ON-time at 10% and 90% of output RF voltage height, was as fast as 1.4 ns. The fall time, t F , was 1.6 ns. IV. D ISCUSSION A. Modeling To confirm the validity of the previously mentioned theory, equivalent circuit analysis has been performed by calculating the S-parameters for both the ON- and OFF-states using unit circuits, as shown in Figs. 5 and 6. The ON-state equivalent circuit model of the demonstrated SPST MMIC switch can be expressed as shown in Fig. 14, with the series FETs in the open-channel state and the shunt FETs in the pinched-off state. This is almost the same circuit as an LPF, with open-channel FETs as series resistors. Thus, low broadband insertion loss can be obtained by using the passband below its cutoff frequency, fCL . Fig. 15 shows the OFF-state equivalent circuit model of the demonstrated MMIC switch with the series pinched-off-state FETs and the shunt open-channel-state FETs. By utilizing the stopband of the HPF below its cutoff frequency, f CH , high isolation can be obtained. The modeled S-parameters are shown in Fig. 9 with dashed lines for both the ON- and OFF-states, of which curves have

B. Dispersion Characteristics and Cutoff Frequency The unit circuit of the proposed HPF/LPF switching concept must satisfy the following equation based on Bloch–Floquet’s theorem, due to its periodic structure [18]:     Vn+1 γ d Vn =e (16) In+1 In where γ is the propagation constant and d is the physical length of the unit cell. Then, (17) is obtained as      A B Vn γ d Vn =e . (17) In In C D From this eigenvalue problem, in order to obtain nontrivial solutions, the following relationship must be fulfilled:    A − eγ d  B  =0 (18) γ d  C D−e  det(FUnitCell ) + e2γ d − (A + D) eγ d = 0.

(19)

Here, due to its reciprocal characteristics det(FUnitCell) = 1.

(20)

Then, the propagation constant can be derived as in [19] γ d = αd + jβd = Cosh−1 (1 + Z Y ).

(21)

C. Cutoff Frequency The cutoff frequency can be derived from the Bloch impedance, Z B [20] Z B = Z0 From (17)

Vn . In

 (A − eγ d )Vn + B In = 0 C Vn + (D − eγ d )In = 0.

(22)

(23)

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Therefore, Z B can be written as Z B = −Z 0

B D − eγ d = −Z 0 . γ d A−e C

From (19) e

γd

=

A+D±



(A + D)2 − 4 . 2

(24)

(25)

Then, the Bloch impedance normalized by Z 0 can be written as Z± B Z0

=−

2B  . A − D ∓ (A + D)2 − 4

(26)

Since this circuit is reciprocal, A = D, normalized Z B will simply be described as Z± B

Z0

= ±√

B A2 − 1

.

ω12 = ω R ωsh κ1 = L e (Ch + CFET,h ).

(33) (34)

(27)

Because the T-type unit circuit is reciprocal in this case, the normalized Bloch impedance can be expressed by using (4) as follows:  Z± Z (Z Y + 2) B =± . (28) Z0 Y While Z B is imaginary, the filter characteristics are in the stopband region. Therefore, when Z B is equal to zero, that frequency stands for the cutoff frequency. D. ON-State For the ON-state, the dispersion relationship is obtained as follows: γ d = Cosh−1 (1 + Z Y )  1 = Cosh−1 1 + (−ω2 L e + j ω RFET,e ) 2   CFET,h + Ch . × 1 − ω2 L via CFET,h

(29)

Theoretically, if RFET,e is not zero in (28), the Bloch impedance has an imaginary part. In order to calculate the cutoff frequency, RFET,e must be zero. If RFET,e cannot be negligible, the filter operates in the stopband region. The cutoff frequency for the ON-state was evaluated as a lossless circuit Z=

Then, the cutoff frequency, f CL , for the ON-state circuit is obtained as follows:     2   4 4 2 ω14 − 16 κ1 + ω 2  κ1 + ω2 ω1 ± sh sh ω1  (32) f CL = 2π 2 where ω R , ωsh , ω1 , and κ1 are defined with the following relationship: 1 ωR = √ L e Ch 1 ωsh =  L via CFET,h

1 j ωL e . 2

(30)

Thus, the normalized Bloch impedance can be written as (31), shown at the bottom of this page.

E. OFF-State For the OFF-state, the dispersion relationship is expressed as follows: γ d = Cosh−1 (1 + Z Y ) ⎡   1 1 −1 ⎣ 2 = Cosh − ω L e Ch 1+ 2 CFET,e   1 ω2 L e − CFET,e L via + 2 2 R + ω2 L via   FET,h ⎫⎤ 1 RFET,h ⎬ ωL e − ωCFET,e ⎦. +j 2 ⎭ RFET,h + ω2 L 2via

To calculate the cutoff frequency, RFET,h has to be zero for the OFF-state, because the Bloch impedance is imaginary. Then, the cutoff frequency for the OFF-state is also evaluated as a lossless circuit   1 1 Z = j ω Le − 2 (36) 2  ω CFET,e  1 Y = j ω Ch − 2 . (37) ω L via Thus, the normalized Bloch impedance for the OFF-state can be written as (38), shown at the bottom of this page. From (37), the cutoff frequency, fCH , for the OFF-state circuit can be written as follows:     2   4 4 2± ω04 − 4 κ + ω κ +  0 ω2L ω2L ω0  (39) f CH = 2π 2

     Le CFET,h 1 2   2 − ω L e Ch + = ±  C Z0 2 1 − ω2 CFET,h L via 2 Ch + 1−ω2 CFET,h L FET,h via        ±  Le − 2 1 ZB 1 1 1 ω C FET,e   − ω2 L e − 2 Ch − 2 +2 = ±  Z0 2 ω CFET,e ω L via 2 Ch − ω2 1L

Z± B

via

(35)

(31)

(38)

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Fig. 18. Equivalent circuit diagram of the three-stage conventional series–shunt configuration switch for the OFF-state.

Fig. 16. Calculated dispersion characteristics for the demonstrated three-stage SPST MMIC switch.

Fig. 17. Equivalent circuit diagram of the three-stage conventional series–shunt configuration switch for the ON-state.

Fig. 19. Calculated frequency response of insertion loss and isolation for both the three-stage conventional series–shunt configuration switch and the demonstrated three-stage MMIC switch with the same parameter values.

where ω R , ωsh , ω1 , and κ1 are defined as

F. Comparison With Series–Shunt Configuration A comparison of the proposed HPF/LPF switching concept and the conventional series–shunt configuration will now be discussed. Figs. 17 and 18 show the ideal equivalent circuit of a three-stage series–shunt configuration switch. The intrinsic difference between a series–shunt switch configuration and the proposed switch circuit is the presence of series and shunt inductors and a shunt capacitor. The series impedance Z and the shunt admittance Y of the unit circuit for the series–shunt configuration can be written using the following equations. For the ON-state RFET,e (42) Z = 2 Y = j ωCFET,h . (43)

ωR = √ ωL =  ω02

1 L e Ch 1 L via CFET,e

= ωR ωL κ = L via Ch + L e CFET,e .

(40) (41)

Fig. 16 shows the calculated dispersion characteristics based on the previously mentioned theory for the demonstrated SPST MMIC switch for both the ON- and OFF-states using the equivalent unit circuit parameters listed in Table I. For the ON -state indicated by a dashed line, the LPF characteristics were observed. The cutoff frequency for the LPF was around 57 GHz. Another linear line in Fig. 16 is the light line in the air. The ON-state curve was observed in the slow wave region on the right side of the light line. Thus, it was found that the ON-state was in the nonradiative transmission region. On the contrary, for the OFF-state, which is indicated by a solid curve, the HPF characteristics were observed from 25 to 39 GHz. The cutoff frequency for the HPF was about 25 GHz. The frequency under 25 GHz was the stopband region. From 38 GHz to beyond 60 GHz, an electromagnetic bandgap region was observed [15]. The region above 38 GHz was at the left-hand side of the light line (dotted line), which was a radiative region. Greater than 10-dB losses were detected at the isolation characteristics shown in Fig. 9. The cutoff frequencies based on the previous theoretical equations were calculated as 24 GHz for the OFF-state and 56.4 GHz for the ON-state. The cutoff frequencies were evaluated by using the extracted equivalent circuit parameters listed in Table I. Their calculated results were in good agreement with the ones obtained from dispersion curves.

For the OFF-state 1 2 j ωCFET,e 1 . Y = RFET,h Z =

(44) (45)

Then, the insertion loss and isolation can be calculated by using (9). The calculated insertion loss and isolation are shown in Fig. 19 with respect to the frequency for both the demonstrated three-stage MMIC switch and the three-stage series– shunt configuration switch. Although the insertion losses had almost the same curves, the isolation curves differed from each other. The curve of the HPF/LPF switching concept indicated better isolation than the series–shunt switch below 25 GHz in this case. For the HPF/LPF switching concept, the series and shunt inductances and the grounded capacitance change its cutoff frequency because of (32) and (39). That is, the HPF/LPF switching concept can control its cutoff frequency in the design so that the switch indicates enough high isolation with low insertion loss according to its specifications. In this

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way, because the HPF/LPF switching concept has a cutoff frequency in spite of having the same number of stages, high broadband isolation characteristics can be achieved by introducing the HPF/LPF switching concept. V. C ONCLUSION This paper presented the analysis of a novel HPF/LPF switching concept. Since the HPF/LPF switching concept has a periodic structure, its equivalent circuit is almost the same as an LPF for the ON-state and is the same as an HPF for the OFF-state. Thus, broadband isolation characteristics with low insertion loss can be achieved by designing its cutoff frequency. A three-stage SPST InGaAs pHEMT MMIC switch based on this HPF/LPF switching concept has been successfully demonstrated with an insertion loss of 10dMO holds automatically in most cases, because the thickness of leaves usually is not higher than a few millimeters, while their length is usually bigger than a

SYDORUK et al.: DESIGN AND CHARACTERIZATION OF MICROWAVE CAVITY RESONATORS

few centimeters, and even for a thick stem, its length is much bigger than its diameter. To explain this dependence at lengths smaller than 10dMO, one can use simple diagrams as shown at the right side of Fig. 9. Here, the vertical alignment of the electric field inside the resonator plays an important role. A parallel oriented (tangential) field, E t , at the side of a sample will not be changed inside the sample, but a perpendicular (normal) field, E n , at the sharp top/bottom borders of a water  column will drop by εwater times. This leads to a change of the field distribution around as well as inside the sample. Due to this decrease, the field inside CMO becomes smaller in either (7a) or (9), leading to a reduction of the related Integral. This effect depends directly on the side-to-top/bottom corners of the sample. We found that it correlates with the (dMO /L MO )1/2 ratio. C. Noise Level and Accuracy The noise level in Fig. 9 is calculated as an error of the estimated resonant frequency multiplied by the total length of a scan (noise level = 0.01 MHz × 300 mm). In the case of a scan over the complete height (see Table I), this value is 3.3 MHz mm for resonator 1 and about 0.01 MHz × 1000 mm = 10 MHz mm for resonator 2. Taking into account the noise level, we can estimate volumes larger than  −1) mL or 200/(ε  −1) mL, using resonator 1 or 2, 1.1/(εMO MO respectively, with an accuracy of (Rmin Rmax )1/2 . (c) Rmin < Z 0 tan θ < Rmax and Z 0 tan θ = (Rmin Rmax )1/2 . (d) Rmin < Z 0 tan θ < Rmax and Z 0 tan θ < (Rmin Rmax )1/2 . (e) Z 0 tan θ ≤ Rmin .

Fig. 5.

Structure of the typical voltage doubler rectifier.

III. A NALYSIS OF THE BASIC R ECTIFIER In rectifier circuits, input power variation results in input impedance change [32]. As addressed in Section II, the variation range of the input impedance can be reduced by using the proposed RCN. For obtaining the variation range and designing the RCN, the input impedance of the subrectifier is analyzed in this section. Meanwhile, the efficiency of the rectifier can also be theoretically calculated. The analysis is helpful to optimize the conversion efficiency. A. Derivation of Closed-Form Equation The schematic of the typical voltage doubler rectifier is shown in Fig. 5 and the equivalent circuit is shown in Fig. 6. It consists of a series resistor (Rs ), a nonlinear junction resistor (R j ), and a nonlinear junction capacitance (C j ). Vbi and Vbr are the forward voltage and breakdown voltage of the diode, respectively. R L is the dc load resistance of the rectifier. The reactive parasitic elements of the physical diode are excluded from this model, because they can be tuned out without affecting the rectifying efficiency. D1 and D2 have identical characteristics, because they are packaged in the same diode Avago HSMS 2822. For RF and dc energy in this circuit, there are two transmission paths, as shown in Fig. 6. The

Fig. 6. Equivalent circuit of the voltage doubler circuit with the Schottky diode model.

RF path is shown as the blue line, which indicates that the two diodes are connected in parallel. For the dc part, the path is indicated by red line. The two diodes are regarded as two dc sources for the dc load. The closed-form expressions for the input impedance and rectifying efficiency are derived from the equivalent circuit with the following assumptions. 1) Voltage V1 and V2 across the diode consists of the dc and the fundamental frequency terms. 2) The forward voltage drop across the intrinsic diode junction (Vbi ) is constant during the turn-ON period. 3) The reverse voltage drop across the intrinsic diode junction (Vbr ) is constant during the breakdown period. Based on the above assumptions, the waveforms on the junctions of the diodes are different before and after breakdown. Thus, the analysis is separated into the two following cases. 1) Analysis of the Rectifier Before Breakdown: When the input power is low, the reverse peak voltage on the diode is less than the breakdown voltage. In this case, the voltage waveforms of D1 (V1 and Vd1 ) are shown in Fig. 7 and

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The current i d1 and i d2 flowing through Rs can be expressed as follows:

Fig. 7. Simplified time-domain waveforms of voltages across (a) D1 and (b) D2 before the reverse voltage reaches Vbr .

expressed as follows [33]: V1 = V10 + V11 cos(ωt) (9)  Vd10 + Vd11 cos(ωt − φ1 ), if D1 is off Vd1 = (10) if D1 is on. −Vbi , The voltage waveform of D2 (V2 and Vd2 ) can be expressed as V2 = −V20 + V21 cos(ωt)  −Vd20 + Vd21 cos(ωt − φ2 ), if D2 is off Vd2 = Vbi , if D2 is on.

(11) (12)

V10 and V20 in (9) and (11) are the dc terms on D1 and D2 . V11 and V21 are the peak voltage of the incident power. Vd10 and Vd11 are the dc and the fundamental frequency components of diode junction voltage Vd1, respectively, when D1 is OFF. Likewise, Vd20 and Vd21 are the dc and the fundamental frequency components of diode junction voltage Vd2 , respectively, when D2 is OFF. For the RF path (in blue line) shown in Fig. 6, the two diodes are connected in parallel. So the fundamental frequency components on the two diodes are equal to the incident power (V11 = V21 = Vin ). For the dc path (in red line) shown in Fig. 6, the output dc voltage is provided by the dc components of the two diodes (V10 = V20 = V0 /2). The term φ1 in (10) is the phase delay between the fundamental components of V1 and Vd1 , and φ2 in (12) is that between V2 and Vd2 . At 915 MHz, since the junction capacitance of most Schottky diodes are small (i.e., HSMS 2822, C j 0 = 0.65 pF), the phase delay is very small as addressed in Appendix A. Thus, they are negligible in the following discussion. With the above relationship, the input impedance of the rectifier can be calculated. The input current i in should be first analyzed. By applying Kirchhoff’s current law as shown in Fig. 6, the relationship between the input current i in and the currents on the diodes i d1 and i d2 is presented as follows: i in = i d1 + i d2 .

(13)

i d1 = Id10 + Id1r cos ωt + Id1i sin ωt

(14)

i d2 = Id20 + Id2r cos ωt + Id2i sin ωt

(15)

where Id10 and Id20 represent the dc component, Id1r and Id2r are the real part of the fundamental component, and Id1i and Id2i are the imaginary part of the fundamental component. The input impedance of the diode at the fundamental frequency is defined as Vin (16) Z in = Iin where Iin represents the fundamental component of i in . According to (13)–(15), Iin is a complex value which consists of Id1r , Id2r , Id1i , and Id2i as follows: Iin = Id1r − j Id1i + Id2r − Id2i .

(17)

By substituting (17) into (16), the input impedance of the rectifier at the fundamental frequency rewritten as follows: Vin . (18) Z in = Id1r + Id2r − j (Id1i + Id2i ) For the diode D1 , Id1r and Id1i are derived by taking an average of i d1 over one full period based on (9) and (10) as follows:  π−θON 1 Id1r = (V1 − Vd1 ) cos ωtdθ π Rs −(π−θON )   π+θON + (V1 + Vbi ) cos ωtdθ π−θON  π−θON 1 Id1i = (V1 − Vd1 ) sin ωtdθ π Rs −(π−θON )   π+θON + (V1 + Vbi ) sin ωtdθ (19) π−θON

where the equation for the current flowing through Rs is written as follows according to Fig. 6 when the diode is OFF: d(C j Vd1 ) = V1 − Vd1 . (20) dt For diode D2 , Id2r , and Id2i are derived by taking an average of i d2 over one full period based on (11) and (12) as follows:  θON 1 Id2r = (V2 − Vbi ) cos ωtdθ π Rs −θON   2π−θON + (V2 − Vd2 ) cos ωtdθ θ  θON ON 1 Id2i = (V2 − Vbi ) sin ωtdθ π Rs −θON   2π−θON + (V2 − Vd2 ) sin ωtdθ (21) Rs

θON

where the equation for the current flowing through Rs is written as follows according to Fig. 6 when the diode is OFF: Rs

d(C j Vd2 ) = V2 − Vd2 . dt

(22)

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From (18)–(22), the rectifier input impedance before the diode breakdown can be derived as Z in1 π Rs Z in1 = (2θON − sin 2θON ) + j Rs C j ω(2π − 2θON +sin 2θON ) (23) where θON is a variable, depending on the input voltage by the following equation: Rs Vbi Rs cos θON + θON cos θON − sin θON = 2π . (24) RL R L Vin Equation (24) is derived from the analysis of dc component of the diode junction voltages, which is included in Appendix B. Since the diode turns ON when Vd reaches the forward drop Vbi of the diode, θON is calculated by 2π

V0 . (25) 2 The nonlinear capacitance of C j can be calculated by

V0 −M C j = C j0 1 + (26) 2Vbi where C j 0 is the diode zero-bias junction capacitance and M is the diode grading coefficient. As observed from (23)–(25), the three variables θON , V0 , and Vin should be solved for calculating the input impedance. As indicated in (23), although the input impedance Z in1 is not directly expressed as the function of Pin , the three variables θon , V0 , and Vin are related to the input power level Pin and the efficiency. Thus, it is necessary to derive the closed-form equations of efficiency by the analysis as follows to obtain the input impedance Z in1 by solving the θon , V0 , and Vin for the given input power level Pin . The rectifier efficiency is determined by the power losses on the diode during one cycle of rectification Vin cos θON = Vbi +

Pdc Pdc = (27) Pin Pdc + PLoss where Pdc is the output dc power and PLoss is the losses on the diodes in the turn-ON and turn-OFF period. They are expressed as follows: ηd =

Pdc = PLoss =

LossON,Rs ,D1 = =

LossON,junction,D1 = = LossOFF,Rs ,D1 = =

V02 (28) RL LossON,Rs ,D1 + LossON,junction,D1 + LossOFF,Rs ,D1 + LossON,Rs ,D2 + LossON,junction,D2 +LossOFF,Rs ,D2 (29) LossON,Rs ,D2

2 1 V0 + Vbi π Rs 2 

 1 3 tan θ × θON 1 + − ON 2 cos2 θON 2 (30) LossON,junction,D2

Vbi V0 + Vbi (tan θON − θON ) (31) π Rs 2 LossOFF,Rs ,D2 Rs C 2j ω2 2π

Fig. 8. Simplified time-domain waveforms of voltages across (a) D1 and (b) D2 when the reverse voltage exceed Vbr .



2

V0 π − θON × + tan θON . + Vbi 2 cos2 θON (32) The detailed derivation of (30)–(32) is included in Appendix B. According to (24)–(32), θON , V0 , and Vin are derived from the given input power level Pin by solving the system of nonlinear equations with the aid of mathematical tools. Therefore, the input impedance and the efficiency of the rectifier before diode breakdown can be calculated. 2) Analysis of the Rectifier After Breakdown: When the input power is high and the diode reverse voltage exceeds the diode breakdown voltage during the OFF period, the waveforms of the voltage on the junctions are shown in Fig. 8. The voltages on the diodes V1 and V2 are expressed as (9) and (11). The voltage waveform on the junction capacitor of D1 (Vd1 ) can be expressed as follows: ⎧ ⎪ V1 ≤ −Vbi ⎨−Vbi , Vd1 = Vd10 + Vd11 cos ωt, −Vbi < V1 < Vbr (33) ⎪ ⎩ Vbr , V1 ≥ Vbr . The voltage waveform on the junction capacitor of D2 (Vd2 ) can be expressed as follows: ⎧ ⎪ V2 ≥ Vbi ⎨Vbi , Vd2 = −Vd20 + Vd21 cos ωt, −Vbr < V2 < Vbi (34) ⎪ ⎩ −Vbr , V2 ≤ −Vbr . As detailed in Appendix A which considers the relationship between the diode voltage and the junction voltage, the dc components of the two diodes are equal to half of the output voltage (Vd10 = Vd20 = V10 = V20 = V0 /2) and the fundamental components of the two diodes are equal to the input incident power (Vd11 = Vd21 = V11 = V21 = Vin ). According to (18), the input impedance is derived by obtaining the real and imaginary parts of the fundamental component of the currents on the diodes (Id1r , Id2r , Id1i , and Id2i ). After

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diode breakdown, the breakdown period should be taken into consideration in the calculation, which is different from that in (19) and (21). For the diode D1 , Id1r and Id1i are derived by taking an average of i d1 over one full period based on (9) and (33) as follows:  −θbr 1 (V1 − Vd1 ) cos ωtdθ Id1r = π Rs −(π−θON )  π−θON + (V1 − Vd1 ) cos ωtdθ θbr θbr

 + + Id1i =

1 π Rs



(V1 − Vbr ) cos ωtdθ

−θbr  π+θON π−θON −θbr

+ +

(V1 + Vbi ) cos ωtdθ (V1 − Vd1 ) sin ωtdθ

−(π−θON )  θbr

+



(V1 − Vbr ) sin ωtdθ

−θbr  π−θON θbr  π+θON π−θON

(V1 − Vd1 ) sin ωtdθ  (V1 + Vbi ) sin ωtdθ .

(35)

For the diode D1 , Id1r and Id1i are derived by taking an average of i d1 over one full period based on (11) and (34) as follows:  π−θbr 1 Id2r = (V2 − Vd2 ) cos ωtdθ π Rs θON  2π−θON (V2 − Vd2 ) cos ωtdθ +  + + Id2i =

1 π Rs



π+θbr θbi

(V2 − Vbi ) cos ωtdθ

−θbi  π+θbr π−θbr π−θbr

θON



+ + +

 (V2 + Vbr ) cos ωtdθ

π+θbr  θbi

(V2 − Vd2 ) sin ωtdθ

(V2 − Vbi ) sin ωtdθ

−θbi  π+θbr π−θbr

 (V2 + Vbr ) sin ωtdθ .

(36)

From (18), (20), (22), (35), and (36), the rectifier input impedance after diode breakdown is expressed in (37), as shown at the bottom of this page.

Z in2 =

According to Appendix C, the relationship of the turn-ON angle θON , the breakdown angle θbr , dc output voltage V0, and the input voltage Vin is determined by π Rs V0 (θON + θbr ) V0 + Vin (sin θON − sin θbr ) = RL 2 + (Vbi θON − Vbr θbr ) V0 Vin cos θON = Vbi + 2 V0 . (38) Vin cos θbr = Vbr − 2 As observed from (26), (37), and (38), the four variables θON , θbr , V0 , and Vin should be solved for calculating the input impedance. It is noted that the input impedance Z in2 cannot be expressed as the function of Pin , but the four variables are related to Pin and the efficiency. Thus, it is necessary to derive the closed-form equations of efficiency by the analysis as follows. To calculate the efficiency defined by (27), the loss of the rectifier considering the breakdown period is determined by PLoss = LossON,Rs ,D1 + LossON,junction,D1 + LossOFF,Rs ,D1 + Lossbr,Rs ,D1 + Lossbr,junction,D1 + LossON,Rs ,D2 + LossON,junction,D2 + LossOFF,Rs ,D2 + Lossbr,Rs ,D2 + Lossbr,junction,D2 . (39) When the diode is in turn-ON period, since the voltage keeps constant as that in low power region, the losses on the intrinsic series resistance and the junction are expressed as the functions of V0 and θON , as indicated by (29) and (30). When the diodes are in OFF and breakdown periods, the losses of the diodes are obtained based on the waveforms in Fig. 8. According to Appendix C, the losses of the diodes in OFF and breakdown period are determined by LossOFF,Rs ,D1 = LossOFF,Rs ,D2 Rs C 2j ω2 Vin2 =   2π 1 × π − θON − θbr + (sin 2θON +sin 2θbr ) 2 (40)

(V2 − Vd2 ) sin ωtdθ

2π−θON

TABLE I PARAMETERS OF THE S CHOTTKY D IODE HSMS 2822

Lossbr,Rs ,D1 = Lossbr,Rs ,D2  2 Vbr − V20 = πR  s × θbr 1 +

1 2 cos2 θbr



3 − tan θbr 2

π Rs (2θON + 2θbr − sin 2θON − sin 2θbr ) + j Rs C j ω(2π − 2θON − 2θbr + sin 2θON + sin 2θbr )

 (41)

(37)

LIN AND ZHANG: DIFFERENTIAL RECTIFIER USING RCN FOR IMPROVING EFFICIENCY OVER EXTENDED INPUT POWER RANGE

Fig. 9.

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Schematic of the source–pull simulation.

Fig. 11. Comparison of input impedance obtained from closed-form formula calculation and ADS simulation.

Fig. 10. Efficiency contours for the rectifier based on source–pull simulation with the input power of 27.5 dBm and the frequency of 915 MHz.

Lossbr,junction,D1 = Lossbr,junction,D2   Vbr Vbr − V20 = [tan θbr − θbr ] (42) π Rs based on the input impedance (23) and (37) with the parameters of the diode shown in Table I. It is noted that the calculation results are obtained without considering mismatch at input port, namely, all incident waves are injected to the. Thus, the efficiency can be calculated from (40)–(42). With these equations, the input impedance and the efficiency of the rectifier after diode breakdown can be calculated, which is similar to the case before diode breakdown. B. Validation To validate the closed-form equations, the input impedance of the voltage doubler rectifier is calculated and a comparison with the simulation results is made. The theoretical calculation is diodes without reflected waves. Moreover, the input impedance Z in is equal to Z in1 in (23) before diode breakdown and equal to Z in2 in (37) after diode breakdown. The simulated input impedance is obtained by utilizing the source–pull technique [2] at different input power levels, as shown in Fig. 9. In the source–pull simulation with a given input power level, the efficiencies for different source impedances can be obtained. For example, when the input power is 27.5 dBm, the constant output RF-dc conversion efficiency contours are shown in Fig. 10. It is observed that, when the source impedance is located inside the smallest contour (138 + j ∗ 41.6), the conversion efficiency can be better than 85%. Thus, the input impedance is obtained from the conjugated source impedance corresponding to the highest conversion efficiency. Using this method, the optimal input impedance at various input power levels is obtained in the simulation. Fig. 11 shows the calculated and simulated input impedance, which agrees well with each other. This verifies the closedform equations. Meanwhile, the efficiency curves obtained in

Fig. 12. Comparison of efficiency obtained from the closed-form formula calculation and ADS simulation.

Fig. 13.

Schematic of the proposed differential rectifier with RCN.

the simulation and calculation are shown in Fig. 12. They also show good agreement. IV. D ESIGN OF THE D IFFERENTIAL R ECTIFIER W ITH RCN Based on the above analytical results of the RCN and voltage doubler rectifier, the proposed differential rectifier with an RCN is designed for differential rectenna applications. Fig. 13 shows the schematic of the proposed differential rectifier. It includes a typical differential rectifier and RCN. The rectifier is matched at the input impedance corresponding to the highest efficiency. The RCN is employed to reduce the variation range of the rectifier input impedance and thus realize better match at various input power levels. To measure the overall circuit, a transmission line Balun is used to convert the differential port into single-ended one. In the design,

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Schematic of the typical differential rectifier. Fig. 16.

Layout of the typical differential rectifier. TABLE II

K EY PARAMETERS OF THE T YPICAL D IFFERENTIAL R ECTIFIER

Fig. 15. Input impedance of the subrectifier. P1 and P1 : −5 dBm.  P2 and P2 : 27.5 dBm. P3 and P3 : 35 dBm.

the simulation is carried out by ADS. The circuit is fabricated on the 30-mil-thick ARLON AD255 substrate (εr = 2.55 and tan δ = 0.0018). The designs are presented in the following. A. Design of the Typical Differential Rectifier The schematic of the typical differential rectifier is shown in Fig. 14. The two double voltage rectifiers are composed of matching network, capacitor C1 (1000 pF) for dc block, Schottky diode HSMS2822, capacitor C2 (100 pF), harmonic suppression network, and dc load R L (2000 ). The initial matching network is designed based on the input impedance obtained by the theoretical analysis in Section III. The input impedances at different input power levels are shown in Fig. 15. P2 (138 − j ∗ 41.6 ) represents the input impedance corresponding to the highest conversion efficiency. By designing and refining the matching network, the impedance is transformed to 50 , which is shown in Fig. 15 at point P2 . Finally, the differential rectifier is designed with the layout shown in Fig. 16. The key parameters are tabulated in Table II. B. Proposed Differential Rectifier With RCN The typical differential rectifier exhibits the highest RF-dc conversion efficiency when the input power level reaches the breakdown voltage, as indicated in the analysis in Section III. However, the efficiency degrades with other input power levels. For a conventional matching network, the input impedance at other input power levels is mismatched with the source. As indicated in Section II, the RCN can provide

a solution to reduce the variation range of the input resistance. It is noted that after diode breakdown, the loss of the rectifier is mainly due to the large reverse current on the diode, which is much higher than the loss due to input impedance mismatch. Thus, it is more effective to reduce the variation range of the input impedance before breakdown. The input impedance has complex impedance, as shown in Fig. 11. However, the differential RCN in Section II is only able to compress resistance. Therefore, the first step is to realize nearly resistive input impedance. The 50- transmission line TL3 is employed to rotate the impedance curve on the Smith chart to the resistance axis as shown in Fig. 17, which nearly eliminates the imaginary part of the complex impedance Z rec . Thus, the resulting resistance Z rot varies from 50 to 250  with the input power range of 0–30 dBm. According to the RCN analysis in Section II, the 5:1 impedance variation range can be compressed to 1.37:1 when Z 0 tan θ = (Rmin Rmax )1/2 according to Table I and (7). Since Z 0 is chosen as 50 , θ is calculated as 65.9°, which determines the length of the shorted stub TL1, as shown in Fig. 2. Then, the electric length of the open stub TL2 is 24.1°. With these parameters, the input impedance of the rectifier Z RCN can be compressed to vary from 42.7 to 58.5 , which provides better matching, as shown in Fig. 17. Based on the above design process, the optimized parameters are L C1 = 40.2 mm (∼0.18λg ), L C2 = 15 mm (∼0.07λg ), and L R = 22.5 mm (∼0.1λg ). C. Results and Comparison The proposed 915-MHz differential rectifier with the RCN is fabricated on the 30-mil-thick ARLON AD255 substrate (εr = 2.55 and tan δ = 0.0018), as shown in Fig. 18. For

LIN AND ZHANG: DIFFERENTIAL RECTIFIER USING RCN FOR IMPROVING EFFICIENCY OVER EXTENDED INPUT POWER RANGE

Fig. 17. levels.

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Input impedance of the proposed rectifier at different input power

Fig. 19. RF-dc conversion efficiency versus different loads of the rectifier with RCN and without RCN.

Fig. 18. Simulated and measured efficiency of the differential rectifiers with RCN and without RCN and calculated efficiency of the differential voltage doubler rectifier. TABLE III C OMPARISON OF THE M EASURED C ONVERSION E FFICIENCY W ITH /W ITHOUT RCN FOR D IFFERENT I NPUT P OWER L EVELS

rectifier. In the theoretical calculation, impedance mismatching is not considered and it is assumed all incident power is injected to the diodes. Thus, it means that the RCN contributes to improving the matching of the overall circuit. Moreover, the peak efficiency of the RCN case is 84.8%, which is nearly the same as that without RCN. Thus, the PCE within lower input range is enhanced without degrading the peak efficiency. The results with different input power levels indicate that the RCN can effectively reduce the variation of the input impedance. Also, the RCN can reduce the impedance variation caused by different dc loads. Fig. 19 shows the RF-dc conversion efficiency of the rectifiers with different dc loads R L . It is observed that the efficiency of the rectifier with RCN is higher than that of the rectifier without RCN. Thus, the RCN also contributes to reducing the impact of the output load change. V. C ONCLUSION

measuring the differential rectifier, a typical Balun is added to the convert the different ports into single-ended one. The measured and simulated results for the differential rectifiers with and without RCN are shown in Fig. 18. Meanwhile, the calculated results for the differential voltage doubler rectifier are also depicted, which are obtained by the theoretical method presented in Section III. It is observed that the measured input power range of the proposed rectifier with the RCN for conversion efficiency >50% is from 5.5 to 33.1 dBm. It is 3.6 dB wider than that of the typical differential rectifier without RCN (8.8–32.8 dBm). An input power range of 13.5–31.3 dBm is obtained for efficiency >70%. The efficiency comparison at the input power levels of −5 to 15 dBm is shown in Table III. It can be seen that the efficiency at lower input power levels with RCN is higher than that without RCN. Besides, the measured RF-dc conversion efficiency with the RCN in lower input power (from −5 to 15 dBm) is closed to the calculated one for the differential voltage doubler

In this paper, a differential RCN-based rectifier operating at 915 MHz has been introduced and experimentally demonstrated. The operation principles of the RCN, rectifier, and matching network have been presented in detail. The input impedance and the RF-dc conversion efficiency of the voltage doubler rectifier have been theoretically analyzed. The analysis provides the guideline for obtaining the optimal input impedance. Compared with a typical rectifier without RCN, the proposed differential rectifier shows the improved RF-dc conversion efficiency, wider input power range, and reduced sensitivity to output load. A PPENDIX A This appendix considers the phase delay between the input voltage and the voltage on the diode. By applying Kirchhoff’s voltage law along the RF path as shown in Fig. 6 (blue line), the equations for the voltage across Rs are written as follows when the diodes D1 and D2 are OFF: d(C j 1 Vd1) = V1 − Vd1 dt d(C j 2 Vd2 ) = V2 − Vd2 . Rs dt

Rs

(43) (44)

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Since C j 1 and C j 2 are a monotonic increasing function, they can be expanded as follows: C j 1 = C0 + C1 cos(ωt − φ1 ) + C2 cos(2ωt − 2φ1 ) + · · · (45) C j 2 = C0 + C1 cos(ωt − φ2 ) + C2 cos(2ωt − 2φ2 ) + · · · (46) Substituting C j 1 and C j 2 into (43) and (44) and neglecting the terms higher than the second harmonic, the equation becomes ω Rs (C1 Vd10 + C0 Vd11) sin(ωt − φ1 ) = Vd10 − V10 + (Vd11 − V11 cos φ1 ) cos(ωt − φ1 ) + V11 sin φ1 sin(ωt − φ1 ) (47) ω Rs (C1 Vd20 − C0 Vd21) sin(ωt − φ2 )

where θON is the phase angle where the diode is turned ON, as shown in Fig. 7. According to (49), (50), (53), and (54), the dc voltage (Vd,dc1 and Vd,dc2 ) is shown as follows:

sin θON θON V0 θON Vd,dc1 = −Vd,dc2 = − Vbi . 1− + Vin 2 π π π (60) By substituting (60) into (57), the relation of input voltage Vin and the output voltage V0 is as follows:

V0 2Rs θON + π + Vbi θON . (61) Vin sin θON = 2 RL Since the switching of the diode occurs when Vd reaches the forward drop Vbi of the diode, θON is calculated by

= Vd20 − V20 + (V21 cos φ2 − Vd21) cos(ωt − φ2 ) − V21 sin φ2 sin(ωt − φ2 ). (48) Since (47) and (48) should hold during the OFF period of the diode, each term should be separately zero. For D1 Vd10 = V10 Vd11 = V11 cos φ1 V11 sin φ1 = ω Rs (C1 Vd10 + C0 Vd11).

(49) (50) (51)

The phase delay φ1 is obtained from (49)–(51) as follows: 

 Vd10 . (52) φ1 = arctan ω Rs C0 + C1 Vd11 Likewise, for D2 , each term of (48) should be zero Vd20 = V20 Vd21 = V21 cos φ2 V21 sin φ2 = ω Rs (C0 Vd21 − C1 Vd20 ).

(53) (54) (55)

The phase delay φ2 is obtained from (53)–(55) as follows: 

 Vd20 . (56) φ2 = arctan ω Rs C0 − C1 Vd21 At 915 MHz, the phase delays φ1 and φ2 are very small as indicated in (52) and (56). Thus, the phase delay is regarded as negligible in the following discussion. A PPENDIX B This appendix considers the conversion efficiency of the voltage doubler rectifier in lower power region before diode breakdown. By applying Kirchhoff’s voltage law along the dc path shown in Fig. 6 (red line), the dc voltages (Vd,dc1 and Vd,dc2 ) of Vd1 and Vd2 are related to the dc output voltage of V0 according to RL V0 = (Vd,dc1 − Vd,dc2). 2Rs + R L

(57)

Vd,dc1 and Vd,dc2 are derived by taking an average of Vd1 and Vd2 over one full period based on (10) and (12) as follows:

θON sin θON θON + Vd1 − Vbi (58) Vd,dc1 = Vd10 1 − π π π

θON sin θON θON − Vd2 + Vbi (59) Vd,dc2 = −Vd20 1 − π π π

Vin cos θON = Vbi +

V0 . 2

(62)

From (61) and (62), the switching angle θON is defined as 2π

Rs Rs Vbi cos θON + θON cos θON − sin θON = 2π . RL R L Vin

(63)

The efficiency is derived from the time-domain waveforms V1 , V2 , i d1, and i d2. For diode D1 , the losses are determined by  π+θON (V1 + Vbi )2 1 LossON,Rs ,D1 = dθ (64) 2π π−θON Rs  π+θON (V1 + Vbi )(−Vbi ) 1 LossON,junction,D1 = dθ (65) 2π π−θON Rs  π−θON 1 LossOFF,Rs ,D1 = i 2 Rs dθ. (66) 2π −(π−θON ) d1 For diode D2 , the losses are determined by  +θON (V2 − Vbi )2 1 dθ LossON,Rs ,D2 = 2π −θON Rs  +θON 1 (V2 − Vbi )Vbi dθ LossON,junction,D2 = 2π −θON Rs  2π−θON 1 2 LossOFF,Rs ,D2 = i d2 Rs dθ. 2π θON

(67) (68) (69)

By substituting the waveforms and relationships of V1 , V2 , i d1, and i d2 , the power losses can be expressed as the functions of Vbi , V0 , and θON , as indicated in (30)–(32). A PPENDIX C This appendix considers the efficiency the voltage doubler rectifier in higher power region after diode breakdown. The dc voltages in high power region are calculated from (33) and (34) as follows: Vd,dc1 = −Vd,dc2

V0 θON + θbr Vin (sin θON − sin θbr ) = 1− + 2 π π 1 (70) − (Vbi θON − Vbr θbr ). π

LIN AND ZHANG: DIFFERENTIAL RECTIFIER USING RCN FOR IMPROVING EFFICIENCY OVER EXTENDED INPUT POWER RANGE

According to the relationship of dc voltage and the output voltage in (57), the input incident power Vin is related to V0 , θON , and θbr as follows: Vin (sin θON − sin θbr ) =

π Rs V0 V0 + (θON + θbr ) RL 2 (71) + (Vbi θON − Vbr θbr ).

Since the build-in switching of the diode occurs when Vd reaches the forward drop Vbi of the diode, θON is calculated by Vin cos θON = Vbi +

V0 . 2

(72)

Likewise, the diode breaks down when Vd reaches the breakdown voltage Vbr of the diode, θbr is calculated by Vin cos θbr = Vbr −

V0 . 2

(73)

The rectifier efficiency is determined by (27). As indicated earlier, when the diode is in turn-ON period, since the voltage keeps at ±Vbi , the losses on the intrinsic series resistance and the junction are expressed as the functions of V0 and θON , as indicated in (30) and (31). However, when the diodes are in OFF and breakdown periods, the losses of the diodes are obtained by the waveforms in Fig. 8. The losses of the diode D1 in OFF and breakdown period are determined by LossOFF,Rs ,D1

1 = 2π +



−θbr

−(π−θON )  π−θON 1

2π 

θbr

2 Id1 Rs dθ 2 Id1 Rs dθ

θbr (V − V )2 1 1 br dθ 2π −θbr Rs  θbr (V1 − Vbr )Vbr 1 = dθ . 2π −θbr Rs

Lossbr,Rs ,D1 = Lossbr,junction,D1

(74) (75) (76)

The losses of the diode D2 in OFF and breakdown period are determined by LossOFF,Rs ,D2 =

Lossbr,Rs ,D2 Lossbr,junction,D2

1 2π



π−θbr θON

2 Id2 Rs dθ

 2π−θON 1 2 + Id2 Rs dθ (77) 2π π+θbr  π+θbr (V2 + Vbr )2 1 = dθ (78) 2π π−θbr Rs  π+θbr (V2 + Vbr )(−Vbr ) 1 = dθ . (79) 2π π−θbr Rs

Based on the waveforms of V1 , V2 , Vd1 , and Vd2 as shown in Fig. 8, the power losses can be expressed as the functions of Vbi , Vbr , V0 , θON , and θbr , as indicated in (30), (31), and (40)–(42). Thus, once V0 , θON and θbr are solved from (71)–(73) with a given input power, the input impedance and the efficiency can be calculated.

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R EFERENCES [1] S. Kim et al., “No battery required: Perpetual RFID-enabled wireless sensors for cognitive intelligence applications,” IEEE Microw. Mag., vol. 14, no. 5, pp. 66–77, Jul. 2013. [2] E. Falkenstein, M. Roberg, and Z. Popovi´c, “Low-power wireless power delivery,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 7, pp. 2277–2286, Jul. 2012. [3] M. Roberg, T. Reveyrand, I. Ramos, E. A. Falkenstein, and Z. Popovi´c, “High-efficiency harmonically terminated diode and transistor rectifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 4043–4052, Dec. 2012. [4] Z. Popovi´c, E. A. Falkenstein, D. Costinett, and R. Zane, “Low-power far-field wireless powering for wireless sensors,” Proc. IEEE, vol. 101, no. 6, pp. 1397–1409, Jun. 2013. [5] J.-H. Chou, D.-B. Lin, K.-L. Weng, and H.-J. Li, “All polarization receiving rectenna with harmonic rejection property for wireless power transmission,” IEEE Trans. Antennas Propag., vol. 62, no. 10, pp. 5242–5249, Oct. 2014. [6] A. Costanzo et al., “Electromagnetic energy harvesting and wireless power transmission: A unified approach,” Proc. IEEE, vol. 102, no. 11, pp. 1692–1711, Nov. 2014. [7] P. Lu, X. S. Yang, J. L. Li, and B. Z. Wang, “A compact frequency reconfigurable rectenna for 5.2- and 5.8-GHz wireless power transmission,” IEEE Trans. Power Electron., vol. 30, no. 11, pp. 6006–6010, Nov. 2015. [8] S. M. Kim, J. I. Moon, I. K. Cho, J. H. Yoon, W. J. Byun, and H. C. Choi, “Advanced power control scheme in wireless power transmission for human protection from EM field,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 3, pp. 847–856, Mar. 2015. [9] M. Del Prete, A. Costanzo, A. Georgiadis, A. Collado, D. Masotti, and Z. Popovi´c, “A 2.45-GHz energy-autonomous wireless power relay node,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 12, pp. 4511–4520, Dec. 2015. [10] D. Masotti, A. Costanzo, M. Del Prete, and V. Rizzoli, “Timemodulation of linear arrays for real-time reconfigurable wireless power transmission,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 2, pp. 331–342, Feb. 2016. [11] S. Ladan, A. B. Guntupalli, and K. Wu, “A high-efficiency 24 GHz rectenna development towards millimeter-wave energy harvesting and wireless power transmission,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 12, pp. 3358–3366, Dec. 2014. [12] J. Guo, H. Zhang, and X. Zhu, “Theoretical analysis of RF-DC conversion efficiency for Class-F rectifiers,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 977–985, Apr. 2014. [13] R. Wang et al., “Optimal matched rectifying surface for space solar power satellite applications,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 1080–1089, Apr. 2014. [14] D. Masotti, A. Costanzo, P. Francia, M. Filippi, and A. Romani, “A load-modulated rectifier for RF micropower harvesting with startup strategies,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 994–1004, Apr. 2014. [15] V. Marian, C. Vollaire, J. Verdier, and B. Allard, “Potentials of an adaptive rectenna circuit,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 1393–1396, 2011. [16] V. Marian, B. Allard, C. Vollaire, and J. Verdier, “Strategy for microwave energy harvesting from ambient field or a feeding source,” IEEE Trans. Power Electron., vol. 27, no. 11, pp. 4481–4491, Nov. 2012. [17] C.-J. Li and T.-C. Lee, “2.4-GHz high-efficiency adaptive power,” IEEE Trans. Very Large Scale Integr. (VLSI) Syst., vol. 22, no. 2, pp. 434–438, Feb. 2014. [18] A. Dolgov, R. Zane, and Z. Popovi´c, “Power management system for online low power RF energy harvesting optimization,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 7, pp. 1802–1811, Jul. 2010. [19] Y. Huang, N. Shinohara, and T. Mitani, “A constant efficiency of rectifying circuit in an extremely wide load range,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 986–993, Apr. 2014. [20] D. Wang and R. Negra, “A 2.3 GHz single-ended energy recovery rectifier with stepped-impedance resonator for improved efficiency of outphasing amplifier,” in Proc. Eur. Microw. Conf., Nuremberg, Germany, Oct. 2013, pp. 920–923. [21] Y. Han, O. Leitermann, D. A. Jackson, J. M. Rivas, and D. J. Perreault, “Resistance compression networks for radio-frequency power conversion,” IEEE Trans. Power Electron., vol. 22, no. 1, pp. 41–53, Jan. 2007.

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[22] K. Niotaki, A. Georgiadis, A. Collado, and J. S. Vardakas, “Dual-band resistance compression networks for improved rectifier performance,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 3512–3521, Dec. 2014. [23] J. Xu and D. S. Ricketts, “An efficient, watt-level microwave rectifier using an impedance compression network (ICN) with applications in outphasing energy recovery systems,” IEEE Microw. Wireless Compon. Lett., vol. 23, no. 10, pp. 542–544, Oct. 2013. [24] T. W. Barton, J. M. Gordonson, and D. J. Perreault, “Transmission line resistance compression networks and applications to wireless power transfer,” IEEE J. Emerg. Sel. Topics Power Electron., vol. 3, no. 1, pp. 252–260, Mar. 2015. [25] Y. Ushijima, T. Sakamoto, E. Nishiyama, M. Aikawa, and I. Toyoda, “5.8-GHz integrated differential rectenna unit using both-sided MIC technology with design flexibility,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3357–3360, Jun. 2013. [26] T. Sakamoto, Y. Ushijima, E. Nishiyama, M. Aikawa, and I. Toyoda, “5.8-GHz series/parallel connected rectenna array using expandable differential rectenna units,” IEEE Trans. Antennas Propag., vol. 61, no. 9, pp. 4872–4875, Sep. 2013. [27] H. Sun, “An enhanced rectenna using differentially-fed rectifier for wireless power transmission,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 32–35, 2016. [28] U. Olgun, C.-C. Chen, and J. L. Volakis, “Investigation of rectenna array configurations for enhanced RF power harvesting,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 262–265, 2011. [29] T. Peter, T. A. Rahman, S. W. Cheung, R. Nilavalan, H. F. Abutarboush, and A. Vilches, “A novel transparent UWB antenna for photovoltaic solar panel integration and RF energy harvesting,” IEEE Trans. Antennas Propag., vol. 62, no. 4, pp. 1844–1853, Apr. 2014. [30] M. J. Nie, X. X. Yang, G. N. Tan, and B. Han, “A compact 2.45-GHz broadband rectenna using grounded coplanar waveguide,” IEEE Antennas Wireless Propag. Lett., vol. 14, pp. 986–989, Dec. 2015. [31] C. Song, Y. Huang, J. Zhou, J. Zhang, S. Yuan, and P. Carter, “A highefficiency broadband rectenna for ambient wireless energy harvesting,” IEEE Trans. Antennas Propag., vol. 63, no. 8, pp. 3486–3495, Aug. 2015. [32] T.-W. Yoo and K. Chang, “Theoretical and experimental development of 10 and 35 GHz rectennas,” IEEE Trans. Microw. Theory Techn., vol. 40, no. 6, pp. 1259–1266, Jun. 1992. [33] C. R. Valenta, M. M. Morys, and G. D. Durgin, “Theoretical energy-conversion efficiency for energy-harvesting circuits under poweroptimized waveform excitation,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 5, pp. 1758–1767, May 2015.

Quan Wei Lin (S’15) was born in Guangdong, China. He received the B.E. degree in information engineering and the M.E. degree in communication and information systems from the South China University of Technology, Guangzhou, China, in 2013 and 2016, respectively. He is going to pursue the Ph.D. degree at the City University of Hong Kong, Hong Kong. His research interests include microwave rectifiers, RF circuits, and antennas. Mr. Lin was recipient of the Third Prize of the Student Innovation Competition of the 2013 IEEE International Workshop on Electromagnetic (iWEM).

Xiu Yin Zhang (S’07–M’10–SM’12) was born in Hubei, China. He received the B.S. degree in communication engineering from the Chongqing University of Posts and Telecommunications, Chongqing, China, the M.S. degree in electronics engineering from the South China University of Technology, Guangzhou, China, in 2006, and the Ph.D. degree in electronics engineering from the City University of Hong Kong, Hong Kong, in 2009. He was with the ZTE Corporation, Shenzhen, China, from 2001 to 2003. He was a Research Assistant from 2006 to 2007 and a Research Fellow from 2009 to 2010 with the City University of Hong Kong. He is currently a Full Professor with the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China. He has authored or co-authored over 100 internationally refereed journal and conference papers. His current research interests include microwave circuits and antennas, LTCC, and wireless power transfer. Dr. Zhang served as a Technical Program Committee Member and Session Organizer/Chair for a number of conferences. He was a recipient of the Top-Notch Young Talents Program of China, the National Science Foundation for Outstanding Young Scholars of China, and the Guangdong Natural Science Fund for Distinguished Young Scholar. He was also the recipient of the Scientific and Technological Award (First Honor) of Guangdong Province. He was the Supervisor of several conference Best Paper Award winners. He has been a regular Reviewer for several international journals, including four IEEE T RANSACTIONS and two IEEE L ETTERS .

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MIMO FMCW Reader Concept for Locating Backscatter Transponders Soenke Appel, Dominic Berges, Dominikus Mueller, Andreas Ziroff, and Martin Vossiek, Fellow, IEEE

Abstract— This paper presents a multiple-input multipleoutput (MIMO) frequency-modulated continuous-wave (FMCW) reader concept for locating backscatter radio-frequency identification (RFID) transponders. The proposed MIMO concept enables transmitter sided digital beamforming after a measurement has been executed. An algorithm is presented to decompose the MIMO response to match a digital beamforming input matrix. A 2.4-GHz experimental reconfigurable localization system based on FMCW technology was set up to demonstrate the concept. Multichannels are implemented to allow the possibility of testing the MIMO concept. A frequency-division multiplexing was implemented into an MIMO FMCW RFID reader and successful tested. Moreover, the system is able to locate multiple transponders simultaneously. Measurements confirm the functionality of the concept. The implementation of a digital beamforming algorithm shows the possibility of transmitting and receiving sided spatial filtering. Angle and distance precision were determined in an antenna chamber as well as in a harsh multipath environment. To help readers to compare the system with others, the mean absolute error of the position shows the system accuracy in a laboratory environment. Finally, an error analysis of the decomposition algorithm verified its robustness. Index Terms— Chirp modulation, frequency-division multiplexing (FDM), multiple-input multiple-output (MIMO), position measurement, radio-frequency identification (RFID) tags, spatial filters, transponders.

I. I NTRODUCTION

L

OCALIZING tagged objects, as it is shown in Fig. 1, is within the scope of numerous research projects. In [1], an overview of wireless local-positioning systems and applications is given. A wide range of localization applications can be addressed using radio-frequency identification (RFID) localization technologies, as described in [2]–[4]. In [1] and [2], the frequency-modulated continuous-wave (FMCW) principle is pointed out as one of the most powerful wireless local-positioning systems for RFID. FMCW reader techniques are mainly used to detect two different types of RFID tags: backscatter transponders and

Manuscript received December 2, 2015; revised June 22, 2016; accepted July 14, 2016. Date of publication August 9, 2016; date of current version September 1, 2016. S. Appel and M. Vossiek are with the Institute of Microwaves and Photonics, University of Erlangen–Nuremberg, Erlangen 91058, Germany (e-mail: [email protected]; [email protected]). D. Berges is with the Lehrstuhl für Hochfrequenztechnik, Technical University of Munich, Munich 80333, Germany (e-mail: [email protected]). D. Mueller and A. Ziroff are with the Department of REE ELE Radio Frequency Technology Germany, Siemens Corporate Technology, Munich 81739, Germany (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2016.2593918

Fig. 1. Typical localization scenario with a single reader and an object tagged with a backscatter RFID tag.

surface acoustic wave (SAW) transponders. A review of SAW transponder tags, application, and theory of function, is presented in [5] and [6]. In [7], the theory of backscattering is presented, while in [8] and [9], local positioning with passive backscatter transponders is explored. Furthermore, [10] shows the localization with active transponders. The manufacturing process of SAW transponder is much more complex than that of backscatter transponder. Therefore, the backscatter transponder is mostly less expensive than the SAW transponder. Moreover, there is a wider range of realizable multiple access functionalities for backscatter transponders. Like all technical systems, localization systems have to contend with a lot of technical limitations. Accuracy, precision, resolution, and range are several key attributes for localization systems, which are hindered by these limitations. A brief overview of FMCW techniques as well as their basic performance limitations can be found in [11], while range limitations and multipath degradation are shown in [12]. Novel concepts are required to overcome these limitations. Using multiple transmitters and receivers as well as multiple transponders enables the usage of multiple-input multiple-output (MIMO) techniques. The diversity technique in [13] set the basis for MIMO concepts in communication systems. A few years after this was published, MIMO concepts broke through the limitation of maximum bit rate in bandwidth limited channels, resulting in reduced bit error rates and transmitted power requirements, as described in [14]. Such a technical leap in MIMO developments opens the possibility to enhance

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Fig. 2. Functional diagram of the chosen MIMO FMCW reader concept. A MIMO FMCW reader in the LOS of a backscatter transponder. The reader consists of a multiple frequency chirp generator, four transmission mixers, four antennas, and a frequency monitor to display the resulting system response.

FMCW localization techniques. Basic ideas are presented in [15], including possible FMCW waveform designs for MIMO channels. One of the first reconfigurable MIMO FMCW systems was realized in [16]. A similar, and commercially available, system is discussed in [17]. Vossiek et al. [18] and Parr et al. [19] proposed an inverse synthetic aperture concept to overcome the multipath degradation. Aside from these developments, the focus of this paper is to present a novel MIMO FMCW reader concept for locating backscatter transponders, where receiving and transmitting digital beamforming after the measurement are taken. Therefore, a test system setup is required to have certain important features. One of those is a multiple physical channel implementation with the capability of individual waveform modulation in each channel. Another is the simultaneous receiving and transmitting of individual channels but also the opportunity for each separate channel to operate at the same time. In Section II, the fundamentals of the MIMO FMCW reader concept for locating backscatter transponders are described. Section III presents a novel decomposition algorithm that converts the MIMO response into the input matrix of a Bartlett beamforming algorithm. In Section IV, an experimental system setup with a detailed explanation of the architecture is presented. Section V demonstrates and evaluates the concept’s performance in a series of measurements. First, the measured MIMO response is discussed. Second, a Bartlett beamformer is used to show that the algorithm works correctly with the MIMO response. Finally, the performance is analyzed in terms of estimation accuracy and precision, as well as for algorithm stability is made.

II. R EADER AND RFID L OCALIZATION C ONCEPT To resolve the task of backscatter transponder localization, an MIMO FMCW reader concept is chosen, as shown in Fig. 2. The reader consists of a multiple frequency chirp generator. It simultaneously generates individual signal waveforms for each channel. Each channel composed of a transmission mixer and an antenna. For one channel, a frequency monitor is delineated to display the resulting system response. The transmission mixers have two purposes. One is routing the generated signal, sTx (t), to the antenna port. The other one is mixing the received signal, sRx (t), from the antenna port with the generated signal, sTx (t), and makes the resulting signal, sIF (t), available at the intermediate frequency (IF) port. The functionality of transmission mixer will be discussed in Section IV in more detail. In line-of-sight (LOS) of the reader, an RFID backscatter transponder is shown. Between reader and transponder, multiple paths are implied. Notice that not all signal paths are sketched, for the reason of simplification. The RFID backscatter transponder modulates the received signals with the backscatter frequency, f B , and scattering the modulated signal back to the reader. The reader’s frequency monitor shows multiple responses and is only sketched for channel 3, for the reason of simplification. Notice that the frequency bin pairs from some channels are interchanged. The reason for that is the fact that the transmission mixers are real value mixers. For complex value mixers, one of each sideband would appear in the negative frequency domain. Real value mixing copies the negative frequencies into the positive frequency domain. The mathematical expression of these multiple responses will be described in this section.

APPEL et al.: MIMO FMCW READER CONCEPT FOR LOCATING BACKSCATTER TRANSPONDERS

In Section II-A, the backscatter response of a single-channel reader system is given. With a single-channel response, only the distance, r , between transponder and reader can be estimated. To estimate both the distance, r , and the direction of arrival (DOA), ζ , a multichannel reader response is used. In Section II-B, a multichannel response is presented. Finally, a novel frequency-division multiplexing (FDM) concept, which separates the channels by frequency, is evolved in Section II-C. A. Intermediate Frequency Signal Expression for a Single-Channel FMCW System in the Presence of a Backscatter Transponder The distance, r , from a reader to a backscatter transponder can be resolved using a single-channel reader. The IF response from a backscatter transponder in the LOS of a single-channel FMCW system is  ATx ARx ± cos 2π ((± f B + μτb ) t (t) = sIF 4    fB μ 2 − τb + f 0 ∓ τb 2 2  + ϕ(t) − ϕ(t − τb ) ± ϕ B (1) + (t) sIF

− sIF (t)

and are, respectively, the upper and lower where sidebands of the response. ATx represents the amplitude of the transmitted signal and ARx of the received signal. The switching frequency of the backscatter is denoted by f B , τb is the round trip duration that the signal needs to travel from the reader to the backscatter and back, t represents the time, f 0 stands for the chirp start frequency, and μ is the chirp slope of the FMCW signal. The chirp slope μ is defined as the bandwidth of the modulation, Bm , divided by the duration of the modulation, Tm . The last terms represent the phase of the transmitting signal, ϕ(t), the phase of the received signal, ϕ(t − τb ), and the phase of the backscatter, ϕ B . B. Intermediate Frequency Signal Expression for a Multichannel FMCW System in the Presence of Multiple Backscatter Transponders To resolve the direction to a backscatter transponder, a multichannel reader is used. For the sake of completeness, the scenario is extended with multiple backscatters, which would result in an IF response ± (t) = sIF,l

M  N  ARx,lmn ATx,l cos{2π f α t + ϕα + ϕe } 4 m n

(2)

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In addition to the terms in (1), l represents a single receiving channel out of a total of L receiving channels, m represents a single backscatter out of a total of M backscatters, and n represents a single transmitting channel out of a total of N transmitting channels. The round trip time, τb , is replaced by the sum of τr,nm , the time of flight (TOF) from the transmitter channel, n, to the backscatter, m, and τr,lm , the TOF from the backscatter, m, to the receiver channel, l. It can be seen that the backscatter modulation frequency, f B,m , has to be greater than the frequency difference between the channels, | f 0,l − f 0,n |, which has to be greater than the beat frequency, μ(τr,nm + τr,lm ), as shown in the following inequality: 2μ(τr,nm + τr,lm ) < | f 0,l − f 0,n | < f B,m ∀ l = n.

C. Frequency-Division Multiplexing for Channel Separation The observation from (6) leads to an FDM channel separation concept. From (3), in order to keep the responses separable, a constant frequency offset between the channels, f δ , is introduced f α = f δ · (l − n) ± f B,m + μ(τr,nm + τr,lm )

(3) f α = f 0,l − f 0,n ± f B,m + μ(τr,nm + τr,lm ),  μ 2 ϕα = 2π − (τr,nm + τr,lm ) + f 0,n (τr,nm + τr,lm ) 2  ∓ f B,m τr,lm ± ϕ B,m (4)

The idea of FDM channel separation has already been proposed in [15] and is called a division by beat frequency offset. The authors content that this operation is not feasible because of crosstalk. This statement would be correct in the case of normal FMCW operations. However, the crosstalk can suppressed by using backscatter transponders. The backscatter transponder modulates the signal into higher frequency bands. Therefore, the crosstalk and the transponder response is separated by frequency in the resulting sIF (t) signal. All crosstalk can be suppressed by high-pass filtering the sIF (t) signal. By taking these steps, the proposed system becomes feasible. Therefore, the conditions from (6) become more strict to

f δ > 4μτr,max (5)

(9)

due to the required transition of the filter. The modulation frequency of a backscatter transponder, f B,m , is freely selectable and, therefore, may be chosen high enough to adhere to the restriction of (9). For a maximum TOF, τr,max , the condition

and ϕe = ϕl (t) − ϕn (t − τr,nm − τr,lm ).

(7)

where the frequency difference between the channels has been replaced by f δ . The phase (4) changes to  μ ϕα = 2π − (τr,nm + τr,lm )2 2 + ( f 0 + n fδ )(τr,nm + τr,lm )  ∓ f B,m τr,lm ± ϕ B,m . (8)

2μ(τr,nm + τr,lm ) < fδ  f B,m /N

with

(6)

needs to be met.

(10)

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source, ζ p , a time difference d sin ζ p (11) c0 between reaching the elements of the antenna array exists. A signal received at any array element with a time difference, τ p , between it and an adjacent element is given by τp =

s p (t − τ p ) ≈ s p (t)e− j 2π f p τ p Fig. 3.

Planar wave arrival at an antenna array.

III. RFID L OCALIZATION A LGORITHM A multistep approach was chosen to determine the position. Direct approaches, as proposed in [20], perform better than two- or multistep approaches in terms of localization accuracy, especially at low signal-to-noise ratios (SNRs). The processing effort of the direct approaches can be immense depending on the size and complexity of the raw data. A multistep approach estimates the frequency and phase in the first step, to obtain distance and DOA in the second step, and to finally determine the position in a final step. Such a sequential strategy takes advantage of the lower processing effort through selective data decimation with a small loss of information, as shown in the following algorithm. Estimating the position directly from the frequency and phase requires 2-D Fourier analyses instead of two serial 1-D Fourier analyses. The processing effort increases quadratically instead of linearly depending on the required position precession. Therefore, the following multistep approach was chosen. Moreover, it demonstrates the performance of the system concept step by step in a more understandable way. Subsequent algorithms for resolving the distance, r , and the DOA, ζ , out of the IF signal, sIF (t), will be proposed. In Section III-A, the most common digital beamforming algorithm is briefly described. It shows in which form the received IF signal, derived in (2), has to be transformed in order to perform a DOA estimation. This information is used to create the proposed decomposition algorithm in Section III-B. The output signal of the decomposition process can be used to estimate the DOA using a digital beamforming algorithm. A by-product of the decomposition process can be used in the estimation of the distance. Matrices will be implied as capital letter, printed in bold and with an underline, and vectors will be implied as small letter and printed in bold.

A. Digital Beamforming Algorithm Basics Relevant to the Proposed Concept Digital beamforming, as described in [21], is based on the principle that a planar wave arrives at a uniform linear array (ULA) antenna, as shown in Fig. 3. A planar wave can be assumed when the far field condition is satisfied. This means that the distance between the ULA antenna and the wave source is much larger than the gap between the elements, d, of the ULA antenna. The number of ULA antenna elements is denoted by G. Depending on the DOA from a particular

(12)

where a phase difference of 2π f p τ p exists between them. According to this phase difference between the elements and the fact that the signal is captured simultaneously, it is equivalent to sampling a wave in space. Frequencies that are sampled in space are called spatial frequencies, ξ p , and are related to the phase difference by ξ p (τ p ) = −2π f p τ p .

(13)

The DOA can be calculated from   ξpλp −1 ζ p (ξ p ) = sin − 2πd

(14)

where |ξ p | ≤ π. A beamforming algorithm transforms the spatial signals from the antenna array into the frequency domain. Therefore, an input matrix ⎞ ⎛ x 1 (1) · · · x G (1) ⎜ .. ⎟ .. (15) X = ⎝ ... . . ⎠ x 1 (K )

···

x G (K )

with K being the number of samples, is converted into the covariance matrix ⎛ ⎞ cov(x 1 , x 1 ) · · · cov(x 1 , x G ) ⎜ ⎟ .. .. .. cov(X) = ⎝ ⎠. (16) . . . cov(x G , x 1 )

···

cov(x G , x G )

A conventional digital beamforming algorithm is the Bartlett beamformer [22], calculated from a† (ξ )cov(X)a(ξ ) . a† (ξ )a(ξ ) The vector a is the steering vector and is defined as  T a(ξ ) = 1 e j ξ e j 2ξ · · · e j (G−1)ξ SBart (ξ ) =

where the T operator transpose and the and transpose any matrix or vector.

†

(17)

(18)

operator conjugate

B. Designed and Implemented Decomposition Process for Converting IF Signals Into the Required Digital Beamforming Input Matrix In order to use the Bartlett beamforming algorithm, the received signals, (2), need to be transformed into the input matrix, (15). Fig. 4 shows the decomposition process. The received IF signals, sIF,1 to sIF,L , are sampled by an analog-to-digital converter (ADC). For simplification and not to confuse the notation, the functions will be left in the continuous domain. However, in order to match the mathematical

APPEL et al.: MIMO FMCW READER CONCEPT FOR LOCATING BACKSCATTER TRANSPONDERS

Fig. 4. Received signals are transformed into an input matrix by the decomposition process.

operations to the process steps of Fig. 4, the continuous functions are put as an argument into the discretization operator, ⊥⊥⊥T {.}. This discretization operator acts as a transformation from the continuous domain into the discrete domain. The sampled IF signals will be Hilbert transformed as follows: sHil,l = H{sIF,l (t)}.

(19)

The Hilbert transformed signals are complex valued vectors, therefore, all of the following processing operations result in complex valued vectors. The Hilbert transformed signals are baseband shifted and low-pass filtered according to    (20) sbb,m,l = sHil,l · e j 2π f B,m t ∗ h LP1 where  f B,m is the backscatter frequency, f B,m , which has been replicated by the decomposition process. The operator ∗ is the convolution operator. The cutoff frequency of the low-pass filter, h LP1 , is chosen, such that no signal information is lost. The baseband signals are separated by shifting the signals again by multiples of the channel offset frequency,  f δ , giving    (21) ssep,o,m,l = sbb,m,l · e− j 2πo f δ t ∗ h LP2 with o ∈ {−N + 1, −N + 2, . . . , N − 1}

(22)

and  f δ is a replication of f δ . The low-pass filter, h LP2 , filters out the unused responses from each response pair. After these steps, the signals were split into (2N − 1) × M × L signal vectors. These signals are reordered by + −∗ + ssep,l−n,m,l sreo,n,m,l = ssep,n−l,m,l

(23)

+ values are the positive frequency parts of where ssep −∗ values are the complex conjugated negative ssep and ssep frequency parts of ssep . Finally, the reordered signals need to be combined and matched to the beamforming input matrix by

sout,m,g =

L  N 

sreo,n,m,l · wg · δn+l−1,g

(24)

l=1 n=1

where g represents each row of the beamforming input matrix and δn+l−1,g is the Kronecker delta. The window functions, wg , weights each row of the input matrix to become uniform.

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The discrete form of the output signal, ⊥⊥⊥T3 {sout,m }, is equal to the beamforming input matrix, X m . Note that the number of signal vectors in sout,m is equal to G = N + L −1. In Fig. 3, it can be seen that the distance from antenna G to the target is (G − 1)d sin ζ times longer than the distance from antenna 1. In the case that the ULA antenna is transmitting as well as receiving, the signal travels these distances twice. Hence, using a ULA antenna, the resulting virtual antenna array is G = N + L − 1 long. The reordered signals, sreo,n,m,l , can be used to estimate the distance, rn,m,l , from the transmitting antenna element n to the backscatter m and back to the receiving antenna element l. The frequency in the signal sreo,n,m,l is proportional to rn,m,l . During the decomposition process, (7) changes to f reo,α,n,m,l = μ(τr,nm + τr,lm )

(25)

and is now independent of the frequencies f B,m and f δ . IV. S YSTEM C ONCEPT AND E XPERIMENTAL S ETUP To test the concept and to verify the proposed algorithm, an experimental system was set up in Section IV-A. Some consideration about feasible architectures was taken to explain the chosen architecture in Section IV-B. A. Experimental Setup In Fig. 5(a), a modular diagram and, in Fig. 5(b), a photograph of the experimental system setup are shown. It consists of a digital processing and controlling part and an analog signal generation and receiving part. The system control and signal processing are integrated in a one chip Xilinx Zynq-7000 field-programmable gate array (FPGA) and a processor system. It ensures capable processing performance and flexibility in terms of programming and reconfigurability. An interconnection board connects the control and signal processing unit to the analog devices. All control and component programming connections are not indicated in the modular diagram [Fig. 5(a)] for the reason of simplification. An Analog Devices AD9514/PLBZ clock board supports the reference clock for the direct digital synthesizer (DDS) and the ADC. These clocks are derived from the same Connor-Winfield PM113-466.56M reference clock achieving good transmitterto-receiver coherence. An Analog Devices AD9959/PCBZ DDS board produces frequency chirps from 96 to 99.34 MHz, which are transformed by the Analog Devices ADF4106 phase-locked loops (PLLs) frequency synthesizer and the Mini Circuits ROS-2536C-119+ voltage-controlled oscillators (VCOs) into the RF band of 2.4–2.4835 GHz. The DDS is a one chip, four individually programmable channel, component. Therefore, it is possible to use individual frequency modulation per channel as well as guaranteeing better coherent performance than individual DDSs per channel. The phase noise at the PLL output is −92 dBc/Hz in the 1–30-kHz frequency offset range; it drops from −92 to about −160 dBc/Hz in the 30 kHz–3-MHz frequency offset range and remains at −160 dBc/Hz for higher frequency offsets.

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Fig. 5.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Experimental system setup shown as (a) module diagram and pictured as (b) photograph.

The transmission frequency mixers are custom designs and marked as TRx mixers in Fig. 5. Four transmission frequency mixers, one for each channel, route the RF signals to the antennas and mix the received signal with the transmitted signals to the IF paths simultaneously. A more detailed schematic of the mixers is shown in Fig. 6. A Xinger II XC2500E-03 3-dB, 90° hybrid coupler transmits the signals. An Infineon BAT24-02LS antiparallel diode pair mixes the received signal to the IF paths. All capacitors are RF short and IF high impedance. The value of the inductances is 15 nH. The value of the resistors is 220 . An Agilent MGA-81563 amplifier drives the transmission frequency mixers with 9 dBm. A signal output power of about 5.5 dBm is available at the antenna port. The conversion loss ranges from 6.8 dB at 2.4 GHz to 5.7 dB at 2.4835 GHz. Four equally spaced Huber and Suhner 1324.19.0002 antennas with a gain of about 9 dBi were used. The antenna gap, d, was set equal to the center wavelength of d = c0 / (2.4 GHz · 2.4835 GHz) ≈ 12.3 cm. The IF signals are connected through custom designed variable gain amplifiers (VGAs) followed by the ADC. The VGAs contain three switchable amplifier stages. Mini Circuits MAR-6SM+, ERA-3SM+, and ERA-5XSM+ monolithic amplifiers were implemented in stages 1–3, respec-

Fig. 6.

Schematic of a transmitting frequency mixer.

tively. A bandpass 300 kHz–80 MHz is included in the VGA design. An additional antialiasing low-pass with a cutoff frequency of 20 MHz was placed at the input of the ADCs. The 4DSP FMC-104 ADC board is connected to the FPGA and processor system. The ADC board is equipped with Texas Instruments ADS62P49 14-b 250-Msps ADCs. The high sampling rate and number of quantization steps were chosen to enable a high degree of freedom. The receiver sensitivity with a minimum SNR of 10 dB is −83 dBm at the antenna. The ADC and an interconnection board are connected with the FPGA and processing system via an FPGA Mezzanine Card-plug, which is a standard connector specified for FPGA boards.

APPEL et al.: MIMO FMCW READER CONCEPT FOR LOCATING BACKSCATTER TRANSPONDERS

B. System Concept The linear frequency chirp’s generation is an essential function of an FMCW system. The influence of frequencysweep nonlinearity and how it degrades the distance estimation are described in detail in [23] and [24]. There are plenty of different concepts in producing a linear chirp. A Delta– Sigma modulation is proposed in [25] and [26]. The ramp generation is based on a fractional divider PLL. The benefit of this concept is a reduction of a DDS. The main drawback is that such systems are not available in multichannel single chip designs so far. A very unusual design is proposed in [27]. It is called a fractional-divider-loop configuration. The DDS is part of the feedback path of the PLL. In this configuration, only a fundamental oscillator is needed. But the linear chirp function, which is implemented in most DDS chips cannot be used. A nonlinear chirp is needed at the output of the DDS in order to achieve a linear chirp at the output of the PLL. Therefore, the read only memory function of the DDS chip can be used. Unfortunately the memory’s depth is limited. The authors also propose an equation to calculate the amount of memory. The longer the sweep duration is and the smaller the phase increments are, the more the memory increases. Hence, the amount of available memory limits the maximum sweep duration as well as the sweep’s phase accuracy. The most common concepts in producing linear chirps are by using a DDS followed by a PLL frequency synthesizer, as in [28], or followed by a frequency mixer in combination with a local oscillator (LO), as in [29]. Those configurations provide sufficient chirp linearity and have the advantage of simple ramp reconfiguration. The chirp linearity produced by a DDS depends on the DDS’s frequency resolution, which is in turn dependent on the number of bits of the phase accumulator, NPA . The minimum possible frequency step size, considering the DDS frequency update rate, f DDS,update, is given by f DDS,update . (26) 2NPA For long sweep durations, this theoretical value is well below the actual frequency step size  f DDS,possible =

 f DDS,actual =

Bm R D f DDS,update Tm

(27)

with R D as the divider ratio between the DDS sweep bandwidth and the output bandwidth of the PLL. As mentioned before, to transform the DDS signal into the RF band, a frequency mixer or a PLL can be used. Applying a PLL to generate the RF chirp, the minimum possible frequency step size from (26), and the actual frequency step size from (27) increases by a factor of R D when measured at the RF output. Furthermore, the phase noise increases by a noise factor Fpn,dB = 20 log (R D )

(28)

assuming that the VCO’s phase noise is below this value. If this is not the case, the overall phase noise is calculated from the VCO’s phase noise. How phase noise affects the measurement accuracy is described in detail in [30] and [31].

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In contrast to a PLL, a frequency mixer in combination with an LO does not add additional phase noise, as long as the phase noise of the local oscillator (LO) is below the output phase noise of the DDS. However, such a circuit produces an upper and a lower sideband at the RF output. Suppressing one of these sidebands with an additional sideband suppression filter is only possible if both sidebands are well separated, because of the finite filter’s roll off. The higher the DDS chirp start frequency, the better the sidebands are separated and the more relaxed are the requirements of the filter’s roll off. To generate a high DDS chirp start frequency, a high DDS bandwidth is needed. Another possibility of sideband suppression is using a quadrature DDS in combination with a quadrature modulator. A quadrature DDS is a DDS with twice as many channels. Therefore, for a four-channel system, an eight-channel DDS is needed. With everything considered, it is more convenient to implement a DDS in combination with a PLL to transform the DDS signal into the RF band due to the relaxed DDS requirements. Additional literature and an overview of linear chirp generation methods can be found in [32]. The interconnection design between transmitter and receiver also has significant influence on the overall performance. Particularly, the maximum reading range is affected by the interconnection design. In monostatic FMCW systems, it is necessary to both transmit and receive at the same time. If the same antenna is used for both the functions, duplexer technology needs to be implemented. Circulators or transmission frequency mixers are commonly used. With both technologies, a significant noise factor affects the system. There is no possibility to implement a low-noise amplifier before the duplexer, because the signal path is bidirectional. Circulators have low transmitter-to-receiver isolation and, therefore, would be inconvenient. A table comparing different circulators can be found in [33]. When evaluating circulators, which fit the desired frequency range, the transmitter-to-receiver isolations range from 15 to 26 dB. The noise factors range from 10.5 to 15.5 dB. During the low transmitter-to-receiver isolation, the amplitude of the transmitter signal in the receiver path is several decades larger than the amplitude of the received signal. Amplifiers in the receiver path saturate due to the amplifier’s limited dynamic range. In contrast to circulators, transmission frequency mixers shown in Fig. 6 broadcast a part of the transmitter signal explicitly to the receiver channel in order to directly drive an antiparallel diode pair into saturation. In this state, the antiparallel diode pair works as a harmonic frequency mixer, as described in [34]. As a consequence, transmitter signals do not affect the amplifier in the receiver path. The design of the IF path defines the system’s dynamic range. A backscatter localization system has the benefit of detecting the target in another signal band rather than detecting the echoes of the environment. Using a bandpass filter in the IF path suppresses all echoes from the environment. Inspecting (9) leads to a bandpass filter with an even higher lower-cutoff frequency to also suppress crosstalk between channels. In order to increase the receiver’s dynamic range, VGAs amplifiers can be used. Depending on the range to the target, the VGAs can be configured to match the amplifier’s

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Fig. 7.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Photograph of the measurement setup in the antenna chamber.

output signal amplitude to the maximum ADC analog input amplitude. At the end of the IF path, the amplified signals have to be sampled using an ADC. The ADC’s bandwidth, BADC, has to be chosen after considering the number of transmitter channels, N, and the number of transponders, M. When BADC > 4μτr,max N M

(29)

every transponder can modulate and be detected in another frequency band. Coherence between the channels of receivers and transmitters influences the system performance significantly in terms of ranging precision, as it can be seen in (5). Therefore, all clocks, sample rates, and transmitted signals should be derived from a single master clock. V. M EASUREMENTS To evaluate the concept’s performance, multiple properties of the system were investigated. Firstly, an MIMO response of the system setup was measured and inspected. Second, a DOA estimation using Bartlett beamformer was performed and the virtual length of the ULA was noted. Third, about a hundred thousand measurements in total were executed to evaluate the standard deviation of the distance and DOA estimation in different scenarios, distances, and angles. Furthermore, the mean absolute error (MAE) of the position is evaluated and plotted. Fourth, an error analysis of the decomposition algorithm was realized to test the algorithm’s stability against errors of the replicated parameters. The measurement setup for the antenna chamber is shown in Fig. 7 and for the laboratory with a harsh multipath environment is shown in Fig. 8. For all measurements, the channels were separated by a frequency offset of 10 kHz. The chirp starts at 2.4 GHz and ends at 2.4835 GHz. The modulation duration was set to 10 ms. The overall antenna’s output power was limited to 25 mW e.i.r.p., including a 9-dBi antenna gain. These parameters are consistent with industrial, scientific, and medical radio bands, as defined in [35]. The

Fig. 8. Photograph of the measurement setup in the laboratory with a harsh multipath environment.

ULA antenna element gap was set to the wavelength of the center frequency between chirp start and stop frequencies. Such a ratio reduces the uniqueness sector range from an angle of −30°–30°. The backscatter transponders used in the measurement campaign are custom designs by Wadim Stein who also designed the backscatter transponder in his publication [36]. The backscatter transponder design is a semipassive antenna base modulation. An Infinion BAR 63-03W p-i-n-diode switches between two different impedances. A Linear Technology LTC6900 oscillator triggers the diode and is powered by an external battery. The reflection coefficient of the design is between −1.2 and −1.4 dB. The backscatter transponders were operated for all following measurements with the same Huber and Suhner 1324.19.0002 antennas as is used for the reader. A. MIMO Response of the System Setup To inspect the MIMO response of the system setup in detail, a measurement with two backscatters was done. The backscatters were placed at a distance of 3.18 m from the localization system. One backscatter had been modulating near the band of 1.7 MHz and the other one had been modulating near the band of 2 MHz. Fig. 9 shows the response in the third receiver channel. In Fig. 9(a), it can be seen that the backscatters are well separated in frequency. Fig. 9(b) shows a version of Fig. 9(a), which is zoomed-in view of the band of the first backscatter. The response bins are marked with the ± . As all the shown bins are from the received notation sn,m,l signal at the third channel, and it is zoomed-in view of the ± first backscatter, then this particular response is sn,1,3 . The axes are scaled, so that the upper and lower sidebands, which come from channel 3, are in the center of the plot. The

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Fig. 10. Bartlett beamforming of a received backscatter response in front of the system compared with an ideal simulated wave source detection with a seven- and four-element ULA. In addition, a received backscatter response lateral to the system is added to the plot.

DOA of a wave source, or in this case of the transponder. The higher the number of ULA elements, the smaller the beam is. This results in a better spatial frequency resolution. With a higher spatial frequency resolution, more transponders with the same modulation frequency can be detected in a closer spatial frequency range. C. Standard Deviation of the Distance and DOA Estimations, and the MAE of the Position

Fig. 9. IF signal response of the multichannel FMCW system in the presence of two backscatter transponders. (a) Frequency segment over both transponder responses. (b) Zoomed-in view of the response of the first backscatter transponder.

pairs on the left and right of center are produced by both channels 2 and 4. Both bins on the extreme ends of the plot are produced by the first channel. These interchange phenomena were mentioned in the concept in Section II, and it is a result of the real value mixing. It can be seen that every response from each transmitter to each receiver is well separated in frequency, so that every signal path of the MIMO channel is mapped to another frequency bin. The measured data was recorded in a laboratory, which corresponds to a harsh multipath environment. Hence, the small side bins result from the multipath environment. For receiver channels other than channel 3, the responses are similar to Fig. 9. B. DOA Estimation Using a Bartlett Beamformer The result of Bartlett beamforming, as described in (17), is shown in Fig. 10. It can be seen that Bartlett beamforming performs with the presented four-channel MIMO setup like an ideal seven-element ULA. The ideal seven-element ULA result is a simulated wave source detection with exact ULA geometry and the absence of any noise. The peak position marks the

Evaluations of the standard deviation of the distance and DOA estimation measurements, both in an antenna chamber and in a laboratory with a harsh multipath environment, were made. The standard deviation of distance and DOA is dependent on the distance and the DOA themselves. Therefore, these parameters were varied in order to obtain multiple sample points. In Fig. 11(a), the standard deviation of the DOA estimation is plotted within confidence intervals. In an angle range from −25° to 25°, at every single 1°, 1000 measurements were made to achieve a smaller confidence interval with a significance level of α = 5%. The distance between transponder and system setup was fixed at 4 m. On average, the standard deviation is around 0.05°. The same measurement was repeated in a harsh multipath environment and is shown in Fig. 12(a). The standard deviation increased by a factor of around 3, depending on distance and angle. It can be seen that the standard deviation tends to increase at higher angles as well as at larger distances. In this measurement setup, the number of samples was reduced to 250 per angle and distance position. Nevertheless, the confidence interval of the sampled standard deviation was 0.92–1.1 times the sample-based point estimation at a significance level of α = 5%. For the evaluation of the standard deviation of the distance, there was less data collected. However, a reasonable confidence interval could still be achieved, as shown in Fig. 11(b). With 100 samples per measurement position and choosing a significance level of α = 5%, the interval had

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Fig. 11. Antenna chamber measurement of the DOA and distance with a reader-to-transponder separation of 4 m. (a) DOA confidence interval of the sampled standard deviation. Based on 1000 samples per angle position and a significance level of α = 5%. (b) Distance confidence interval of the sampled standard deviation. Based on 100 samples per angle position and a significance level of α = 5%.

Fig. 12. Laboratory measurement of the DOA and distance with a harsh multipath environment and four different reader-to-transponder separations. (a) Sample-based point estimate of the standard deviation of the DOA. Based on 250 samples per angle position and a significance level of α = 5%, the confidence interval of the sampled standard deviation is 0.92–1.1 times the sample-based point estimation. (b) Sample-based point estimate of the standard deviation of the distance. Based on 20 samples per angle position and a significance level of α = 5%, the confidence interval of the sampled standard deviation is 0.76–1.46 times the sample-based point estimation.

a width of approximately 1 mm at a fixed distance of 4 m. An average standard deviation of 2.5 mm around 0° and an average standard deviation of 3.5 mm at both extreme angles can be noted. The reason for increasing standard deviations at the extreme angles was the antenna’s focus. In Fig. 12(b), this measurement was repeated in the laboratory with a harsh multipath environment. With only 20 samples per position, the confidence interval of the sampled standard deviation increased and is 0.76–1.46 times the point estimation at a significance level of α = 5%. However, it can be seen that the standard deviation of the distance increases with higher distances and also at the extreme angles. At no position does the standard deviation of the distance reach 1 cm. In addition, the MAE was evaluated in the laboratory. The MAE is defined as MAE y =

I 1 | yi − y| I

(30)

i=1

where I is the number of iterations at a single position, i represents the actual measurement iteration,  y stands for the estimated and derived position, and y is for the real position along the ULA antenna. The same applies for the MAE z , with  z and z representing the position orthogonal to the

ULA antenna. The result of the MAE evaluation is shown in Fig. 13. The evalution is based on 20 samples per position and carried out in a harsh multipath environment and at four different reader-to-transponder distances. The MAE reaches 0.3 m for the measurement at 2.35 m. At this distance, the backscatter transponder was located close to the metal wall. Multipath is the obvious reason for the estimation degradation. Multipath effects the distance estimation much more than the DOA estimation. Therefore, the MAE in the z-direction, shown in Fig. 13(b), is higher than in the x-direction, shown in Fig. 13(a). The multipath degradation of the estimated distance results for rising absolute values of the DOA in an increasing MAE in the y-direction and a decreasing MAE in the z-direction. The MAE for the remaining measurements is in the same magnitude as the sample-based point estimate of the standard deviation. Finally, the maximum range of the system setup was evaluated at 30 m and a DOA of 0°. The standard deviation is 0.58 m and 6.8° and the MAE is 1.76 m in the y-direction and 0.65 m in the z-direction. D. Error Analysis of the Decomposition Algorithm Simulations of how stable the decomposition algorithm performs an analysis were made. Two parameters are replicated

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Fig. 14. Simulation of the absolute spatial frequency error depending on backscatter modulation frequency error and channel offset frequency error.

Fig. 13. Laboratory measurement and derivation of the position. Carried out in a harsh multipath environment and at four different reader-to-transponder distances. Based on 20 samples per position. (a) MAE of the estimated position y along the ULA antenna. (b) MAE of the estimated position z orthogonal to the ULA antenna.

by the decomposition process. The channel offset f δ and the backscatter frequency, fδ , is replicated with  modulation frequency, f B , is replicated with  f B . A mismatch between the real parameter and the replicated parameter results in a channel offset frequency error fδ − fδ e(  fδ ) = 

(31)

and a backscatter modulation frequency error fB − fB. e(  fB) = 

(32)

Depending on these errors, Fig. 14 shows an absolute spatial frequency error e(ξp ) = ξp − ξ p .

(33)

All white fields are valued exactly zero. The reason for this is that the beamforming algorithm produces higher side lobes at higher frequency error. The main lobe does not deviate from its original position. At a particular boundary, the side lobes become higher than the main lobe and the resulting spatial frequency calculation becomes false. The decomposition process operates spatial frequency error free in a wide range of channel offset frequency errors. Especially regarding

the ADC and the DDS having the same reference source, a common error can be expected in the subhertz range. The decomposition process operates spatial frequency error free in an almost full range of backscatter modulation frequency errors. Further investigations have shown that a backscatter frequency error that is higher than the Nyquist frequency also results in a spatial frequency error. VI. C ONCLUSION A novel MIMO FMCW reader concept for locating backscatter transponders was presented. Based on this, an experimental system setup was realized. A well-separated MIMO response demonstrated the concept. This response was successfully performed with a Bartlett beamformer. Moreover, the presented four-element ULA performs like a seven-element ULA. In both scenarios, an antenna chamber and a harsh multipath environment, the DOA standard deviation is less than the angle process noise of numerous applications, e.g., wind and vibration affected applications. Moreover, because of its small value, the resulting standard deviation of the distance opens a wide spectrum of applications where a high precision is demanded. Accuracy considerations based on evaluated MAE show the high impact of multipath degradation. Compared with the system in [16], an MIMO localization system with fully active transponders, the proposed system yields a comparable accuracy, except for the 2.35-m laboratory measurement where the backscatter transponder was placed close to the metal wall. A simulated error analysis shows that the presented decomposition algorithm operates in a wide

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range of channel offset frequency errors and in almost the full range of modulation frequency errors. To improve the localization accuracy or even extend the maximum range, direct approach or 2-D Fourier analyses should be investigated. The 2-D Fourier analyses, where the phase correlation of the time steps relating to the spatial ULA elements was interpreted, might improve the distance estimation robustness against multipath degradation. Besides localization, further investigations could be made into finding concepts to identify the orientation of a tagged object. Finally, a similar concept for SAW transponders would be attractive to investigate. R EFERENCES [1] M. Vossiek, L. Wiebking, P. Gulden, J. Wieghardt, C. Hoffmann, and P. Heide, “Wireless local positioning,” IEEE Microw. Mag., vol. 4, no. 4, pp. 77–86, Dec. 2003. [2] R. Miesen et al., “Where is the tag?” IEEE Microw. Mag., vol. 12, no. 7, pp. S49–S63, Dec. 2011. [3] M. Bouet and A. L. dos Santos, “RFID tags: Positioning principles and localization techniques,” in Proc. IEEE 1st IFIP Wireless Days, Nov. 2008, pp. 1–5. [4] T. Sanpechuda and L. Kovavisaruch, “A review of RFID localization: Applications and techniques,” in Proc. IEEE 5th Int. Conf. Elect. Eng./Electron., Comput., Telecommun. Inf. Technol. (ECTI-CON), vol. 2. May 2008, pp. 769–772. [5] L. Reindl, G. Scholl, T. Ostertag, H. Scherr, U. Wolff, and F. Schmidt, “Theory and application of passive SAW radio transponders as sensors,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 45, no. 5, pp. 1281–1292, Sep. 1998. [6] V. P. Plessky and L. M. Reindl, “Review on SAW RFID tags,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 57, no. 3, pp. 654–668, Mar. 2010. [7] P. V. Nikitin and K. V. S. Rao, “Theory and measurement of backscattering from RFID tags,” IEEE Antennas Propag. Mag., vol. 48, no. 6, pp. 212–218, Dec. 2006. [8] J. Heidrich, D. Brenk, J. Essel, G. Fischer, R. Weigel, and S. Schwarzer, “Local positioning with passive UHF RFID transponders,” in Proc. IEEE MTT-S Int. Microw. Workshop Wireless Sens., Local Positioning, RFID, Sep. 2009, pp. 1–4. [9] P. Heide, J. Ilg, R. Roskosch, K. Hofbeck, W. Piesch, and M. Vossiek, “Anti-theft protection system for a motor vehicle, and a method for operating an anti-theft protection system,” U.S. Patent 6 946 949, Sep. 20, 2005. [10] M. Vossiek and P. Gulden, “The switched injection-locked oscillator: A novel versatile concept for wireless transponder and localization systems,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 4, pp. 859–866, Apr. 2008. [11] G. M. Brooker, “Understanding millimetre wave FMCW radars,” in Proc. 1st Int. Conf. Sens. Technol., Palmerston North, New Zealand, 2005, pp. 152–157. [12] G. Li, D. Arnitz, R. Ebelt, U. Muehlmann, K. Witrisal, and M. Vossiek, “Bandwidth dependence of CW ranging to UHF RFID tags in severe multipath environments,” in Proc. IEEE Int. Conf. RFID, Apr. 2011, pp. 19–25. [13] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [14] A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications—A key to gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004. [15] J. J. M. de Wit, W. L. van Rossum, and A. J. de Jong, “Orthogonal waveforms for FMCW MIMO radar,” in Proc. IEEE Radar Conf., May 2011, pp. 686–691. [16] R. Gierlich, J. Huettner, A. Ziroff, R. Weigel, and M. Huemer, “A reconfigurable MIMO system for high-precision FMCW local positioning,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3228–3238, Dec. 2011. [17] A. Stelzer, K. Pourvoyeur, and A. Fischer, “Concept and application of LPM—A novel 3-D local position measurement system,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 12, pp. 2664–2669, Dec. 2004. [18] M. Vossiek, A. Urban, S. Max, and P. Gulden, “Inverse synthetic aperture secondary radar concept for precise wireless positioning,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 11, pp. 2447–2453, Nov. 2007.

[19] A. Parr, R. Miesen, and M. Vossiek, “Inverse SAR approach for localization of moving RFID tags,” in Proc. IEEE Int. Conf. RFID, Apr./May 2013, pp. 104–109. [20] A. J. Weiss, “Direct position determination of narrowband radio frequency transmitters,” IEEE Signal Process. Lett., vol. 11, no. 5, pp. 513–516, May 2004. [21] P. S. Naidu, Sensor Array Signal Processing, 2nd ed. Boca Raton, FL, USA: CRC Press, 2009. [22] M. S. Babtlett, “Smoothing periodograms from time-series with continuous spectra,” Nature, vol. 161, no. 4096, pp. 686–687, 1948. [23] M. Pichler, A. Stelzer, P. Gulden, and M. Vossiek, “Influence of systematic frequency-sweep non-linearity on object distance estimation in FMCW/FSCW radar systems,” in Proc. IEEE 33rd Eur. Microw. Conf., Oct. 2003, pp. 1203–1206. [24] S. O. Piper, “Homodyne FMCW radar range resolution effects with sinusoidal nonlinearities in the frequency sweep,” in Proc. Rec. IEEE Int. Radar Conf., May 1995, pp. 563–567. [25] T. Musch, I. Rolfes, and B. Schiek, “A highly linear frequency ramp generator based on a fractional divider phase-locked-loop,” IEEE Trans. Instrum. Meas., vol. 48, no. 2, pp. 634–637, Apr. 1999. [26] M. Pichler, A. Stelzer, and C. Seisenberger, “Modeling and simulation of PLL-based frequency-synthesizers for FMCW radar,” in Proc. IEEE Int. Symp. Circuits Syst., May 2008, pp. 1540–1543. [27] S. Scheiblhofer, S. Schuster, and A. Stelzer, “High-speed FMCW radar frequency synthesizer with DDS based linearization,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 5, pp. 397–399, May 2007. [28] A. Stelzer, E. Kolmhofer, and S. Scheiblhofer, “Fast 77 GHz chirps with direct digital synthesis and phase locked loop,” in Proc. IEEE Asia–Pacific Microw. Conf., vol. 3. Dec. 2005, pp. 1–4. [29] M. Jankiraman, Design of Multi-Frequency CW Radars, 1st ed. Raleigh, NC, USA: SciTech Pub. Inc., Jul. 2007. [30] J. Hüttner, “Concepts for ultra-wideband impulse-radio localization and ranging,” Ph.D. dissertation, Inst. Electron. Eng., Faculty Eng., Univ. Erlangen-Nuremberg, Erlangen, Germany, 2010. [31] R. Ebelt, D. Shmakov, and M. Vossiek, “The effect of phase noise on ranging uncertainty in FMCW secondary radar-based local positioning systems,” in Proc. IEEE 9th Eur. Radar Conf., Oct./Nov. 2012, pp. 258–261. [32] M. Pichler, A. Stelzer, P. Gulden, C. Seisenberger, and M. Vossiek, “Phase-error measurement and compensation in PLL frequency synthesizers for FMCW sensors—I: Context and application,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 5, pp. 1006–1017, May 2007. [33] S. K. Cheung, T. P. Halloran, W. H. Weedon, and C. P. Caldwell, “MMIC-based quadrature hybrid quasi-circulators for simultaneous transmit and receive,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 3, pp. 489–497, Mar. 2010. [34] M. Cohn, J. E. Degenford, and B. A. Newman, “Harmonic mixing with an antiparallel diode pair,” IEEE Trans. Microw. Theory Techn., vol. MTT-23, no. 8, pp. 667–673, Aug. 1975. [35] European Communications Office (ECO), The European Radiocommunications Committee (ERC) of the Electronic Communications Committee (ECC) of the European Conference of Postal and Telecommunications Administrations (CEPT) Relating to the use of short range devices (SRD) 70-03, ERC recommendation, document ERC Rec. 70-03, May 2015. [36] W. Stein, A. Aleksieieva, S. Roehr, and M. Vossiek, “Phase modulated 61 GHz backscatter transponder for FMCW radar-based ranging,” in Proc. German Microw. Conf. (GeMIC), Mar. 2014, pp. 1–4.

Soenke Appel was born in Hamburg, Germany, in 1984. He received the M.Sc. degree in microelectronic systems in a cooperation degree course from the Hamburg University of Applied Sciences, Hamburg, Germany, the West Coast University of Applied Sciences, Heide, Germany. He is currently pursuing the Dr.Ing. (Ph.D.) degree at the Institute of Microwaves and Photonics, University of Erlangen– Nuremberg, Erlangen, Germany. From 2012 to 2016, he consulted with the RF Technology Group, Siemens Corporate Technology, Munich, Germany, as a Ph.D. Researcher. In 2016, he joined the Active and Passive Antennas Group, Airbus Defense and Space Electronics and Border Security, Ulm, Germany. Researcher, from 2012 to 2016. His research concerns RF identification localization technologies.

APPEL et al.: MIMO FMCW READER CONCEPT FOR LOCATING BACKSCATTER TRANSPONDERS

Dominic Berges was born in Starnberg, Germany, in 1987. He received the M.Sc. degree in electrical engineering and information technology from the Technical University of Munich, Munich, Germany, in 2014, where he is currently pursuing the Dr.Ing. (Ph.D.) degree at the Lehrstuhl für Hochfrequenztechnik. He consults the RF Technology Group, Siemens Corporate Technology, Munich, Germany, as a Ph.D. Researcher. His current research interests include RF/microwave transceiver systems and circuits.

Dominikus Mueller was born in Munich, Germany, in 1979. He received the Dipl.-Ing. (M.S.E.E.) degree in electrical and electronic engineering from the Karlsruhe Institute of Technology, Karlsruhe, Germany, in 2006, and the Ph.D. degree in electrical engineering from the University of the Federal Armed Forces, Munich, Germany, in 2011. He has been with Siemens Corporate Technology, Munich, Germany, since 2011.

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Andreas Ziroff received the Ph.D. degree from the University of Ulm, Ulm, Germany, in 2007. His Ph.D. work focused on the field of passive integrated circuits at millimeter-wave frequencies in low-temperature co-fired ceramic. He is currently performing application-oriented research on RF sensing and radio location with the Department of Corporate Technology, Siemens Corporate Technology, Munich, Germany. His responsibilities include the transfer of respective technologies form research state into system concepts and systems ready for product development. His current research interests include system concepts and front-end concepts for RF systems, preferably at millimeter-wave frequencies. Martin Vossiek (M’96–SM’05–F’16) received the Ph.D. degree from Ruhr-Universität Bochum, Bochum, Germany, in 1996. He joined Siemens Corporate Technology, Munich, Germany, in 1996, where he was the Head of the Microwave Systems Group from 2000 to 2003. Since 2003, he has been a Full Professor with the Clausthal University of Technology, Clausthal-Zellerfeld, Germany. Since 2011, he has been the Chair of the Institute of Microwaves and Photonics, University of Erlangen–Nuremberg, Erlangen, Germany. He has authored or co-authored approximately 190 papers in his research fields. His research has led to over 85 granted patents. His current research interests include radar, transponder, RF identification, and locating systems. Prof. Vossiek is a Member of the German IEEE Microwave Theory and Techniques (MTT)/Antennas and Propagation Chapter Executive Board. He was the Founding Chair of the MTT IEEE Technical Committee MTT-27 Wireless-Enabled Automotive and Vehicular Application. He has been a Member of the organizing committees and Technical Program Committees for international conferences and he has served on the Review Boards of numerous technical journals. He was a recipient of several international awards. From 2013 to 2015, he was an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES .

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Potential and Practical Limits of Time-Domain Reflectometry Chipless RFID Maximilian Pöpperl, Andreas Parr, Student Member, IEEE, Christian Mandel, Member, IEEE, Rolf Jakoby, Member, IEEE, and Martin Vossiek, Fellow, IEEE Abstract— This paper provides a fundamental analysis of the maximum possible information content and reading range of chipless time-domain reflectometry (TDR) radio frequency identification (RFID). Bits encoded on a tag as well as bits that can be decoded by the reader need to be considered to estimate the information content on a tag. This needs an approach that deals with the entire RFID system. Factors such as insertion loss, reader signal, reading range, and channel properties impact the number of bits that can be decoded by the reader. Taking these parameters into account on their own, we can model the signal properties at the decoder by equations from radar theory such as the Cramer–Rao lower bound. The overall performance of these systems cannot be satisfactorily described by the radar theory, as today’s chipless RFID systems use different modulation schemes to increase the information content. Modulation theory can provide a detailed analysis of the modulation schemes depending on the channel and signal properties. This theory influences the tag design and the demodulation algorithms on the decoder, but is not suited to describe the RFID systems as a whole. This paper provides an approach that combines the radar and modulation theories to provide an exhaustive description of chipless TDR RFID communication. The maximum information content obtainable in practice can be estimated with this analysis. The introduced methodology is applied to surface-acoustic wave and ultra-wideband delay-line-based TDR tags. We present the simulations and measurements taken with different tags to show the practical importance of the theoretical findings. Index Terms— Chipless radio frequency identification (RFID), RFID, RFID tags.

I. I NTRODUCTION ECENT advances in the field of time-domain reflectometry (TDR) radio frequency identification (RFID) show that chipless TDR tags offer several attractive features over and above those offered by common semiconductor-based RFID systems [1], [2]. Their suitability for operating in harsh environments especially opens up a wide range of applications. Despite their chipless architecture, tags with up to 83-b [3] and 128-b data capacity [4] have been successfully implemented. Surface-acoustic wave (SAW) tags with 96- data capacity are commercially available [5].

R

Manuscript received December 20, 2015; revised June 28, 2016; accepted July 14, 2016. This work was supported by the German Research Foundation (DFG) under Grant JA 921/38-1 and Grant VO 1453/15-1. M. Pöpperl, A. Parr, and M. Vossiek are with the Institute of Microwaves and Photonics, University of Erlangen–Nürnberg, Erlangen 91054, Germany (e-mail: [email protected]; [email protected]; [email protected]). C. Mandel and R. Jakoby are with the Institute for Microwave Engineering and Photonics, Technische Universität Darmstadt, Darmstadt 64289, Germany (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2593722

The current research in TDR RFID focuses on SAW technology, as information densities on SAW are higher than in other TDR tags [6]. A number of different geometric structures and designs have been developed to maximize the data density [7], [8]. The applicable types of modulation have been investigated. Pulse-position modulation (PPM) is used on SAW tags in the majority of cases [4]. Phase-shift keying (PSK), amplitude-shift keying (ASK), or a combination of modulation schemes have been evaluated as well [3], [9]. The main disadvantages of SAW tags are the complex technology deployed, the high cost of printing individual codes, and the high attenuation levels due to the conversion of electromagnetic into acoustic waves and vice versa via piezoelectric transducers [10]. Ultra-wideband (UWB) delay-line-based tags can overcome some of the limitations as there is no need for wave-type conversion with them and printing can be more cost-effective than in SAW technology [11]. Research on UWB microwave tags is focused on reducing the physical dimensions. A typical approach for UWB delay-line-based tags is the use of meandered transmission lines [12]. Approaches with left-handed delay lines [13] or filter structures [14] are pursued as well. Different modulation schemes, such as PSK [15], PPM, ASK, and quadrature amplitude modulation (QAM), can be applied in UWB delay-line-based tags, similar to SAW TDR tags. The data capacity on UWB microwave tags is generally lower than on SAW tags due to the short delays on UWB tags. This drawback can be counteracted by employing high bandwidths and multidimensional modulation schemes. Research into chipless TDR RFID, as mentioned above, focuses mainly on the tag design. Hence, it is common practice to analyze the tag and the reader separately. A more holistic approach is needed to improve chipless TDR RFID communication due to the high reader–tag interdependency. We introduce such a holistic approach here that describes the entire TDR RFID communication in detail. Our analysis applies to both SAW and UWB delay-line-based tags, as they are very similar. The approach enables a detailed look at the physical limits of TDR RFID systems. We evaluate the design space of TDR tags by applying the general bounds of the communication, modulation, and radar theories, which are combined into an overall approach for TDR RFID systems. We subsequently simulate our theoretical findings and validate them in tests. II. P HYSICAL L IMITS An analysis of the fundamental physical limits of chipless TDR RFID is key to the development of tags with improved

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We neglect the losses introduced by an antenna-tag or antennareader mismatch as they are small compared with the on-tag losses. The signal-to-noise-ratio (SNR) resulting from AWGN can be derived by  4 λ0 2 2 G G tag reader 4πr PRX PTX = (4) SNR = N0 B S L tag k B T0 B S F Fig. 1.

Principle of a TDR RFID communication setup.

performance. The upper bound of the data capacity and reading range can provide a reference point for new concepts as well as for systems that are currently available. A. Communication Principle Fig. 1 shows the fundamental communication principle of chipless TDR RFID systems in a monostatic setup. The reader transmits an interrogation signal to the tag. This signal is received by the tag, reflected at fixed reflectors on the tag, and retransmitted. The reader receives the signal from the tag and demodulates it. We assume an additive white Gaussian noise (AWGN) channel for the communication between the reader and the tag. There is no need to consider the influence of a multipath, which could be described with a more complex channel model [16], as we shall determine the absolute maximum values. Multipath propagation decreases the information content and reading range more significantly than AWGN. Hence, we need to use this simple channel model in our analysis of the absolute maximum ratings. We calculate the power loss from the reader output to the tag input L fs for a farfield scenario, in which the reader and tag antennas are placed in each other’s main beams and for negligible polarization mismatch, as in [17]   4πr 2 1 (1) L fs = G reader G tag λ0 where G reader and G tag are the reader gain and tag antenna gain, respectively. The parameter λ0 is the free-space wavelength of the used center frequency and r is the distance between the tag and the reader. The attenuation on the tag L tag has to be considered in addition to the free-space loss. This parameter is dominated, in the case of a SAW tag, by the conversion loss L con (conversion of electromagnetic wave into acoustic wave and vice versa) and by the propagation loss of the acoustic wave on the tag. There is no conversion loss with UWB delay-line-based tags. However, there is a higher attenuation on the transmission line, which depends on the line length, which is described by the delay on the tag Ttag , and the attenuation constant α of the transmission line. In addition, each reflector stage comes with a small insertion loss L rs . Assuming N reflectors, the loss on the tag L tag can be calculated as follows: L tag = L con eα·Ttag L rs N.

(2)

The overall attenuation L from the reader to the tag and back can be calculated by 4  L con eαTtag L rs N 4πr λ0 L = L 2fs L tag = . (3) G 2reader G 2tag

where PRX is the received power, PTX the transmitted power, N0 the noise power density, B S the signal bandwidth, and F the noise figure of the receiver. The constant k B denotes the Boltzmann constant and T0 the ambient temperature. B. Shannon–Hartley Theorem The equations derived above are commonly used in radar applications. We apply the information theory to the channel to determine the physical limits of information content and reading range. The channel capacity determines the maximum data rate that can be transmitted via a communication channel in mobile communication systems. The maximum possible channel capacity Cmax = B S ld(1 + SNR)

(5)

is given by the Shannon–Hartley theorem [18], where ld denotes the logarithm to the basis 2. This depends on the bandwidth of the signal B S and the SNR, which is mainly determined by the reading range (4). The maximum information content Imax of chipless TDR RFID systems can be calculated with the product of the channel capacity and the signal propagation time Ttag on the tag Imax = Cmax Ttag = B S Ttag ld(1 + SNR).

(6)

The information content consequently depends on the timebandwidth product B S Ttag and the SNR. Typical values for SAW tags are a bandwidth of about 40 MHz and a signal propagation time of 3 μs [5], which leads to a time-bandwidth product of 120. A UWB microwave tag with a bandwidth of 1 GHz and a signal propagation time of 8 ns, in contrast, yields a time-bandwidth product of 8 [19]. The SNR in (4) decreases with the fourth power of the distance between the reader and the tag. By inserting (4) into (6), the relationship between the reading range and the information content becomes apparent. III. P RACTICAL L IMITATIONS The Shannon–Hartley theorem sets the physical limit of the maximum information content a tag can store. This approach considers the reader performance only in terms of the antenna gain and noise figure. In the following, practical limitations to the reader and the influence of modulation schemes are analyzed in detail. A. Modulation Schemes The basic principle behind chipless TDR tags is the coding of information on a fixed number of reflections N. Each reflector has different parameters, so that the reflected signals

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can be modulated. The position of a reflector, in addition to the reflection amplitude and phase, is variable. Amplitude modulation is used in modulation schemes such as pulseamplitude modulation or on–off keying , and the like. PSK or QAM, which combines amplitude and phase modulation, are applicable phase modulation schemes. The most commonly used modulation in SAW, and in UWB delay-line-based tags as well, is PPM, where the reflector positions encode the data. The bit error rate (BER) is a common metric in practical systems. It is determined by statistical analyses of different modulation schemes. We highlight M-PPM, M-PSK, and M-QAM modulation schemes as they are applicable in chipless TDR RFID tags. M denotes the order of the modulation. PPM is the most popular modulation in TDR RFID. Some PPM implementations on SAW tags are described in [4] and [8], while a statistical analysis is presented in [20]. The symbol error rate (SER) can be limited by a lower bound for M-PPM, which can be calculated by [21] SER ⎧PPM (M)   ES 1 ES − ld(M) ⎪ 2 N0 ld(M) −2 ln(2) ⎪ , for > 4 ln(M) ⎪ ⎨e N0 

2 ≤ √ ES ⎪ −ld(M) ES ⎪ N0 ld(M) − ln(2) ⎪2e ⎩ , for ln(M) < ≤ 4 ln(M) N0 (7) where E S is the energy per symbol. The BER of M-PPM is calculated from the SER with BERPPM (M) =

2ld(M)−1 SERPPM (M). 2ld(M) − 1

(8)

The maximum information content is a function of E S /N0 , assuming a fixed BER. A modular M-PPM is deployed in [9], i.e., the modulation is done in sections and a minimum distance δ S is kept between each section. This is needed to separate the reflections at the reader. This implementation guarantees that the minimum distance is kept for all the reflector constellations. However, this leads to a huge amount of unused space on the tag. We can deal with this issue and increase the data density by setting the minimum distance between the modulation sections to less than δ S or we can allow the modulation sections to intersect. Some reflector constellations need to be avoided to keep the minimum distance δ S between the reflectors and thus assure correct demodulation. Space is utilized more efficiently and the data density is increased despite some unusable reflector constellations. Fig. 2 shows exemplarily a schematic of two M-PPM constellations on a tag. The solid rectangles indicate used reflectors with a minimum interspacing δ S . The dashed ones are not used but are possible states, where the minimum interspacing is σr . The information content of the 4-PPM in the sections at the top is 8 b, as there are four reflectors. The intersected PPM at the bottom has a smaller delay and reaches 11 b, which results from determining all the possible combinations to place the four reflectors at 21 states and subtract the number of constellations that do not have the minimum distance δ S between two reflectors.

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Fig. 2.

M-PPM setup (a) in sections and (b) intersected.

The commonly used M-PSK [9] is an alternative to M-PPM. Gray coding is assumed when calculating the BER for M-PSK. The SER and BER are described in [18] and are given by 

π  ES sin SERPSK (M) = erfc (9) N0 M and BERPSK (M) =

SERPSK (M) . ld(M)

(10)

Tight manufacturing tolerances often prevent the use of M-PSK in chipless TDR RFID systems. Hence, the differential PSK (DPSK) is favored. BER and SER double here [21] as the phase shift between two symbols, and not the absolute phase, is modulated. DPSK is the only way to implement phase modulation in many applications. Note that the amplitude of the reflected pulses can be modulated in some TDR RFID systems. ASK is of special interest as the combination with phase modulation, which is QAM, significantly increases the information content. Furthermore, M-QAM can be deployed at a rather low SNR, which becomes apparent when the SER is calculated [18]

   3ld(M)E S 1 . (11) SERQAM (M) = 2 1 − √ erfc 2(M − 1)N0 M Gray coding is assumed, as in M-PSK. Hence, (10) is also valid for M-QAM. The combination of PSK and ASK is not the only technique available for increasing the information content. Another approach would be to combine PSK and PPM or QAM and PPM. The information content in bits for the described modulations can be calculated by I = Nld(M).

(12)

We need to consider the BER increase at higher orders of modulation M at a constant SNR to maximize the information content. Moreover, the SNR decreases if the number of reflections N is increased. We look for the best tradeoff between the modulation order and the number of reflections. The optimization problem can be described by inserting (2) into (4) and substituting E S /N0 with SNR/N, which is valid if the signal energy E S is distributed equally among the reflections.

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B. Demodulation Constraints We can perform a stochastic analysis based on the modulation theory to obtain the BER for a received signal. The demodulation process by itself is limited by radar constraints such as radar resolution and radar precision. Reflectors and modulation states need to be adequately interspaced for accurate demodulation (see Fig. 2). The distance between two reflectors δ S must be maintained to avoid intersymbol interference (ISI), which is linked to the signal bandwidth by c1 (13) δS ≈ 2B S where c1 is the speed of propagation on the tag [22]. Another constraint that determines the minimum distance between the modulation states is the statistical uncertainty associated with determining the position, phase, or amplitude of a modulated signal. We can calculate the minimum standard deviation of the position in PPM using the Cramer–Rao lower bound (CRLB) [23], [24]. The minimum distance σr between the states of a modulated reflector on a tag in TDR systems with PPM is   N0 c1 N0 σr ≥ 1+ (14) 2π B S E S ES using the CRLB. The minimum average phase estimation error σϕ in phase modulation schemes can be calculated by a modified Cramer–Rao bound (MCRB) [18], [25] 1 N0 · . (15) σϕ ≥ 2 ES The minimum distance or phase between the modulation states is crucial for the maximum order of a modulation scheme. Furthermore, the minimum distance between the reflectors δ S and the maximum modulation order determine the maximum information content that can be decoded by the reader. C. Combined Constraints of Radar and Information Theory Having explored the modulation theory, we analyzed the radar constraints in order to describe the performance of a chipless RFID system. Each approach has its own restrictions, such that the only way to estimate the overall system performance, information content, and reading range is to combine the modulation and demodulation constraints. We propose a two-step approach to combine these. The first step is exemplarily done for clearness for M-PPM with nonintersecting modulation sections [Fig. 2(a)]. The results for the other modulation schemes that are applicable to chipless RFID can be derived in a similar manner. The delay on the tag Ttag is assumed to be a fixed value that can be calculated from Ttag = TS N + N (M − 1) Tr

(16)

where TS and Tr are the minimum delays that result from the distances given in (13) and (14). We assume that the signal energy is equally distributed to the reflections. Hence,

E S /N0 can be substituted by SNR/N. The SNR additionally depends on N, which can be seen in (3). The remaining system parameters are treated as constants, so that the SNR can be simplified to a constant parameter k divided by N. Therefore, the E S /N0 can be written as k/N 2 . Applying this to (16) and inserting TS and Tr leads to  √   π k 2Ttag B M< −1 + 1. (17) N N Solving (17) for N and eliminating this parameter in (12) gives an expression for the maximum tag information content with a fixed tag delay and constant system parameters. The information content can be calculated by   √ √ ld(M)k 1/4 π kπ + 8Ttag B(M − 1) − πk 1/4 . I < 2(M − 1) (18) This step in the approach eventually combines the radar equation that was derived for chipless RFID systems with the CRLB and the radar resolution. We combine the modulation theory with the radar equation exemplarily for M-PSK in the following. We choose M-PSK for the sake of simplicity. We can derive the results for M-PPM and M-QAM in the same way. We assume the system parameters are known and constant so that we can substitute E S /N0 again with k/N 2 . Furthermore, the maximum tolerable BERPSK,max has to be defined to calculate the maximum information content. Inserting these assumptions into (10) yields √  π  erfc Nk sin M . (19) BERPSK,max ≥ ld(M) We can eliminate this parameter in (12) by solving (19) for N. Solving the resulting equation for the information content I , we can express I as √ π kld(M) sin M I ≤ . (20) erfc−1 (ld(M)BERPSK,max ) The inverse of the complementary error function can be calculated numerically. Both the steps in the approach give an upper bound for the information content depending on the order of the modulation scheme, which is variable. The order of modulation can be eliminated by inserting (18) into (20) or vice versa. Solving the resulting expression for the information content I would give an equation for the information content, which does not depend on the order of the modulation and the number of reflections, but on the system parameters and the distance between the tag and the reader, which is contained in the parameter k. The maximum of the information content has to be found numerically as none of the derived equations can be solved analytically for M. We calculate this maximum for exemplary TDR RFID systems in the following section.

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Fig. 3.

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TABLE I

TABLE II

SAW-RFID S YSTEM PARAMETERS

UWB M ICROWAVE S YSTEM PARAMETERS

Theoretical information content of a SAW-RFID system.

IV. A PPLICATION TO TDR RFID We derived the practical limits of the modulation and radar theories that influence chipless TDR RFID systems in the previous section. We can calculate the maximum of the information content in practical systems as a function of the reading range and by considering these bounds. We apply the theoretical findings to SAW- and UWB delay-line-based tags in a frequency modulated continuous wave (FMCW) setup in this section. The interrogation units in the test setup are based on this principle. We assume FMCW in the simulations as well to compare the experimental and simulation results. The noise bandwidth is typically given as 1 divided by the sweep duration Tsweep . The BER is fixed at 10−3 . A. SAW RFID We use the parameters in Table I to calculate the physical limits of the information content in SAW-RFID systems. Limitations of the information content on SAW tags are determined by the defined tolerable BER, as the delays on SAW tags are long. Fig. 3 shows the maximum information for the different modulation schemes and the Shannon–Hartley theorem versus the reading range. The phase modulation schemes outperform PPM, which performs better at a low SNR, in short-range scenarios. Information content of more than 100 b is theoretically possible at a reading range of 10 m. The plots show the huge potential of SAW tags. The high information content and also the gap between the modulation

Fig. 4. Theoretical information content of a UWB microwave RFID system.

schemes and the Shannon bound offer much room for further developments.

B. UWB Microwave Tags The parameters for simulating the UWB delay-line-based system are given in Table II. The center frequency and antenna gains are increased compared with the SAW tag to such an extent that the antenna size is approximately the same as that on the SAW tag. The bandwidth is also much higher for this system than for the SAW system. This is because a free frequency band for UWB applications is available in many countries. The main disadvantage of the UWB delay-linebased tags is the short delay on the tag [19]. The maximum information content is thus not limited primarily by the BER, but the CRLB and MCRB strongly influence the information content in the simulations. Fig. 4 shows the calculated information content for the discussed modulation schemes and the Shannon bound. The shapes of the curves are clearly different from the SAW RFID system curves. This is because of the different bounds that limit the information content in both the scenarios. PSK, and possibly QAM, have the edge over PPM for UWB delay-linebased tags. All the modulation schemes theoretically reach more than 50 b at a reading range of 10 m. These systems can be further improved by changing the system parameters and not the modulation schemes, as there is not such a large gap between the modulation schemes and the Shannon bound.

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Fig. 7. Fig. 5.

Test setup of the SAW RFID system on top of a building outdoors. TABLE III

Measured SER of a SAW-RFID system.

M EASURED R EADING R ANGE OF SAW RFID S YSTEMS

Fig. 6. Measured response signal from a SAW tag outdoors in front of a building at a distance of 1.1 m.

V. P RACTICAL E VALUATION We tested two different RFID systems to validate the theoretical analysis done so far. We took many single measurements with both the systems to estimate the SER for different distances between the reader and the tag. We compared the measured values for the SER with the theoretically estimated ones. A. SAW RFID The test setup for a SAW system consists of a 20-b Epcos tag and a reader developed by Siemens Ltd. (see Fig. 5) [26]. The tag is modulated by a PPM with intersecting modulation sections. The reader unit uses FMCW radar signals to determine the pulse positions. The measurement parameters are the same as those used in the simulations (see Table I). We conducted measurements in an anechoic chamber to realize an AWGN channel. We tested the system outdoors in front of a building, on top of a building, and indoors in a lab as well. We varied the reading range between 1 and 4 m to ensure the far-field conditions that are available at distances above 0.93 m in this setup according to [27]. We recorded up to 10 000 measurements for each position to evaluate the SER. Fig. 6 shows that the reconstruction in the anechoic chamber fairly matches the theoretical analysis. There are slight deviations from the ideal, in the form of systematic errors, such as the cable attenuation and length, antenna gain inaccuracy

and noise figure, and the imperfect attenuation of the anechoic chamber. Other measurement campaigns show that the SER is lower indoors than outdoors. The reason for this is that the multipath is scattered from many objects located very near the reader in the indoor scenarios. The tag delay is high, so that the multipath is highly attenuated when the tag’s response is received and the multipath is diffuse. Multipath outdoors, in contrast, is generated by buildings or other objects located at greater distances from the reader and with a high radar cross section. Reflections from the objects can produce delays that are similar to those produced with the SAW tag. A response signal from the tag with a disturbing reflection from a building at the beginning of the signal is shown in Fig. 7. The tag reflections are marked and the plot shows that the first one is strongly interfered by the multipath. The response signal is similar for both the outdoor scenarios. We found no such high reflections with similar delays to that of the tag’s response in the anechoic chamber and indoors. The trend of the data from all the scenarios matches the theoretical analysis. The data show that the constraints discussed above are applicable to an AWGN channel, which we can assume for an anechoic chamber. Table III shows the achieved reading ranges for an SER of 1%. The reading range in the anechoic chamber differs from that indoors by about 0.3 m. The outdoor reading range is about 0.7 m lower than indoors. B. UWB Microwave Tags Fig. 8(a) shows the 12-b meander structure we used to evaluate the theoretical results with a UWB delay-line-based tag.

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Fig. 8. UWB RFID system. (a) Meandered delay-line-based tag. (b) UWB FMCW interrogation unit.

Fig. 10.

Test setup of the UWB microwave system.

note that the information content of the measured SAW tag is much higher than that of the delay-line-based tag. Increasing the information content of the UWB delay-line-based tag will also reduce the reading range. C. Influence of Multipath Propagation

Fig. 9.

Measured SER of a UWB microwave tag.

The tag is based on three transmission paths that provide separable delays. A 16-PSK was used for the tests. The dimensions of the tag without antennas are 11.1 cm × 8.7 cm. The antennas are attached by SMA connectors to the tag. Hence, the transmission lines as well as the reader are both matched to 50 . Measurements were made in an anechoic chamber. We designed and built a UWB FMCW radar module as the reading unit [Fig. 8(b)] [28]. We chose the same set of system parameters used in the simulations (see Table II). To address practical reading scenarios, we avoid any calibration measurements, as it is often prevented in realistic applications. Data from the tests conducted in the anechoic chamber are best suited to verify the theoretical analysis owing to low multipath during the impulse-response delay with the delay-line-based tag. Fig. 9 shows the measurement setup. We conducted more than 1000 measurements per reading range to estimate the SER. The maximum reading range in the anechoic chamber is 3.8 m due to the chamber dimensions. The minimum reading range is at least 1.5 m to achieve farfield conditions using the high gain reader antenna according to [27].The maximum number of errors at the maximum reading range is 7. This equates to an SER of 0.7% at a reading range of 3.8 m. The results of the tests in the anechoic chamber and the theoretical SER are plotted in Fig. 10. The measurement results in the anechoic chamber clearly match well with the theory. Figs. 6 and 10 clearly show that the delay-line-based tag outperforms the SAW tag. The lower measured SER for the delay-line-based tag is due to the higher reading range, which in turn is due to the lower attenuation on the tag. However,

Due to the backscatter functionality of the tag, multipath propagation affects TDR RFID communication. One problem in multipath channels is of course fading, which can be modeled for the described scenarios using Ricean fading distributions. The effects on the BER of conventional modulation schemes can be found in [21]. For TDR RFID, such a channel model is not sufficient anyway, as influences due to fading are not the main disturbances. The more critical factors are the impulse responses of the channel and the received unwanted reflections, respectively. The channel impulse responses are heavily dependent on the environment. In indoor scenarios, reflections often arrive early with a high signal strength at the receiver. After some time the reflection power reduces drastically. This behavior is critical for UWB RFID systems, as the tag response also has small delays. The reflections from the environment directly interfere with the tag response. SAW tags provide much higher delays. The typical multipath in the indoor scenarios fades away during the delay period and no strong, single reflection appears, so that the tag response is not disturbed heavily. A different situation is provided by the outdoor scenarios. The overall multipath is lower compared with the indoor scenarios. This is an advantage for UWB RFID systems, as buildings and other objects are generally far away, so that few reflections arrive in the delay range of the tag response. Using SAW tags, high reflections from buildings and other big objects are received with a high delay that can match with the delay of the tag response. Such single reflections usually have a similar attenuation and delay as the tag response. Thus the impact on the tag response is high. These two scenarios already show the strong dependency of disturbances of the tag responses by multipath on the environmental conditions. Hence, a general estimation of the performance of the TDR tags in realistic scenarios is not possible and a detailed analysis is necessary for each environment. However, the performance

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of TDR RFID is reduced in multipath scenarios, so that channel models that consider multipath propagation are not suitable to describe the maximum information content. VI. C ONCLUSION We have shown the maximum information content of TDR RFID tags through fundamental analysis. The information content of several hundred bits can be achieved with passive chipless TDR tags for ranges in the region of about 1 m. Reading ranges greater than 10 m can be achieved with lower data capacities. The information content and reading range do not depend solely on the tag, but also on the reader setup and the modulation schemes deployed. We also showed that the limitation of the information content can be described by combining the radar, modulation, and information theories. High-order modulation schemes can serve to significantly increase the information content. Furthermore, the environment has great influence on the performance of TDR RFID systems. Tests show that multipath effects due to the environment and reflections on the tag reduce the reading range and information content. The impact of multipath on different modulation schemes needs to be investigated next in order to increase the maximum information content. Reading ranges greater than 10 m bring localization algorithms into the spotlight. Note that high reading ranges are challenging in scenarios with several tags, so that multiple access methods need to be developed. In summary, we demonstrated that in terms of information content and reading, passive chipless TDR tags are suitable for many different applications, but further detailed studies are needed to cut the manufacturing costs in SAW systems and to reduce the dimensions of UWB delay-line-based tags, as well as multiple reflections on them. R EFERENCES [1] S. Preradovic and N. C. Karmakar, “Chipless RFID: Bar code of the future,” IEEE Microw. Mag., vol. 11, no. 7, pp. 87–97, Dec. 2010. [2] V. P. Plessky and L. M. Reindl, “Review on SAW RFID tags,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 57, no. 3, pp. 654–668, Mar. 2010. [3] X. Huang, Z. Chen, M. Wang, H. Xu, and P. Chen, “Large capacity SAW tag,” in Proc. Joint UFFC, EFTF, PFM Symp., 2013, pp. 779–782. [4] C. S. Hartmann, “A global SAW ID tag with large data capacity,” in Proc. IEEE Ultrason. Symp., Oct. 2002, pp. 65–69. [5] P. Hartmann. RFSAW. RF SAW Inc., accessed on Dec. 18, 2015. [Online]. Available: http://www.rfsaw.com/Pages/default.aspx [6] N. C. Karmakar, Chipless RFID Reader Architecture. Boston, MA, USA: Artech House, 2013. [7] L. Reindl, G. Scholl, T. Ostertag, H. Scherr, U. Wolff, and F. Schmidt, “Theory and application of passive SAW radio transponders as sensors,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 45, no. 5, pp. 1281–1292, Sep. 1998. [8] S. Harmä, V. P. Plessky, C. S. Hartmann, and W. Steichen, “Z-path SAW RFID tag,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 55, no. 1, pp. 208–213, Jan. 2008. [9] S. Harmä, W. G. Arthur, C. S. Hartmann, R. G. Maev, and V. P. Plessky, “Inline SAW RFID tag using time position and phase encoding,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 55, no. 8, pp. 1840–1846, Aug. 2007. [10] G. Scholl et al., “SAW-based radio sensor systems for short-range applications,” IEEE Microw. Mag., vol. 4, no. 4, pp. 68–76, Dec. 2003. [11] B. Shao, Fully Printed Chipless RFID Tags towards Item-Level Tracking Applications. Royal Institute of Technology (KTH): Stockholm, Sweden, 2014.

[12] A. Chamarti and K. Varahramyan, “Transmission delay line based ID generation circuit for RFID applications,” IEEE Microw. Compon. Lett., vol. 16, no. 11, pp. 588–590, Nov. 2006. [13] C. Mandel, M. Schüßler, M. Maasch, and R. Jakoby, “A novel passive phase modulator based on LH delay lines for chipless microwave RFID applications,” in Proc. IEEE MTT-S Int. Microw. Workshop Wireless Sens., Local Positioning, RFID, Sep. 2009, pp. 1–4. [14] M. Nickel, C. Mandel, M. Schüßler, and R. Jakoby, “Filter-based slow wave structures for application in chipless microwave RFID,” in Proc. German Microw. Conf. (GeMiC), Mar. 2015, pp. 68–71. [15] M. Schüßler, C. Mandel, M. Maasch, A. Giere, and R. Jakoby, “Phase modulation scheme for chipless RFID- and wireless sensor tags,” in Proc. Asia–Pacific Microw. Conf., Dec. 2009, pp. 229–232. [16] D. Arnitz, Tag Localization in Passive UHF RFID. Graz, Austria: Graz Univ. Technol., 2011. [17] T. S. Rappaport, “Mobile radio propagation,” in Wireless Communications: Principles and Practice. Upper Saddle River, NJ, USA: Prentice-Hall, 2002, pp. 177–254. [18] S. Haykin, Communication Systems. New York, NY, USA: Wiley, 2001. [19] C. Mandel et al., “Higher order pulse modulators for time domain chipless RFID tags with increased information density,” in Proc. Eur. Microw. Conf., Sep. 2015, pp. 100–103. [20] H. Nikookar, Introduction to Ultra Wideband for Wireless Communications. Dordrecht, The Netherlands: Springer Science+Business Media B.V., 2009. [21] J. G. Proakis, Digital communications. Boston, MA, USA: McGraw-Hill, 2008. [22] R. Miesen et al., “Where is the tag?” IEEE Mircow. Mag., vol. 12, no. 7, pp. S49–S63, Dec. 2011. [23] S. Lanzisera and K. S. J. Pister, “Burst mode two-way ranging with Cramer–Rao bound noise performance,” in Proc. IEEE GLOBECOM, Nov./Dec. 2008, pp. 1–5. [24] H. L. Van Trees, Detection, Estimation, and Modulation Theory. New York, NY, USA: Wiley, 2001. [25] N. Noels, H. Steendam, and M. Moeneclaey, “The Cramer–Rao bound for phase estimation from coded linearly modulated signals,” IEEE Commun. Lett., vol. 5, no. 7, pp. 207–209, May 2003. [26] C. C. Ruppel and T. A. Fjeldly, Advances in Surface Acoustic Wave Technology, Systems and Applications, vol. 2. Singapore: World Sci., 2001, pp. 277–326. [27] C. A. Balanis, “Region separation,” in Antenna Theory: Analysis and Design. New York, NY, USA: Wiley, 1997, pp. 145–150. [28] M. Pöpperl, C. Carlowitz, C. Mandel, M. Vossiek, and R. Jakoby, “An ultra-wideband time domain reflectometry chipless RFID system with higher order modulation schemes,” in Proc. German Microw. Conf., Bochum, Germany, 2016, pp. 401–404.

Maximilian Pöpperl was born in Würzburg, Germany, in 1990. He received the B.Sc. and M.Sc. degrees in electrical engineering and information technology from the Ilmenau University of Technology, Ilmenau, Germany, in 2014. He has been a Research Assistant with the Institute of Microwaves and Photonics, University of Erlangen–Nürnberg, Erlangen, Germany, since 2014. His current research interests include chipless radio frequency identification tag and system design.

Andreas Parr (S’11) was born in Fürth, Germany, in 1983. He received the Diploma degree in electrical engineering, electronics, and information technology from the University of Erlangen–Nürnberg, Erlangen, Germany, in 2011. He has been a Research Assistant with the Institute of Microwaves and Photonics, University of Erlangen–Nürnberg, since 2011. His major field of study is UHF-RFID localization. Mr. Parr was a recipient of the IEEE RFID Best Poster Award in 2014.

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Christian Mandel (S’11–M’15) received the Diploma degree in electrical engineering and information technology from the Technische Universität Darmstadt (TU Darmstadt), Darmstadt, Germany, in 2009. He defended his Ph.D. thesis in 2014. He is currently a Post-Doctoral Researcher with the Institute for Microwave Engineering and Photonics, TU Darmstadt. His current research interests include chipless radio frequency identification and chipless wireless sensors.

Rolf Jakoby (M’97) received the Dipl.-Ing. and Dr.Ing. degrees in electrical engineering from the University of Siegen, Siegen, Germany, in 1985 and 1990, respectively. He joined the Research Center of Deutsche Telekom, Darmstadt, Germany, in 1991. Since 1997, he has a Full Professor with the Technische Universität Darmstadt, Darmstadt, Germany. His current research interests include radio frequency identification, microwave and millimeter-wave detectors and sensors for various applications, and in particular on reconfigurable RF passive devices by using novel approaches with metamaterial structures, liquid crystal, and ferroelectric thick/thin film technologies. Mr. Jakoby is a Member of the Society for Information Technology of the VDE and a Member of the IEEE MTT and AP societies. He is the Editor-inChief of Frequenz. He is the Organizer of various workshops and a Member of various TPCs. He was the Chairman of the European Microwave Conference 2007 and the German Microwave Conference 2011. He holds nine patents. Over the last six years, he was the recipient of 11 awards. In 1992, he was the recipient of an award from the CCI Siegen and, in 1997, the ITG-Prize for an excellent publication in the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION.

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Martin Vossiek (M’96–SM’05–F’16) received the Ph.D. degree from Ruhr University Bochum, Bochum, Germany, in 1996. He joined Siemens Corporate Technology, Munich, Germany, in 1996, where he was the Head of the Microwave Systems Group from 2000 to 2003. Since 2003, he has been a Full Professor with Clausthal University, ClausthalZellerfeld, Germany. Since 2011, he has been the Chair of the Institute of Microwaves and Photonics, University of Erlangen–Nürnberg, Erlangen, Germany. He has authored or co-authored approximately 190 papers. His research has led to over 85 granted patents. His current research interests include radar, transponder, radio frequency identification, and locating systems. Prof. Vossiek is a Member of the German IEEE Microwave Theory and Techniques (MTT)/Antennas and Propagation Chapter Executive Board. He was the Founding Chair of the MTT IEEE Technical Committee MTT-27 Wireless-Enabled Automotive and Vehicular Application. He has been a Member of organizing committees and Technical Program Committees for international conferences and has served on the Review Boards of numerous technical journals. From 2013 to 2015, he was an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES . He was a recipient of several international awards.

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Toward a Reliable Chipless RFID Humidity Sensor Tag Based on Silicon Nanowires Arnaud Vena, Etienne Perret, Senior Member, IEEE, Darine Kaddour, and Thierry Baron

Abstract— This paper presents a chipless radio frequency identification (RFID) sensor tag having both identification and sensing capabilities. It is based on one resonant scatterer operating as a signal processing antenna in the 3–7.5 GHz band. The scatterer is used to monitor a physical parameter variation, as well as to identify the remote sensor. To make a resonator sensitive to the humidity, silicon nanowires are deposited on the tag surface using a simple process. The tag needs only one conductive layer so that it can be directly printed on the product to sense and to identify. Measurements done using a bistatic radar configuration in the frequency domain validate this concept. To demonstrate the reliability of such an application, two chipless RFID sensors placed in various environments are simultaneously detected using an anticollision technique based on spectral separation.

Fig. 1. RFID sensor using (a) classical passive or active approach with dedicated sensor components and (b) without dedicated sensor components. In the latter case, the sensitive matching circuit can be easily obtained by adding a sensing material directly upon the tag antenna. This principle can also be used in chipless RFID.

Index Terms— Anticollision, chipless, humidity, radio frequency identification (RFID), sensor, silicone nanowire.

I. I NTRODUCTION

S

ENSING a physical parameter remotely is required in numerous monitoring applications such as weather stations or sensor networks. Using the passive radio frequency identification (RFID) technology for these applications could be interesting, because it does not need to be powered by a battery cell since energy is directly extracted from the electromagnetic (EM) field of the reader [1]–[5]. This gives a key advantage to sensor RFID technology, because maintenance of the remote sensor is largely reduced. To detect a physical parameter variation in the classical RFID, additional discrete components providing a voltage output proportional to a physical parameter, such as temperature or gas concentration, are needed. The signal obtained from these dedicated sensors is then sampled by the chip

Manuscript received October 9, 2015; revised April 20, 2016 and June 20, 2016; accepted June 25, 2016. This work was supported in part by the Laboratoire des Technologies de la Microéletronique and the Grenoble Institute of Technology and in part by the French National Research Agency under Grant ANR-09-VERS-013. A. Vena is with the Centre National de la Recherche Scientifique, Institut d’Electronique et Systèmes, Université de Montpellier, Montpellier 34090, France (e-mail: [email protected]). E. Perret is with the Laboratoire de Conception et d’Intégration des Systèmes, Grenoble Institute of Technology, Valence 26000, France, and also with the Institut Universitaire de France, Paris 75000, France (e-mail: [email protected]). D. Kaddour is with the Laboratoire de Conception et d’Intégration des Systèmes, Grenoble Institute of Technology, Valence 26000, France (e-mail: [email protected]). T. Baron is with the Laboratoire des Technologies de la Microéletronique, Centre National de la Recherche Scientifique, Grenoble 38054, France (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.2016.2594229

and stored in a register as shown in Fig. 1(a) [1]. More recently, UHF RFID chip with integrated sensors are available on the market [6]. In contrast, to reduce the sensor-tag cost, other techniques much simpler based on the measuring of the backscatter power (in magnitude or phase) at the receiving reader antenna or the threshold transmitted power of the tag [see Fig. 1(b)] are introduced [3], [4], [7]–[10]. In this case, the antenna is made sensitive to one of the environment parameters such as temperature [3], humidity [8]–[10], or a mechanical stress to detect a strain [11]. However, such very simple RFID sensor tags are still not implemented in real applications. Indeed, the large-scale production of these sensors is inhibited by the lack of high sensor-to-sensor reproducibility. The major limitation of these RFID sensor tags without dedicated sensors comes from the variability in the chip’s response. Indeed, for these very low-cost sensors, where detection is based on the transmitted activation power or the backscattered power at the receiving antenna, a strong versatility is generally observed concerning the threshold power of the chips as well as the connection between the chip and the antenna. Moreover, an RFID chip is a quite complex electrical component, that is to say, nonlinear and frequency dependent. All these aspects have to be taken into account in order to realize accurate sensor measurements. Consequently, with such an approach, making an RFID sensor tag without dedicated sensors but with a repeatable behavior is a hard task. Chipless RFID technology has given a new paradigm in the field of RF identification [12]. A chipless tag does not embed any silicon chip so that no classical modulation scheme is needed for communication. Instead, simply its EM signature, from a radar point of view, is used to extract a unique ID [2], [13]–[15]. The two major strengths of

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chipless RFID tags are their unit cost, which could be very low [16], [17], and their high levels of reliability even if they operate in harsh environments [15], [18]. As in the classical RFID, the issue of integrating a sensor function to chipless RFID has been raised, though more recently. Indeed, some smart functionalities such as sensing can be implemented into a chipless RFID [19]–[25], allowing to differentiate this technology from barcode. Contrary to conventional RFID sensors, the major advantages of chipless RFID sensors are their potential to have good sensitivity and repeatability and of course an incomparable lifetime. In the literature, some chipless sensors can be found that are able to detect gases. Among them, we can find ethylene gas sensor [19], nitrogen oxides (NOx) gas sensor using a carbon nanotube (CNT) deposit [20], or hydrogen chipless sensor based on a surface acoustic wave substrate [21]. In [22], the use of a resonator for which the resonant frequency is correlated with the temperature is presented. Besides, in the state of the art, silicon nanowires (SiNWs) have shown strong sensitivity to humidity [26] and temperature variations [27]. One of the first totally integrated chipless tag sensors, with remote (wireless) reading of the identifier and the value to be measured, was experimentally validated in [24]. By a chipless tag sensor we mean a compact chipless tag with identification and sensor functions that are totally integrated and compatible with the spirit of chipless RFID (that is, simple in design and low in cost, as well as potentially fully printable, similar to barcodes in that there is no discreet element connected to the tag). SiNWs have also been used to realize humidity sensors based on retransmission chipless tags encoding information with group delay variations. However, the reproducibility and repeatability study, which is crucial for sensing, has never been addressed for this type of extremely low-cost fully printable sensor tags. In this work, a small deposit of SiNW is placed on one of the resonators, to transform a chipless tag into a humidity sensor. The basic idea is to monitor its resonant frequency, which becomes significantly dependent on the humidity variation [24]–[26]. The data are encoded in the frequency domain using a classical frequency shift encoding [14]. As a result with only four resonators, a coding capacity of 13 b is achieved within a small surface (6 × 2.5 cm2 ). As for [19], the tag could be directly printed on the object since only one conductive layer is needed for realization. Moreover, the deposit process of SiNWs (melted in a solvent) is very basic and can be realized with the classical inkjet printer, together with the conductive pattern deposit. This is why very low-cost chipless sensor tags with a large coding capacity can be performed. Even if the literature concerning chipless RFID sensors is growing, only few works have demonstrated a real wireless measurement of the chipless sensor remotely or the presence of a sensor and ID function together inside a tag. Moreover, to the author’s knowledge, there is no work focused on the evaluation of the repeatability of chipless tag sensing measurement. We will show that contrary to RFID sensors based on the same idea (fully analogic tag, without dedicated sensor components), very good repeatability could be obtained

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Fig. 2. Principle of a chipless RFID sensor tag based on the supervision of (a) conductivity variation and (b) effective permittivity variation.

with these homemade chipless sensor tags. This also suggests that such types of sensor tags realized with the automatized process would present higher performances. These points will be developed in detail in the following sections. II. C ONCEPT AND A PPLICATIONS In chipless RFID, the only way possible to detect a physical parameter variation and consequently modify the EM response of a tag consists in using some sensitive materials providing variable conductivity, permittivity, or permeability. A change in conductivity will induce a variation in the tag response level, while a change in permeability or permittivity will affect the resonant frequency or the phase of the scatterer. Finally, from the reading system point of view, a chipless RFID sensor can be seen as multiple antennas loaded with complex load sensitive to various physical parameters. Fig. 2(a) shows the frequency response of a chipless sensor for which the resonant peak at f 0 is sensitive to the surrounding material resistivity. The sensor is made of a dipole loaded by a variable resistance. The resistance variation produces a change in the reflected power. On the contrary, when only the imaginary part of the load is modified, a resonant frequency shift is observed, as shown in Fig. 2(b). In this last case, the load could be a capacitor made by a dielectric for which the permittivity is sensitive to the physical parameter to measure. Since there is no transmission protocol, simultaneous detection of several chipless sensors is a difficult task. However, each EM signature could be discriminated in the frequency domain using spectral separation or in the time domain using gating if the sensors are sufficiently separated. In this paper, we experiment the spectral separation to handle more than one sensor at a time. Indeed, as shown in Fig. 3(a)–(c), we allocate a frequency window for each sensor in which their resonance frequency is free to move when subject to a physical parameter variation. Fig. 3(a) shows the case when two resonators are present

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Fig. 4. Microscopic image of SiNWs deposited upon classical RF substrate. The arrow represents a dimension of 20 μm.

Fig. 3. Chipless RFID tag with sensing capability: principle of anticollision technique based on spectral separation to handle several sensors at a time. (a) Frequency windows associated to each tag. (b) RH effect on the resonant frequency of tag 1 interrogated separately. (c) RH effect on the resonant frequency of tags 1 and 2 interrogated at the same time.

in the environment but no variation occurs. In the second case shown in Fig. 3(b), only one chipless sensor is detected. Its resonance frequency evolves in the channel for which the center frequency is F1 . Thus, it is considered as the sensor with ID 1. Around the center frequency F1 , its resonance frequency depends on the relative humidity (RH). A practical implementation of this configuration is detailed in Section III. In the last case shown in Fig. 3(c), in addition to the chipless sensor with ID 1 presented in Fig. 3(b), we have a second chipless sensor (ID 2) for which the resonance frequency evolves in channel 2, around the center frequency F2 . These two sensors are independent and are not supposed interfering with each other, so that different RH values can be detected on both the channels. This last case is investigated in Section IV. III. S ILICON NANOWIRE -BASED C HIPLESS S ENSOR A. Silicon Nanowires Behavior The main difference between a chipless tag and a chipless sensor is that its resonant frequency depends on sensitive material dielectric properties. For a chipless sensor, a frequency

Fig. 5. (a) Multiple coupled loops resonator geometry. (b) Distribution of the electrical field on the resonator at the resonance frequency. The length of the resonator is equal to ls = 34 mm and the resonant frequency is close to 3.3 GHz. The height H is equal to 12 mm.

shift is due to a dielectric constant variation correlated with a physical quantity, whereas the resonance frequency of a chipless tag is not intended to change. For the first case, a specific sensitive material should be used. In this work, the material under consideration is able to change its electrical properties, depending on the humidity. It can be found in the state of the art that SiNWs make a good candidate to achieve this goal [26], [27]. For this work, the SiNWs have been elaborated by catalytic chemical vapor deposition [28]. A typical scanning electron microscopy picture of the fabricated nanowires is presented in Fig. 4. The deposition process is compatible with printing techniques since nanowires can be mixed with a solvent that evaporates at ambient temperature. Thus to make a sensitive resonator, drops containing nanowires are deposited directly on the metallic strip in the area where the electric field is important, as shown in Fig. 5(b). When the humidity evolves, SiNWs electrical properties change (nanowires capture water molecules, and the effective permittivity is locally modified [26]) so that the field lines are

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Fig. 7. Measurement setup for hygrometry variation. (a) View of the box in front of the antennas. (b) Photo of the inner box containing the probe and the chipless sensor under test. The water level is about 1 cm.

Fig. 6. RCS simulation results of a multiple coupled loop chipless sensor of length ls = 34 mm (see Fig. 5). The four curves correspond to various parameters (conductivity σ and permittivity εr) of the dielectric properties of the sensitive deposit.

rearranged around the resonator and a small frequency shift can be observed. If we focus on the frequency deviation observed in [10], one can notice that when the RH decreases, the frequency rises. The phenomenon behind this behavior could be understood by the following explanation. With less water molecules, the effective permittivity due to a mixing involving air, dielectric, and water is lower. Therefore, the resonant frequency rises. Besides, in the state of the art, some experiments revealed a sensitivity of the SiNWs to the temperature due to a conductivity change [27]. In our case, such a variation was not observed since we did not notice a strong contrast concerning the magnitude of the resonant peak. B. Chipless Sensor Design For the chipless tag, we seek to design a scatterer with a quality factor as high as possible to increase the sensitivity to a permittivity change of the material. This scatterer is intended to have a large enough radar cross section (RCS) value to be properly detected by the reading system. These constraints lead to the design of the tag presented in Fig. 5(a). It is based on several squared loops of the same dimensions placed close to each other. The quality factor of the overall resonator is controlled by the distance between the loops, whereas the RCS magnitude depends on the number of loops. The wire width (L − h)/2 is thin (250 μm) allowing a high quality factor. The main difference between the behavior of one bigger squared loop resonator and the proposed resonator is the level of the structural mode of reflection, which is lower in this case and allows a better detection. As a result, compared with the tag used in [24], the selectivity is increased. The use of several identical loops allows increasing the RCS level at the resonant frequency. In Fig. 6, we observe the simulated resonance frequency of resonator of length ls = 34 mm. The sensitive material placed in the middle as shown in Fig. 5(a) is modeled by a thin layer of a lossy dielectric material for which we can vary both the permittivity and the conductivity. In reality, the thickness of the deposit depends on the size of the particles.

SiNWs have a diameter from 10 to 100 nm and a length that can be several microns. For the sake of computation time and meshing size, we approximated this deposit with a thin layer of dielectric material having a thickness equal to copper (17.5 μm). Indeed, according to the operating frequency and the related wavelength, we can state the hypothesis that this deposit behaves like a homogeneous dielectric having an effective permittivity resulting from a mixing between SiNWs, water, and the dielectric underneath. To model the effect of RH on the spectral response of the sensor, both the permittivity εr and tanδ of the sensitive material are varying. The chipless sensor is placed on a flat polycarbonate (εr = 2.9)-based support as is the case when measured practically on the plastic box shown in Fig. 6. When the sensitive layer has both low permittivity and conductance, which are εr = 2 and σ = 0.1 S/m, we obtain a sharp peak resonating at 3.32 GHz. These dielectric values are intended to model the case when the RH is low. As we will see later, to model the case when the RH is at its maximum value, both the permittivity and the conductance are increased. In Fig. 6, this case is modeled with εr above 12 and σ above 1 S/m. We obtained a resonance peak shifted toward a lower frequency and an attenuation of several decibels. These simulation results are compared with the measurement results in the following section. C. RF Measurements The experiment first introduced in [10] has been used to detect the sensitivity of the new chipless sensor tag in Fig. 5. A plastic box partially filled with water has been used. When the box is closed, the RH rises from the ambient humidity to 100% in half an hour for the box used of size 40 × 20 × 30 cm3 . The box is placed in front of the two antennas in an anechoic chamber as shown in Fig. 7(a). The two antennas are directly connected to the vector network analyzer. The sensor is put inside the box as shown in Fig. 7(b). To supervise the value of the RH and the temperature during the measurement cycle, a probe connected to a hygrometer/thermometer is inserted inside the box. The calibration technique used is explained in detail in [14]. The measurement of the empty chamber and a known reference object is done before any sensor measurement. Fig. 5(b) shows the electric field distribution at the resonance. We notice that the maximum value is located in the middle of

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Fig. 8. Photos of different realized sensor tags operating at around 3.3 GHz having a SiNW deposit located in the middle of the scatterer. The length ls is 34 mm according to Fig. 5(a).

Fig. 9. Recorded frequency responses when RH rises from 74% to 98%. Zoomed-in-view of a portion of the curves showing the maximum of each curve (crosses). The sensor tag used is depicted in Figs. 5 and 7.

the loop, thus the deposition of SiNWs N doped is located in this area as shown in Fig. 8. We realized four tags having the same dimensions as shown in Fig. 8, to evaluate the reproducibility of the fabrication process. Additionally; we limited the sensitive area to a surface of 0.4 × 1.3 cm2 . The nanowires mixed in alcohol solutions are manually deposited in the center of the scatterer, with the help of a micropipette. The alcohol evaporates at ambient temperature, whereas the nanowires remain fixed on the scatterer. We observed in each case a random distribution of nanowires within the specified area (see Fig. 4). The curves plotted in Fig. 9 show the evolution of the frequency shift as a function of the time for a multiple coupled loop resonator of length ls = 34 mm and height H = 12 mm. To correlate this result with the humidity, we superimposed the value recorded using a hygrometer TES-1396. During this measurement cycle, the temperature is controlled to be constant. The temperature has been recorded and a very slight variation lower than 0.1 °C was noticed. Overall, all the changes observed in the measurement results are due to the strong sensitivity of the SiNWs to the humidity. A maximum frequency shift of 35 MHz (from 3.295 to 3.330 MHz) can be observed from 74% to 98% of RH. The sensitivity is close to 1.5 MHz per 1% of RH. It is clear from Fig. 9 that the observed

Fig. 10. Measurement process reproducibility on the same sample separated by several hours. (a) Measurement cycle 1. (b) Measurement cycle 2.

RH changes can be characterized by different approaches. For instance, we can consider the following: 1) magnitude variation of the RCS for a given frequency (for example, the resonance frequency of the tag for a specified value of RH); 2) process frequency; 3) magnitude variation of this maximum [the peak position of each curve (marked by crosses in the zoomed-in-view of Fig. 9)]. These three approaches have been tested; the measurements results showed that approaches 1) and 2) exhibit better results. However, technique 2) (see Fig. 9) looks more promising when considering the detection reliability as regards the measurement distance and calibration versatility. In the following section, we provide details regarding the reproducibility of these novel sensors. D. Reproducibility The measurement process reproducibility is first evaluated. Measurements on the same sample but after having extracted and replaced the sensor tags in the box have been done several times and for some of them several hours after. The results plotted in Fig. 10(a) and (b) are compared with the simulation results. We notice a very similar behavior between these two runs. For the sake of detection, we are interested only in the magnitude and the frequency variation of the resonance peak.

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Fig. 11. (a) Typical recorded frequency responses as a function of the time when RH rises. Only two measurement results are shown. (b) RCS magnitude variation at f = 3.3325 GHz as a function of the recorded RH for four measurements cycles. (c) Frequency shift of the maximums of the RCS values as a function of the recorded RH for four measurement cycles.

Thus, the measurement results have been normalized to the initial maximum RCS value. To model these variations with the simulation model aforementioned, when RH is below 80%, the couple of parameters εr = 2 and σ = 0.1 S/m can be used. For an RH value close to 98%, we need to define a permittivity εr = 12 and a conductivity σ = 1 S/m. Fig. 11(a) presents the already mentioned case 1), which is to say, the variation of the RCS magnitude for a specific frequency. For each measurement, the same variation of 4.5 dB is observed for a change in the RH value between 78% and 98%. Only a small constant change in the RCS level differentiates the four measured curves. These differences typically correspond to misalignments that can occur during the measurement process between the tag, the reference object, and the antennas of the reading system. In Fig. 11(b), the variation of the resonance frequency is studied [case 2)] for four different measurements of the same tag. A shift of 25 MHz is observed. The recorded points provide a linear characteristic between the frequency deviation and the RH in the range of 84%–98%. After that, to evaluate the reliability of manufacturing process of the sensor tag, three samples having the same dimensions have been realized. A similar deposit of SiNWs in concentration and quantity at the same location (see Fig. 8) has been performed. The extraction of the relationship between

Fig. 12. Reliability of manufacturing process: characterization of three sensor tags with the same SiNWs deposit and dimensions. (a) RCS magnitude variation at f = 3.3325 GHz as a function of the recorded RH. (b) Frequency shift of the maximums of the RCS values.

the magnitude and the RH variations is done for a given frequency as shown in Fig. 12(a). Additionally, in Fig. 12(b), the frequency shift as a function of the RH for the three samples is plotted. For the sake of comparison, we also measured and plotted the relationship obtained for a scatterer with no deposit. In Fig. 12(a), we notice a similar shape for the three samples except for one showing an initial RCS magnitude of 1 dB below the others. As observed, the magnitude is decreasing as the RH rises from 75% to 98%. For the reference sample with no deposit, the curve shows a smooth slope with a variation of less than 0.8 dB to be compared with the 3.5–6 dB difference recorded with the three samples having SiWNs. Moreover, we observe that the sensitivity of these three samples is getting higher for a high RH; that is, above 90%. The frequency shift of the resonant peak shows a much more repeatable behavior between the different samples having sensitive material. The difference with the reference sample is even more important, because in this case the curve of this last one shows a flat response, meaning that it is not affected at all by the humidity variation. With SiNWs, a frequency shift

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Fig. 14. Photo of the two sensors. The internal and the external sensors have a length, respectively, equal to 20 and 34 mm according to Fig. 5(a).

Fig. 13. Effective permittivity deviation as a function of RH for several lengths ls . The initial effective permittivities are, respectively, equal to 1.752 and 1.494 for the resonator depicted in Fig. 5(a) of lengths equal to 34 and 20 mm. The curves have been extracted from a humidity measurement cycle described in Section III, using (2).

from 15 to 35 MHz is obtained. With the help of the measured values obtained for the three sensitive resonators, we extract a relationship between RH and resonance frequency. The dashed curve shown in Fig. 12(b) is based on a second-order polynomial fitting. We observe maximum and standard errors of 9.58 % and 2.78%, respectively, on the RH value given by the extrapolated curve. Consequently, these sensors are not intended to give an absolute value of the RH values because the variability between every sample may be still high for some applications. However, they can be used to provide a threshold value as the green/red sensor does. E. Characterization of a SiNW Deposit To extract a model of the sensor, we can use (1), which makes a link between the resonant frequency of a halfwavelength resonator and its physical length ls . If we redefine the effective permittivity as in (2), the term εeff0 corresponds to the case where there is no deposit, while the term εeff(ls,RH) represents the effective permittivity variation depending on the RH c (1) fr = √ 2 · ls · εeff c2 . (2) εeff0(ls ) + εeff(ls ,RH) = 4 · ls2 · fr2 It is interesting to plot in Fig. 13, the effective permittivity deviation εeff(ls,RH) as a function of the RH for two sensor tags of lengths, respectively, equal to 34 and 20 mm. For a 30% of RH variation, the maximum deviation on the effective permittivity is equal to 0.025 and 0.01, respectively, for the resonator of lengths 34 and 20 mm. The difference between the two chipless sensors could be explained by an electric field strength variation. A probable explanation is the versatility of the deposit between the two sensors. Besides, according to the simulation results, we did not observe a lower sensitivity for

Fig. 15. Frequency shift as a function of RH when sensor 1 of length ls = 34 mm is inside the box.

the resonator of length 20 mm when subject to the same dielectric variations. Indeed, the quantity as well as the repartition is not really controlled with the previously described deposition technique. Despite this issue, this variation range gives us an idea of the frequency deviation that could be expected for other resonators of length ls . IV. H ANDLING OF T WO S ENSORS AT A T IME For a wireless sensor application, it is often required that more than one sensor can be detected simultaneously. However, in the state of the art, the anticollision topic for chipless RFID tags is rarely discussed. Some techniques are presented in [29], but are best suited for the surface acoustic wave chipless tags. The following experiment shows that it is possible to detect two sensors at a time using the technique of anticollision based on spectral separation [30]. For this purpose, we need to use two resonators having different resonant frequencies. We modify our measurement setup to insert one resonator inside the box and a second one having a different length, outside the box as shown in Fig. 14. So, only the sensor inside the box deviates its resonant frequency as the humidity rises. The frequency responses, respectively, around the resonant peaks of sensors 1 and 2 are recorded. As sensor 1 resonating around 3.3 GHz is inside the box, we observe a frequency shift when the humidity rises. Meanwhile, the resonant frequency of sensor 2 (5.2 GHz) does

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not change significantly since the RH is quite stable in the anechoic chamber. The extracted curves between the deviation frequency and the RH are shown in Fig. 15. This confirms that two sensors placed in separated environments can be detected simultaneously. To confirm this behavior, we inverted the two sensors and observed that only the sensor inside the box sees its resonant frequency varying. V. C ONCLUSION It was demonstrated that in the case of a single detection, a chipless sensor tag can be realized with only one resonator within 3.5 × 1.5 cm2 . Since the tag has no ground plane and the sensitive material is deposited by drops at ambient temperature, its realization is fully compatible with the printing techniques. The measurement setup used confirmed the sensitivity of SiNWs to detect a change in the RH by monitoring the resonance frequency deviation on the resonator. A quasilinear behavior has been observed between the frequency deviation of the sensing resonator and the RH, and the best sensitivity was found for a high RH. Measurements made on the same sample with a few hours of delay proved the measurement repeatability, which is a real advantage compared with the passive analog UHF RFID sensor technology. The crucial task of evaluating the reliability of the manufacturing process has been done by characterizing several sensor tags fabricated with the same technique. Better repeatability could be obtained with tags produced with the automated fabrication process, such as inkjet printing. The measurement of two RFID chipless sensor tags at the same time was performed using an anticollision technique based on spectral separation. This shows the potential of chipless sensors for wireless sensing applications. ACKNOWLEDGMENT The authors would like to thank the Grenoble Institute of Technology and the LTM Laboratory for their support of this project. R EFERENCES [1] K. Finkenzeller and D. Müller, RFID Handbook: Fundamentals and Applications in Contactless Smart Cards, Radio Frequency Identification and Near-Field Communication, 3rd ed. New York, NY, USA: Wiley, 2010. [2] E. Perret, Radio Frequency Identification and Sensors: From RFID to Chipless RFID, 1st ed. New York, NY, USA: Wiley, 2014. [3] C. Occhiuzzi, S. Caizzone, and G. Marrocco, “Passive UHF RFID antennas for sensing applications: Principles, methods, and classifcations,” IEEE Antennas Propag. Mag., vol. 55, no. 6, pp. 14–34, Dec. 2013. [4] G. Marrocco, L. Mattioni, and C. Calabrese, “Multiport sensor RFIDs for wireless passive sensing of objects—Basic theory and early results,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2691–2702, Aug. 2008. [5] G. Marrocco, “Pervasive electromagnetics: Sensing paradigms by passive RFID technology,” IEEE Wireless Commun., vol. 17, no. 6, pp. 10–17, Dec. 2010. [6] S. Manzari, T. Musa, M. Randazzo, Z. Rinaldi, A. Meda, and G. Marrocco, “A passive temperature radio-sensor for concrete maturation monitoring,” in Proc. IEEE RFID Technol. Appl. Conf. (RFID-TA), Tampere, Finland, Sep. 2014, pp. 121–126. [7] R. Bhattacharyya, C. Floerkemeier, S. Sarma, and D. Deavours, “RFID tag antenna based temperature sensing in the frequency domain,” in Proc. IEEE Int. Conf. RFID, Orlando, FL, USA, Apr. 2011, pp. 70–77. [8] J. Siden, X. Zeng, T. Unander, A. Koptyug, and H.-E. Nilsson, “Remote moisture sensing utilizing ordinary RFID tags,” in Proc. IEEE Sensors, Atlanta, GA, USA, Oct. 2007, pp. 308–311.

[9] Y. Jia, M. Heiß, Q. Fu, and N. Gay, “A prototype RFID humidity sensor for built environment monitoring,” in Proc. Int. Workshop Geosci. Remote Sens. Edu. Technol. Training, Shanghai, China, Dec. 2008, pp. 496–499. [10] S. Manzari, C. Occhiuzzi, S. Nawale, A. Catini, C. Di Natale, and G. Marrocco, “Polymer-doped UHF RFID tag for wireless-sensing of humidity,” in Proc. IEEE Int. Conf. RFID, Orlando, FL, USA, Apr. 2012, pp. 124–129. [11] C. Occhiuzzi, C. Paggi, and G. Marrocco, “RFID tag antenna for passive strain sensing,” in Proc. 5th Eur. Conf. Antennas Propag. (EUCAP), Rome, Italy, Apr. 2011, pp. 2306–2308. [12] S. Tedjini, N. Karmakar, E. Perret, A. Vena, R. Koswatta, and R. E-Azim, “Hold the chips: Chipless technology, an alternative technique for RFID,” IEEE Microw. Mag., vol. 14, no. 5, pp. 56–65, Jul. 2013. [13] I. Preradovic, I. Balbin, N. C. Karmakar, and G. F. Swiegers, “Multiresonator-based chipless RFID system for low-cost item tracking,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 5, pp. 1411–1419, May 2009. [14] A. Vena, E. Perret, and S. Tedjini, “Chipless RFID tag using hybrid coding technique,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3356–3364, Dec. 2011. [15] A. Vena, E. Perret, and S. Tedjni, “A depolarizing chipless RFID tag for robust detection and its FCC compliant UWB reading system,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 8, pp. 2982–2994, Aug. 2013. [16] A. Vena et al., “Design of chipless RFID tags printed on paper by flexography,” IEEE Trans. Antennas Propag., vol. 61, no. 12, pp. 5868–5877, Dec. 2013. [17] D. Betancourt et al., “Square-shape fully printed chipless RFID tag and its applications in evacuation procedures,” in Proc. 9th Eur. Conf. Antennas Propag. (EuCAP), Lisbon, Portugal, May 2015, pp. 1–5. [18] A. Vena, E. Perret, B. Sorli and S. Tedjini, “Toward reliable readers for chipless RFID systems,” in Proc. 31st URSI Gen. Assemb. Sci. Symp. (URSI GASS), Beijing, China, 2014, pp. 1–4. [19] L. Yang, R. Zhang, D. Staiculescu, C. P. Wong, and M. M. Tentzeris, “A novel conformal RFID-enabled module utilizing inkjet-printed antennas and carbon nanotubes for gas-detection applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 653–656, 2009. [20] S. Shrestha, M. Balachandran, M. Agarwal, V. V. Phoha, and K. Varahramyan, “A chipless RFID sensor system for cyber centric monitoring applications,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 5, pp. 1303–1309, May 2009. [21] W. Jakubik and M. Urbanczyk, “Hydrogen detection in surface acoustic wave gas sensor based on interaction speed,” in Proc. IEEE Sensors, vol. 3. Oct. 2004, pp. 1514–1517. [22] T. T. Thai et al., “Design and development of a millimetre-wave novel passive ultrasensitive temperature transducer for remote sensing and identification,” in Proc. Eur. Microw. Conf. (EuMC), Paris, France, Sep. 2010, pp. 45–48. [23] A. Guillet, A. Vena, E. Perret, and S. Tedjini, “Design of a chipless RFID sensor for water level detection,” in Proc. 15th Int. Symp. Antenna Technol. Appl. Electromagn. (ANTEM), Toulouse, France, Jun. 2012, pp. 1–4. [24] A. Vena, E. Perret, S. Tedjini, D. Kaddour, A. Potie, and T. Barron, “A compact chipless RFID tag with environment sensing capability,” in IEEE MTT-S Int. Microw. Symp. Dig., Montreal, QC, Canada, Jun. 2012, pp. 1–3. [25] R. S. Nair, E. Perret, S. Tedjini, and T. Baron, “A group-delay-based chipless RFID humidity tag sensor using silicon nanowires,” IEEE Antennas Wireless Propag. Lett., vol. 12, pp. 729–732, 2013. [26] H. Li, J. Zhang, B. Tao, L. Wan, and W. Gong, “Investigation of capacitive humidity sensing behavior of silicon nanowires,” Phys. E, Low-Dimensional Syst. Nanostruct., vol. 41, no. 4, pp. 600–604, 2009. [27] C.-P. Wang, C.-W. Liu, and C. Gau, “Silicon nanowire temperature sensor and its characteristic,” in Proc. IEEE Int. Conf. Nano/Micro Eng. Molecular Syst. (NEMS), Kaohsiung, Taiwan, Feb. 2011, pp. 630–633. [28] F. Dhalluin, T. Baron, P. Ferret, B. Salem, P. Gentile, and J.-C. Harmand, “Silicon nanowires: Diameter dependence of growth rate and delay in growth,” Appl. Phys. Lett., vol. 96, no. 13, p. 133109, 2010. [29] C. Hartmann, P. Hartmann, P. Brown, J. Bellamy, L. Claiborne, and W. Bonner, “Anti-collision methods for global SAW RFID tag systems,” in Proc. IEEE Ultrason. Symp., vol. 2. Montreal, QC, Canada, Aug. 2004, pp. 805–808. [30] E. Perret, R. S. Nair, E. B. Kamel, A. Vena, and S. Tedjini, “Chipless RFID tags for passive wireless sensor grids,” in Proc. 31st URSI General Assembly Sci. Symp. (URSI GASS), Beijing, China, Aug. 2014, pp. 1–4.

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Arnaud Vena received the Dipl.-Ing. degree in electrical engineering from the Institut National Polytechnique de Grenoble (Grenoble-INP), Grenoble, France, in 2005, and the Ph.D. degree from the Université de Grenoble, Grenoble, France, in 2012. He joined ACS Solution France SAS, Montreal, QC, Canada, where he was responsible for the development of radio frequency identification (RFID) contactless card readers. In 2009, he started his research with Grenoble-INP, where he focused on the design of chipless RFID systems. From 2012 to 2013, he held a post-doctoral position with the Tampere University of Technology, Tampere, Finland, where he was involved in the field of conventional and chipless RFID sensors. Since 2013, he has been an Associate Professor of electrical engineering with the Institut d’Electronique et Systèmes, Université de Montpellier, Montpellier, France. His current research interests include wireless sensors, RFID systems, and printed electronics. Etienne Perret (S’02–M’06–SM’13) received the Dipl.-Ing. degree in electrical engineering from the Ecole Nationale Supérieure d’Electronique, d’Electrotechnique, d’Informatique, d’Hydraulique, et des Télécommunications, Toulouse, France, in 2002, and the M.Sc. and Ph.D. degrees from the Toulouse Institute of Technology, Toulouse, France, in 2002 and 2005, respectively, both in electrical engineering. He held a post-doctoral position with the Institute of Fundamental Electronics, Orsay, France, from 2005 to 2006. Since 2006, he has been an Associate Professor of Electrical Engineering with the Grenoble Institute of Technology, Valence, France. In 2014, he was elevated to Junior Member of the Institut Universitaire de France, Paris, France, an institution that distinguishes professors for their research excellence, as evidenced by their international recognition. He has authored or co-authored over 130 technical conferences, letters and journal papers, and books. His research activities cover the electromagnetic modeling of passive devices for millimeter- and submillimeter-wave applications. His current research interests include wireless communications, especially radio frequency identification (RFID) and chipless RFID, and advanced computer aided design techniques based on the development of an automated co-design synthesis computational approach. Dr. Perret is a Technical Program Committee Member of the IEEE International Conference on RFID. He was a recipient of the French’s Innovative Techniques for the Environment Award in 2013. He was a Keynote Speaker and the Chairman of several international symposiums. He was named one of the MIT Technology Review’s French Innovators under 35 in 2013 for his work on chipless RFID.

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Darine Kaddour was born in Mechmech, Lebanon, in 1982. She received the B.S. degree in physics from the Faculty of Sciences, Lebanese University, Tripoli, Lebanon, in 2003, and the M.S. and Ph.D. degrees from the Institut National Polytechnique de Grenoble, Grenoble, France, in 2004 and 2007, respectively. She has been an Associate Professor with the Grenoble Institute of Technology, Valence, France, since 2009. Her current research interests include microwave circuits and radio frequency identification antennas.

Thierry Baron is currently the Director of Research with the Centre National de la Recherche Scientifique, Paris, France. He is the Head of the Laboratoire des Technologies de la Microelectronique, Centre National de la Recherche Scientifique–LETI, Grenoble, France. He is involved in the field of nanomaterials elaboration and their integration in Si CMOS devices. He has authored or co-authored over 150 scientific publications in peer-reviewed journals.

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Reliable Orientation Estimation of Vehicles in High-Resolution Radar Images Fabian Roos, Student Member, IEEE, Dominik Kellner, Jürgen Dickmann, and Christian Waldschmidt, Senior Member, IEEE

Abstract— With new generations of high-resolution imaging radars, the orientation of vehicles can be estimated without temporal filtering. This enables time-critical systems to respond even faster. Based on a large data set, this paper compares three generic algorithms for the orientation estimation of a vehicle. An experimental MIMO imaging radar is used to highlight the requirements of a robust algorithm. The well-known orientated bounding box and the so-called L-fit are adapted for radar measurements and compared with a brute-force approach. A quality function selects the best fitted model and is a key factor to minimize alignment errors. Moreover, the reliability of the estimation is evaluated with respect to the aspect angle, the distance to the target, and the number of sensors. An approach to estimate the reliability of the current orientation estimation is introduced. It is shown that the root mean square error of the orientation estimation is 9.77° and 38% smaller compared with the common algorithm. In 50% of the evaluated measurements the orientation estimation error is smaller than 3.73°. Index Terms— Automotive radar, bounding box estimation, dimension extraction, Doppler radar, MIMO radar, orientation estimation, radar imaging, radar signal processing, reliability.

I. I NTRODUCTION

C

AR manufacturers offer driver assistance and safety systems to prevent accidents and to enhance the comfort of driving. Especially radar sensors are commonly used for environment perception, because unlike the camera, they can operate under severe weather conditions. To operate such systems in urban scenarios a more precise position, orientation, and dimension estimation is required compared with highways. This is due to a more dynamic environment with many different obstacles and fast changing situations. The driving path of approaching vehicles, e.g., from side roads, needs to be determined as fast as possible for real-time decision making. With the availability of high-resolution image radars, the possibility to estimate the orientation of vehicles arises as shown in [1]. Reasonable boundary conditions for automotive

Manuscript received August 14, 2015; revised December 13, 2015 and June 23, 2016; accepted June 24, 2016. Date of publication July 21, 2016; date of current version September 1, 2016. F. Roos and C. Waldschmidt are with the Institute of Microwave Engineering, Ulm University, 89081 Ulm, Germany (e-mail: [email protected]; [email protected]). D. Kellner is with the Institute of Measurement, Control and Microtechnology, Ulm University, 89081 Ulm, Germany (e-mail: dominik.kellner@ uni-ulm.de). J. Dickmann is with the Group Research and Advanced Engineering, Daimler AG, 89081 Ulm, Germany (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.2016.2586476

radar sensors to estimate the contour and thus the orientation of vehicles are investigated in [2]. A bandwidth larger than 1 GHz and an azimuth resolution better than 1° should be used. In this paper, we show that a bandwidth of only 500 MHz is sufficient for orientation estimation. To estimate the orientation of a target vehicle, different approaches, such as the Doppler distribution, box models, orthogonal line pairs, or radar response models, can be used and are explained in the following. Only algorithms using single measurements are considered, because the estimated orientation is used as input for a tracking not covered by this paper. This reduces the latency in dynamic maneuvers. In [3], the Doppler distribution of the target vehicle is used to set up the velocity vector and to determine the orientation if a linear motion is present, e.g., the vehicle travels along a straight line. However, turning vehicles in urban scenarios are a key factor and are not covered by the approach. The algorithms used in this paper are inspired from approaches utilized for laser scanners. The (orientated) bounding box (OBB) approach is often selected, which is based upon the determination of the convex hull. A cost-effective algorithm to find the minimal area box is the so-called rotating calipers algorithm presented in [4]. A laser scanner can extract a detailed contour of vehicles, but if only the area of the enclosing rectangle is minimized, the resulting model can be misaligned. Therefore, Kmiotek and Ruichek [5] introduce a symmetry assumption as an enhancement. It is assumed that the visible and the invisible part of the contour are symmetrical, which reduces the alignment error. Often two visible sides of the vehicle are present, which can be modeled with two perpendicular lines. As the two sides are perpendicular and of different lengths, this fitted model is called L-fit. This approach is applied to laser scanner data in [6]. The two orthogonal lines of the L-fit can be found using the Hough transform, which is known from image analysis. The required geometric relations for detection are listed in [7]. Zhao and Thorpe [8] apply them to laser scanner data as well. Using a radar response model as determined in [9] or [10], the different scattering centers of a vehicle are modeled and matched to the detected ones in the radar measurement. These models are determined from stationary targets and typically depend on the used radar sensors and the type of vehicle. However, during motion the varying Doppler velocities of the targets can be exploited to resolve significantly more reflections and to separate the target from close clutter.

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Fig. 1. For a part of the scene the trajectory is shown as ( ) and for six single measurements (a)–(f), the radar reflections are shown as ( ). For each ) and the fitted model using the brute-force approach is shown as ( ). For each measurement the step the actual vehicle orientation is shown as ( corresponding video image is presented. The error of each fit is (a) ϕerr = 0.50°, (b) ϕerr = 4.84°, (c) ϕerr = −1.21°, (d) ϕerr = −0.81°, (e) ϕerr = 29.02°, and (f) ϕerr = 0.49°.

In this paper, in addition to [11], a radar-specific orientation estimation algorithm is proposed, which is applied to measurement data gathered from an experimental MIMO radar sensor. In addition, the dimension estimation of the vehicle is presented together with a study of how the results depend on the number of used sensors and on the range to the target. Concepts to identify the quality of the current estimation are also presented. This paper is organized as follows. In Section II, the experimental setup and the preprocessing of radar data are explained. The requirements for the algorithms are discussed in Section III. This includes the description of how vehicles appear in the radar data. Three different orientation estimation algorithms are presented in detail in Section IV. Afterward the scene is evaluated in Section V. The dependency on the introduced aspect angle, the range, and the number of sensors is shown. The possibility to extract the target vehicle’s dimensions is presented as well as its limitations. In addition, an approach to estimate the reliability of the estimation is introduced, followed by a conclusion in Section VI. II. M EASUREMENT S ETUP AND S IGNAL P REPROCESSING At the front of the vehicle two experimental MIMO radar sensors using the 77-GHz frequency band are mounted at a height of 30–40 cm with an orientation directly toward the driving path as shown in Fig. 2(a). As stated in [12], a number of different antenna types have been used in the past: 1) folded reflectarrays; 2) lens antenna systems; 3) mechanical rotating antennas; and 4) planar patch configurations. The sensor for this paper uses a planar antenna system based on patches in an MIMO configuration with 10 Rx and 2 Tx channels. The placement of the transmit and receive antennas

Fig. 2. Mounting position of the radar sensor in (a), the placement of the antenna elements in (b), and the used chirp-sequence modulation scheme for the two transmit antennas in (c).

is shown in Fig. 2(b). While current available radar sensors mostly use the FMCW modulation technique as mentioned in [13], the transmit antennas emit alternatively a linear chirpsequence modulation with 128 frequency ramps as shown in Fig. 2(c). This results in a time-division multiplexing and

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a nearly doubled aperture length using 256 ramps. The time delay between the two consecutive chirp blocks results in a Doppler-dependent phaseshift that must be known for the angle extraction. One antenna element in the virtual aperture is overlapping and the phaseshift between the two chirp blocks can be determined. The receive antennas are equally spaced at a distance of 0.545λ and are evaluated in parallel with analog-to-digital converters with a resolution of 12 bit. The element distance is chosen as large as possible for a long aperture, but so small that no ambiguities are present within the field of view. As a field of view, ±45° can be evaluated. The transmit antennas consist of 12 patches in the vertical direction and the receive antennas use 18. This leads to an elevation beamwidth of ±3.5°. With a chirp duration Tc of 20.48 μs and a chirp repetition time Tr of 27.015 μs, a pause of roughly 7 μs is required for the return and stabilization of the frequency ramp, which is generated by a voltage controlled oscillator and a fractional N phase-locked loop with a reference frequency of 50 MHz. These timings yield an unambiguous maximal velocity of 36 m/s and a Doppler resolution of 0.28 m/s. Compared with the radar sensors of series-production vehicles [13], the velocity resolution is enhanced by 50% due to the longer observation time. The adjustable bandwidth is set for this measurement to 500 MHz, leading to a range resolution of 0.30 m. The internal low-pass filter limits the maximal distance to 53 m. The radar sensor outputs the time-domain data, which are stored for signal processing. The typical angular resolution of the state-of-the-art sensors is between 3° and 4°, using three to four channels as stated in [12]. With ten receive channels and with the help of linear prediction as presented in [14], the angular resolution is enhanced to 1°. In contrast to the state-of-theart automotive sensors that typically only measure a couple of scattering centers per target vehicle, the experimental MIMO radar sensor can usually register five to fifteen scattering centers. The number of resolvable centers is dependent on the orientation of the target car, the distance to it, and its velocity due to a better separability in the range and velocity. For a peak detection the ordered statistic constant false-alarm rate (OS-CFAR, [15]) algorithm is used. After applying the OSCFAR, there are multiple reflections of stationary and moving objects. To select only those reflection points of the vehicle, the clustering algorithm density based spatial clustering of applications with noise [16] is used. The Doppler velocity is taken into account to identify moving targets. The target vehicle as well as the vehicle with the radar sensors both use an inertial measurement unit with differential GPS assistance to estimate the actual orientation and position to enable a precise error evaluation. III. P ROBLEM F ORMULATION Using radars, it is important to apply a robust approach as shown in Fig. 1. The target vehicle is driving in this part of the scene a figure eight, which is shown with the trajectory ( ). For several measurements, the radar reflections ( ) after applying the signal processing steps are shown together with a video

image and the ground truth rectangle ( ) representing the actual orientated vehicle. The east–north–up coordinate system is used, which means the driving direction of the car with the radar sensors is the x-axis and the y-axis is aligned to the left of it. In contrast to laser scanners, a radar is much more susceptible to detect multipath reflections. The transmitted electromagnetic wave can bounce off the ground and can get reflected from the underbody of a vehicle. This is clearly visible in Fig. 1(a), where the parts of the contour opposite to the viewing direction are detected. If the visible contour is orthogonal to the line of sight of the radar sensor, the reflections may outshine and lead to detections in front of the actual target. This effect can be observed in Fig. 1(e) where the rear part is cluttered. The algorithm used should also cope with the clustering faults, which result in reflection points that do not support the vehicular model. Such measured points are called outliers and in Fig. 1(c) an example is shown. The algorithm should compensate such outliers so that the orientation is not flawed. The gathered contour of the vehicle has, in contrast to that obtained from laser scanners, stronger deviations and is dependent on the incident angle. As multipath reflections are present, the symmetry assumption of [5] cannot be assumed anymore [see Fig. 1(d)]. IV. O RIENTATION E STIMATION A LGORITHM After the clustering, every target consists of several points that are used for the model fitting. The presented algorithms are all iterative ones, so that a quality function is required. Two algorithms are taken from the literature and are enhanced to fulfill the requirements introduced in Section III. For comparison, a brute-force algorithm is shown, which also uses the quality function and should indicate the best possible outcome. A. Two Perpendicular Lines (L-fit) The idea to use two perpendicular lines, as presented in [6], must be adapted before it can be applied to radar data. The obtained reflections are cluttered and, as mentioned in Section III, measured points are also present inside the vehicle. In [6], the assignment of reflection points to one of the two lines is done using a corner candidate calculation. This approach cannot be used with radar data due to the fact that an assignment of points to one of the two lines is not trivial. There are points inside the vehicle that originate from neither side. The proposed algorithm picks three points at random in each iteration. The first and the second points are assigned to the first line, the second and the third belong to the perpendicular line. With this definition, two perpendicular lines are calculated using linear regression as presented in [6]. For each iteration, the current fit is evaluated using a quality function as presented in Section IV-D. Fig. 3 shows two possible iterations, where Fig. 3(b) shows the final accepted model. The three picked reflection points ( ) are connected as ( ) and are in most cases not orthogonal. Calculating the perpendicular fit results in the model ( ).

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Fig. 3. Two possible iterations of the L-fit algorithm. Three points are randomly chosen ( ) and connected ( ). The resulting perpendicular lines are expanded as ( ).

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Fig. 5. Rotating the main direction in (a) and different resulting rectangles in (b). Figure taken from [11].

Fig. 4. Procedure of the enhanced orientated bounding box algorithm (EOBB) taken from [11].

B. Enhanced Orientated Bounding Box (EOBB) The fundamental principle of this algorithm, which is introduced in [11], is the rotating calipers approach in [4]. After determining the convex hull the basic rectangle is created using the extreme values of the convex hull. In each iteration step, the rectangle is rotated and aligned with one convex hull side (see iteration loop 1 in Fig. 4). The number of convex hull points is directly linked to the number of iterations. To cope with the outlier points as shown in Fig. 1(c), in further iterations every hull point will be ignored and the rectangles are rotated and evaluated once again (see iteration loop 2 in Fig. 4). Ignoring only one outlier point is not sufficient as outlier points commonly appear in groups, and ignoring a single point often does not alter the convex hull essentially. C. Brute-Force Approach as Best Case Scenario (MainDir) The possible rotations of the rectangle of the enhanced OBB (EOBB) algorithm is determined by the convex hull. Although one hull point is ignored, the best orientation may not be found. To allow a fine angular rotation step size the following brute-force approach is applied, which obviously is computationally expensive. See [11] for a detailed description. In the radar reflections, a strong distinctive contour is visible, which is the main direction and is detected using the random sample consensus algorithm in [17]. This main direction can be rotated in a small range to enhance the step size as shown in Fig. 5(a). For each main direction, the maximal enclosing rectangle is set up and shrunk step by step. To select the best model, the quality function of Section IV-D is applied.

Fig. 6.

Corridors are used to evaluate the current L-fit.

D. Evaluating Each Fit: The Quality Function By picking randomly three points using the proposed L-fit algorithm, a misaligned line pair as shown in Fig. 3(a) is down-weighted using a quality function. The fitted L should be shifted as far as possible toward the radar sensor position in order to ensure an alignment on the contour points. This is done using a corridor weighting function as shown in Fig. 6. Points in the inner corridor support the current model, in contrast to points between the inner and the outer corridors. Reflections outside of the outer corridor are probably from the other line or inside the vehicle and should only result in a constant error. The EOBB and the brute-force algorithm use the same quality function to rate each iteratively calculated model. An error value needs to be minimized, which is further called an optimization variable. The optimization variable consists of several important factors instead of just using the area of the rectangle as in [5]. First of all, the area of the box model should be as small as possible since a misaligned bounding box typically consumes more space. Clustered points that lie on the contour of the

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TABLE I

results. Compared with the EOBB, the rms error drops again by 19% and the mae by 12%.

A LGORITHM C OMPARISON

A. Dependency of Aspect Angle Accurate results can be achieved if two contour sides of the vehicle are visible as they increase the justification of the model. This dependency is shown with the introduced aspect angle as ϕasp := ϕGT − ϕaz bounding box support the current chosen model as can be seen in Fig. 1. Hence, each point is associated to a correspondent side as described in [11]. The sum of the distances from each point to the respective side is the second criteria. Points inside the rectangle are considered to support the current model due to possible multipath reflections and are therefore called inliers. Points outside the bounding box do not confirm the model and are rated as outliers. The third important factor is the ratio between the number of outliers and inliers. To summarize, the resulting area and the distances to the corresponding sides need to be minimized, and as many points as possible should be inside the box. V. E XPERIMENTAL R ESULTS Target ranges from 12.7 to 40.2 m are analyzed in a scene with 4600 single measurements. In contrast to the scene presented in [11], the target vehicle is nearly at the edge of the field of view. It is shown in Section V-B that the distance to the target has a significant influence and thus the overall results are inferior. To cover all the possible orientations, the target vehicle is driving in circles and figure eights. The vehicle orientation is estimated with every algorithm and is named as ϕest , while the actual orientation ϕGT is labeled after the ground truth. The error of the estimation is defined as ϕerr := ϕest − ϕGT .

(1)

The following two statistical parameters are applied to analyze the error: 1) the root mean squarer (rms) error and 2) the median absolute error: mae = median {abs (ϕerr )} .

(2)

The statistical parameters are listed in Table I. The wellknown OBB is sensitive to outliers since no compensation is available, which increases the rms error. Outlier points can be compensated by introducing a quality function (OBB + QF) and both the mae and the rms error go down significantly by 47% and 30%, respectively. Applying the EOBB approach, the implemented outlier treatment reduces the rms error again by 11% and the mae by 10%. The adapted L-fit is by 10% better in the rms error, but the mae is higher by 7% compared with the EOBB. The L-fit has less outliers as can be seen in the smaller rms error compared with the EOBB, because it is like the MainDir, not limited in the angular rotation steps. In exchange, the mae is higher due to the fact that the L-fit is challenged by the front views of the vehicle. The brute-force algorithm (MainDir) is not restricted to the rotation steps by the convex hull and therefore leads to optimal

(3)

with the azimuth angle ϕaz to the target vehicle. For ϕasp = 0°, only the short contour of the rear view [see Fig. 1(b)] is visible. The radar reflections from multipath propagation do not provide the whole vehicle’s dimensions, hence gaining a valid fit is challenging. This also holds for the front view with ϕasp = 180°, where the backscatter area is also smaller due to the front view of the vehicle. For the long side [see Fig. 1(f)] with ϕasp = 90°, the errors are smaller compared with the short side. With one large side visible, a stable contour alignment is provided. The algorithms yield optimal results if two contours are detected while the car is oncoming [see Fig. 1(a)] with ϕasp = 135° or departing [see Fig. 1(c) and (d)] with ϕasp = 45°. In Fig. 7, the dependency of the aspect angle for the different algorithms is shown. It can be clearly seen that for the front and rear views, the errors are higher than for the long side and the L shape, respectively. B. Dependency of Range The radar sensor has an azimuth resolution of 1° after applying the linear prediction. This leads to a specific crossrange at a given distance. The farther away a target is, the larger the crossrange grows. A target at the edge of the field of the view has therefore less reflections than a nearby target. This has an influence on the orientation estimation as shown in Fig. 8. If less reflections are available, the estimation gets less reliable, which can be seen in the increasing rms error. C. Dependency of Number of Sensors For target detection, two radar sensors are used as mentioned in Section II. The lateral distance between the two sensors leads to different reflection centers due to the different aspect angles. This leads for some measurements to two separate strong reflection centers on the contour, which enhances the model fitting as shown in Fig. 9. Using only one sensor, both the rms error and the mae increase by 17% for the EOBB. Particularly, the L-fit requires a second strong reflection center, which is why the rms error increases by 32% and the mae by 21%. D. Dimension Extraction The presented algorithms allow the extraction of the dimensions of the target vehicle. The part of the scene shown in Fig. 1 is evaluated with respect to the box dimensions Fig. 10. The target vehicle, a Mercedes-Benz E-class T model, has a length of 4.905 m and a width of 1.854 m. In the

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Fig. 9. Radar reflections for one measurement using both the sensors in (a) and only one sensor in (b). In this measurement, the long side is visible and the algorithm benefits from the second reflection center.

Fig. 7. Function of the rms error (a) and the mae (b) from the aspect angle.

Fig. 10. Extracted dimensions of the target vehicle for the part of the scene shown in Fig. 1. For comparison, the actual dimensions are shown. The marked positions correspond to the measurements shown in Fig. 1.

visible, the box dimensions agree with the actual values [see Fig. 10(a), (c), (d), and (f)]. If the rear or front view is visible at least the length extraction is distorted [see Fig. 10(b) and (e)]. E. Reliability of the Current Fit Fig. 8.

Function of the rms error and the mae from the range to the target.

presented scene, the mean values are 4.86 and 2.06 m, respectively. This results in an rms error of 0.76 and 0.48 m, respectively. If both the sides or the long side of the target is

Once the model for the actual measurement is found, the question of the reliability of the actual fit arises. The approach to use the dimensions does not provide a valid rating due to the fact that, if only the rear or the front is visible, the error can be small as well, as can be seen in the example in Fig. 1(b). If the radar reflections are unambiguous as in Fig. 11(a), the optimization variable, which results from applying the quality function for each discrete angle, should only have one distinct

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VI. C ONCLUSION For the orientation estimation of target vehicles in high-resolution radar images, three different algorithms are presented using single measurements. Two state-of-the-art approaches of the OBB and the L-fit from the literature are adapted. The orientation error is less than 3.73° in 50% of the evaluated measurements applying the EOBB. An rms error of 9.77° is achieved, which is a reduction by 38% compared with the well-known OBB algorithm. The improved outcome is due to the consideration of possible outliers in the radar data. The precise target vehicle dimensions are extracted with an rms error of 0.76 and 0.48 m, respectively. The results deteriorate by 20% to 30% if only one sensor is used instead of two sensors. Evaluating the trend of the optimization variable, resulting from the introduced quality function, enables an evaluation of the reliability of the orientation estimation. Using this reliability value, the orientation estimation can be enhanced by an integration in a temporal filter in the future work. R EFERENCES

Fig. 11. Fitted models in radar reflections in (a) and (b), and the corresponding evaluation of the optimization variable resulting from the introduced quality function in (c) and (d). The extracted main direction is at 0° and is rotated.

minimum. In Fig. 11(c), the determined main direction of the brute-force approach is at 0°, which gets rotated. Afterward, the rotation with the minimal value is selected. In this example the trend is unambiguous. The radar reflections in Fig. 11(b) are cluttered and hence the trend in Fig. 11(d) has two minima indicating the chosen fit might not be ideal. The proposed approach is to evaluate the trend of the optimization variable. If there are several local minima or the global minimum is not distinctive, this indicates a nonreliable fit.

[1] M. Andres, P. Feil, W. Menzel, H.-L. Bloecher, and J. Dickmann, “3D detection of automobile scattering centers using UWB radar sensors at 24/77 GHz,” IEEE Aerosp. Electron. Syst. Mag., vol. 28, no. 3, pp. 20–25, Mar. 2013. [2] H.-L. Blöecher, M. Andres, C. Fischer, A. Sailer, M. Goppelt, and J. Dickmann, “Impact of system parameter selection on radar sensor performance in automotive applications,” Adv. Radio Sci., vol. 10, pp. 33–37, Sep. 2012. [3] D. Kellner, M. Barjenbruch, K. Dietmayer, J. Klappstein, and J. Dickmann, “Instantaneous lateral velocity estimation of a vehicle using doppler radar,” in Proc. 16th Int. Conf. Inf. Fusion, Jul. 2013, pp. 877–884. [4] G. Toussaint, “Solving geometric problems with the rotating calipers,” in Proc. Medit. Electrotech. Conf., May 1983, pp. 1–8. [5] P. Kmiotek and Y. Ruichek, “Representing and tracking of dynamics objects using oriented bounding box and extended Kalman filter,” in Proc. 11th Int. IEEE Conf. Intell. Transp. Syst., Oct. 2008, pp. 322–328. [6] H. G. Jung, Y. H. Cho, P. J. Yoon, and J. Kim, “Scanning laser radarbased target position designation for parking aid system,” IEEE Intell. Transp. Syst., vol. 9, no. 3, pp. 406–424, Sep. 2008. [7] C. R. Jung and R. Schramm, “Rectangle detection based on a windowed Hough transform,” in Proc. 17th Brazilian Symp. Comput. Graph. Image Process., Oct. 2004, pp. 113–120. [8] L. Zhao and C. Thorpe, “Qualitative and quantitative car tracking from a range image sequence,” in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., Jun. 1998, pp. 496–501. [9] K. Schuler, D. Becker, and W. Wiesbeck, “Extraction of virtual scattering centers of vehicles by ray-tracing simulations,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3543–3551, Nov. 2008. [10] M. Bühren and B. Yang, “Automotive radar target list simulation based on reflection center representation of objects,” in Proc. Workshop Intell. Transp. (WIT), Hamburg, Germany, Mar. 2006, pp. 161–166. [11] F. Roos et al., “Estimation of the orientation of vehicles in highresolution radar images,” in Proc. IEEE MTT-S Int. Conf. Microw. Intell. Mobility (ICMIM), Apr. 2015, pp. 1–4. [12] W. Menzel and A. Moebius, “Antenna concepts for millimeter-wave automotive radar sensors,” Proc. IEEE, vol. 100, no. 7, pp. 2372–2379, Jul. 2012. [13] J. Hasch, E. Topak, R. Schnabel, T. Zwick, R. Weigel, and C. Waldschmidt, “Millimeter-wave technology for automotive radar sensors in the 77 GHz frequency band,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 3, pp. 845–860, Mar. 2012. [14] C. Fischer, M. Andres, H.-L. Bloecher, J. Dickmann, and W. Menzel, “Adaptive super-resolution with a synthetic aperture antenna,” in Proc. 9th Eur. Radar Conf. (EuRAD), Oct./Nov. 2012, pp. 250–253. [15] H. Rohling, “Radar CFAR thresholding in clutter and multiple target situations,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 4, pp. 608–621, Jul. 1983.

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[16] M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for discovering clusters in large spatial databases with noise,” in Proc. 2nd Int. Conf. Knowl. Discovery Data Mining, 1996, pp. 226–231. [17] M. A. Fischler and R. Bolles, “Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM, vol. 24, no. 6, pp. 381–395, 1981.

Fabian Roos (S’15) received the M.Sc. degree from the Karlsruhe Institute of Technology, Karlsruhe, Germany, in 2014. He is currently pursuing the Ph.D. degree with the Institute of Microwave Engineering, Ulm University, Ulm, Germany. He is a Research Assistant with the Institute of Microwave Engineering, Ulm University. He is currently involved in adaptivity for chirp sequence radar. His current research interests include automotive radar signal processing, especially for chirp sequence radar.

Dominik Kellner was born in Munich, Germany, in 1985. He received the Dipl.-Ing. degree (equivalent to the M.Sc.) in mechatronics and information technology from the Technical University of Munich, Munich, Germany, in 2010. He is currently pursuing the Ph.D. degree at Ulm University, Ulm, Germany. His current research interests include automotive radar and environmental perception with a focus on motion state estimation of extended objects, radar alignment, and ego motion state estimation using high-resolution Doppler radar.

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Jürgen Dickmann received the Diploma degree in electrical engineering from the University of Duisburg–Essen, Essen, Germany, in 1984, and the Dr.-Ing. degree from RWTH Aachen University, Aachen, Germany, in 1991. He started his career with the AEG Research Center, Ulm University, Ulm, Germany, in 1986, where he conducted research on III/V processing techniques, millimeter-wave devices, and millimeterMMICs. From 2005 to 2009, he was in charge of teams developing sensor technologies, sensor fusion, and situation analysis concepts. In 2015, he was also in charge of the transfer into series for the complete environmental perception, including all sensors and navigation/localization components for the next S-Class. He is a Manager of Active Sensors with the Group Research and Advanced Engineering, Daimler AG, Ulm, Germany. He is responsible for the development of active sensors (radar and scanner) and environmental perception in driver assistanceand active safety systems. His current focus is on sensor-technology, such as radar, lidar, and laser scanner technology for highly automated functions/selfdriving cars. Dr. Dickmann was the recipient of an award from VDE/ITG in 1992. Christian Waldschmidt (S’01–M’05–SM’13) received the Dipl.-Ing. (M.S.E.E.) and the Dr.-Ing. (Ph.D.E.E.) degrees from the University Karlsruhe (TH), Karlsruhe, Germany, in 2001 and 2004, respectively. From 2001 to 2004, he was a Research Assistant at the Institut für H¨ochstfrequenztechnik and Elektronik (IHE), Universit¨a t Karlsruhe (TH). Since 2004 he has been with Robert Bosch GmbH, in the business units Corporate Research and Chassis Systems. He was heading different research and development teams in high-frequency engineering, EMC, and automotive radar. His research topics include integrated radar sensors, radar system design, millimeter-wave technologies, antennas and antenna arrays, UWB, and EMC. In 2013 he returned to academia. He was appointed as the Director of the Institute of Microwave Engineering with Ulm University, Ulm, Germany. The research topics focus on radar and RF sensing, mm-wave and submillimeter-wave engineering, antennas and antenna arrays, MIMO, array signal processing, modulation techniques, and wave propagation. He has authored or co-authored over 100 scientific publications and holds over 20 patents. Prof. Waldschmidt is the Vice Chair of the IEEE MTT-27 Technical Committee (wireless enabled automotive and vehicular applications), the Executive Committee Board Member of the German MTT/AP joint chapter, and a Member of the ITG Committee Microwave Engineering (VDE). He is a Reviewer for multiple IEEE T RANSACTIONS and L ETTERS . In 2015, he served as a TPC Chair of the IEEE MTT-S International Conference on Microwaves for Intelligent Mobility and an Associate Guest Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES .

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 9, SEPTEMBER 2016

Self-Calibration of a 3-D-Digital Beamforming Radar System for Automotive Applications With Installation Behind Automotive Covers Marlene Harter, Jürgen Hildebrandt, Andreas Ziroff, Member, IEEE, and Thomas Zwick, Senior Member, IEEE

Abstract— Given the complexity of a radar system with digital beamforming, a number of error influences can appear due to hardware imperfections, temperature drift, and aging. These errors can result in a reduced target detection probability or even lead to false detections. For automotive driver assistance applications, there is a growing demand for hidden integration of radar systems. When installed behind a painted bumper or design radome, additional performance degradation may occur. First, the hardware imperfections according to their appearance in the presented radar system and their effects are analyzed with respect to the angular spectrum estimation. A software-based self-calibration algorithm is proposed and evaluated by means of simulation and measurement. Finally, the extended performance capability of the self-calibration procedure to compensate additional error components from automotive covers is investigated by means of a painted bumper in front of the radar sensor. Index Terms— Antenna arrays, automotive applications, calibration, microwave sensors, MIMO radar.

I. I NTRODUCTION

T

ODAY, there is a growing demand for hidden integration of automotive radar systems for driver assistance applications [1]. When installed behind painted bumpers or designoriented radomes, the shape, material, and coating properties of such covers may lead to a significant radar performance degradation. Therefore, a tradeoff between electromagnetic and car body design constraints has to be found, or new compensation methods are required [2], [3]. Given the complexity of a radar system with digital beamforming (DBF) itself, a number of error influences can appear due to hardware imperfections, temperature drift, and aging. These errors can result in a reduced target detection probability or even lead to false detections. For the compensation of the

Manuscript received August 14, 2015; revised January 10, 2016 and June 27, 2016; accepted July 4, 2016. Date of publication August 8, 2016; date of current version September 1, 2016. M. Harter was with the Institut für Hochfrequenztechnik und Elektronik, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany. She is now with Robert Bosch GmbH, Leonberg 71229, Germany (e-mail: [email protected]). J. Hildebrandt is with Robert Bosch GmbH, Leonberg 71229, Germany (e-mail: [email protected]). A. Ziroff is with Siemens AG, Munich 81739, Germany (e-mail: [email protected]). T. Zwick is with the Institut für Hochfrequenztechnik und Elektronik, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany (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.2016.2593731

Fig. 1.

Photograph of the 3-D-DBF radar system [8].

hardware tolerances, a radar sensor is, in general, initially calibrated at the production site. In the literature, there is a distinction between calibration methods with calibration targets at known and at unknown positions [4]. For the so-called off-line methods, the calibration data are determined from a number of known reference positions [5], [6]. Due to their high effort, such methods are only used for an initial calibration. In contrast to that, a self-calibration with a target at unknown position would allow an initial calibration as well as an automatic recalibration during the radar sensor’s lifetime, so that even the errors due to the sensor’s covers as well as temperature drift and aging effects could be compensated. In this paper, an analysis of the relevant errors affecting the angle determination according to the briefly introduced DBF radar system with 3-D imaging capability [7], [8] is given first. In Section III, the errors affecting the DBF-based angular spectrum estimation are discussed and analyzed. Afterward, a software-based self-calibration method is introduced in Section IV, which allows the acquisition of the calibration data of a target at unknown position [9]. For the first demonstration, the proposed self-calibration is applied to a measurement of trihedrals with the 3-D-DBF radar system in Section V. Finally, in Section VI, the extended performance of the selfcalibration procedure is evaluated by the second measurement, in which the 3-D-DBF radar system is covered with a silvermetallic painted bumper. II. 3-D-DBF R ADAR S YSTEM The 3-D-DBF radar is a 24-GHz frequency-modulated continuous wave DBF radar system, based on eight transmit (Tx) and eight receive (Rx) channels, which is shown in Fig. 1. It combines the classical target range and velocity measurement with the additional capability of angle measurement

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HARTER et al.: SELF-CALIBRATION OF 3-D-DBF RADAR SYSTEM

Fig. 2. Antenna arrangement of the inverted T-shaped antenna array [7], [8].

in 2-D. Due to an orthogonal arrangement of the Tx and Rx antenna arrays in the form of an inverted T (see Fig. 2), in the following called T-array, the elevation and azimuth information can be provided. In this radar system, the transmitters are operated in sequential mode, whereas the receivers allow the parallel reception of the reflected signal. The Tx and Rx antenna arrays consist of patch antennas, which have a distance of dTx = 14.2 mm and dRx = 14.5 mm, respectively. More detailed information about the 3-D-DBF radar system can be found in [7] and [8]. III. I NFLUENCES OF E RRORS Amplitude and phase differences between the antennas within the antenna array or between channels cause an erroneous array factor and consequently an erroneous angular spectrum. They can result in a beam pointing error and broadening of the main beam, a rise in the sidelobes, or the deviation of the gain of the antenna array [10]. According to [11], the errors due to component tolerances can be assumed to be random and uncorrelated. Systematic errors, which lead to correlated distortions in the signal paths, have to be corrected in advance. Under these conditions, the error influences in the antenna array factor can be described by statistical methods. Within the signal model, the amplitude and phase errors for each channel are described by a diagonal matrix with a respective error coefficient An e j ϕn = (1 + δn )e j ϕn of channel n out of n = 1, . . . , N channels. Thereby, the amplitude factor An contains the amplitude error δn , whereas ϕn describes the phase error. For simplification, the amplitudes of all the channels are normalized to one. The beam pointing error is caused by phase errors and leads to an erroneous angle determination. The standard deviation σψ of the beam pointing error for a linear antenna array with an equidistant antenna spacing of d can be estimated by    N/2 2 kdn) (w n n=−N/2 (1) σψ = σϕ  N/2 2d 2n2 w k n n=−N/2 according to [12], where wn describes the amplitude weighting of the nth antenna element, k is the wave number, and σϕ is the standard deviation of the phase error. From (1), the standard deviation of the beam pointing error can be calculated to

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Fig. 3. Results of the performed Monte-Carlo simulations for evaluation of (a) the sidelobe level caused by amplitude and phase errors and (b) the influence of coupling within the antenna array.

σψ = 0.49° for an antenna array with eight antennas in a distance of d = 14.5 mm. Thereby, a standard deviation of the phase error of σϕ = 20° was assumed, which corresponds to the analysis of the realized 3-D-DBF radar system described in [7]. For amplitude weighting, a Dolph–Chebyshev window with 30-dB sidelobe suppression was chosen. In order to verify this calculated value, a Monte-Carlo simulation with 100 000 trials was performed, from which a standard deviation of σψ = 0.48° could be determined. Thus, (1) is well suited for an estimation of the resulting beam pointing error while involving less computation effort. From this analysis, it could be shown that the relatively high phase errors, which are present in the realized system, do only result in comparatively small beam pointing errors. Amplitude and phase errors can cause increased and irregular sidelobes, which may lead to false detections. In order to quantitatively estimate the influence of such errors on the sidelobes, another Monte-Carlo simulation was performed. The same antenna array parameters as before were used and the effect on the maximum and mean sidelobe deviation was evaluated. According to the measured amplitude and phase errors of the real 3-D-DBF radar, the standard deviations of the amplitude and phase errors were assumed to be σδ = 2.5 dB and σϕ = 20°, respectively. The result of the Monte-Carlo simulation with 100 000 trials is shown by the angular spectrum in Fig. 3(a). According to Fig. 3(a), a maximum sidelobe level of approximately −7 dB and a mean sidelobe level of −18 dB can be determined for an azimuth angle of ψ = 73°. In addition, in this Monte-Carlo simulation, a Dolph–Chebyshev window with 30-dB sidelobe suppression was applied before angular processing. Due to the small antenna distances, coupling effects occur between the antennas. According to [13], coupling combines several electromagnetic mechanisms, such as direct electromagnetic radiation, indirect coupling caused by nearfield scattering, and coupling inside the distribution network. All these effects could lead to degraded antenna patterns and altered antenna impedances. The strength of the coupling is dependent on the antenna type and the antenna spacing used. The coupling between the directly adjacent patch antennas within the Tx and Rx antenna arrays of the 3-D-DBF radar

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was determined to be less than −32 dB, whereas the coupling of nondirectly neighboring antennas is significantly lower. In order to characterize the influences of the coupling on the antenna array, a Monte-Carlo simulation was performed. Therefore, a fully occupied matrix was used, with ones on the main diagonal and the coupling coefficients on the other diagonals. The amplitudes and the phases of the coupling coefficients were chosen to be normally distributed with a standard deviation of 3 dB for the amplitudes and 10° for the phases. The mean values were chosen in the order of the performed coupling simulations and measurements. According to the angular spectrum in Fig. 3(b), only a slight deviation and increase in the sidelobes can be observed in comparison to the ideal array factor, whereas no beam pointing error occurs. Compared with the influences of the amplitude and phase errors in the 3-D-DBF radar, the antenna coupling can be regarded as negligible. IV. S ELF -C ALIBRATION P ROCEDURE According to the performed analysis of the realized 3-D-DBF radar system in Section III, the amplitude and phase differences of the Tx and Rx channels lead to increased sidelobes and misalignment of the main lobe. In the following, the steps of the self-calibration procedure are presented for the T-array with linear equally spaced antennas and uniform amplitudes under far-field conditions [9]. During a measurement cycle, in which the transmitters are switched sequentially, the measurement scenario is considered to be static. 1) After the reception of the signals, the measurement data of each transmitter and receiver combination are range processed with a fast Fourier transform. 2) In each range spectrum, the complex data x˜mn of the unknown calibration target are extracted by means of a maximum search with respect to each combination of m = 1, . . . , M transmitter and n = 1, . . . , N receiver channels. Calculating the mean over all signal amplitudes |x| ¯ =

M N 1  |x˜mn | MN

(2)

m=1 n=1

allows to estimate the erroneous amplitude gain of each transmitter and receiver combination by ¯ Aˆ mn = |x˜mn |/|x|.

(3)

3) Assuming a linear antenna array with N equidistant antennas under far-field conditions, the phases of the incident wave behave linearly or constantly in the case of a frontal incident wave. This information can be used in order to estimate the error-free phases by the method of least squares, which is applied as follows: min a,b

N 

( f (x n ; a, b) − arg(x˜n ))2 .

(4)

n=1

As a best fit model function, a straight line f (x; a, b) = a · x + b with x = n is chosen. By minimizing the

TABLE I P OSITIONS AND RCS VALUES OF THE T HREE T RIHEDRALS

error squares, the unknown parameters a and b can be estimated. The phase errors can be determined via ϕˆn = φˇ n − arg(x˜n )

(5)

where φˇ n describes the estimated phase by the least squares method of antenna n. For the presented T-array, the phase errors of the receivers with respect to each transmitter are identical as well as those of the transmitters with respect to each receiver are identical. Thus, the phase corrections of the transmitters and the receivers can be considered independently. That means, the phase errors of the transmitters can just be calculated similar to (5) by ϕˆ m = φˇ m − arg(x˜m )

(6)

where the ideal phases φˇm are estimated by the least squares regression like in (4). 4) Finally, the measured data are corrected with the phase errors ϕˆm for the transmitters and ϕˆn for the receivers and with the estimated amplitudes Aˆ mn . The proposed calibration algorithm was evaluated by a Monte-Carlo simulation with 100 000 trials by using an antenna array consisting of eight antennas with a spacing of d = 14.5 mm. The amplitude and phase errors were both chosen normally distributed with a standard deviation of σδ = 2.5 dB and σφ = 20°. With this simulation, it could be determined that the increased sidelobes could be fully corrected by means of the calibration method and the standard deviation of the resulting beam pointing error was reduced to σψ = 0.43°. V. V ERIFICATION OF S ELF -C ALIBRATION BY M EASUREMENTS The applicability of the presented self-calibration procedure for amplitude and phase correction was verified by measurements with the 3-D-DBF radar system [9]. Therefore, the measurement setup shown in Fig. 4 with three trihedrals at different positions is chosen. The measurement data were acquired with the 3-D-DBF radar system placed on a table at a height of 1 m above ground. In Table I, the radar cross sections (RCSs) of the trihedrals as well as their positions are given relative to the radar system. Usually, the positions of the targets used for self-calibration are unknown; however, here they serve as a reference. In the first step, the three trihedrals are detected in the range processed data by means of a maximum search. According to Section IV, the amplitude errors can be determined for all the transmitter and receiver combinations regarding each trihedral.

HARTER et al.: SELF-CALIBRATION OF 3-D-DBF RADAR SYSTEM

Fig. 4.

Photograph of the measurement setup with three trihedrals.

Fig. 5.

Mean amplitude errors estimated by the three trihedrals.

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Fig. 7. Measured slices of the uncalibrated and the calibrated 2-D angular spectra of the trihedrals. (a) Trihedral 1 uncalibrated. (b) Trihedral 1 calibrated. (c) Trihedral 2 uncalibrated. (d) Trihedral 2 calibrated. (e) Trihedral 3 uncalibrated. (f) Trihedral 3 calibrated. TABLE II M EASURED U NCALIBRATED AND C ALIBRATED A NGULAR P OSITIONS OF T RIHEDRALS Fig. 6. Estimated phase errors of the transmitter and receiver channels and their calculated mean value. (a) Transmitter. (b) Receiver.

The measured amplitude errors estimated from the mean of the three trihedrals are shown in the color-coded plot in Fig. 5. Comparing the transmitters, a reduced Tx power for transmitters Tx 1, Tx 2, and Tx 5 can be observed. Furthermore, the calculated phase errors are plotted in Fig. 6(a) for the transmitters and in Fig. 6(b) for the receivers. The estimated phase errors of the receivers are almost identical for each trihedral. Some of the transmitters show a slight phase error variation dependent on the measured trihedral. In the following, the plotted mean phase errors out of the measured phase errors from the trihedrals are used for calibration. In Fig. 7, the uncalibrated and the calibrated 2-D angular plots can be seen for each trihedral. Prior to angular processing, a Dolph–Chebyshev window with 30-dB sidelobe suppression was applied for amplitude tapering. The increased sidelobes can be extremely reduced by the applied calibration procedure. For trihedrals 1 and 3, similar results are obtained. This is due to the fact that both the trihedrals are located nearly in the same angular direction.

A numerical comparison of the measured angular positions of the trihedrals without and with calibration is given in Table II. It can be seen by comparing the measured angles with the true angles in Table I that the measured azimuth angles can be improved, whereas the measured angles in elevation are not affected by the proposed self-calibration method of the 3-D-DBF radar system. Whether the measured angles are corrected or not by the self-calibration algorithm depends on the size and distribution of the existing phase errors within the antenna array of the respective radar system. For a more detailed investigation, the angular spectra in elevation and azimuth are shown in Fig. 8 for trihedral 3. The shown angular spectra are calibrated either by means of the estimated amplitude and the phase errors of each trihedral

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Fig. 9. Sketch of the measurement scenario with a trihedral and the bumper in front of the 3-D-DBF radar system.

Fig. 8. Measured uncalibrated and different calibrated angular spectra of trihedral 3. (a) Elevation cut. (b) Azimuth cut.

or by the mean amplitude and phase errors of all the three trihedrals. If the calibration is performed by the estimated amplitude and phase errors of trihedral 3 itself, all sidelobes are completely below −30 dB according to the applied window function. By using only the calibration data determined by trihedrals 1 or 2, the sidelobes could be decreased, too. However, in this case, the calibrated sidelobes appear to be more irregular and cannot be suppressed completely down to −30 dB. Therefore, the mean of the estimated calibration data of all the three trihedrals is a good compromise. For a robust calibration, it is recommended to use the mean of the estimated calibration data of several unknown targets in different angular directions for amplitude and phase correction. Due to the relatively small phase errors of the transmitters and receivers in the presented 3-D-DBF radar, the beam error after calibration is