IEEE MTT-V055-I12 (2007-12A). [55, 12A ed.]

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NOVEMBER 2007

VOLUME 55

NUMBER 11

IETMAB

(ISSN 0018-9480)

PAPERS

Linear and Nonlinear Device Modeling Mildly Nonquasi-Static Two-Port Device Model Extraction by Integrating Linearized Large-Signal Vector Measurements . ......... ........ ......... ......... ........ ......... ......... ...... A. Cidronali, C. Accillaro, and G. Manes Behavioral Thermal Modeling for Microwave Power Amplifier Design . ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ..... J. Mazeau, R. Sommet, D. Caban-Chastas, E. Gatard, R. Quéré, and Y. Mancuso Active Circuits, Semiconductor Devices, and Integrated Circuits 3-D Integration of 10-GHz Filter and CMOS Receiver Front-End ....... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ... T. Choi, H. Sharifi, H. H. Sigmarsson, W. J. Chappell, S. Mohammadi, and L. P. B. Katehi Design of Cryogenic SiGe Low-Noise Amplifiers ....... ......... ... ....... ........ .. S. Weinreb, J. C. Bardin, and H. Mani A New Compact Load Network for Doherty Amplifiers Using an Imperfect Quarter-Wave Line ....... ......... ......... .. .. ........ ......... ......... ... H. Park, J. Van, S. Jung, M. Kim, H. Cho, S. Kwon, J. Jeong, K. Lim, C. Park, and Y. Yang Linearization of CMOS Broadband Power Amplifiers Through Combined Multigated Transistors and Capacitance Compensation .. ......... ........ ......... ......... ........ ......... ......... .. C. Lu, A.-V. H. Pham, M. Shaw, and C. Saint Millimeter-Wave and Terahertz Technologies Demonstration of a 311-GHz Fundamental Oscillator Using InP HBT Technology ..... ......... ........ ......... ......... .. .. ..... V. Radisic, D. Sawdai, D. Scott, W. R. Deal, L. Dang, D. Li, J. Chen, A. Fung, L. Samoska, T. Gaier, and R. Lai Field Analysis and Guided Waves Some Properties of Generalized Scattering Matrix Representations for Metallic Waveguides With Periodic Dielectric Loading ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . S. S¸ims¸ek and E. Topuz Power Transfer in a Large Parallel Array of Coupled Dielectric Waveguides ... ......... ......... ........ ......... . J. S. Wei Generalized Impedance Boundary Condition for Conductor Modeling in Surface Integral Equation ... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. Z. G. Qian, W. C. Chew, and R. Suaya Rigorous Mode-Matching Method of Circular to Off-Center Rectangular Side-Coupled Waveguide Junctions for Filter Applications ... ......... ........ ......... ......... ........ ... ....... ......... ........ ......... ......... ..... J. Zheng and M. Yu

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) CAD Algorithms and Numerical Techniques Passivity Enforcement With Relative Error Control ...... ......... ......... ........ ......... . S. Grivet-Talocia and A. Ubolli An Efficient Time-Domain Simulation Method for Multirate RF Nonlinear Circuits .... ... J. F. Oliveira and J. C. Pedro

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Filters and Multiplexers Design of Bandpass Elliptic Filters Employing Inductive Windows and Dielectric Objects ..... ... ...... F. J. Pérez Soler, M. Martínez Mendoza, F. D. Quesada Pereira, D. Cañete Rebenaque, A. Alvarez Melcon, and R. J. Cameron A 25–75-MHz RF MEMS Tunable Filter ......... ..... .... ........ K. Entesari, K. Obeidat, A. R. Brown, and G. M. Rebeiz A Dual-Band Coupled-Line Balun Filter . ......... ........ ......... ......... ........ ......... ....... L. K. Yeung and K.-L. Wu A Microstrip Ultra-Wideband Bandpass Filter With Cascaded Broadband Bandpass and Bandstop Filters ..... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... C.-W. Tang and M.-G. Chen

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Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements A Parallel-Strip Ring Power Divider With High Isolation and Arbitrary Power-Dividing Ratio . ..... L. Chiu and Q. Xue Slow-Wave Line Coupler With Interdigital Capacitor Loading .. .. L. Li, F. Xu, K. Wu, S. Delprat, J. Ho, and M. Chaker A Symmetrical Four-Port Microstrip Coupler for Crossover Application ........ ......... ......... ... Y. Chen and S.-P. Yeo Modified Wilkinson Power Dividers for Millimeter-Wave Integrated Circuits .. ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... .... S. Horst, R. Bairavasubramanian, M. M. Tentzeris, and J. Papapolymerou

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Instrumentation and Measurement Techniques Inverse Synthetic Aperture Secondary Radar Concept for Precise Wireless Positioning ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... .. M. Vossiek, A. Urban, S. Max, and P. Gulden

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Biological, Imaging, and Medical Applications Using a priori Data to Improve the Reconstruction of Small Objects in Microwave Tomography ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... A. Fhager and M. Persson A Real-Time Exposure System for Electrophysiological Recording in Brain Slices ..... ......... ........ ......... ......... .. .. ........ ...... A. Paffi, M. Pellegrino, R. Beccherelli, F. Apollonio, M. Liberti, D. Platano, G. Aicardi, and G. D’Inzeo

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

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Digital Object Identifier 10.1109/TMTT.2007.913518

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007

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A Wideband and Scalable Model of Spiral Inductors Using Space-Mapping Neural Network Yazi Cao and Gaofeng Wang, Senior Member, IEEE

Abstract—A wideband and scalable model of RF CMOS spiral inductors by virtue of a novel space-mapping neural network (SMNN) is presented. A new modified 2- equivalent circuit is used for constructing the SMNN model. This new modeling approach also exploits merits of space-mapping technology. This SMNN model has much enhanced learning and generalization capabilities. In comparison with the conventional neural network and the original 2- model, this new SMNN model can map the input–output relationships with fewer hidden neurons and have higher reliability for generalization. As a consequence, this SMNN model can run as fast as an approximate equivalent circuit, yet preserve the accuracy of detailed electromagnetic simulations. Experiments are included to demonstrate merits and efficiency of this new approach. Index Terms—Modeling, neural networks, space mapping, spiral inductor.

I. INTRODUCTION N silicon-based RF integrated circuits (RFICs), on-chip spiral inductors are widely used due to their low cost and ease of process integration [1]. However, lack of fast, accurate, and scalable models becomes one of design bottlenecks for RFICs. Various modeling approaches for spiral inductors on silicon substrate have been reported [2]–[7]. Numerical methods such as the finite-difference time-domain (FDTD) method, the method of moments (MoM), and the finite-element method (FEM) have been used for accurately solving spiral inductor problems [2], [3]. However, these numerical approaches are very time consuming and a small change in the geometrical or material parameters could require a completely new simulation run. The pitfalls of the numerical approaches prevent their direct application in design implementation. Instead, these numerical approaches can be effectively utilized for design verification. The existing equivalent-circuit models suffer two major drawbacks: limited applicable bandwidth and poor scalability. The modified -models have very limited applicability due to their narrow frequency range or less flexibility in design parameters [4]. A modified T-model demonstrated a promising improvement in broadband accuracy [5]. However, this model suffers

I

poor scalability because its circuit elements were expressed in terms of port-parameters (e.g., -parameters) instead of the actual geometrical dimensions and technology parameters. A 2- model was proposed to improve the scalability and accuracy over a broad bandwidth [6]. Unfortunately, this 2model has a complicated circuit topology with double element numbers, which leads to difficulty in parameter extraction and yields longer circuit simulation time. Recently, a study of applying artificial neural networks (ANNs) to model spiral inductors has been reported [7]. The ANN can map complex nonlinear input–output relationships at a much higher speed than numerical models, yet with the same accuracy as the numerical simulation [8]. However, the widely used multilayer perception (MLP) neural network is essentially a black-box model, and all the information about the input–output relationships must be learned from the training data. Consequently, the training data set and the hidden neurons must be large enough to characterize the problem to be modeled. In this study, a novel space-mapping neural network (SMNN) is proposed to model RF CMOS spiral inductors. This new modeling approach exploits merits of space-mapping technology [9]. It is composed of a three-layer perceptron network and a new modified 2- model. The simple three-layer perceptron network, which is easily realizable in the SPICE sub-circuit format, can be used for calculating the values of circuit elements according to the geometrical dimensions and technology parameters. The new modified 2- model is utilized to account for various parasitic effects of spiral inductors on a lossy substrate. This SMNN model has much enhanced learning and generalization capabilities. It is as fast as an approximate equivalent circuit and preserves the accuracy of detailed electromagnetic (EM) simulations. This modeling approach offers a wideband and scalable compact circuit for on-chip spiral inductors, which can be easily used in circuit simulations and optimization design. II. SMNN MODEL A. Modified 2- Model of On-Chip Spiral Inductors

Manuscript received February 15, 2007; revised August 5, 2007. This work was supported by the National Natural Science Foundation of China under Grant 60376031, Grant 90307017, and Grant 60444004. Y. Cao is with the School of Electronic Information, Wuhan University, Wuhan, Hubei 430072, China. G. Wang is with the Institute of Microelectronics and Information Technology, Wuhan University, Wuhan, Hubei 430072, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.909602

The physical layout of on-chip spiral inductor is shown in Fig. 1. It is a two-port device, which can be characterized by -parameters, inductances, input impedance, and quality factor. There are four types of parasitic effects that must be considered in modeling spiral inductors on a lossy substrate, which are: 1) the effect of eddy currents induced within the substrate due to magnetic coupling; 2) the skin and proximity effects; 3) the capacitive substrate coupling; and 4) the conductor loss originated

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Fig. 3. Structure of the SMNN model for on-chip spiral inductor.

TABLE I TRAINING DATA AND VALIDATION DATA

Fig. 1. Physical layout of on-chip spiral inductor.

TABLE II COMPARISON BETWEEN 24-ELEMENT 2- MODEL AND THIS MODEL

Fig. 2. Modified 2- model of on-chip spiral inductor.

from the lossy substrate return path. All these effects can introduce loss and degrade the characteristics of spiral inductors. Fig. 2 is equivalent circuits of the new modified 2model. In this model, two single loops and are used to model the effect of eddy currents induced within the substrate [10]. As for heavily doped substrates, those elements need to be carefully modeled. However, they will be excluded herein because the current RF CMOS processes usually use a lightly doped Si-substrate and such eddy-current effect is normally insignificant. – branches and Moreover, two series and two inductors and are employed to model the skin and proximity effects. Herein, a semiempical method is utilized as follows [6]: or

The three-element – – oxide–substrate models account for the capacitive substrate coupling [11]. in the three-element model denotes the oxide capacitance. The overall oxide cacan be partitioned into the three parts pacitance and as follows: (2) where and . Similarly, the overall substrate capacitance can be written into the following three elements:

(1) and are two empirical coefficients that can be where held as constant within a wide range. Specifically, and .

(3) where tors

and and

. The substrate resisrepresent the effect of displace-

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Fig. 4. Comparison of real parts of input impedances Re[Z (! )] by this SMNN model, the EM simulation, and the 2- model for D2, D4, and D6.

ment currents in the substrate. There exists a relation between the substrate capacitances and substrate resistors (4) where is the so-called relaxation time constant of the Si substrate. is included to account for the caA parallel capacitance pacitive effects between metal windings and underpass of the on-chip spiral inductor. Note that the inter-winding capacitance is important for capturing the distributed characteristics of the metal windings. is added into the conventional 2Finally, a new element circuit model to model the conductor loss arising from the magnetic field generated through the substrate return path under high frequency [12]. Eddy currents in the coil metal are induced by the magnetic field generated through the substrate return path, which have a significant impact on the characteristics at broadis an important mechband. The loss due to the new element anism to account for the broadband characteristic degradation. Herein, the number of independent elements, which completely determine the characteristics of a spiral inductor, is reduced from 25 in the conventional 2- circuit model to only six ). Other in this modified model (i.e., circuit elements can be derived according to (1)–(4). These six independent circuit elements in this modified 2- model will be computed using the three-layer perceptron network in the SMNN model. As pointed out by Burghartz and Rejaei [1], in practice, the measurement RF ground may not coincide with the true physical RF ground. Although the measurement RF ground is not included in this new modified 2- model, a simple alternative approach could be as follows: after the SMNN model is successfully established, the measurement RF ground (e.g., a planar

substrate contact) can be readily added to the SMNN model by and simply extending a port from the node between through a parasitic impedance of the planar substrate contact, which also implicitly treats the problem that the planar substrate contact may not be uniformly distributed.

B. Structure of the SMNN Model -parameters are usually preferable for the model outputs due to ease of measurement. Since and are equal for a passive two-port reciprocal network like a spiral inductor, only is included in the model output. Herein, the -parameters are computed using the full-wave EM simulation tools—ADS Momentum from Agilent Technologies, Palo Alto, CA [13]. These real and imaginary parts of -parameters are used as the de. The structure of the SMNN model for spiral sired output inductors is illustrated in Fig. 3. Six geometrical parameters of an on-chip spiral inductor used as the model input vector are: 1) linewidth ; 2) spacing ; 3) inner radius ; 4) ; 5) length of coplanar-pass ; and 6) number of turns . Some technology parameters such length of under-pass as thicknesses of metal line and substrate are not incorporated in model input parameters because they usually cannot be controlled by circuit designers for a given fabrication process. In this SMNN model, the mapping from the geometrical pato the six independent circuit elements of the rameters modified 2- model is implemented by a simple three-layer perceptron neural network . This mapping is achieved by training the neural network to minimize the error between the available and output of the modified 2simulated training data model through an optimization in space [14] (5)

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Fig. 5. Comparison of inductances L11 (top) and L22 (bottom) by this SMNN model, the EM simulation, and the 2- model for D2, D4, and D6 (Herein SRF11 and SRF22 denote the two self-resonance frequencies).

Fig. 6. Comparison of quality factors Q11 (top) and Q22 (bottom) by this SMNN model, the EM simulation, and the 2- model for D2, D4, and D6. TABLE III COMPARISON BETWEEN THE SMNPO 2- MODEL AND THIS MODEL

where denotes the weights of the three-layer perceptron netis the work, is the total number of learning samples, and error vector given as

with (6) is the number of training base points for the input where design parameters, and is the number of frequency points per frequency sweep. Once this simple three-layer perceptron neural network is trained, an SMNN model for the on-chip spiral inductor is established.

C. Training of the SMNN Model The SMNN model can be developed using supervised [15]. The weight training to determine the weight vectors

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Fig. 7. Comparison of real parts of input impedances Re[Z (! )] by this SMNN model, the EM simulation, and the SMNPO 2- model for D2, D4, and D6.

vectors are adjusted to reduce the difference until achieving the minimization of the mean square error (MSE) defined as

(7) where is the number of training samples and is the error vector. Partial composite design of experiments [16] is used to determine the size of the training samples. All of the training samples will be presented to the SMNN in one batch. The outputs computed by the SMNN model are compared with the desired outputs obtained from the full-wave EM simulation. An antiunitary Bayesian method is utilized as the training algorithm in the three-layer perceptron network. This approach can achieve much better generalization performance than the conventional Levenberg–Marquardt method. III. EXPERIMENTS AND RESULTS The proposed SMNN is applied to model RF silicon-based spiral inductors for Chartered Semiconductor Manufacturing’s 0.13- m RF CMOS six-metal process with the top metal made of Cu. In this process, the inductor’s intrinsic windings are strapped by a large number of vias. Table I lists all the training and validation data. It can be seen that the ranges of input parameters differ from one another. To evaluate the proposed SMNN model, comparisons with other typical modeling approaches over various experiments are included in Sections III-A–C. A. Comparison With EM Simulation and 2- Model In this experiment, the training and test data in Table I is used. The simple three-layer perceptron neural network with eight

tansig neurons in the hidden layer is constructed. In neural network literature, tansig is defined as the hyperbolic tangent sigmoid transfer function that is mathematically equivalent to the hyperbolic tangent tanh. Using the results from the full-wave EM simulator as the standard reference, Table II lists the errors of the SMNN in comparison with those of the 24-element 2model [6]. From Table II, one can clearly observe that not only the training error, but also the test error of the SMNN model has been reduced nearly by half compared with the 24-element 2- model. The SMNN model preserves the accuracy of the full-wave EM simulation, yet reduces the CPU times. Moreover, the SMNN model keeps reliable generalization capability by incorporating the prior knowledge. Consider three devices with the following geometrical paramm, m, m, eters: (a) D2: m, and m; (b) D4: m, m, m, m, and m; m, m, m, (c) D6: m, and m. Fig. 4–6 depict real parts , inductances, and quality facof input impedances tors of three devices D2, D4, and D6 versus frequency over a broad bandwidth up to 20 GHz, computed by using this SMNN model, the EM simulation, and the 24-element 2- model. Fig. 4 illustrates that this SMNN model can accurately capover the entire ture the characteristic behaviors of frequency range up to 20 GHz, including the dramatic growth prior to resonance, the peak at resonance, and the sharp drop after resonance, where the 24-element 2- model produces a significant large error. Good agreement between the inductances computed using the full EM simulation and those obtained using this SMNN model can be observed in Fig. 5 for operating frequency up to 20 GHz, which is in sharp contrast to the results from the 24-element 2model. In addition, the transition from inductive to capacitive

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mode as the frequency increases beyond the self-resonance frequency has been accurately reproduced by the SMNN model, as shown in Fig. 5. Note that the 24-element 2- model has much less accuracy in the transition range. When the operating frequency is higher than the self-resonance frequency, the effective inductance becomes negative and the spiral inductor actually behaves like a capacitor. In practical designs, the spiral inductors may not be used for the frequency beyond the self-resonance. Nevertheless, as a broadband model for spiral inductors, it will still be very desirable for such a model to give accurate prediction for the behavior beyond the self-resonance in some occasions. For instance, in order to study the dependency of the self-resonance frequency on design parameters during the design phase, the model needs to be accurate near and even beyond the self-resonance frequency so that a reliable sensitivity analysis of the self-resonance frequency can be performed. Finally, Fig. 6 reveals good agreement between quality factors computed using the EM simulation and those obtained from the SMNN model over a broad bandwidth up to 20 GHz. The close identification of the peak and accurate capture of the characteristic behaviors of quality factors over the entire frequency range up to 20 GHz suggests salient merits of this SMNN model over the 2- model. B. Comparison With the Space-Mapping Neural Network Plus the Original (SMNPO) 2- Model To illustrate the effects of different equivalent circuits for spiral inductors, the SMNN plus the new modified 2- model (i.e., this model) and the SMNPO 2- model are both studied and compared in this experiment. The data in Table I is again used as the training and test data in this experiment. In both the cases of the new modified 2- model and the original 2- model, the neural networks take the same eight tansig neurons in the hidden layer. Table III compares the results obtained by the SMNN plus the new modified 2- model and those obtained by the SMNPO 2- model. From Table III, it can be observed that the SMNN plus the new modified 2- model can considerably reduce not only the training error, but also the test error with the same hidden neurons in comparison with the SMNPO 2- model. , Fig. 7–9 depict the real parts of input impedances inductances, and quality factors of D2, D4, and D6 versus the frequency over broad bandwidth up to 20 GHz, computed by using the SMNN plus the new modified 2- model, the EM simulation, and the SMNPO 2- model. In this new modified 2- model, an additional circuit element is introduced to model the effect of the eddy current in the coil metal arising from the magnetic field generated through the substrate return path. Using the EM simulation results as the standard reference, Fig. 7–9 demonstrate that the SMNN plus the new modified 2model exhibits a better accuracy than the SMNPO 2- model. apparently In particular, the additional circuit element improves the accuracy of inductances, input impedances, and quality factors near the self-resonance frequencies, and has significantly contributed to the degradation of quality factors over wideband.

Fig. 8. Comparison of inductances L11 (top) and L22 (bottom) by this SMNN model, the EM simulation, and the SMNPO 2- model for D2, D4, and D6.

C. Comparison With the Conventional Four-Layer Perceptron Network To compare performance of different ANN models for spiral inductors, the conventional four-layer perceptron network [7] and this SMNN model are trained by the same training sample set in Table I. The size of the hidden layer is determined experimentally. The conventional four-layer perceptron network, in which the frequency is also incorporated in model input parameters, has 25 tansig neurons in each hidden layer, whereas this SMNN needs only eight hidden tansig neurons. Table IV compares the results obtained by using the conventional four-layer perceptron network and those computed by using this SMNN model. From Table IV, it can be observed that this SMNN model can map the input–output relationships with

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thors’ best knowledge, it is the first time to combine the spacemapping technique with the equivalent circuits for modeling such complex passive components in the RFICs. Instead of applying the stiff and time-consuming parameter extraction, this model uses a simple three-layer perceptron neural network for evaluating the circuit component values in the equivalent circuit in terms of the geometrical dimensions and technology parameters. This model preserves accuracy of the full-wave EM simulation, yet greatly reduces the CPU requirement. Moreover, a new modified 2- equivalent circuit has been introduced to account for parasitic effects of spiral inductors on a lossy substrate. The total number of independent parameters, which completely determine the characteristics of a spiral inductor, has been reduced from 25 in the original 2- circuit model to only six in this new modified 2- equivalent circuit. This SMNN model keeps reliable generalization capability by incorporating such a new modified 2- model. Numerical results demonstrated that this SMNN model has very promising wideband performance and good scalability for the full set of model parameters, which suggests that this model is extremely useful in pre-layout simulation and optimization for physical design of on-chip spiral inductors. The nature of easy interface with the standard SPICE circuit simulators makes this SMNN model easy to use in circuit element design and optimization in RFICs.

Fig. 9. Comparison of quality factors Q11 (top) and Q22 (bottom) by this SMNN model, the EM simulation, and the SMNPO 2- model for D2, D4, and D6. TABLE IV COMPARISON BETWEEN FOUR-LAYER PERCEPTRON NETWORK AND THIS MODEL

much fewer hidden neurons and has higher reliability for generalization than the conventional four-layer perceptron network. IV. CONCLUSION A wideband and scalable compact model of RF on-chip spiral inductors using a novel SMNN has been introduced. To the au-

REFERENCES [1] J. N. Burghartz and B. Rejaei, “On the design of RF spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 50, no. 3, pp. 718–729, Mar. 2003. [2] S. H. Song, H. B. Lee, H. K. Jung, S. Y. Hahn, K. S. Lee, C. Cheon, and H. S. Kim, “Spectral domain analysis of the spiral inductor on multilayer substrates,” IEEE Trans. Magn., vol. 33, no. 2, pp. 1488–1491, Mar. 1997. [3] B. Rejaei, “Mixed-potential volume integral-equation approach for circular spiral inductors,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1820–1829, Aug. 2004. [4] D. Melendy, P. Francis, C. Pichler, K. Hwang, G. Srinivasan, and A. Weisshaar, “A new wideband compact model for spiral inductors in RFICs,” IEEE Electron Device Lett., vol. 23, no. 5, pp. 273–275, May 2002. [5] J. C. Guo and T. Y. Tan, “A broadband and scalable model for on-chip inductors incorporating substrate and conductor loss effects,” IEEE Trans. Electron Devices, vol. 53, no. 3, pp. 413–421, Mar. 2006. [6] W. Gao and Z. Yu, “Scalable compact model and synthesis for RF CMOS spiral inductors,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1055–1064, Mar. 2006. [7] T. Liu, W. Zhang, and Z. Yu, “Modeling of spiral inductors using artificial neural network,” in Proc. Int. Joint Neural Networks Conf., Aug. 2005, vol. 4, pp. 2353–2358. [8] Q. J. Zhang, K. C. Gupta, and V. K. Devabhaktuni, “Artificial neural networks for RF and microwave design-from theory to practice,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1339–1350, Apr. 2003. [9] J. W. Bandler, M. A. Ismail, J. E. Rayas-Sánchez, and Q. J. Zhang, “Neuromodeling of microwave circuits exploiting space-mapping technology,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2417–2427, Dec. 1999. [10] A. C. Watson, D. Melendy, P. Francis, K. Hwang, and A. Weisshaar, “A comprehensive compact modeling methodology for spiral inductors in silicon-based RFICs,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 849–857, Mar. 2004. [11] C. P. Yue and S. S. Wong, “Physical modeling of spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 47, no. 3, pp. 560–568, Mar. 2000. [12] A. M. Niknejad and R. G. Meyer, “Analysis of eddy-current losses over conductive substrate with applications to monolithic inductors and transformers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 166–176, Jan. 2001. [13] Momentum Agilent EESOF EDA. Agilent Technol., Palo Alto, CA, 2003.

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[14] V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, and Q. J. Zhang, “Advanced microwave modeling framework exploiting automatic model generation, knowledge neural networks and space mapping,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1822–1833, Jul. 2003. [15] Q. J. Zhang and K. C. Gupta, Neural Networks for RF and Microwave Design. Norwood, MA: Artech House, 2000. [16] S. R. Schmidt and R. G. Launsby, Understanding Industrial Designed Experiments. Colorado Springs, CO: Air Force Academy, 1992.

Yazi Cao received the B.S. degree in electrical engineering from Wuhan University, Hubei, China, in 2004, and is currently working toward the Ph.D. degree in electrical engineering at Wuhan University. His research interests include RFIC design, modeling, and simulation.

Gaofeng Wang (S’93–M’95–SM’01) received the Ph.D. degree in electrical engineering from the University of Wisconsin–Milwaukee, in 1993, and the Ph.D. degree in scientific computing from Stanford University, Stanford, CA, in 2001. From 1988 to 1990, he was with the Department of Space Physics, Wuhan University, Hubei, China. From 1990 to 1993, he was a Research Assistant with the Department of Electrical Engineering, University of Wisconsin–Milwaukee. From 1993 to 1996, he was a Scientist with Tanner Research, Inc., Pasadena, CA. From 1996 to 2001, he was a Principal Engineer with Synopsys Inc., Mountain View, CA. In Summer 1999, he was a consultant to Bell Laboratories, Murray Hill, NJ. From 2001 to 2003, he was the Chief Technology Officer (CTO) with Intpax Inc., San Jose, CA. Since 2004, he has been the Chief Technology Officer (CTO) with Siargo Inc., San Jose, CA. He is also currently a Professor and the Head of the CJ Huang Information Technical Research Institute, Wuhan University. He has authored or coauthored over 120 publications. He holds six U.S. patents. His research and development interests include integrated circuit (IC) and microelectromechanical systems (MEMS) design and simulation, computational electromagnetics, electronic design automation, and wavelet applications in engineering.

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Design and Experimental Verification of Compact Frequency-Selective Surface With Quasi-Elliptic Bandpass Response Guo Qing Luo, Wei Hong, Senior Member, IEEE, Qing Hua Lai, Ke Wu, Fellow, IEEE, and Ling Ling Sun, Senior Member, IEEE

Abstract—A compact frequency-selective surface (FSS) having a quasi-elliptic bandpass response is presented in this paper. This was realized by vertically cascading substrate integrated waveguide (SIW) cavities. A single SIW cavity FSS has been fully studied and approximate analytical formulas are introduced to calculate its resonant frequencies. Two different resonances can be excited by a plane wave in the single SIW cavity FSS. According to theories of the cascading cavity filter and dual-mode filter, cross coupling can be realized in cascading SIW cavity FSSs, thus a compact FSS with a quasi-elliptic bandpass response is imple-band sample was fabricated by a printed circuit mented. A board (PCB) process. Experiments were carried out to validate this design method. Measured results are in agreement with predicted ones. The proposed quasi-elliptic FSS presents a number of advantages, namely, high selectivity, stable performance, and much reduced volume. Index Terms—Cascading cavities, dual-mode filter, frequency-selective surface (FSS), substrate integrated waveguide (SIW), quasi-elliptic.

I. INTRODUCTION REQUENCY-SELECTIVE surfaces (FSSs) can be viewed as generalized spatial filters in radom, sub-reflectors with dual frequency antennas, and they can also be used as insertions in waveguides to form waveguide filters. Most FSSs have been realized thus far by planar layered structures. Single-layer planar FSS performance is susceptible to the change in incident angles and polarization states [1]–[3]. To improve FSS performance, many design methods have been proposed and demonstrated. Conventional dielectric loading planar FSSs with stable bandwidth and incident angle insensitivity has been reported, for example, in [4]. High-order bandpass FSS structures are usually designed by using multilayer planar topology such as double-layer planar FSSs, as shown in [5]. The FSS layers serving as individual resonators are stacked together using thick dielectric slab spacers

F

Manuscript received March 15, 2007; revised July 12, 2007. G. Q. Luo and L. L. Sun are with the Key Laboratory of RF Circuits and Systems, Ministry of Education, Microelectronic CAD Center, Hangzhou Dianzi University, Hangzhou 310018, China (e-mail: [email protected]). W. Hong and Q. H. Lai are with the State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210096, China. K. Wu is with the Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montreal, Montreal, QC, Canada H3C 3A7. Digital Object Identifier 10.1109/TMTT.2007.910085

(roughly a quarter-wavelength), which act as impedance inverters, to form multipole filtering FSSs [6]. Using this concept, waveguide filters with Chebyshev and elliptic responses have been obtained by using three single FSS cascading as insertions [7], [8]. However, the problem is related to the fact that each cascading unit is approximately one quarter-wavelength. In order to generate a high-selectivity performance, the total thickness of such an insertion becomes bulky, which makes it unsuitable for many practical applications. Another design method of the high-order filtering FSS was proposed in [6] and [9], in which the FSSs are composed of metal patches at the top and bottom layers, which are coupled by a nonresonant slot or coplanar-waveguide resonators in the middle layer. Using these structures, a highly selective FSS with a multipole can be obtained by adding a number of coupling resonators in the middle layer. Different filtering responses can thus be obtained by adjusting the coupling level. Thus far, rarely reported work can be found about monolithic FSSs with an elliptic or a quasi-elliptic response because the cross coupling needed in the elliptic filter is hardly realized in planar layered FSSs. In this paper, we propose a novel FSS with a quasi-elliptic response by using cascading substrate integrated waveguide (SIW) cavities, which are based on the theories of a cascading cavity filter and dual mode filter [10]–[12]. The SIW technique was firstly applied to a bandpass FSS design, as described in [13]–[15], in which the SIW cavity is constructed by setting a metallic via array around each periodic slot. cavity resonance is introduced in the Unique distorted SIW cavity FSS with conventional aperture resonance generated by the periodic slot remaining. A highly selective FSS with a steep transition band can easily be realized by adjusting the coupling level between these two different resonances. Since these two resonances are independent of dielectric thickness, each layer thickness of the SIW cavity FSS can be set far less than one quarter-wavelength and the total volume of SIW cavity FSSs can greatly be reduced. Fig. 1 shows the geometrical configuration of a basic FSS unit of three vertically cascading SIW cavities. Square loop slots at the top and bottom metal surfaces have the same dimensions and . Coupling square loop slots at two middle metal surand . Some paramefaces have the same dimensions ters of the SIW cavity FSS discussed in this paper are fixed for and for the purpose of simplicity. The unit of all the geometrical parameters in this paper is millimeters.

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Fig. 3. Electric field distribution diagrams of the aperture resonance (left) and cavity resonance (right) in which the white blank corresponds distorted TE to zero electric field and black corresponds to the strongest electric field.

TABLE I APERTURE RESONANCE FREQUENCY IN SIW CAVITY FSS (P = C; w = 0:2)

Fig. 1. Geometrical configuration of three SIW cavities cascading FSS with square loop slots etched at four metal surfaces.

are plotted in Fig. 3. The pole at 29.9 GHz corresponds to an aperture resonance because, at this time, the main part of the , which is the same as the incident electric field vector is electric field. This resonance occurs when the perimeter of the square loop slot is a multiple of wavelength. Its resonant frequency can be estimated by the approximate formula in (1). This revised formula is derived from the approximate formula in [1], which was only suitable for calculating the aperture resonant frequency of a conventional planar layered FSS with thick dielectric loading at one or two sides. Calculated results of aperture resonant frequency by (1) and the High Frequency Structure Simulator (HFSS) are listed in Table I. It can be found that this revised analytical formula can give an adequate precision over a wide frequency range (1)

Fig. 2. Typical frequency response of single SIW cavity FSS. (P = = 2:39; w = 0:2).

C

=

7:14; l

II. SINGLE SIW CAVITY FSS In order to design a high-order filtering FSS using the cascading SIW cavities, the characteristics of the single SIW cavity -band FSS must be studied thoroughly. An example of a single SIW cavity FSS with a square loop slot is presented. Its typical response is shown in Fig. 2 in which a normal incident . plane wave is at the -axis with Two poles in Fig. 2 correspond to two different resonant modes in the SIW cavity FSS of which electric field profiles

The pole at 30.2 GHz corresponds to a cavity resonance introduced by the SIW cavity. Its resonant field profile is shown in Fig. 3, in which the main electric field is , which is perpendicular to the incident plane and incident field. It is a distorted cavity mode similar to that of a conventional cavity resonance. Based on the analytical calculation formula of cavity resonant frequency in [16], an approximate equation for the discavity mode can be derived as follows, in which torted and are plotted in Fig. 3: equivalent (2) (3) (4)

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TABLE II FREQUENCY OF CAVITY RESONANCE IN SIW CAVITY FSS (P = C; w = 0:2)

where is a constant value of 0.54. Calculated results of the cavity resonant frequency by (2) and HFSS are given in Table II. The largest discrepancy between these two methods is less than 5% over a frequency range from 20 to 40 GHz. With the above analytical equations, initial geometrical parameters’ values of the SIW cavity FSS for a given specification can be derived quickly, and then full-wave electromagnetic (EM) simulation can be deployed for the final design and optimization. According to the dual-mode filter theory [13], the coupling coefficient between two modes can be calculated by and denote the upper and lower resonant fre(5), where quencies, respectively,

Fig. 4. Single SIW cavity FSS frequency responses with transmission zero at the lower and upper sidebands.

Fig. 5. Coupling mechanism of cascading SIW cavity FSS.

(5) From (1) and (2), it can be found that the aperture resonant frequency is mainly controlled by and the cavity resonant and . The FSS frequency is primarily determined by with a given specification can easily be realized by adjusting the coupling magnitude between these two resonances. Fig. 4 shows examples of a single SIW cavity FSS with a transmission zero at the upper and lower sidebands. These two FSSs can easily be realized by adjusting the parameters of and . III. QUASI-ELLIPTIC CASCADING SIW CAVITY FSS Two different resonances can be excited by the plane wave in the single SIW cavity FSS. According to the theories of a cascading cavity filter and dual-mode filter [10]–[12], a highselectivity SIW cavity FSS with a quasi-elliptic bandpass response can be realized with a proper design because the FSS is a generalized filter. The proposed quasi-elliptic SIW cavity FSS structure is shown in Fig. 1, which is composed of three vertically cascading SIW cavities. The middle SIW cavity is used to achieve coupling between the top and bottom cavities and its role is similar to that of a coupling waveguide used in [12]. This filter topology is shown in Fig. 5, where the arrows denote electric field polarization of the four resonances. Since the two resonances in one cavity are two different types of resonances, these two different resonances in each of the two cavities can be assumed to be uncoupled.

Fig. 6. Single SIW cavity FSS response with different cavity size 7:2; l = 2:4; w = 0:2).

C

. (P =

Coupling between two different resonances ( and ) in the same cavity has been partially discussed in Section II. and are mainly determined by cavity size and slot dimensions and . The aperture resonance is mainly determined by and is nearly unaffected by the variation of (see Fig. 6). From the FSS theory [1], we know that if the periodic spacing (pitch) is smaller, the relative location of grating lobe moves farther away from the passband. In the SIW cavity FSS, with a given so is always set the minimum is in our design. The required coupling and identical to are mainly obtained by tuning and .

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Fig. 8. Structure of the calculated inter-cavity coupling k of cavity resonance with probe excitation.

Fig. 7. Inter-cavity coupling k of aperture resonance with the variation of coupling slot l and w . (P = 7:2; C = 6:4; l = 2:5; w = 0:2).

A. Coupling Between Aperture Resonances In a conventional dual-mode elliptic filter, the cross coupling is generated between degenerated modes. Its coupling magnitude is difficult to calculate explicitly because they cannot be discriminated. In our proposed SIW cavity FSS, the aperture resonance and cavity resonance can be excited individually with a proper setting. From Fig. 6, it can be found that the aperture resonance occurs at a fixed frequency with the frequency of cavity resonance greatly increased as decreases. The coupling coefficient of aperture resonance in a different cavity is shown in Fig. 7. The coupling coefficient is more sensitive to the variation of coupling slot length than that of . This suggests that the coupling coefficient of slot width aperture resonance in a different cavity is mainly controlled by the length of coupling slot . B. Coupling Between Cavity Resonances resonance The coupling coefficient of the distorted in a different cavity cannot directly be calculated when it is excited by a plane wave. Aperture resonance cannot be avoided because energy should be coupled into the cavity by the aperture resonance. From the filter theory, we know that the coupling coefficient is mainly determined by the geometrical structure, and it is independent of the excitation. In order to calculate of the cavity resonance between different cavities, probes are used as input and output, which are placed near the center point of the cavity edge. The equivalent modeling structure and the excited cavity resonance are shown in Fig. 8. From this figure, it can be found that the cavity resonance excited in this structure is identical to that shown in Fig. 3. Inter-cavity coupling coefficient of the distorted cavity resonance with the variation of length and width of the coupling slot is shown in Fig. 9. The coupling is more sensitive to the variation of slot length than that of slot width. When is set to 2.1, the coupling coefficient coupling slot length reaches a maximum value.

Fig. 9. Inter-cavity coupling k of cavity resonance with the variation of coupling slot l and w . (P = C = 7:2; l = 2:5; w = 0:2).

C. Quasi-Elliptic Bandpass SIW Cavity FSS When the coupling mechanism has fully been examined, a filter synthesis procedure [10], [11] can be carried out. The network is assumed to be symmetrical so elements of coupling maare symmetrical along the main diagonal. The filter trix specifications are given as follows: center frequency: 30 GHz, bandwidth: 500 MHz, stopband attenuation: 15 dB, and passband return loss: 20 dB. A fourth-order network (shown in Fig. 5) is used to realize the given specifications. Normalized is achieved by an optimized process, which coupling matrix is listed in (6) as follows with :

(6) Once the coupling matrix is obtained, the circuit simulated response can be calculated. Previous approximate formulas are used to generate the initial values of geometrical parameters, and then an optimization process on the basis of a full-wave method is carried out to obtain their final values. After a properly optimized process, the SIW cavity FSS with the given

LUO et al.: DESIGN AND EXPERIMENTAL VERIFICATION OF COMPACT FSS WITH QUASI-ELLIPTIC BANDPASS RESPONSE

Fig. 10. Simulated frequency response of the equivalent-circuit model and proposed SIW cavity FSS full-wave simulated results. (P = C = 7:2; l = 2:58; w = 0:24; l = 1:39; w = 0:15).

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Fig. 11. Fabricated three SIW cavity cascading FSS with quasi-elliptic response.

specification has been reached and its full-wave simulated response is presented in Fig. 10, in which only dielectric loss is taken into account. From this figure, it can be found that these two results are in good agreement with each other over the passband. An evident discrepancy existed at the rejection band, especially the upper rejection band, because only two main resonances have been taken into account in the circuit model. In full-wave analysis, we find there has a spurious resonance at 35.6 GHz, of which resonant domain is restricted in the inner area between two square loop slots at the outer and inner metal surfaces. IV. MEASURED RESULTS The proposed SIW cavity FSS with a quasi-elliptic response is fabricated by using a Rogers 5880 substrate, whose permittivity is 2.2, loss tangent is 0.001, and conductivity S/m. Experiments are carried out in unbounded is space and the measurement setup is the same as that of [13, Fig. 10]. The measurement sample is shown in Fig. 11, and its entire dimensions are 150 mm 150 mm and the total thickness is less than 0.85 mm. Measured results are shown in Figs. 12 and 13. Measured and simulated transmission coefficients with a plane wave normal incidence are presented in Fig. 12. Conductor loss and dielectric loss are taken into account in this full-wave simulation. Insertion loss due to the conductor loss is 1.4 dB, which is greater than that of the simulation result (0.7 dB) in which only the dielectric loss has been counted. -band, the roughness of the metal surface can also Over the lead to visible insertion loss. The measured insertion loss of 2.5 dB is higher than the simulated insertion loss of 2.1 dB, and the measured bandwidth is wider than the simulated counterpart. Such discrepancies between the measured and simulated results can mainly be attributed to the roughness of the metal surface, fabrication tolerances, material parameter errors, and test error since, at

Fig. 12. Simulated and measured transmission coefficients of the proposed cascading SIW cavities FSS with a plane wave normal incidence.

band, a small change of the above considered factors would give rise to a great degradation on performance. It should be noted that the proposed FSS with a given specification is a narrowband filter with a fractional bandwidth of 1.67%, which leads to a high simulated insertion loss. If its bandwidth is set to be large, the insertion loss can be reduced significantly. The proposed cascading SIW cavity FSS performances at different incident angles and polarization states are shown in Fig. 13. As the sample size is not large enough compared with the aperture of the horn antenna, the maximum incident angle in our experiments is restricted to less than 20 . From these measured results, it can be found that the proposed SIW cavity FSS performance is stable with the variation of incident angle and polarization state. The upper stopband with TM polarization is obviously degraded because the grating lobe moves closer to the passband when the incident angle increases. In order to achieve more stable performances, some improvements, such as the addition of dielectric layers at the top and bottom metal surfaces, may be considered.

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proposed FSS is less than , which is far less than in a conventional planar multilayer FSS, and its total thickness has greatly been reduced. REFERENCES

Fig. 13. Measured results of the proposed cascading SIW cavity FSS at different polarization states and incident angles. (a) TE. (b) TM.

[1] B. A. Munk, Frequency Selective Surfaces. New York: Wiley, 2000. [2] T. K. Wu, Frequency Selective Surfaces and Grid Arrays. New York: Wiley, 1995. [3] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces-a review,” Proc. IEEE, vol. 76, no. 12, pp. 1593–1615, Dec. 1988. [4] R. J. Luebbers and B. A. Munk, “Some effects of dielectric loading on periodic slot arrays,” IEEE Trans. Antennas Propag., vol. AP-26, no. 4, pp. 536–542, Jul. 1978. [5] R. J. Luebbers and B. A. Munk, “Mode matching analysis of biplanar slot arrays,” IEEE Trans. Antennas Propag., vol. AP-27, no. 3, pp. 441–443, May 1979. [6] A. Abbaspour, K. Sarabandi, and G. M. Rebeiz, “Antenna-filter-antenna arrays as a class of bandpass frequency selective surfaces,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1781–1789, Aug. 2004. [7] C. A. Kyriazidou, H. F. Contopanagos, and N. G. Alexopoulos, “Monolithic waveguide filters using printed photonic bandgap materials,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 297–307, Feb. 2001. [8] M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, “Novel waveguide filters with multiple attenuation poles using dual-behavior resonance of frequency-selective surfaces,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3320–3326, Nov. 2005. [9] R. Pous and D. M. Pozar, “A frequency-selective surface using aperture coupled microstrip patches,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1763–1769, Dec. 1991. [10] J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [11] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters: Impedance Matching Networks and Coupling Structures. New York: McGraw-Hill, 1964. [12] H. C. Chang and K. A. Zaki, “Evanescent-mode coupling of dual-mode rectangular waveguide filters,” IEEE Trans Microw. Theory Tech., vol. 39, no. 8, pp. 1307–1312, Aug. 1991. [13] G. Q. Luo et al., “Theory and experiment of novel frequency selective surface based on substrate integrated waveguide technology,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4035–4043, Dec. 2005. [14] G. Q. Luo, W. Hong, H. J. Tang, and K. Wu, “High performance frequency selective surface using cascading substrate integrated waveguide cavities,” IEEE Microw. Wireless Compon Lett., vol. 16, no. 12, pp. 648–650, Dec. 2006. [15] G. Q. Luo et al., “Filtenna consisting of horn antenna and substrate integrated waveguide cavity FSS,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 92–98, Jan. 2007. [16] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961.

V. CONCLUSION A design concept of a compact FSS with a quasi-elliptic bandpass response has been proposed and presented in this paper. This novel FSS has been realized by using a cascading SIW cavity FSS. The single SIW cavity FSS has fully been studied and an approximate formula have been derived for calculating the aperture resonant frequency and distorted resonant frequency. According to the theories of the cascading cavity filter and dual-mode filter, three vertically cascading SIW cavities are used to realize a quasi-elliptic response FSS. The coupling mechanism has been fully investigated and a filter synthesis procedure has been used to design the SIW cavity FSS with a given specification. Experiments have been carried out, and measured results are in agreement with the simulated ones. The proposed FSS performance remains unchanged with the incident angle and polarization state. Each layer thickness of the

Guo Qing Luo was born in Jiangxi Province, China, on April 3, 1979. He received the B.S. degree in material science from the China University of Geoscience, Wuhan, China, in 2000, the M.S. degree in material science from Northwest Polytechnical University, Xi’an, China, in 2003, and the Ph.D. degree in radio engineering from Southeast University, Nanjing, China, in 2007. Since April 2007, he has been Microelectronics Computer-Aided Design (CAD) Center, Hangzhou Dianzi University, Hangzhou, China, where he is currently a Lecturer. He has authored or coauthored 13 technical papers. His current research interests include RF, microwave and millimeter-wave passive devices, antennas, FSSs, and numerical algorithms in electromagnetics. Dr. Luo was the recipient of the Best Research Paper Prize in Natural Science (second-class) of Nanjing, China, in 2007.

LUO et al.: DESIGN AND EXPERIMENTAL VERIFICATION OF COMPACT FSS WITH QUASI-ELLIPTIC BANDPASS RESPONSE

Wei Hong (M’92–SM’07) was born in Hebei Province, China, on October 24, 1962. He received the B.S. degree from the Zhenzhou Institute of Technology, Zhenzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, where he is currently a Professor with the Department of Radio Engineering. In 1993, 1995, 1996, 1997, and 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and University of California at Santa Cruz, respectively. He has been engaged in numerical methods for EM problems, millimeter-wave theory and technology, antennas, EM scattering, inverse scattering and propagation, RF front-ends for mobile communications, and the parameters extraction of interconnects in very large scale integration (VLSI) circuits. He has authored or coauthored over 200 technical publications. He authored Principle and Application of the Method of Lines (Southeast Univ. Press, 1993, in Chinese). He has served as a reviewer for many technique journals such as Proceedings of the IEE (Part H) and Electronics Letters. Dr. Hong is a Senior Member of the Chinese Institute of Electronics (CIE). He is the vice president of the Microwave Society and Antenna Society, CIE. He has served as the reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the two-time recipient of the First-Class Science and Technology Progress Prize presented by the State Education Commission in 1992 and 1994, respectively. He was the recipient of the Fourth-Class National Natural Science Prize in 1991, and the Third-Class Science and Technology Progress Prize of Jiangsu Province. He was also the recipient of the Foundation for China Distinguished Young Investigators presented by the National Science Foundation (NSF) of China.

Qing Hua Lai received the B.S. degree in radio engineering from University of Electronic Science and Technology of China, Chengdu, China, in 2004, and is currently working toward the Ph.D. degree in radio engineering from Southeast University, Nanjing, China. His current research interests include the design of microwave passive components, active devices and high performance antenna.

Ke Wu (M’87–SM’92–F’01) is Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering with the École Polytechnique de Montréal, Montréal, QC, Canada. He also holds a Cheung Kong endowed chair professorship (visiting) with Southeast University, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China, and the City University of Hong Kong. He has been the Director of the Poly-Grames Research Center. He has authored or coauthored over 515 referred papers and several books/book chapters. He has served on the Editorial/Review Boards of numerous technical journals, transactions, and letters, including being an Editor and Guest Editor. His current research

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interests involve substrate integrated circuits (SICs), antenna arrays, advanced computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors. He is also interested in the modeling and design of microwave photonic circuits and systems. Dr. Wu is a Fellow of the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is a member of the Electromagnetics Academy, Sigma Xi, and URSI. He has held key positions in and has served on various panels and international committees including the chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. He is currently the chair of the joint IEEE Chapters of the Microwave Theory and Techniques Society (MTT-S)/Antennas and Propagation Society (AP-S)/Lasers and Electro-Optics Society (LEOS), Montréal, QC, Canada. He is an elected IEEE MTT-S Administrative Committee (AdCom) member for 2006–2009 and serves as the chair of the IEEE MTT-S Transnational Committee. He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award.

Ling Ling Sun (SM’97) was born in Zhejiang, China, in 1956. She received the B.S. degree from Nanjing University of Posts and Telecommunications, Nanjing, China, in 1982, and the M.S. degree from the University of Electronic Science and Technology, Chengdu, China, in 1985. She then joined the Electronics and Information Department, Hangzhou Dianzi University (HDU), as an Assistant Professor, and became a Professor in 1999. Since 1997, she has been with the Microelectronic Computer-Aided Design (CAD) Center, HDU, as the Vice Director and Directer. She has also been the Director of the System-on-Chip (SOC) Key Laboratory, Chinese Ministry of Information Industry, and the Very Large Scale Integration (VLSI) Key Laboratory of Zhejiang Province. In 2002, she was a Visiting Scholar with the University of California. From 2000 to 2004, she was also the Dean of the School of Electrical and Information Engineering, HDU. She is currently Vice President of HDU and the President of the Micro CAD Center, HDU, and the leader of the very important subject in circuits and system of Zhejiang Province and Ph.D. supervisor of Zhejiang University and Dublin City University, Dublin, Ireland. Her major research interests includes the design and CAD of very deep submicrometer (VDSM)/RF/microwave integrated circuits (ICs) and ICs and systems, such as RF/microwave device modeling, the RF/microwave broadband power amplifier design, research of RFIC CAD technology, and the development of the electronic design automatic (EDA) tool for VDSM ICs. She is also very active in the RF identification (RFID) and medical electronic equipment areas. She has been in charge of over 30 national and Ministry or Province scientific research projects, which includes two 863 (national High Technology Project of China) projects and several projects of the National Nature Science Foundation of China (NSFC). She has authored or coauthored over 60 papers in journals and conferences. She is an Editor of several academy journals including the Electronic Journal of China and the Microwave Journal of China. Prof. Sun is a Senior Member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Circuits and Systems Society (IEEE CAS). She is the chair of the IEEE EDS Hangzhou Chapter. She is a committee member of the Microwave Society and the Chinese Institute of Electronics, vice-chief committee member of the Microwave Integrated Circuit and Wireless Communication Professional Committee of China. She was the recipient of a SecondGrade Award of Science and Technology Progress of the Zhejiang Province for her excellent academy achievements in 2004.

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Reflection Coefficient Shaping of a 5-GHz Voltage-Tuned Oscillator for Improved Tuning Alan Victor, Senior Member, IEEE, and Michael B. Steer, Fellow, IEEE

Abstract—Negative resistance voltage-controlled oscillators (VCOs) are systematically designed to operate with loaded resonator networks that permit stable steady-state oscillation over a specified tuning bandwidth. Circuit parasitics, however, significantly affect tuning behavior and complicate straightforward design. This paper introduces a scheme that compensates for the effect of parasitics by introducing an embedding network that modifies the effective active device reflection coefficient and thus enables conventional one-port oscillator design techniques to be used. A common-base SiGe HBT VCO operating from 4.4 to 5.5 GHz demonstrates the technique. Phase noise is better than 85 dBc/Hz at 10-kHz offset from the carrier and the second harmonic is less than 20 dBc, while higher order harmonics are less than 40 dBc. The voltage-tuned oscillator demonstrates an oscillator figure-of-merit of at least 182 dBc/Hz over a 800-MHz tuning range. The phase-noise-bandwidth (in megahertz) product is 159 dBc/Hz. Index Terms—Negative resistance, oscillator, resonator, varactor tuning, voltage-controlled oscillator (VCO).

I. INTRODUCTION

D

ESIGN OF stable negative resistance oscillators traditionally uses the one-port oscillator stability requirement outlined by Kurokawa [1]. In applying the criterion, each of the networks—the active device, resonator load, and device termination—are characterized as one-ports. When a device with adis connected to a loaded resonator of admittance , the voltage amplitude and frequency mittance of the resulting equilibrium oscillation are determined when and . In this procedure, the assumption is that the device admittance at a single frequency is a strong function of voltage amplitude while the resonator admittance is a function only of angular frequency. This condition can be represented graphically by first denoting the locus of the negative of the device’s complex admittance as (also referred to as the inlocus) and the locus verse device reflection coefficient or of the resonator admittance as . For stable single-frequency oscillation, the intersection of these loci in the complex plane then occurs at a single point. Multiple intersections and inappropriate angular intersection of these loci Manuscript received April 23, 2007; revised August 13, 2007. A. Victor is with Harris Stratex Networks, Morrisville, NC 27560 USA (e-mail: [email protected]). M. B. Steer is with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695-7911 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.909140

are directly related to several key performance-limiting parameters including spurious or multiple oscillations, oscillator startup problems, and excess noise [2]. In a tunable oscillator designed to cover a large tuning range, the device admittance is also frequency dependent. Therefore, the device admittance is more ac. curately described as be maximized while Resonator design requires that the achieving the desired admittance change with tuning voltage. Furthermore, for a voltage-controlled oscillator (VCO), voltage tuning of the resonator must satisfy the specific stability criteria, including single point of intersection and appropriate angle of intersection, over the tuning range. With emphasis on these characteristics and the presence of parasitic elements, achieving a proper stable resonator–device interface is troublesome. An alternative and equally viable approach to stability analysis of a broad class of oscillators, particularly for those using three-terminal devices, is application of the two-port criteria developed for amplifier stability assessment. However, the one-port approach is preferred by designers because the one-port connection is closer to the intended operation. The one-port wave assessment of oscillator stability is not unlike the Bode criteria applied to two-port feedback systems [3], [4]. However, unlike the two-port open-loop assessment of stability, the one-port characterization technique is conveniently aligned with measurements made by a vector network analyzer (VNA) [5], [6]. As well, the nonlinear limiting effect of the active device is readily measured. In the oscillator design approach presented in this paper, the design objective is the generation of a frequency-dependent with a prescribed reflection negative conductance using a three-terminal active device in coefficient shape a common-base (series-inductive feedback) configuration. Reactive loading modifies the effective device conductance so that it becomes frequency dependent. Modifications can be incorporated in the resonator load, but then it is seen that the frequency-dependent behavior of the tank circuit is inappropriate, resulting in multiple oscillations and other instabilities. is compromised. Another issue In addition, the resonator is that small-signal -parameters are generally good indicators of oscillator operation, particularly for the frequency of oscillation [7]; however, they do not provide sufficient information to determine if stable oscillation will occur. This paper introduces circuit modifications that facilitate design for correct operation of the active device–resonator combination. The technique uses measured reflection coefficients, and compensates for the effect of parasitics at the interface between the active device and resonator. In Section II, series feedback oscillators are discussed and the design criterion for oscillator is presented in terms of the device startup

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complex reflection coefficient. Section III presents and demonstrates a device-mapping technique to modify the active device characteristic. The mapping is achieved using a combination of additional capacitive reactive loading at the emitter–base terminals and at the collector. The net result is an effective active device characteristic that is largely a function of signal amplitude, while the frequency-dependent characteristics are properly modified. These modifications consider the resonator plus parasitic elements at the device interface. As such, a conventional approach to oscillator design can be used. Section IV then documents the performance of a VCO designed using the technique presented here. It is seen that the required device mapping is achieved over the frequency range of the VCO. II. SERIES FEEDBACK OSCILLATORS A negative resistance oscillator is normally realized using a series capacitor in the emitter, and a negative conductance oscillator is realized using a series inductor in the base lead. Both oscillator types use feedback to obtain a negative real component. In [8], the value of series feedback reactance required is found in terms of device impedances and, in general, this can be extended for all passive terminations and applied to any terminal of the active device. An interesting observation for both configurations is that the resulting reflection coefficient is optimum over a restricted region of the Smith chart. Here, optimum is in the sense that the resulting real part of the resonator series resistance (or shunt conductance) for a series-tuned (or shunt-tuned) resonator is minimized (maximized) to meet the criteria for oscillator startup. This criteria simply stated is that or . Compliance with these requirements requires that the complex reflection coeffibe greater than unity. Furthermore, cient of the active device there is a specific angular range of active device reflection coefficient that is found to assist in providing these conditions. However, it is not sufficient to simply have large values of . The reflection coefficient angle must be constrained to minimize the losses associated with the resonator, at least to assure oscillator startup. Thus, a specific angular range of active device reflection coefficients is found to provide these conditions. Fig. 1 plots the of the resonator as a function equivalent parallel resistance for several values of . of the reflection coefficient angle Also shown in Fig. 1 is the equivalent oscillator expressed as for . Returning to the curves, it is seen that the point where families of values converge for a reasonable is approximately 140 . Angles of the rerange of device flection coefficient which are less would require resonators with , in order to satisfy oscillator starting higher unloaded , conditions. If a large tuning range is required, then reflection coefficient angles greater than 100 are desired. Thus, it is clear that the design of the tank circuit (or resonator) and the active device interface is a methodical process to provide appropriate admittance (or impedance) variation over the tuning bandwidth of the VCO. It is not possible, therefore, to simply embed parasitics in the tank circuit and design an oscillator with the required attributes. The common base configuration used here is shown in Fig. 2. The resonator, to the left of ( – ) in Fig. 2, uses a tapped transmission line to improve the loaded and series

Fig. 1. Resistance R of a parallel (or shunt-tuned) resonator required to satisfy the condition of oscillation for: (a) j0j = 1:4, (b) j0j = 2, and (c) j0j = 4 versus the reflection coefficient angle 0. Curve (d) is the oscillator equivalent Q for j0j = 2.

Fig. 2. Common base oscillator configuration. Capacitors at (a) and (b) are modifications of the network compensating for parasitic inductances.

back-to-back varactors to increase the ac breakdown voltage [9] and the unloaded . The series feedback inductance, to the right of the ( – ) interface, includes device mounting pads, printed board traces, and film inductors. The capacitors at (a) and (b) in Fig. 2 are auxiliary compensating capacitors whose selection and function will now be described. The values of the capacitors at (a) and (b) are derived using an iterative approach that involves finding the complex load required to obtain the necessary frequency and amplitude dependence of , the reflection coefficient required of the device network presented at the ( – ) cut. Referring to Fig. 2, the development begins by assigning to be the -parameter matrix of the active circuit to the right of the ( – ) line. This -parameter network comprises the small-signal parameters of the transistor modified by the addition of series feedback and normalized to 50- source and load terminations. The network at this point , and does does not include the effects of a complex load not include the capacitors at (a) and (b). The device input reflection looking to the right of the ( – ) cut is then modified . is similar to the to reflection coefficient used in the oscillator design approach of Gonzales and Sosa [8], except that the terminations are not reto be conveniently stricted to 50 . This enables the loci of plotted as the values of the capacitors at location (a) and (b) and vary. The effects of the capacitors are incorporated in

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Fig. 3. Resonator, left of cutaway line (x–x), is separated from the active network to the right. Fig. 5. Small-signal 1=S measurement of the active device on a compressed Smith chart. (a) Proper calibration and delay. (b) Delay not applied. Marker 1: 133 j 155 (uncalibrated) at 3.8 GHz, marker 2: 55:2+ j64:9 at 4.4 GHz, marker 3: 14:1 + j 47:1 at 4.8 GHz, marker 4: 20:95 + j 29:8 at 5.4 GHz.

0

Fig. 4. Negative conductance network loaded with 50- termination. Active device and series feedback are centered on the card. A 35-ps delay is required to reference measurement at the circuit card edge. Scale is 5 : 1.

the source termination. The reflection locus curve that has the proper dependence on amplitude and frequency sets the values becomes , the device reflection of the capacitors and then coefficient with the required attributes. As previously discussed, the oscillator design process uses Fig. 1 as a guide to selecting the required input reflection coefficient of the active circuit. The next step in design is using a VNA to measure the reflection coefficient of the active device network to the right of ( – ) in Figs. 2 and 3, shown again in its measurement configuration in Fig. 4. Measurement of the active device is through the tapered tap line and includes the emitter return resistor and bypass capacitor. Here it is imperative that the correct reference plane be established. Use of short-open-load (SOL) calibration permits the reference plane to be set correctly for a 3.5-mm subminiature A (SMA) connector. The connector center pin is located right of the center cut line ( – ), as shown in Fig. 3, requiring that 35 ps of additional delay be incorporated

in calibration. This delay accounts for the offset location of the SMA open and the length of the connector center pin. The small value of corrective delay is significant, as it represents a major shift in the reflection coefficient phase required of the resonator. active device The resulting inverse reflection coefficient or locus is curve (a) in Fig. 5. From this curve, it is seen that the required resonator load is capacitive. The resonator or tank circuit is shown to the left of the ( – ) line in the oscillator schematic of Figs. 2 and 3. Measurement of the tank circuit, using a similar procedure to that described above for the active device, yielded the resonator locus shown in Fig. 6. Again, SOL calibration and correct delay adjustment is required. The resonator locus is seen to have significant parasitic series inductance, which is attributed to the varactor interconnection PC traces, as well as the pads. It is this parasitic that prevents straightforward VCO design. Design of course proceeds by matching the characteristics of the active device (see Fig. 5) and that of the tank circuit should provide (see Fig. 6). First, the small-signal device of the resonator for all tuned frequencies. Second, the rotation of should be positive and in the locus. As device self-limopposite direction of the iting occurs with an increase in the drive signal to the active and should sum to 0 . device, the argument of This should be unique for each tuning voltage and, thus, oscillation frequency. Finally, the trajectory of the limiting locus should intersect at right angles to minimize phase noise [1], [6]. In this study, these requirements are referred to as a “complement” relationship between the active device and the resonator reflection coefficient locus. Inspection of curve (a) in Fig. 5 and the resonator locus in Fig. 6 illustrates the problem in achieving the single-frequency stable oscillation condition at all tuning voltages, i.e., as limiting occurs, the trajectory of the negative conductance of the device intersects the resonator locus at multiple points, particularly around 5 GHz (marker points 3 and 4 on curve (a) in Fig. 5).

VICTOR AND STEER: REFLECTION COEFFICIENT SHAPING OF 5-GHz VOLTAGE-TUNED OSCILLATOR FOR IMPROVED TUNING

Fig. 6. Resonator locus on a compressed Smith chart showing that the resonator is dominantly inductive over the voltage tuning range. Varactor voltage increases in the direction of the arrow with increasing frequency marked: (a) from 4.5 GHz to (b) 5.3 GHz.

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Fig. 7. Modified active device reciprocal reflection 1=S curve, which rotates counterclockwise as the device limits. The incident power measurement is at +10 dBm. Marker 1: 10:5 j 95:3 at 3.5 GHz, marker 2: 679 + j 535 at 4.5 GHz, marker 3: 157 + j 335 at 4.8 GHz, and marker 4: 42:6 + j 148:5 at 5.3 GHz.

0

These conditions lead to multioscillation. A technique for addressing this problem is presented in Section III. III. REFLECTION COEFFICIENT SHAPING A technique for modifying the active device network that enables straightforward design of a single-frequency stable wideband VCO is presented here. Previously it was pointed out that the input reflection coefficient of the active device of the active network can be represented as a mapping of device as a function of collector termination. Next, additional device modification is used to modify the map. The input termination is next added to the device. The objective here then is to find the appropriate terminations at the collector and emitter terminals of the active device for a given series feedback that results must yield impedance. The corresponding versus frequency, i.e., , which a locus provides the proper interface to the resonator. If possible, the network modification should position the reflection coefficient of the modified active device in the region of the chart in Fig. 1 above 100 . The trajectory of the negative conductance as just intersects the unit device limiting occurs and where circle must complement the argument of the resonator. This is the situation shown in Fig. 7 where the new modified device characteristic was achieved by adding capacitive terminations to the collector and the emitter–base terminals. Here, unlike the conventional common base series feedback oscillator situation, the input of the active device network is now capacitive (see Fig. 7). Consequently, a portion of the parasitic inductance of the resonator is successfully absorbed. Thus, the small-signal one-port reflection coefficient is initially inductive. Normally, with a common-base oscillator, limiting at inlocus moving creasing power levels results in the device’s along lines of constant susceptance as the negative conductance of the active device decreases. Instead, with the modified network here, there is a counterclockwise rotation of the active network’s inverse reflection coefficient as limiting occurs. The discussion can now return to the oscillation condition as determined by matching the resonator locus in Fig. 6 to the mod-

Fig. 8. Multioscillation at 5.1 GHz prior to reflection coefficient shaping. Resolution BW: 3 MHz, video BW: 1 MHz, ref: 10 dBm, ATT: 20 dB.

ified active device characteristic shown in Fig. 7. Oscillation occurs when a point on the resonator locus in Fig. 6 corresponds to the point of the same frequency on the modified-device locus in Fig. 7. Under small-signal conditions, the loci may not coincide, but the important point is that they do when limiting occurs, as well as providing for the startup of oscillation. The counterclockwise rotation of the modified active device locus, as described above, assures stable single-frequency oscillation. In particular, oscillation over the frequency range from 4.5 to 5.3 GHz follows the trajectory from Point (a) to Point (b) in Fig. 6. Multioscillation, as demonstrated in Fig. 8, is suppressed in this technique. Note that in effect the resonator is operated as a shunt tunable inductance, as opposed to a tunable capacitive reactance. Here is a case where the use of two-port small-signal -parameters to manage the resonator design would not be appropriate providing little useful design insight. IV. OSCILLATOR PERFORMANCE The oscillator design procedure outlined above was followed in implementing a VCO operating from 4.5 to 5.5 GHz using an SiGe HBT. The oscillator schematic is shown in Fig. 2 and

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Fig. 9. Tuning curve showing oscillation frequency and VCO sensitivity as a function of tuning voltage.

Fig. 11. Phase noise measured at the top and bottom of the tuning range, as well as at 5.1 GHz where phase noise is optimum. Minimum phase noise floor 116 dBc/Hz at 1-kHz offset, 160 dBc/Hz at 10-MHz offset.

0

0

voltage of 0 V) and 5.3 GHz (a tuning voltage of 9 V), as well as at 5.1 GHz where the best phase noise was obtained. Phase noise was measured using a Rohde & Schwarz FSUP26 signal source analyzer and a test set loop bandwidth of 5 kHz. The phase noise is approximately the same across the tuning range noise corner frequency of 30 kHz. The phase noise with a (10 kHz), is better than 85 dBc/Hz, at 10-kHz offset, (1 MHz), is better than 130 dBc/Hz. while at 1 MHz The best measured phase noise near band center (5.1 GHz) is 135 dBc/Hz. Comparison of different oscillators requires that phase-noise measurements be normalized to the same offset frequency, shape of the phase noise which can be done assuming a where is the offset frequency so that Fig. 10. Output power and harmonics, demonstrating low-level harmonic content.

includes the active device modified by additional capacitors (a) and (b). Device characterization and circuit operation was at 5 V and 30-mA bias current. In characterizing the oscillator, the varactor tuning voltage was verified against the desired frequency sweep of range by comparing the resonator locus with the the active device. Open-loop one-port measurements were done with 10 dBm of incident power. The resonator tuning characteristics are trimmed against those of the active device. This ultimately sets the oscillator tuning gain . Additional tuning gain adjustment is controlled by the coupling between the varactor stack and the microstrip line. Average tune gain is 120 MHz/V. The tuning performance of the oscillator is shown in Fig. 9. The tune characteristic is monotonic with no jumps or discontinuities in the tuning curve as the oscillator was tuned over the full voltage tuning range. Fig. 10 presents the fundamental output power and harmonics. The fundamental output varies by less than 2 dB over the full tuning range and the harmonic levels are relatively low. The measured phase noise is shown in Fig. 11 at the ends of the tuning range, i.e., 4.5 GHz (corresponding to a tuning

MHz

MHz

(1)

Another commonly used quantitative assessment of oscillator performance is provided by the oscillator figure-of-merit (FOM), which accounts for dc power consumed [10] as follows: (2) is conventionally taken as 1 mW. For Si monolithic where VCOs, it is conventional to use just the power drawn by the VCO core, while for other technologies, including hybrid VCOs, it also is not possible to separate out a VCO core. While does not include weightings for tuning bandwidth and RF output power, it serves as a useful metric to compare like VCOs. Another FOM providing bandwidth weighting is (3) is the tuning bandwidth and is the reference where bandwidth taken here as 1 MHz. A number of tunable oscillators operating in the range 1–10 GHz are compared in

VICTOR AND STEER: REFLECTION COEFFICIENT SHAPING OF 5-GHz VOLTAGE-TUNED OSCILLATOR FOR IMPROVED TUNING

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TABLE I COMPARISON OF RF VCOS OPERATING AT POWER LEVELS GREATER THAN 10 dBm (APPROXIMATELY). PHASE NOISE AND HARMONIC SUPPRESSION ARE WORST CASE, AND RF OUTPUT POWER IS MINIMUM, ACROSS THE TUNING RANGE. THE SYMBOL (C) INDICATES THAT CORE POWER ONLY (BUFFERS ARE ADDITIONAL). ALL OSCILLATORS ARE HYBRIDS, EXCEPT AS INDICATED

0

Table I. Harmonic suppression is an important parameter with these oscillators, which are designed for direct generation of required RF power levels without subsequent buffering. For the VCO described in this paper, the conventional FOM, i.e., , is equal to or better than 182 dBc/Hz. Averaged over the 800-MHz tuning range, 0–9 V, and phase noise at 10-kHz, 100-kHz, and 1-MHz carrier offsets, the average is 184 dBc/Hz. This is among the best reported metrics for VCOs operating between 1–10 GHz. With bandwidth , the oscillator reported here is weighting, captured by the best reported for oscillators producing more than 10 dBm operating in the range of 1–10 GHz, as far as the authors are aware. The performance of oscillators in the 1–20-GHz range designed as on-chip oscillators was recently surveyed in [10]. V. CONCLUSION The standard oscillator design procedure matches the inverse of the active device to the reflection reflection coefficient coefficient of a tank circuit. Design, however, is often complicated by resonator parasitics so that the effective negative admittance of the active device satisfies the condition of oscillation at multiple frequency points. The Kurokawa oscillator condition establishes that for stable oscillation at the operating point of a negative conductance oscillator that (4) where subscript refers to the operating point. In the standard approach to oscillator design, the device susceptance is assumed

, to be independent of signal amplitude, i.e., and the loaded resonator conductance to be independent of fre, so that the stability condition bequency, i.e., comes the much simpler (5) The focus of this paper was managing the third term of (4), , while the fourth term was addressed by proper design of the resonator. This paper has introduced reactive compensation elements at the device–resonator interface that resulted in the reflection coefficient of the augmented active device having the necessary frequency dependence to compensate for the nonideal resonator characteristic. That is the technique that results in both the effective negative resistance (conductance) and susceptance of the device properly complementing the frequency-dependent admittance of the resonator including parasitics. Equally important is that the standard one-port approach to stable oscillator design can be used. The topology developed is suited to realizing stable spurious-free wideband VCOs using three-terminal devices in a common-base configuration. However, the general concept of an introduced augmentation network should be applicable to the broad class of oscillators using three-terminal active devices. The design of a 4.5–5.3-GHz voltage-tunable oscillator was presented as an example. The conventional FOM of the VCO considered is 182 dBc/Hz. Furthermore, the VCO produces a minimum output power of 0 dBm and has good harmonic suppression exceeding 20 dB over an 800-MHz bandwidth and exceeding 47 dB over a 500 MHz bandwidth.

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REFERENCES [1] K. Kurokawa, “Some basic characteristics of broadband negative resistance oscillator circuits,” Bell Syst. Tech. J., vol. 48, no. 6, pp. 1937–1955, Jul.–Aug. 1969. [2] C. Rauscher, “Large-signal technique for designing single-frequency and voltage-controlled GaAs FET oscillators,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 4, pp. 293–304, Apr. 1981. [3] R. W. Jackson, “Criteria for the onset of oscillation in microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 3, pp. 566–569, Mar. 1992. [4] D. J. H. Maclean, Evaluating Feedback in Amplifiers and Oscillators: Theory, Design and Analogue Applications. Baldock, U.K.: Res. Studies Press Ltd., 2004. [5] J. W. Boyles, “The oscillator as a reflection amplifier: An intuitive approach to oscillator design,” Microw. J., vol. 29, no. 6, pp. 83–98, Jun. 1986. [6] D. Esdale and M. Howes, “A reflection coefficient approach to the design of one-port negative resistance oscillators,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 8, pp. 770–776, Aug. 1981. [7] P. J. Topham, A. Dearn, and G. Parkinson, “GaAs bipolar wideband oscillators,” in IEE Characterization of Oscillators Design and Meas. Colloq., Feb. 3, 1992, pp. 2/1–2/4. [8] G. Gonzalez and O. J. Sosa, “On the design of a series-feedback network in a transistor negative-resistance oscillator,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 1, pp. 42–47, Jan. 1999. [9] A. P. Knights and M. J. Kelly, “Laterally stacked varactor formed by ion implantation,” Electron. Lett., vol. 35, no. 10, pp. 846–847, May 13, 1999. [10] P. Kinget, Integrated GHz Voltage Controlled Oscillators. Norwell, MA: Kluwer, 1999, pp. 353–381. [11] S.-S. Myoung and J.-G. Yook, “Low-phase-noise high-efficiency MMIC VCO based on InGaP/GaAs HBT with the LC filter,” Microw. Opt. Technol. Lett., vol. 44, no. 2, pp. 123–126, Jan. 20, 2005. [12] C.-H. Lee, S. Han, B. Matinpour, and J. Laskar, “Low phase noise X -band MMIC GaAs MESFET VCO,” IEEE Microw. Guided Wave Lett., vol. 10, no. 8, pp. 325–327, Aug. 2000. [13] Z. Q. Cheng, Y. Cai, J. Liu, Y. Zhou, K. M. Lau, and K. J. Chen, “A low phase-noise X -band MMIC VCO using high-linearity and Ga N=GaN low-noise composite-channel Al Ga N=Al HEMTs,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 1, pp. 23–28, Jan. 2007. [14] H. Zirath, R. Kozhuharov, and M. Ferndahl, “Balanced Colpitt oscillator MMICs designed for ultra-low phase noise,” IEEE J. Solid-State Circuits, vol. 40, no. 10, pp. 2077–2086, Oct. 2005. [15] Y.-K. Chu and H.-R. Chuang, “A fully integrated 5.8-GHz U-NII band 0.18- m CMOS VCO,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 7, pp. 287–289, Jul. 2003. [16] C. C. Meng, Y. W. Chang, and S. C. Tseng, “4.9-GHz low-phase-noise transformer-based superharmonic-coupled GaInP/GaAs HBT QVCO,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 6, pp. 339–341, Jun. 2006. [17] C. C. Meng, C. H. Chen, Y. W. Chang, and G. W. Huang, “5.4 GHz–127 dBc/Hz at 1 MHz GaInP/GaAs HBT quadrature VCO using stacked transformer,” Electron. Lett., vol. 41, no. 16, pp. 906–908, Aug. 2005. [18] T. M. Hancock and G. Rebeiz, “A novel superharmonic coupling topology for quadrature oscillator design at 6 GHz,” in Proc. IEEE Radio Freq. Integr. Circuit Symp., Jun. 2004, pp. 285–288. [19] S.-W. Yoon, S. Pinel, and J. Laskar, “A 0.35- m CMOS 2-GHz VCO in wafer-level package,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 229–231, Apr. 2005. [20] J.-H. Yoon, S.-H. Lee, A.-R. Koh, Kennedy, P. Gary, and N.-Y. Kim, “Optimized phase noise of LC VCO using an asymmetric-inductance tank in InGaP/GaAs HBT technology,” Microw. Opt. Technol. Lett., vol. 48, no. 6, pp. 1035–1040, Jun. 2006.

[21] O. T. Esame, I. Tekin, A. Bozkurt, and Y. Gurbuz, “Design of a 4.2–5.4 GHz differential LC VCO using 0.35 m SiGe BiCMOS technology for IEEE 802.11a applications,” Int. J. RF Microw. Computer-Aided Eng., vol. 17, no. 2, pp. 243–251, Mar. 2007. [22] A. P. M. Maas and F. E. van Vliet, “A low-noise X -band microstrip VCO with 2.5 GHz tuning range using GaN-on-SiC p-HEMT,” in Proc. GaAs Conf.—13th Eur. Gallium Arsenide and Other Compound Semiconduct. Applicat. Symp., Oct. 2005, pp. 257–260. [23] J.-H. Yoon, S.-H. Lee, A.-R. Koh, B. Shrestha, S.-H. Cheon, G. P. Kennedy, and N.-Y. Kim, “A novel harmonic noise frequency filtering VCO for optimizing phase noise,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1805–1808. [24] C. C. Meng, S. C. Tseng, Y. W. Chang, J. Y. Su, and G. W. Huang, “4-GHz low-phase-noise transformer-based top-series GaInP/GaAs HBT QVCO,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1809–1812.

Alan Victor (S’68–M’72–SM’07) received the B.S.E.E. degree from the University of Florida, Gainesville, in 1971, the M.S.E. degree from Florida Atlantic University, Boca Raton, in 1981, and is currently working toward the Ph.D. degree at North Carolina State University, Raleigh. He is currently a Senior Scientist with Harris Stratex Networks (formerly Harris Microwave), Morrisville, NC, where he is involved with millimeter-wave transceiver design. Prior to joining Harris Stratex Networks, he was with Motorola Communications and IBM Microelectronics. He also cofounded a wireless local area network (LAN) manufacturing company providing data communication products to the auto identification industry. His main interests are power oscillators and the application of ferroelectric materials. Mr. Victor is a member of Eta Kappa Nu.

Michael B. Steer (S’76–M’82–SM’90–F’99) received the B.E. and Ph.D. degrees in electrical engineering from the University of Queensland, Brisbane, Australia, in 1976 and 1983, respectively. He is currently the Lampe Professor of Electrical and Computer Engineering with North Carolina State University, Raleigh. In 1999 and 2000, he was Professor with the School of Electronic and Electrical Engineering , The University of Leeds, where he held the Chair in Microwave and Millimeter-Wave Electronics. He was also Director of the Institute of Microwaves and Photonics, The University of Leeds. He has authored over 360 publications on topics related to RF, microwave and millimeter-wave systems, high-speed digital design, and RF and microwave design methodology and circuit simulation. He coauthored Foundations of Interconnect and Microstrip Design (Wiley, 2000). Prof. Steer is active in the IEEE Microwave Theory and Technique Society (IEEE MTT-S). In 1997, he was secretary of the IEEE MTT-S, and from 1998 to 2000, he was an elected member of the IEEE MTT-S Administrative Committee (AdCom). He was the Editor-in-Chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (2003–2006). He was a 1987 Presidential Young Investigator (USA). In 1994 and 1996, he was the recipient of the Bronze Medallion presented by the U.S. Army Research Office for “Outstanding Scientific Accomplishment.” He was also the recipient of the Alcoa Foundation Distinguished Research Award presented by North Carolina State University in 2003.

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A Low-Loss 74–110-GHz Faraday Polarization Rotator Neal R. Erickson, Member, IEEE, and Ronald M. Grosslein

Abstract—We have developed a switchable Faraday polarization rotator for 74–110 GHz with a typical room-temperature insertion loss of 0.6 dB and a maximum of 1.3 dB. The device uses a cylindrical ferrite rod in the HE11 mode with an axial magnetic field applied by a solenoid. The ports are square metallic waveguides transitioning to ceramic tapers, which then couple to the ferrite rod. The device is designed for use at a temperature of 20 K, where typical loss is 0.5 dB. The design rotation is 45 about the zero-bias value, but up to 90 is possible. The switching time is 10 s. Index Terms—Faraday rotator, ferrite devices, ferrite switch, millimeter wave.

I. INTRODUCTION IGH electon-mobility transistor (HEMT) amplifiers are now available that cover full waveguide bands in the millimeter-wave range with noise temperatures low enough to enable very sensitive radiometry and spectroscopy. However, fluctuations in their gain [1], particularly at the large cryogenic temperatures, lead to much degraded sensitivity over wide bandwidths unless their input can be switched at 1 kHz or they are used in a correlation mode. Mechanical optical beam choppers are limited to 50 Hz, far too low to be effective. Fast switches using p-i-n diodes are available for -band, but their bandwidths are quite limited, and their loss is fairly high. Ferrite switches based on a Y-junction circulator geometry have low loss and sufficient speed, but also limited bandwidth. The correlation receiver geometry [2] is complex, and the required hybrids, phase switches, and phase matching limit the bandwidth. In the application motivating this study, a dual-polarized receiver using monolithic microwave integrated circuit (MMIC) HEMT amplifiers was required to cover 74–110 GHz with 31-MHz spectral resolution [3], and achieve true radiometric noise across the full band. None of the existing technologies seemed adequate to assure this performance. A fast switch may be built using linear polarization rotation in a ferrite in a dual-mode waveguide. If the polarization is switched 45 , then the addition of a polarization splitter such as a wire grid or an ortho-mode transition (OMT) turns the device into a two-way switch [4]. An advantage to this approach is that a dual-polarized beam may be switched with no increase in complexity. Thus far, ferrite rotators have been limited to

H

Manuscript received May 10, 2007; revised August 20, 2007. This work was supported in part by the National Science Foundation under Grant AST-0096854. The authors are with the Astronomy Department, University of Massachusetts, Amherst, MA 01003 USA (e-mail: [email protected]; [email protected]. umass.edu). Digital Object Identifier 10.1109/TMTT.2007.909871

relatively narrow bands. Most research has been done using ferrite rods inside a metallic waveguide [4], but a filled metallic waveguide has sufficient dispersion in the rotation angle to limit bandwidths to 15% for 20-dB cross-polarization isolation [5]. A partially filled waveguide has similar limitations, but a quadruple ridged waveguide [6]–[8] increases bandwidth to 25%, and a triple-ridged waveguide [9] is somewhat better. While potentially promising for wider bandwidth, this type of waveguide is very poorly suited to millimeter-wave construction. Rotators in free space [10], [11] can easily achieve the needed bandwidth, but are best suited for fixed bias using permanent magnetization of the ferrite. Switching the field over the large area of the ferrite plate would require a very large solenoid. The use of a dielectric waveguide within the rotator almost entirely eliminates dispersion, and very wide bandwidths may be achieved with a very compact device. One such device in the -band [12] with a single polarization input and a dual polarized output achieved 24% bandwidth, limited by resonances with high loss, while a moderate bandwidth dual-polarization rotator has been built in the -band [13]. The need for 40% bandwidth in this application appears best met by the dielectric waveguide approach, making the device very compact so that it can be rapidly switched with a small solenoid. In this work, we have made a very careful study of mode control in coupling into a ferrite rod, and have designed a transition with very little mode conversion over a full waveguide band. In related work, we have developed a full-band OMT [14], and together these make a dual-polarization two-way switch with a switching speed of 10 kHz. The switch uses a low-inductance solenoid for the magnetic bias, and the entire assembly is designed for operation at 20 K. II. ROTATOR MICROWAVE DESIGN hybrid mode in a dielectric Faraday rotation using the rod has been used for many years in the construction of isolators and modulators [15], [16]. Uniform rotation is obtained over a full waveguide band if the ferrite rod diameter is large enough so mode is largely contained inside. The that the field of the degree of confinement within a dielectric rod shows quasi-cutoff behavior, leading to a critical diameter for a given mode [17]. By choosing the diameter properly, the degree of confinement can be selectively controlled between the lowest mode and higher modes. The much more difficult problem is exciting only the desired mode over a wide band, and selectively attenuating the unwanted modes so that they do not lead to resonances with attendant loss of power. The input metallic waveguide to a rotator can be either square or round, and for this work, we required that the walls would be smooth (not corrugated). The general method of coupling from

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Fig. 1. Model of waveguide transition used to design the rotator. Port 1 is round or square, both tapers are simple linear tapers, and port 2 is a circular dielectric waveguide, with no metal boundary. Port 2 connects to the ferrite rod.

this waveguide to the ferrite rod is shown in Fig. 1, which is modeled after the method used in isolators, which works extremely well over a full waveguide band. A high dielectric constant tapered rod is inserted into the metal waveguide, and as the rod increases in diameter, the metal walls decrease in size until all of the field is in the rod. This rod then passes out the open end of the waveguide and the mode becomes a pure dielectric waveguide mode. The taper then contacts the ferrite rod where the rotation occurs, and the process is reversed at the other end of the rod. The ferrite was chosen to be an Ni ferrite, type TransTech TT2-111,1 since this is one of very few low-loss millimeterwave materials. The rod diameter was chosen to be 1.2 mm for good confinement at 75 GHz. The equation relating rotation to field and length is (1) is the axial magnetic field, is the gyromagnetic where ratio rad/s Oe (where is 2.11 for TT2-111), is the ferrite length, is the speed of light, and is the dielectric constant of the ferrite 12.5. This formula is correct only in the limit where the frequency is well above resonance as follows: (2) which at saturation is 14 GHz. From (1), the minimum length for 45 rotation at a saturated field of 5000 G is 1.35 mm. We chose a length approximately twice this long so that a very high field is not needed. The matching taper should have a similar dielectric constant, and 10 (alumina) is a close match. For this ferrite, 1 since the frequency is well above resonance so the ferrite may be treated as a simple dielectric for purposes of impedance matching. The presence of a bias field produces a relatively small change in the permeability tensor, and impedance matching is only slightly affected [18], but the detailed transmission behavior will change with bias. For simplicity of construction, only linear tapers were considered for both the waveguide and dielectric. This geometry was chosen because it can be built accurately and because slow tapers usually make efficient mode converters. At these high frequencies, parts become so small that fabrication tolerances limit the choice of shapes. A 3-D electromagnetic (EM) simulator [19] was used to calculate the coupling efficiency of the waveguide mode into the dielectric using a taper of variable length. The computer model used a metallic waveguide port on one side and a dielectric port on the other, as in Fig. 1. The goal of this study was to design a complete rotator with a loss 0.5 dB over a band of 74–110 GHz (consistent with the performance of the WR10 isolators). This 1Ferrite

TT2-111, TransTech Corporation, Adamston, MD.

requires transitions with a mode conversion loss 0.1 dB once other losses are considered. The rotator transition appears essentially the same as the one used in the isolator, yet the coupling is found to be much poorer when the metal waveguide has a square cross section, rather than rectangular. The essential problem may be seen with one simple study. Considering first a uniform rectangular outer cross section (2.54 1.27 mm), and an inner taper with , we find 0.5-dB loss from 74–108 GHz in coupling from the single metal waveguide mode to the lowest order dielectric waveguide mode. If we change just the outer waveguide to add a taper in width over the length of the dielectric taper (ending at 1.27-mm square), we find this modification is actually beneficial to the performance, increasing bandwidth (for the same loss) to 75–112 GHz. If we then further modify the transition to include a height taper (as well as width) so that the outer cross section varies from 2.54- to 1.27-mm square, we find the bandwidth (for the same loss) degrades to 78–106 GHz with 0.1-dB loss across the entire band, and 3-dB loss at 110 GHz. In each of these cases, the loss is dominated by mode conversion. In each case, the same mode is used in the ferrite rod and the same modes are present in the transition region where the metal waveguide is partially filled. Even the higher mode cutoff frequencies in the most filled part of the transition are relatively little affected by the metal waveguide cross section. , there are three Above 103 GHz, with a taper having propagating modes in the partially filled cross section having the proper - and -plane symmetry to be excited with two modes above 85 GHz. Thus, it appears that the very efficient transition in the case of the rectangular (isolator) cross section is fortuitous. No taper length with simple geometry could be found that worked well over the full band so other options were considered. Lowering the dielectric constant of the taper has little effect on the cutoff frequencies of the higher modes in the most filled part of the transition because the final diameter must increase. However, lowering does have the effect of reducing the excitation amplitude for these modes because there is a lower dielectric contrast between the taper and surrounding air. was reduced in steps, and at a value of 7, coupling could be made very high with 0.1-dB loss over a full waveguide band. However, the lower increases the reflection at the interface with the ferrite, requiring the addition of an impedance transformer between the faces. This is done with a quarter-wave-thick disk having 9.4. Simulations using a metal waveguide with circular cross section gave no better results, and even the lower taper did not work acceptably. More complex waveguide cross sections were rejected because of the difficulty in fabrication. Mode matching alone is insufficient to ensure low loss through the complete structure because it neglects the effect of resonant enhancement of loss due to trapped modes. When the complete structure was simulated, treating the ferrite as a simple dielectric, large resonances were observed with losses 3 dB. These are due to modes excited within the dielectric waveguide, having little coupling amplitude to the metal waveguide. The typical means to eliminate such resonant modes is to add an absorber around the ferrite to selectively add loss to these modes since they have poorer confinement within the

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Fig. 2. Cross section through the rotator. All parts are round, except the waveguides, which are square.

Fig. 3. Insertion loss (solid line) and return loss (dashed line) predicted for WR10 rotator at zero magnetic bias.

rod. However, some of the higher mode fields do not extend very far outside the rod, and any significant loss for them adds some loss to the desired mode. Acceptable mode suppression required modifying the geometry, making the metal waveguide ends very thin to minimize reflections from the surfaces, and to scatter power outward. The ferrite diameter was also made smaller relative to that of the taper, reducing the confinement of the higher modes within it. Ideally, EM simulations should be run with magnetic bias in the ferrite so that all of the effects of Faraday rotation may be simulated. Long solution times and large memory requirements make this full solution impractical. With magnetic bias, no symmetry boundaries can be used, increasing the solution volume by a factor of four, and magnetic materials double the size of field calculations relative to simple dielectrics. Supports for the rod and cone structure are two thin Mylar disks. Simulations show little effect by their addition. Mylar has very low thermal conductivity so the rod may not dissipate any significant power in operation or be exposed to thermal radiation loading. While this study covered many design options, it is certain that a better design is possible. No computer-aided optimization was used due to the long solution times. The best overall geometry is shown in Fig. 2, and simulation results in Fig. 3 show that the loss can be less than 0.5 dB from 72 to 111 GHz, with an input return loss exceeding 20 dB over nearly this full band. However, as will be discussed in comparison with measurements, the results obtained are sensitive to materials properties, which were not well known at the time of the design. The solenoid coil used to bias the ferrite was constrained to have low enough inductance to allow switching with a rise time 10 s with a drive voltage 30-V peak. Using a magnetics design program [20], a solenoid coil was designed having the optimum aspect ratio to bias the ferrite with the least number of ampere turns of a given size wire. The resulting inductance is 0.33 mH and the required current is 0.65 A for 45 rotation. This current is close to predictions, but because the magnetization in a short ferrite cylinder is quite inhomogeneous, it is not possible to accurately determine the effective field for a given current. The very fast switching can cause significant eddy-current heating of the surrounding metal structure. Assuming a square wave modulation, this heating scales linearly with frequency, and also increases as 1/rise time. Eddy-current heating is most

important in high field regions, and the waveguide ends, which extend into the solenoid have the highest field. If these were made of copper, the heating would be very large at 20 K since its resistance becomes very small. However, if the waveguide is made of stainless steel, then the resistance remains high, even at low temperature, and with thin walls, the heating should be small. It is necessary to plate the inside of the waveguide with a good conductor to keep the RF loss low, but a skin depth of 1 m produces little extra heating. It is also Cu plating necessary to keep other metal parts relatively far from the coil to minimize distortions of the field during fast switching. It is difficult to predict this heating given poorly known low-temperature conductivities. Measurements of a complete unit at 18 K driven with a 0.65-A 1-kHz square wave with 10- s rise time indicate a heating of 0.11 W. The resistance of the coil (using copper wire) is 0.1 and dissipation is very small at 20 K, but thermal runaway is a potential problem since the resistance of Cu is a steep function of temperature. Thermal runaway is prevented by heat sinking the windings of the coil. Both ends of each winding layer are brought out and soldered to the top metal on thin 1-mm squares of AlN, which are glued to the copper support block. The coil was tested to twice the required current (at 20 K) with no evidence of runaway. Heat sinking is particularly important because the entire structure is cooled by conduction to the outer surface. The only thermal path for heat to enter or leave the ferrite rod is via the solenoid, and the rod must be as cold as possible to minimize the noise added to the receiver. At room temperature, because of the greatly increased dissipation, the coil cannot be operated continuously, although relatively short periods of operation have caused no problems. III. CONSTRUCTION DETAILS The final WR10 model required some unconventional materials. The tapers are made of a forsterite ceramic,2 with , and a measured loss tangent 0.0005 at 100 GHz. This material is somewhat soft and weak, and the tapers are fragile. The matching disks are made from a ceramic loaded circuit board material,3 with . This material is machinable and it is relatively easy to make the small disks. The loss is unknown at 2TransTech

Corporation, Dielectric material DS-6, Adamstown, MD.

3TMM-10,

Rogers Corporation, Chandler, AZ.

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TABLE I MATERIALS AND DIMENSIONS OF PARTS USED IN ROTATOR

Fig. 4. Room-temperature insertion loss of rotator #1 at rotation of 0 (heavy curve), and 45 (light dashed curve) with an OMT on one port and a horn on the other. The loss is nearly the same in both rotations.

100 GHz, but the disks are quite thin. The absorber surrounding the ferrite rod is carbon-loaded epoxy made by adding lamp black carbon to a low-viscosity epoxy. The microwave properties of this material are not well known, but it seems to be a better matched load at 100 GHz than any commercial material. Table I lists the materials used and their dimensions. All of the parts were assembled using an epoxy able to survive wide thermal cycling. The taper-ferrite assembly was aligned on axis to within 0.05 tilt. The stainless-steel tapered waveguides were then optically aligned relative to the tapers to within 5- m offset. At the ends of the square waveguides, removable step transitions were used to convert to circular waveguide for some tests. The entire assembly is 2.4-cm long and 2.0-cm square. IV. PERFORMANCE Switching speed (at room temperature) was measured in a transmission setup using a pulse generator and power amplifier able to supply a pulse with a current rise time of 8 s. The measured response time for 90 change of rotation was 15 s (to 90% of final amplitude), indicating that the ferrite itself has a switching time of 10 s. At 18 K, the switching time, including a faster driver, is 8–10 s. Testing of the microwave properties is complicated because of the difficulty in launching a mode into a square or circular waveguide with high modal purity and low reflectivity in any higher mode. The problem is that two higher modes propagate within the square waveguide above 83.4 GHz, and these modes couple to higher modes within the taper-ferrite assembly. Any asymmetric discontinuity in the square waveguide (including typical offsets at flange joints) excites these modes, leading to the appearance of resonances that are artifacts of the setup. Ideally the rotator should be designed to suppress these modes, but the typical solution, i.e., a quad-ridged waveguide, is impractical at these frequencies. Initial tests used a pair of full-band OMTs with the rotator between them, but many narrow resonances appeared in the transmission that could possibly be attributed to the these higher modes, as well as the voltage standing-wave ratio (VSWR) of the OMTs. An improved setup used one OMT as a rectangular to square waveguide transition for the signal input with a corrugated circular feed horn on the output. After

20 cm to attenuate higher modes, the signal was re-coupled into a second identical horn. This setup still had some narrow features, but they were much reduced in amplitude. The advantage of the free-space coupling is that it can be made lossy for all modes and rotation can be tested by simply rotating the receiving horn. These tests can only be performed at room temperature. The insertion loss measured in this way for zero bias and 0.65-A bias (45 rotation) is shown in Fig. 4. The setup was normalized with the OMT in place and then the rotator was added. The minimum midband loss of 0.5 dB is believed to be entirely due to waveguide and dielectric losses since the predicted mode conversion is very small. All significant features in the loss are unchanged when the rod/taper assembly is rotated by 45 within the complete block, and a second rod/taper assembly shows the same general features. 20 dB from 78 to 110 GHz The return loss (unbiased) is and 15-dB worst case at 76 GHz, but because of the need for transitions from the square to rectangular waveguide (with a circular section between), the data may not represent the true performance of the rotator. The polarization purity of the rotator output was determined by measuring the best power null as the receiving horn is rotated. This null is 20-dB deep across the band with no bias, meaning that some circular component is present. Circular polarization arises because the unbiased ferrite retains a random magnetization, and any component at a right angle to the axis induces a phase shift between linear polarizations. With high bias, the residual field is largely eliminated, leading to the results shown in Fig. 5. At 0.65-A bias (45 rotation), the best null is 25 dB, except at the two resonances noted before. The high isolation level across the 75–114-GHz band demonstrates that the frequency dispersion of the rotation is 4 , as is expected for this type of device. Positive and negative bias give somewhat different results, probably because of slight residual fields. While the rotator is designed for 20-K operation, it is difficult to measure its properties with much accuracy at this temperature. In a practical sense, the most important measurement is the degradation of the noise at the input to the wideband low-noise receiver with which it is to be used. A MMIC HEMT amplifier with an input isolator (operating at 16 K) was characterized over the 75–110-GHz band with a circular corrugated feed horn and an OMT used to transition to the waveguide of the horn. The amplifier alone had a noise temperature of 40–60 K over the band, which increased to 55–70 K with the OMT and feed horn.

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Fig. 5. Cross polarization isolation measured at 0.65-A bias corresponding to 45 rotation. The two curves are for the different directions of rotation.

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Fig. 6. Insertion loss of rotator #1 at 20 K, with 45 rotation, as derived from the noise added to a radiometer. Resolution is 0.1 GHz.

When the unbiased rotator was added the noise temperature increased by 10 K at the worst spots, but over most of the band, the increase was 5 K. Biasing the rotator to 0.65 A for 45 rotation increased the noise 10 K, except at two resonances (as seen at room temperature, but shifted somewhat up in frequency), where the increase was 20–30 K. Fig. 6 shows the loss inferred from added noise using the measured rotator temperature of 16 K. A second rotator was built with the metal waveguides each 0.12 mm closer toward the center, and the room-temperature resonances are significantly reduced, but the loss above 109 GHz increases. The overall noise temperature of a second amplifier with this rotator (biased to 45 rotation) on the input is shown in Fig. 7, and the inferred peak loss below 108 GHz is 0.5 dB. While the bandwidth is as predicted, the most striking difference between measurements and the simulation in Fig. 3 is in the presence of deeper resonances, which increase in depth substantially at low temperature. These features result from insufficient loss in the absorber surrounding the ferrite. This material was initially simulated as and , but further simulations to fit the data indicate that its loss is much lower, at room temperature and at 20 K.

Fig. 7. Noise temperature of a complete receiver with rotator #2 on the input. Rotator is biased to 45 . The noise includes the OMT, feed horn, and vacuum window loss. Noise is measured at 100-MHz resolution.

Fig. 8. Rotator predicted insertion loss with materials properties adjusted to fit measurements at room temperature and 20 K. See text for values. Ferrite is at zero magnetic field. Return loss is for room temperature only.

The loss of this material, being due to carbon, is expected to decrease at low temperature. The dielectric constant and loss of the principal materials were adjusted in simulations to bring the frequencies of resonances and the overall loss into rough agreement. Resonant frequencies were found to be primarily sensitive to of the ferrite and its dimensions. The best fit to the room-temperature data is and , and at 20 K, and . Varying for the tapers produced only poorer was known for this material from actual fits to the data and measurements. The results of simulations at zero rotation are shown in Fig. 8 with the best fit values noted above for room temperature and 20 K. The fit is generally good, but the resonance near 89 GHz is not actually observed at room temperature. While the resonances can be reduced by increasing the loss of the absorber, the overall loss of the rotator begins to increase so it may be difficult to find an approrapidly for priate material. An alternative is to shorten the ferrite and shift the resonances out of band. Simulations of the identical structure with a ferrite rod 1.65-mm long show 0.2-dB resonances, even with an absorber with . This would require a much higher magnetic field (and coil current), which is practical

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ferrite rod. These higher modes may originate at some distance from the rotator, leading to fine-scale ripple. This loss may also be caused in part by inhomogeneities within the ferrite material, including residual magnetization. VI. RADIOMETRIC PERFORMANCE

+

0

Fig. 9. Differential loss between the 45 and 45 rotation states for both polarizations passing through the same rotator, as inferred from radiometric measurements. The rotator temperature is 18 K.

This rotator is used as the input beam switch for a very wideband radiometer [3] used for astronomical spectral observations on the FCRAO 14-m radome enclosed telescope. With the switch rate at 1 kHz, the receiver continuum offset is 2 10 of the total power, and such a small total power offset is expected due to the radome environment. The dome should have no spectral character, and spectra taken over an 24-GHz band with 31-MHz resolution show no spectral features to within the noise after 2 h of integration. This demonstrates that in a real environment where the background level may be as much as 100 K, the switch still is completely effective in noise in a millimeter-wave HEMT amplifier. eliminating VII. CONCLUSIONS

with the current design, although eddy-current heating would increase by a factor of 2.8. V. VARIATION IN LOSS WITH ROTATION This device was designed for use as a switch on the input to a radiometer so an important measure of performance is its differmode should show ential loss in the two states. While the no change in behavior with rotation, the higher modes make the behavior more complicated. Since the overall loss does depend on rotation, it is essential to operate it in a symmetric mode where the rotation switches 45 . In this mode, there is still some difference in loss, which has been determined through a radiometric measurement, viewing an unpolarized load through the rotator. In these tests, the complete receiver (amplifier, OMT, rotator, and horn) is operated at 18 K. With the load at the same temperature as the rotator, there should be no signal, but any other input temperature will produce an offset. With the load at a temperature of 77 K, the differential signal was measured for both polarizations and converted to the loss shown in Fig. 9. Note that both polarizations show similar general behavior, but with substantial differences. As expected, the differential signal increases by a factor of 4 with the load at 295 K, consistent with loss, and not some other effect. This loss is most important when the input background temperature is high, and in the intended astronomical application the background temperature is typically 30 K, producing an offset signal 0.3 K. This offset should be quite stable. The differential loss is sensitive to the symmetry of the two bias states chosen and to any residual field in the ferrite so the data were taken with an adjustable offset, which was tuned to produce the smallest average differential loss. The differential loss significantly depends on external components, and it is not certain how much is inherent to the rotator. In tests, it was observed that changing or even just rotating the OMT that launched the mode into the square input waveguide significantly affected the differential loss, particularly at frequencies above 83.5 GHz where higher modes may propagate. This loss arises mostly from power coupled into higher modes, which lack circular symmetry or are poorly confined within the

We have developed a full-waveguide-band polarization rotator with extremely low loss and a switching speed 10 s. It can rotate a linear polarization by any amount up to 90 . The device is intended for 20-K operation, but could be operated at room temperature with some modifications. It is well suited for polarimetry or the current application as a two-way switch. This fast switch can entirely overcome the gain fluctuation noise present in millimeter-wave HEMT amplifiers. REFERENCES [1] N. C. Jarosik, “Measurements of the low-frequency-gain fluctuations of a 30-GHz high-electron-mobility-transistor cryogenic amplifier,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 2, pp. 193–197, Feb. 1996. [2] J. J. Faris, “Sensitivity of a correlation radiometer,” J. Res. Nat. Bureau Standards Sect. 71C (Eng. and Instrum.), pp. 153–170, Apr.–Jun. 1967. [3] N. Erickson, G. Narayanan, R. Goeller, and R. Grosslein, “An ultrawideband receiver and spectrometer for 74–110 GHz,” in From Z -Machines to ALMA: (Sub)Millimeter Spectroscopy of Galaxies, 2007, vol. 375, pp. 71–81, Astron. Soc. Pacific Conf. Ser. [4] C. R. Boyd, Jr., “High power reciprocal ferrite switches using latching Faraday rotators,” in IEEE S-MTT Int. Microw. Symp. Workshop, Philadelphia, PA, Jun. 2003, Issues in Ferrites and Dielect. High-Power Applicat. [5] W. E. Hord, F. J. Rosenbaum, and J. A. Benet, “Theory and operation of a reciprocal Faraday-rotation phase shifter,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 2, pp. 112–119, Feb. 1972. [6] H. N. Chait and N. G. Sakiotis, “Broad band ferrite rotators using quadruply-ridged circular waveguide,” IRE Trans. Microw. Theory Tech, vol. MTT-7, no. 1, pp. 38–41, Jan. 1959. [7] B. M. Dillon and A. A. P. Gibson, “Finite element solution for Faraday rotation section,” in 8th Int. Antennas Propag. Conf., Edinburgh, U.K., 1993, pp. 226–229. [8] A. Baird and W. T. Nisbet, “Ferrite polar selectors for signal separation in satellite receivers,” in IEE Ferrite Materials, Devices, Applicat, Colloq., Edinburgh, U.K., 1989, pp. 10/1–10/4. [9] B. M. Dillon and A. A. P. Gibson, “Broadband Faraday rotation sections using three ridges,” IEEE Microw. Guided Wave Lett., vol. 4, no. 1, pp. 83–85, Jan. 1994. [10] G. F. Dionne, J. A. Weiss, G. A. Allen, and W. D. Fitzgerald, “Quasioptical Faraday rotator for millimeter waves,” in IEEE MTT-S Int. Microw. Symp. Dig., 1988, pp. 127–130. [11] G. M. Smith, C. P. Unsworth, M. R. Webb, and J. C. G. Lesurf, “Design, analysis and application of high performance permanently magnetized, quasi-optical, Faraday rotators,” in IEEE MTT-S Int. Microw. Symp. Dig., 1994, pp. 293–296. [12] C. E. Barnes, “Further developments in dielectric waveguide devices for millimeter wavelengths,” in IRE Professional Group on Microw. Theory Tech. Microw. Symp., 1962, pp. 107–111. [13] B. G. Keating, P. Ade, J. J. Bock, E. F. Hivon, W. L. Holzapfel, A. E. Lange, and H. Nguyen, “A large angular scale CMB polarimeter,” Proc. SPIE, vol. 4843, pp. 284–295, 2003.

ERICKSON AND GROSSLEIN: LOW-LOSS 74–110-GHz FARADAY POLARIZATION ROTATOR

[14] G. Narayanan and N. Erickson, “A novel full waveguide band orthomode transducer,” in 13th Int. Space Terahertz Technol. Symp., Mar. 2002, pp. 505–514. [15] C. E. Barnes, “Broad-band isolators and variable attenuators for millimeter wavelengths,” IRE Trans. Microw. Theory Tech., vol. MTT-9, no. 6, pp. 519–523, Nov. 1961. [16] N. R. Erickson, “Very low loss wideband isolators for mm-wavelengths,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, May 2001. [17] G. C. Southworth, Principles and Applications of Waveguide Transmission. Princeton, NJ: Van Nostrand, 1950. [18] C. R. Boyd, “Impedance matching considerations for ferrite Faraday rotators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2371–2374, Jul. 2005. [19] High Frequency Structure Simulator (HFSS). Ansoft Corporation, Pittsburgh, PA, 2003. [20] Maxwell SV. Ansoft Corporation, Pittsburgh, PA, 2003. Neal R. Erickson (M’78) received the B.S. degree from the California Institute of Technology, Pasadena, in 1970, and the Ph.D. degree from the University of California at Berkeley, in 1979. Since 1979, he has been with the Astronomy Department, University of Massachusetts at Amherst, where he is currently a Research Professor. He has been extensively involved in the field of low-noise millimeter- and submillimeter-wave receiver systems and astronomical observations with these systems. He has also designed Schottky diode mixers and

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varactor frequency multipliers, quasi-optical systems, and low-noise HEMT amplifiers, and much of the receiver system for the Submillimeter Wave Astronomy Satellite. In 1982, he cofounded Millitech, and in 2000, he founded Erickson Instruments LLC.

Ronald M. Grosslein received the B.A. degree (cum laude) in physics from Amherst College, Amherst, MA, in 1978. In 1979, he joined the Five College Radio Astronomy Observatory, University of Massachusetts at Amherst, where he is currently a Research Engineer. He assists with the design, construction, system integration, testing, installation, troubleshooting, and maintenance of millimeter-wave cooled receivers. His current interests include Dewar design and extending conventional machining techniques to smaller size scales.

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Finite-Difference Time-Domain Modeling of Periodic Guided-Wave Structures and Its Application to the Analysis of Substrate Integrated Nonradiative Dielectric Waveguide Feng Xu, Member, IEEE, Ke Wu, Fellow, IEEE, and Wei Hong, Senior Member, IEEE

Abstract—The finite-difference time-domain (FDTD) method incorporating an equivalent resonant cavity model is presented for the modeling and analysis of guided-wave propagation characteristics of complex periodic structures. By transforming electromagnetic field variables into a new set of periodic variables, which can also be resolved from the Maxwell’s equations, one can convert a periodic guided-wave problem into an equivalent resonator problem. Thus, the FDTD method used for a resonant cavity problem can be adopted to simulate periodic guided-wave structures. In addition, the proposed FDTD algorithm can be extended to model lossy periodic propagation problems. In this study, the substrate integrated nonradiative dielectric waveguide, which is a special type of periodic guided-wave structure subject to a potential leakage loss due to its periodic gaps, is investigated as a showcase. The proposed method is first validated and is then used to analyze the guided-wave characteristics of substrate integrated nonradiative dielectric waveguides. It is shown that the substrate integrated nonradiative dielectric waveguide structure, which can easily be fabricated in planar form, has a well-behaved propagation property suitable for high-performance millimeter-wave circuit design. Index Terms—Complex propagation constant, equivalent resonant cavity model, finite-difference time-domain (FDTD) method, periodic guided-wave structure, quality factor, substrate integrated nonradiative dielectric waveguide.

I. INTRODUCTION

T

HE finite-difference time-domain (FDTD) method, first introduced by Yee [1], is a powerful, robust, and popular modeling algorithm based on the direct numerical solution of the differential Maxwell’s equations in the time domain. For its application in the modeling of guided-wave structures, a compact 2-D FDTD method has been proposed and widely applied to calculate the propagation characteristics of guided-wave structures [2]–[6]. The 2-D FDTD approach takes the advantages of CPU time reduction and memory saving over its 3-D Manuscript received May 18, 2007; revised August 9, 2007. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and in part by the National Science Foundation of China under Grant 60621002. F. Xu and K. Wu are with the Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montreal, Montreal, QC, Canada H3C 3A7 (e-mail: [email protected]; [email protected]). W. Hong is with the State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910059

counterparts. The propagation constant is given as an input parameter and the eigenfrequencies of interest are found via discrete Fourier transform (DFT). Some authors have tried to extract attenuation constants by introducing a damping factor [7]–[9]. However, the simulation accuracy largely depends on the samples of transient signals. In some cases, it is difficult to obtain accurate results. As a result, an iterative technique is proposed to improve the simulation accuracy [10]. An extension of the 2-D FDTD method was applied to the analysis of periodic guided-wave structures in [11]. Besides, a spatially looped time-domain algorithm was also used for the analysis of periodic structures [12]. Subsequently, these types of methods has been widely used in the simulation of periodic structures, especially antenna feeds, phase shifters, and photonic-bandgap materials [13]–[17]. Recently, the research on negative refractive index metamaterials has further motivated the applications of the FDTD method [18]. A common characteristic of this type of FDTD method is that the propagation constant is given as an input parameter for solving the eigenfrequencies. According to Floquet’s theorem for periodic structures [19], the periodic boundary condition is introduced by setting the phase difference (based on the propagation constant ) between field components at the periodic boundaries of one cell. This makes it difficult to apply the FDTD scheme to the case of open or lossy guided-wave structures, as one cannot simultaneously use two initial values (phase and attenuation constants) to calculate the eigenfrequencies. Recently, a solution to this problem was developed that extracts the attenuation constant in one periodic cell by using the FDTD method [20], [21]. However, the proposed approach lacks a rigorous mathematical proof and derivation. In particular, [20, eq. (1)] or [21, eq. (5)] used for extracting the complex propagation constant are taken from the case of 2-D transmission lines. This representation is inaccurate because the electromagnetic fields in a 2-D transmission line can be written as if is denoted as the propagation direction ( is the complex propagation constant), whereas the fields in a periodic guided-wave structure can only be regarded as , where is a periodic function [11], [19]. Only when the distance between two sampling points is equal to the periodic length, the equation is accurate. However, this is controversial with the periodic boundary conditions of one periodic cell simulation domain in [20] and [21]. If the simulation domain or window is extended to a two periodic structure to bypass this problem, the effect of a pseudoresonant mode due to the periodicity is significantly increased.

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In this paper, we make use of an equivalent resonant cavity model to transform the periodic guided-wave problem into an equivalent resonance problem [22]. Once the electromagnetic fields are transformed into a new set of variables, which can be resolved from the Maxwell’s equations, the existing FDTD method for modeling resonant cavities can be adopted to calculate the complex propagation constants of periodic guided-wave structures. In the propagation direction, the proposed equivalent resonant cavity is enclosed by the periodic boundary conditions, and the size of the equivalent resonant cavity is equal to that of a single periodic cell. With these assumptions, the eigenfrequency extracted by the FDTD method for resonators yields the desired solution in terms of frequency and quality factor. The attenuation constant can then be obtained from the extracted quality factor. This suggests that we obtain two groups of results, namely, eigenfrequencies and attenuation constants, from one group of initial phase constants. In this paper, some periodic guided-wave structures with complex geometry will be considered as examples for the study and demonstration of the proposed scheme, which is related to hybrid integrated circuits proposed in [23]. Among them, the substrate integrated nonradiative dielectric waveguide, which was developed from the original nonradiative dielectric waveguide by drilling a pattern of air holes or slots on a dielectric substrate, represents one of the most low-cost and high-performance millimeter-wave planar integrated guided-wave structures [24]–[28]. The authors’ previous study on substrate integrated nonradiative dielectric waveguides was developed by using a static effective dielectric constant method [24]–[26]. However, this approach does not allow for explaining the offset behaviors of the propagation constant observed for different air–hole patterns if the obtained static effective dielectric constants remain unchanged. As a result, the physical conception and explanation of substrate integrated nonradiative dielectric structures and circuits become vague, and the existing electromagnetic-bandgap (EBG) phenomena cannot be investigated by using this static method. To solve this problem, an accurate equation of effective dielectric constants should be introduced based on the full-wave simulation. In this paper, the proposed FDTD method is used to model and analyze the propagation characteristics of the substrate integrated nonradiative dielectric guide, and the results show that the substrate integrated nonradiative dielectric guide with via-slot geometry has a very low leakage loss, as well as a stable propagation behavior. This paper is organized as follows. Section II introduces the transformation of the periodic guided-wave structure into an equivalent resonant cavity and the existing FDTD analysis procedure. Section III verifies the proposed approach considering a long rectangular waveguide periodically loaded with lossy dielectric blocks. Section IV finally applies the novel method to the case of substrate integrated nonradiative dielectric guides.

is the attenuation constant and is the where phase constant. In the -direction, according to the Floquet’s theorem [19], we have (2a) (2b) where is the period. Since the propagation velocity of energy for the guided-wave signal equals to its group velocity , we can transform the attenuation constant related to the transmission distance to a time-dependent variable [22] (3a) (3b) Assuming (4) (5a) (5b) we have (6a) (6b) In (6), the loss related to the attenuation constant has been transformed into a complex frequency, while the phase difference related to the propagation constant has been integrated in the deformed electromagnetic field components. These field components can now be solved with the help of the Maxwell’s equations. As such, the modeling of a periodic guided-wave structure has been transformed to the modeling of an equivalent resonant cavity. As the loss is now included in the radian frequency , we can easily set up the periodic boundary condition of the equivalent resonant cavity only by means of the phase constant . The equivalent size of the resonant cavity is that of one periodic cell. Now we can use the FDTD method to calculate the eigenand quality factor of the resonant cavity. The frequency is the operation frequency of the periodic eigenfrequency guided-wave structure corresponding to the initial phase constant . For a lossy resonant cavity, the relation between comand quality factor is plex frequency (7) Thus, we can make use of the quality factor attenuation constant . From (4), we have

to calculate the (8)

II. DEFORMED ELECTROMAGNETIC FIELD VARIABLES The electromagnetic field components in a periodic guidedwave structure for any complex propagation constant in the -direction satisfy

where

(9) (1a) (1b)

When a group of phase constants is obtained, it can be used to calculate a group of attenuation constants conveniently.

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The time-domain difference equations of the deformed electromagnetic field components for the equivalent resonant cavity are similar to those of the conventional electromagnetic field components, e.g.,

FDTD time-domain response of an eigenvalue problem can be expressed in terms of a superposition of the resonant modes

(12) where is one of the six electromagnetic field components, is the complex residue, is the damping factor, and is the resonant frequency of the th resonant mode. By using transient time samples , one can extract and . Once the damping factor and resonant frequency have been determined, the quality factor can easily be obtained as (13)

(10a)

However, a good selection of the time sampling conditions is very important since some authors report a relation with the accuracy of these types of methods [32], [33]. B. Padé Approximation Among the second type of techniques, the Padé model combined with the fast Fourier transform (FFT) is often utilized to improve the spectral response for a faster and more accurate extraction of the resonant frequencies and quality factors from a smaller time window [32], [33]. The spectral response can be obtained from the time-domain signals with the help of an FFT. Subsequently, the Padé approximation is used to improve the spectral response from which one can calculate the quality factor (14)

(10b) where and are electric conductivity and permittivity of the and are magnetic conductivity and permedium, while denotes the discrete meability of the medium, respectively. time step. If the periodic length is discretized into segments along the -direction the periodic boundary conditions of the equivalent resonant cavity model can be derived from (2) and (5) as (11a) (11b) There are two common FDTD techniques to simulate a microwave resonator for an efficient extraction of the resonant-frequency and quality factor . They make use of impulse excitation sources and are described as follows. A. Time-Domain Signal Samples The first technique makes use of Prony’s method or the general pencil of function (GPOF) technique, both theoretically capable of calculating the quality factors and resonant frequencies directly from transient time samples very rapidly [29]–[31]. The

where is the 3-dB bandwidth and is the resonant frequency. It is to be mentioned that the simulation accuracy of this technique depends on the initial spectral response and its samples. This paper mainly focus on the extraction of the attenuation constant based on the novel FDTD simulation outlined above, which corresponds to a damped oscillation in the equivalent resonant cavity. Thus, one can consider a different approach: performing the FDTD iterative calculation until the transient signals vanish. The spectral responses can then be obtained via a DFT. Since all the time-domain signals are used, an accurate spectral response can be obtained. The resonant frequency can then be extracted from these data and inserted into (14) to determine the quality factor. In the remaining portion of this paper, this procedure will be verified and demonstrated by a number of examples. III. VERIFICATION AND NUMERICAL RESULTS In order to validate the proposed method, at first an infinitely long rectangular waveguide periodically loaded with a series of lossy dielectric blocks, as shown in Fig. 1, is analyzed. The mm, mm, dimensions of the waveguide are mm, and mm. in the waveguide, the In the case of a dominant mode complex propagation constant can be analytically calculated. When the transfer matrix of one periodic cell is obtained by

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Fig. 1. Infinitely long rectangular waveguide loaded periodically with a series of lossy dielectric blocks. (a) Parts of infinite periodic guided-wave structure. (b) Single periodic cell structure model. Fig. 3. Spectral response of transient signals H with = 563:44 rad/m, tan  = 0:01.

Fig. 2. Imaginary part of the transient signals H with = 800:35 rad/m, tan  = 0:01.

using the Floquet’s theorem [19], [22], the complex propagation constant can be derived as (15) In the presented example using the FDTD method to simulate the periodic guided-wave structure in Fig. 1, the discretization mm, which yields cell is selected to numbers of 17, 3, and 8 in the - - and -directions, respectively. Although a distribution source favors accurate simulation results, in this example, a point source excitation is adopted in order to verify the applicability of the presented method in genwith a Gaussian pulse eral cases. A random dipole source is excitation is selected. A random transient time response then chosen to perform the DFT. At first, the case of dielectric and loss tangent is considered. constant Here, the simulation duration is selected to 20 000 time steps. As shown in Fig. 2, the transient signals for a phase constant rad/m vanish after approximately 16 500 time steps.

Fig. 4. Comparison between the FDTD method and the analytical method for the attenuation constants and phase constants of the waveguide filled with lossy dielectric blocks (tan  = 0:01).

Fig. 3 shows the spectral responses obtained from the imagby means inary part of the selected transient time response rad/m. With of the DFT with phase constants of the help of (14), we obtain the quality factor from these frequency responses. The small figure in Fig. 3 shows the details of the spectral response in the vicinity of the resonant frequency. In Fig. 4, the results of both phase and attenuation constants are shown, and the proposed method is compared to the analytical method in (15). When the loss tangent of the dielectric block is increased to , we can obtain accurate results after only a short time-domain simulation of 5000 time steps. In Fig. 5, the results of both phase and attenuation constants are again shown and compared to the analytical approach (15). The results of the proposed simulation technique show excellent agreement with the analytical method. It is worth pointing out that the simulation procedure of the FDTD method proposed here is similar to that of the previous FDTD method described in Section I. However, the simulation process of the latter ends when the eigenfrequency is found,

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where and are electric conductivity and permittivity of the medium, respectively. is the permeability of the medium. When the loss tangent is small enough, such as , the phase constant is almost independent of the loss tangent, whereas the attenuation constant can be approximated as (17) As a result, we can conclude that for a sufficiently small loss tangent, the phase constant of the substrate integrated nonradiative dielectric waveguide is independent of the loss tangent, which is shown in our simulations. Furthermore, the attenuation constant can be written as (18) Fig. 5. Comparison between the FDTD method and the analytical method for the attenuation constants and phase constants of the waveguide filled with lossy dielectric blocks (tan  = 0:05).

whereas in the method presented in this study, the procedure is extended by calculating the quality factor and, in turn, the attenuation constant from the spectral responses or time sample signals. This procedure is valid as it has been mathematically proven that the attenuation constant can be obtained from the quality factor of the equivalent resonator. Thus, the existing FDTD code can be easily used to calculate the complex propagation constant by implementing only a small extension. We believe that this is probably the biggest advantage of the proposed FDTD model. IV. SUBSTRATE INTEGRATED NONRADIATIVE DIELECTRIC WAVEGUIDE The architecture of the substrate integrated nonradiative dielectric waveguide structure is described here, which uses the numerical FDTD approach to determine its propagation characteristics. The substrate integrated nonradiative dielectric guide is implemented in a high dielectric constant substrate Duroid/RT6010, which has a loss tangent of and a relative dielectric constant . Obviously, such a small loss tangent will result in very long iteration time if the leakage loss is also very small due to a high-quality via-hole or via-slot pattern. When the dielectric loss is supposed to be zero in order to allow for determining leakage loss, such a long simulation time will result in the failure of the above described damped oscillation technique. Thus, we propose a two-step simulation process that still allows for using the damped oscillation method to accurately extract the complex propagation constant. When electromagnetic wave propagates in a lossy infinitely extended dielectric and the loss tangent is very small, the attenuation constant and phase constant can be written as

(16a) (16b)

where the first term is proportional to the loss tangent of the diimplies that the leakage loss electric, and the second term is a function of the radian frequency. As long as the field patterns keep unchanged and the loss tangent is very small, (18) is accurate. In the simulation, two different loss tangents are assumed that are small enough to satisfy the conditions of (18). On the other hand, they should be large enough to allow for a rapid vanishing of the damped oscillation and, therefore, reducing iteration time. By means of (18), the attenuation constant of a substrate integrated nonradiative dielectric waveguide with an arbitrary small loss tangent can be calculated at the same time with the attenuation constant related to leakage loss. We call this procedure the two-step damped oscillation simulation. With the help of advanced manufacturing techniques such as laser cutting or drilling, novel hybrid integrated structures can be fabricated. As shown in Fig. 6, a novel type of substrate integrated nonradiative dielectric waveguide is constructed by using via-slots instead of via-holes [24]–[28]. The via-slots array technique has been used in the development of substrate integrated waveguide structures and has resulted in a new class of hybrid integrated waveguides with better propagation characteristics [34]. Here we will first discuss the substrate integrated nonradiative dielectric guide to outline its excellent performance, and in a second step verify the proposed techniques applied to such structures. As shown in Fig. 6, the air via-slots are directly cut in the dielectric substrate Duroid/RT6010. The dimensions in Fig. 6 are given as follows. The width is 2.8 mm, the height is 2.54 mm, the length of air slot is 1.4 mm, the gap is 0.2 mm, the width of air slot is 0.8 mm, the distance between two columns of slots is 0.2 mm, and the relative dielectric constant is 10.2. Three columns of air slots in each side of the dielectric strip are used. In the FDTD simulation, the spatial step is selected to 0.2 mm and perfectly matched layer (PML) absorbing boundary conditions are introduced in the two laterals of the substrate integrated nonradiative dielectric guide [35]. Since the main nonradiative dielectric guide mode is the mode, the analysis concentrates mainly on this mode. Considering the properties of the mode, a random dipole source with a Gaussian pulse excitation is adopted. A transient time response is then selected to perform the DFT. The FDTD simulation is performed with loss tangents of of 0.015, 0.02, and 0.025, respectively. The relative simulation times, for which the damped oscillation vanishes, result in 15 000, 11 000, and 8000

XU et al.: FDTD MODELING OF PERIODIC GUIDED-WAVE STRUCTURES

Fig. 6. Exploded view of the substrate integrated nonradiative dielectric waveguide constructed by using air via-slot arrays.

Fig. 7. Comparison of the attenuation constants of the substrate integrated nonradiative dielectric guide LSM mode calculated from the FDTD simulation and with the help of (18) (tan  = 0:015).

time steps, respectively. Obviously, these loss tangents are small enough to satisfy (18). , Fig. 7 shows the attenuation constants for which are obtained both from FDTD simulation and (18) ( and ). Excellent agreement between the analytical solution in (18) and the presented FDTD solution verifies the applicability of the proposed two-step damped oscillation technique. When the loss tangent is reduced to (Duroid/ RT6010), a 90 000 time-step iterative simulation is needed. Especially if the loss tangent is supposed to be zero in order to allow for calculating the attenuation constant related to leakage loss, a much longer iteration time is necessary. Therefore, in this case, the two-step damped oscillation technique becomes more

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Fig. 8. Comparison of the attenuation constants of the substrate integrated nonradiative dielectric guide in LSM mode calculated from the FDTD simulation directly, from (18), and from HFSS with tan  = 0:0023, as well as the attenuation constant a of the leakage loss.

efficient. Fig. 8 shows the results for the attenuation constants of the substrate integrated nonradiative dielectric guide in the mode, which are obtained both directly from the FDTD simulation and calculated from (18) ( and ). In addition, a commercial software [Ansoft’s High Frequency Structure Simulator (HFSS)] is used to verify the attenuation constants , which is also shown in Fig. 8. When the HFSS package is employed, two substrate integrated nonradiative dielectric guides with different lengths should be used to model the guided-wave properties from -parameters. In this scheme, a numerical calibration technique, as discussed in [36], is used to extract the phase and attenuation constants. The attenuation constant related to leakage loss (when the dielectric loss is zero), which is calculated from (18), is also shown in Fig. 8. Excellent agreement between the results verifies the proposed FDTD method and the related damped oscillation technique. It can be seen from Fig. 8 that the attenuation constant due to the leakage loss is very small compared to the attenuation constant caused by the dielectric loss, which illustrates that the substrate integrated nonradiative dielectric guide with air via-slots array shows very good propagation characteristics. It is obvious that the air via-slots can reduce the effective dielectric constant of the substrate, which is related to their shapes, locations, and patterns. When the dielectric strip of the nonradiative dielectric guide remains unchanged, the effective dielectric constant of the equivalent substrate on the laterals of the strip can be found. The propagation characteristics of the equivalent nonradiative dielectric guide are similar to those of the substrate integrated nonradiative dielectric guide. The fact that the dielectric strip of the equivalent nonradiative dielectric guide remains unchanged simplifies the analysis and physical interpretation. When the gaps between slots are small, EBG phenomena will not appear in the single mode operating region. Obviously,

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Fig. 9. Comparison of the propagation constants of the LSM mode in the substrate integrated nonradiative dielectric guide calculated from the FDTD simulation (tan  = 0:015 and tan  = 0:0023) and from HFSS (tan  = 0:0023), as well as propagation constant of the nonradiative dielectric guide with an effective dielectric constant of 2.406 for the equivalent dielectric substrate on the laterals of the strip.

an approximate mathematical equation of the effective dielectric constant can be found. When the gaps increase, EBG phenomena will appear and can be utilized in some cases. Fig. 9 shows the phase constants of the substrate integrated nonradiative dielectric guide in the mode, which are obtained from both FDTD simulation ( and ) and commercial software (Ansoft’s HFSS) , as well as the phase constant of an equivalent nonradiative dielectric guide with effective dielectric constant of 2.406. The excellent agreement between the two FDTD data curves verifies the independency of the phase constant of the loss tangent in the substrate integrated nonradiative dielectric guide in the case of a small loss tangent. Furthermore, the excellent agreement between FDTD and HFSS simulations verifies the accuracy of the proposed method. The agreement between the data of equivalent nonradiative dielectric guide and the substrate integrated nonradiative dielectric guide shows that, in single mode operation, the propagation characteristics of the substrate integrated nonradiative dielectric guide are similar to those of an equivalent nonradiative dielectric guide. Thus, if an approximate mathematical equation for the effective dielectric constant can be found, the analysis and design of substrate integrated nonradiative dielectric guide becomes significantly simplified. In the via-slots substrate integrated scheme, three columns of air slots on each side of the dielectric strip are deployed. In the following procedure, the entire analysis is performed for a loss tangent by using the previously outlined two-step damped oscillation technique ( and ). Figs. 10 and 11 show the eigenfrequencies and quality factors of the substrate integrated nonradiative dielectric schemes with two, three, and four columns, respectively, given the dimensions in Fig. 6 as follows: mm, mm

Fig. 10. Comparison of the eigenfrequencies of the substrate integrated nonradiative dielectric guide in LSM mode when 2–4 columns of air-via slots are used, respectively.

Fig. 11. Comparison of the equivalent quality factors of the substrate integrated nonradiative dielectric guide LSM mode when air via-slots are 2–4 columns, respectively.

mm, and mm. Considering the results given in Figs. 10 and 11, we can conclude that a selection of three via-slot columns are a reasonably good choice in terms of performance. Figs. 12 and 13 illustrate the eigenfrequencies and quality factors when the slot length is varied from 0.2 to 1.8 mm and the propagation constant is equal to 350 and 700 rad/m, respectively. When the slot length increases, the results tend to saturate. Considering these results, as well as structure stability, we can select the slot length to 1.4 mm. As a final step after an extensive discussion and analysis of the air via-slots based substrate integrated nonradiative dielectric guide, a set of readily available measurement results in [24] to verify the proposed FDTD method is used. The type of substrate integrated nonradiative dielectric guide used here is constructed by an air via-holes pattern, as shown in Fig. 14. The dimensions are given as follows: mm,

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Fig. 12. Eigenfrequencies of the substrate integrated nonradiative dielectric guide in LSM mode when the length of slot changes from 0.2 to 1.8 mm and the propagation constant is equal to 350 and 700 rad/m, respectively.

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Fig. 15. Cuflon substrate integrated nonradiative dielectric simulated and measured dispersion characteristics for the LSM mode.

and a maxCuFlon with a relative permittivity of imum peak dissipation factor of 0.00045 at 18 GHz. To measure the propagation constant of the mode, two guides with different length were constructed. The difference of length was 26.289 mm. In addition, using a Duroid/RT6002 substrate of 0.254-mm thickness, microstrip-to-nonradiative dimode were designed electric guide transitions for the [24]. Experimental results are presented in Fig. 15 along with FDTD simulation results. In the presented FDTD simulation, conformal FDTD techniques are used in order to obtain accurate results [37], [38]. Fig. 15 illustrates that the prediction of dispermode is sufficiently accurate. sion characteristics for the V. CONCLUSION

Fig. 13. Quality factors of the substrate integrated nonradiative dielectric guide LSM mode when the length of slot changes from 0.2 to 1.8 mm and the propagation constant equals to 350 and 700 rad/m, respectively.

Fig. 14. CuFlon substrate integrated nonradiative dielectric guide with air viaholes.

mm, height

mm, the width of strip mm, and the mm. The dielectric used for this structure is

The FDTD method based on an equivalent resonant cavity model is proposed and demonstrated to rapidly and accurately simulate the guided-wave characteristics of closed/open periodic structures. By means of the resonant frequency and quality factor obtained from these simulations, both the eigenfrequency and attenuation constant can be extracted from a given phase constant. This implies a greatly extended application range of the FDTD method. Moreover, the presented simulation procedure of the proposed FDTD model suggests that the algorithms of the conventional FDTD method can be very easily extended to calculate the complex propagation constant. A two-step damped oscillation technique is proposed and used for a rapid and accurate extraction of the quality factor of the lossy equivalent resonant cavity and subsequently the calculation of the attenuation constant of periodic guided-wave structures. Finally, the proposed technique is used to analyze and demonstrate a new type of substrate integrated nonradiative dielectric guides, which are constructed by air via-slots arrays, and shows excellent dispersion characteristics. Simulated data and measured results are compared and verify the excellent prediction capabilities of the proposed FDTD method.

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REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value involving Maxwell’s equations problems in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 5, pp. 302–307, May 1966. [2] S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guided Wave Lett., vol. 5, no. 5, pp. 165–167, May 1992. [3] A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” Electron. Lett., vol. 28, no. 15, pp. 1451–1452, Jul. 1992. [4] S. Xiao and R. Vahldieck, “An efficient 2-D FDTD algorithm using real variables,” IEEE Microw. Guided Wave Lett., vol. 3, no. 5, pp. 127–129, May 1993. [5] A. C. Cangellaris, “Numerical stability and numerical dispersion of a compact 2-D/FDTD method used for the dispersion analysis of waveguides,” IEEE Microw. Guided Wave Lett., vol. 3, no. 1, pp. 3–5, Jan. 1993. [6] I. P. Hong and H. K. Park, “Dispersion characteristics of a unilateral fin-line using 2-D FDTD,” Electron. Lett., vol. 32, no. 21, pp. 1992–1994, Oct. 1996. [7] J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, “A treatment of magnetized ferrites using the FDTD method,” IEEE Microw. Guided Wave Lett., vol. 3, no. 5, pp. 136–138, May 1993. [8] M. Fujii and S. Kobayashi, “Compact two-dimensional FD-TD analysis of attenuation properties of lossy microstrip lines,” in IEEE MTT-S Int. Microw. Symp. Dig., 1995, pp. 797–800. [9] M. Fujii and S. Kobayashi, “Accurate analysis of losses in waveguide structures by compact two-dimensional FDTD method combined with autoregressive signal analysis,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 6, pp. 970–975, Jun. 1996. [10] B.-Z. Wang, W. Shao, and Y. Wang, “2-D FDTD method for exact attenuation constant extraction of lossy transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 289–291, Jun. 2004. [11] A. C. Cangellaris, M. Gribbons, and G. Sohos, “A hybrid spectral/ FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microw. Guided Wave Lett., vol. 3, no. 10, pp. 375–377, Oct. 1993. [12] M. Celuch-Marcysiak and W. K. Gwareck, “Spatially looped algorithms for time-domain analysis of periodic structures,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 4, pp. 860–865, Apr. 1995. [13] R. Coccioli, F. Yang, K. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrate,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2123–2130, Nov. 1999. [14] F. Yang, K. Ma, Y. Qian, and T. Itoh, “A novel TEM waveguide using uniplanar compact photonic-bandgap (UC-PBG) structure,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2092–2098, Nov. 1999. [15] F. Yang, K. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonicbandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1509–1514, Aug. 1999. [16] V. Radisic, Y. Qian, and T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microw. Guided Wave Lett., vol. 8, no. 1, pp. 13–14, Jan. 1998. [17] V. Radisic, Y. Qian, R. Coccioli, and T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microw. Guided Wave Lett., vol. 8, no. 2, pp. 69–71, Feb. 1998. [18] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic finitedifference time-domain analysis of loaded transmission-line negativerefractive-index metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1488–1495, Apr. 2005. [19] R. E. Collin, Field Theory of Guided Waves. New York: McGrawHill, 1960. [20] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Efficient finite-difference time-domain (FDTD) modeling of periodic leaky-wave structures,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 317–320. [21] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic FDTD analysis of leaky-wave structures and applications to the analysis of negative-refractive-index leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1619–1630, Apr. 2006. [22] F. Xu, K. Wu, and W. Hong, “Equivalent resonant cavity model of periodic guided-wave structures and its application in finite difference frequency domain algorithm,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 4, pp. 697–702, Apr. 2007.

[23] K. Wu, “Integration and interconnect techniques of planar and nonplanar structures for microwave and millimeter-wave circuits—Current status and future trend,” in Proc. Asia–Pacific Microw. Conf., Taipei, Taiwan, R.O.C., 2001, pp. 411–416. [24] Y. Cassivi and K. Wu, “Substrate integrated non-radiative dielectric waveguide,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp. 89–91, Mar. 2004. [25] Y. Cassivi and K. Wu, “Substrate integrated NRD guide (SINRD) on high dielectric constant substrate for millimeter wave circuits and systems,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 1639–1642. [26] Y. Cassivi and K. Wu, “Substrate integrated circuits concept applied to the nonradiative dielectric guide,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 152, no. 6, pp. 424–433, Dec. 2005. [27] N. Grigoropoulos and P. R. Young, “Low cost non radiative perforated dielectric waveguides,” in Proc. 33th Eur. Microw. Conf., Munich, Germany, Oct. 2003, pp. 439–442. [28] S. W. H. Tse and P. R. Young, “Photonic crystal non-radiative dielectric waveguide,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1079–1081. [29] J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, “Computation of resonant frequencies and quality factors of open dielectric resonators by a combination of the finite-difference time-domain (FDTD) and Prony’s methods,” IEEE Microw. Wireless Compon. Lett., vol. 2, no. 11, pp. 431–433, Nov. 1992. [30] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–233, Feb. 1989. [31] P. Kozakowski, A. Lamecki, and M. Mrozowski, “Provisional model technique in the FDTD analysis of high- resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 501–503, Nov. 2004. [32] S. Dey and R. Mittra, “Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation,” IEEE Microw. Guided Wave Lett., vol. 8, no. 12, pp. 415–417, Dec. 1998. [33] W. Guo, W. Li, and Y. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 5, pp. 223–225, May 2001. [34] F. Xu, X. Jiang, and K. Wu, “FDFD modeling of substrate integrated waveguide without phase-bias,” in Proc. 35th Eur. Microw. Conf., Paris, France, Oct. 2005, pp. 853–856. [35] D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microw. Guided Wave Lett., vol. 4, no. 8, pp. 268–270, Aug. 1994. [36] F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 66–73, Jan. 2005. [37] S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1737–1739, Sep. 1999. [38] W. Yu and R. Mittra, “A conformal FDTD technique for modeling curved dielectric surfaces,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 1, pp. 25–27, Jan. 2001.

Q

Feng Xu (M’05) was born in Jiangsu, China. He received the B.S. degree in radio engineering from Southeast University, Nanjing, China, in 1985, the M.S. degree in microwave and millimeter-wave theory and technology from the Nanjing Research Institute of Electronics and Technology, Nanjing, China, in 1998, and the Ph.D. degree in radio engineering from Southeast University, Nanjing, China, in 2002. From 1985 to 1996, he was with the Nanjing Research Institute of Electronics and Technology, where he conducted research in the areas of antenna and RF circuits design. Since 2002, he has been with the Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC, Canada, where he has been a Post-Doctoral Researcher and is currently a Research Associate. His current research interests include numerical methods for electromagnetic field problems and advanced microwave and millimeter-wave circuits and components.

XU et al.: FDTD MODELING OF PERIODIC GUIDED-WAVE STRUCTURES

Ke Wu (M’87–SM’92–F’01) is Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering with the École Polytechnique de Montréal, Montréal, QC, Canada. He also holds a Cheung Kong endowed chair professorship (visiting) with Southeast University, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China, and the City University of Hong Kong. He has been the Director of the Poly-Grames Research Center. He has authored or coauthored over 515 referred papers and several books/book chapters. He has served on the Editorial/Review Boards of numerous technical journals, transactions, and letters, including being an Editor and Guest Editor. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors. He is also interested in the modeling and design of microwave photonic circuits and systems. Dr. Wu is a Fellow of the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is a member of the Electromagnetics Academy, Sigma Xi, and URSI. He has held key positions in and has served on various panels and international committees including the chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. He is currently the chair of the joint IEEE Chapters of the Microwave Theory and Techniques Society (MTT-S)/Antennas and Propagation Society (AP-S)/Lasers and Electro-Optics Society (LEOS), Montréal, QC, Canada. He is an elected IEEE MTT-S Administrative Committee (AdCom) member for 2006–2009 and serves as the chair of the IEEE MTT-S Transnational Committee. He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award.

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Wei Hong (M’92–SM’07) was born in Hebei Province, China, on October 24, 1962. He received the B.S. degree from the Zhenzhou Institute of Technology, Zhenzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, where he is currently a Professor with the Department of Radio Engineering. In 1993, 1995, 1996, 1997, and 1998, he was a short-term Visiting Scholar with the University of California at Berkeley and University of California at Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter-wave theory and technology, antennas, electromagnetic scattering, inverse scattering and propagation, RF front-ends for mobile communications, and the parameters extraction of interconnects in very large scale integration (VLSI) circuits. He has authored or coauthored over 200 technical publications. He authored Principle and Application of the Method of Lines (Southeast Univ. Press, 1993, in Chinese). He has served as a reviewer for many technique journals such as Proceedings of the IEE (Part H) and Electronics Letters. Dr. Hong is a Senior Member of the Chinese Institute of Electronics (CIE). He is the vice president of the Microwave Society and Antenna Society, CIE. He has served as the reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the two-time recipient of the First-Class Science and Technology Progress Prize presented by the State Education Commission in 1992 and 1994, respectively. He was the recipient of the Fourth-Class National Natural Science Prize in 1991, and the Third-Class Science and Technology Progress Prize of Jiangsu Province. He was also the recipient of the Foundation for China Distinguished Young Investigators presented by the National Science Foundation (NSF) of China.

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Design of Synthetic Quasi-TEM Transmission Line for CMOS Compact Integrated Circuit Meng-Ju Chiang, Student Member, IEEE, Hsien-Shun Wu, Member, IEEE, and Ching-Kuang C. Tzuang, Fellow, IEEE

Abstract—This paper presents the design guidelines of the synthetic quasi-TEM transmission line (TL) based on standard 0.18- m one-poly six-metal complementary metal–oxide–semiconductor (CMOS) technology. The synthetic quasi-TEM TL, also called the complementary-conducting-strip transmission line (CCS TL), is composed of five structural parameters to synthesize its guiding characteristics. Twenty-four designs of CCS TL are reported, with the following unique attributes. First, a characis yielded. Second, the teristic impedance range of 8.62–104.0 maximum value of the slow-wave factor is 4.79, representing an increase of 139.5% over the theoretical limit of the quasi-TEM TL. Third, the ratio of the area of the CCS TL to its corresponding quality factor ( factor) can help to estimate the cost of the loss for the circuit miniaturizations. Additionally, the important CMOS manufacturing of metal density is for the first time involved in the reported TL designs. By following the proposed design method-band CMOS rat-race ologies, a practical design example of a hybrid is reported and experimentally examined in detail to reveal the feasibility of the proposed design guidelines to synthesize the CMOS CCS TL. The chip size without contact pads is 420.0 m 540.0 m. The measured loss and isolation of the hybrid at 36.3 GHz are 3.84 and 58.0 dB, respectively.



Index Terms—Complementary metal–oxide–semiconductor (CMOS), hybrid, quasi-TEM, thin-film microstrip (TFMS), transmission line (TL).

I. INTRODUCTION OMPLIMENTARY metal–oxide–semiconductor (CMOS) technology promises a higher level of integration and lower cost than the III–V compounds, enabling the production of multifunction RF transceivers and RF system-on-chip (SOC) designs [1]–[5]. These multifunction circuits on silicon generally adopted lumped or lumped- distributed equivalence, which is faced with not only operation at ever-increasing frequencies, but also saving power consumption and chip area when the processes are scaled down. On the other hand, the transmission line (TL) frameworks, such as microstrip (MS) and coplanar waveguide (CPW), have gained popularity in CMOS RF circuit designs [6]–[12]. Moreover, such TLs can be made to establish a 3-D monolithic microwave integrated circuit (MMIC) in order to save chip area [13]–[19]. The concept of the synthetic quasi-TEM TL was recently reported and successfully employed to miniaturize RF integrated circuits [20]–[23]. Such

C

Manuscript received May 15, 2007. This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC 96-2752-E-002009-PWE and Grant NSC 95-2221-E-002-084-MY2. The authors are with Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]. ntu.edu.tw). Digital Object Identifier 10.1109/TMTT.2007.910089

successes are mainly due to an efficient meandering of the TL to achieve the highest degree of integration. Extensive studies indicate that the synthetic quasi-TEM TL, also called the complementary-conducting-strip transmission line (CCS TL), has better guiding properties than those of the conventional MS when the signal trace is meandered in a 2-D plane [21]. Following the same concept of miniaturization, this study focuses on the design of the CCS TL using standard 0.18- m one-poly six-metal (1P6M) CMOS technology, which is available from most standard silicon foundry services in the world. Fig. 1 illustrates two approaches to designing a CMOS CCS TL. As shown in Fig. 1, the signal traces of the CCS TL are meandered by obeying two basic rules. First, the length of the signal trace is fixed. The length is defined as 270.0 m in this study. Second, the CCS TL is meandered by at least four bends in the square area. By following the winding course defined above, the CCS TLs are designed for the characteristics impedance ( ) of 88.1, 50.7, and 22.7 . The CCS TLs in the bottom row of Fig. 1 are designed by referring to the concept of thin-film microstrip (TFMS), which is regarded as a special limiting case. Although all the limiting cases are meandered with a line space of 2.0 m, which is the minimum value defined in this study, the corresponding area increases when decreases. For example, the area of 22.7- CCS TL in Fig. 1(f) is 10152.0 m , representing 10.25 occurrences of 88.1CCS TL in Fig. 1(d). However, the CCS TL is designed with stacked metal by manipulating the advantage of multilayer CMOS technology. Fig. 1(a) and (c) shows the 22.7- and 88.1- CCS TLs, which can be designed with areas of 6750.0 and 1019.7 m , respectively. Conversely, the quality factor ( factor) of the 88.1- CCS TL in Fig. 1(a) at 10.0 GHz is 1.97, which is 6.8% lower than that in Fig. 1(d). The slow-wave factor (SWF) of the 22.7- CCS TL in Fig. 1(b) at 10.0 GHz is 2.67, which is 33.5% higher than the theoretical limit of the quasi-TEM TL. Furthermore, by applying the stacked metal to the designs of 22.7- and 50.7- CCS TLs in Fig. 1(a) and (b), two CCS TLs automatically meet the metal density requirement without inserting additional dummy metals. The metal density, which denotes the ratio of the total metal layout area to the TL area, is strongly required by the foundry to manage the variation of chemical–mechanical polishing (CMP) in the wafer manufacture, maintaining the wafer yield and design reliability [24]. The design approaches, which lead to different guiding characteristics of the CCS TL, are extensively investigated in Section III after the validity check of the full-wave EM simulation, which is presented in Section II for extracting the guiding characteristics of the CMOS CCS TL. Section IV reports a

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Fig. 1. CMOS synthetic quasi-TEM TL. The length of all TLs is 270.0 m. (a) Z = 88:1 . (b) Z = 50:7 . (c) Z = 22:7 . (d) Z = 88:1 . (e) Z = 50:7 . (f) Z = 22:7 .

practical example of a -band rat-race hybrid realized by the CCS TL, based on the design guidelines summarized at the end of Section III to reveal the superior performance in terms of low loss and compact area of the CMOS integration. Finally, Section V presents a brief summary. II. CMOS MULTILAYER SYNTHETIC QUASI-TEM TL: CCS TL The proposed CCS TL is constructed by the unit cell on the silicon substrate. As shown in Fig. 2(a), the unit cell, whose dimensions are much smaller than the guiding wavelength at the operating frequency, is the smallest element in the TL. The signal trace is composed of a central patch and four connecting arms, with the latter used to connect adjacent cells. Fig. 2(a) only displays two arms for simplicity. However, the central patch and mesh ground plane can be constructed by using solid vias to link metals in a multilayer structure. As shown in Fig. 2(b), the thickness of the mesh ground plane is increased by stacking to in order to decrease the series resistance of the TL, thus factor of the CCS TL [21]. The unit cell in enhancing the Fig. 2(a) shows a periodicity of , and alternately combines two types of TLs shown in Fig. 2(b) and (c) to form a quasi-TEM TL. Fig. 2(b) displays a cross-sectional view of the – cut, and clearly shows the well-known MS structure, which is locally a capacitive region from the circuit point of view. In contrast, Fig. 2(c) displays an MS with a tuning septa [25], which can be regarded as an elevated CPW [26], and is a high-impedance inductive region alongside the – cut. Both microminiaturized guiding frameworks support the quasi-TEM mode, alternating to guide the electromagnetic (EM) energy, thus allowing arbitrary syntheses of the quasi-TEM TL with required characteristic impedance [21]. The central patch with a dimension and the mesh ground plane of an inner slot with a dimension

Fig. 2. CMOS CCS TL. (a) 3-D view. (b) Cross-sectional view of A–A cut in (a). (c) Cross-sectional view of B –B cut in (a).

form the complementary conducting surfaces. The term denotes the width of the connecting arm, thus forming the and , then the CCS so-called CCS TL. If TL is regarded as the conventional TFMS, forming a special limiting case in Fig. 2. Furthermore, the values of the strucand the number of metal tural parameters, namely, layer are restricted by the capability of the CMOS technology, which defines the minimum and maximum values of line space, linewidth, and number of metal layers. Hence, the proposed CCS TL design is scalable by following the continuing improvement of the semiconductor technology. Fig. 3(a) shows a chip photograph of a practical design example of the proposed CCS TL in Fig. 2(a). The signal , and the mesh ground plane is made trace is realized by to . As shown in Fig. 3(a), the of metal layers from prototype is designed with the following structural parameters: m, m, m, and m. The relative dielectric constants of the inter-media-dielectric (IMD) and silicon substrate are 4.0 and 11.9, respectively. The thickness and conductivity of the silicon substrate are 482.6 m

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calibration procedures have been carried out to eliminate the parasitics of the signal–ground (SG) pads. After the two-port -parameters are obtained, the complex propagation constant ( , where is the attenuation constant and is the phase constant) and characteristic impedance ( ) are extracted by the well-documented procedures in [21]. Parallel to the on-wafer measurements, the CCS TL shown in Fig. 3(a) is also theoretically examined by Ansoft’s commercial software package High Frequency Structure Simulator (HFSS) with the structural and material parameters mentioned above. The simulated results are also compared to those of the extracted results based on the measurements to verify the validity of full-wave EM simulations. Fig. 3(b) and (c) shows the comparisons. The maximum deviation of 5.7% in the real part of characteristics impedance ( ) is achieved in the range from 5.0 to 30.0 GHz. Two imagare nearly identical. Furthermore, the meainary parts of sured normalized phase constants, denoted by / , indicate a maximum difference of 8.0%, as opposed to the HFSS simulations, and two normalized attenuation constants are nearly identical. The normalized phase constant is 2.37 at 10.0 GHz, which of the quasi-TEM TL. is higher than the theoretical limit The value is the relative dielectric constant of the IMD. Two sets of curves in Fig. 3(b) and (c) shows excellent agreement in the range from 5.0 to 30.0 GHz, confirming the validity of the on-chip CCS TL characteristics using full-wave EM simulations. Section III presents the analysis of the CMOS CCS TL shown in Fig. 2(a) with various structural parameters via Ansoft’s commercial software HFSS. The design guidelines for CCS TL to synthesize the specific guiding properties are also summarized based on the extensive EM simulations. III. DESIGN OF MEANDERED CMOS CCS TLS A. Simulation Setups for Theoretical CCS TL Designs

Fig. 3. Validity checks of the simulated results of CCS TL against measurement. (a) Chip photograph of the prototype fabricated by 0.18-m 1P6M CMOS technology. (b) Complex characteristic impedance of prototype. (c) Complex propagation constant of prototype. Structural and material parameters are W = 5:0 m, S = 4:0 m, P = 30:0 m, and W = 28:0 m. The relative dielectric constants of the IMD and silicon substrate are 4.0 and 11.9, respectively. The thickness and conductivity of the silicon substrate are 482.6 m and 11.0 S/m, respectively. The thickness and resistivity of M layer are 2.0 m and 37 m /sq, respectively. The thickness and resistivity of the layers M –M are 0.55 m and 79 m /sq, respectively.

and 11.0 S/m, respectively. The thickness and resistivity of layer are 2.0 m and 37 m /sq, respectively. The thickness and resistivity of the layers – are 0.55 m and 79 m /sq, respectively. The characteristics of the CCS TL are gained from the on-wafer measurements. The two-port -parameters of the CCS TL are measured after the short-open-load-thru (SOLT)

The design of the CCS TL by using standard 0.18- m 1P6M CMOS technology is focused upon here. The material parameters, including the substrate thicknesses and relative dielectric constant, for the HFSS simulations, are set up by following the of all TLs definitions reported in Section II. Furthermore, in this study are designed with the maximum and minimum linewidths of 30.0 and 2.0 m, respectively. The minimum line is 2.0 m. Both the minimum linewidth and line space of space for layers – are 0.5 m. It is to be noted that the design rules for all these metal layers mentioned above conform to the standard foundry rules defined by most manufacturers. Conversely, before performing the HFSS simulations, the CCS TL is meandered by following two basic rules reported in Section I. The physical length of the TL is 270.0 m, and the TL is meandered by at least four bends in each square area. The guiding properties of the CMOS CCS TL at 10.0 GHz, namely, characteristic impedance ( ), SWF, and factor are extracted by the same procedures reported in Section II. The SWF is defined as the normalized phase constant ( / ) of the CCS TL, and the factor is the ratio of the phase constant to twice of the attenuation constant. Table I summarizes the extracted results of varying the corresponding structural parameters , and the metal layers for discussions in the following sections.

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TABLE I REFERENCE DESIGNS OF 10.0-GHz CCS TLS USING STANDARD 0.18-m 1P6M CMOS TECHNOLOGY

The metal layer, which is applied to the CCS TL design, is highlighted in Table I, which also shows the corresponding metal density, defined as the ratio of the total metal layout area to the TL area. The metal layers with the metal density below and above 30.0% are shown in gray and black, respectively. This work on CMOS TL design is the first to take the process issue of the metal density into consideration. Such a process issue, which is specifically defined by the manufacturer, dominated the yield of the CMOS circuit. B. Special Limiting Designs of CCS TL The CCS TL is identical to the conventional TFMS when and . Conversely, as indicated in [27, Fig. 5], the factor of the TFMS significantly decreases if the effective thickness between the signal trace and ground plane decreases. to the ground Hence, the CCS TL in this category applied to the signal trace to achieve low loss. However, plane and the drawback of the low-loss design is that the metal densities of to , are zero. Additional the rest of the metal layers, from chip area is stipulated to accommodate the dummy metal inserts. Additionally, due to the limiting designs of the CCS TL reported is set to 2.0 m to minimize the in Table I, the line space of area of the layout. Based on the closed-form expressions of the limiting designs of the CCS TL, and complex propagation constant are determined by the linewidth once

the material parameters and the thickness between the signal trace and ground plane are fixed [28]. Therefore, as indicated in Table I, characteristic impedance ( ) increased from 22.7 decreased from 30.0 to to 88.1 when the linewidth of 2.0 m. It is to be noted that 30.0 and 2.0 m are the maximum and minimum linewidths defined in Section III-A, which limit syntheses of the CCS TL. Moreover, at 10.0 GHz, the the factor of the 88.1- CCS TL is 2.07, which is over 100% lower than that of the 22.7- CCS TL. The SWF of five speciallimiting designs is below 2, which is the theoretical limit of the quasi-TEM TL on the substrate with a relative dielectric constant of 4. C. CCS TL With The CCS TLs with in Table I show the following design characteristics. The characteristic impedance can be elevated above 88.1 simply by decreasing the ratio of to . As shown in Fig. 2(c), and determine the effective area of the high-impedance region in CCS TL. Therecan be raised by adjusting and without varying fore, and , which determine the effective linewidth of the CCS below 22.7 , the metal layer TL. Conversely, to synthesize of central patch is vertically extended from to . Such an extension enlarges the overlapping area between the signal trace and ground plane, resulting in an increase of capacitance

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Fig. 5. Normalized area (A ) against real part of the characteristic impedance (Z ) for 270.0-m-long CCS TLs at 10.0 GHz. The symbols in Fig. 5 are identical to those shown in Fig. 4.

Fig. 4. AMD against the SWF for CCS TLs at 10.0 GHz.

per unit length of the CCS TL. By applying the design guidesyntheses, the value of ranges from 8.62 up to lines to ratio of 12.06 (104.0 /8.62 ). This 104.0 , showing a ratio is significantly wider than that of the TFMS designs reported in Section III-B. Moreover, the SWF of the CCS TL can be raised by the following two design guidelines. The first guideline is to reduce the ratio of to . This approach is applied to designing the from 88.1 to 104.0 . As indicated in Table I, CCS TL with is reduced the SWF increased from 0.97 to 2.03 when / from 1.14 to 1.02. The second approach is to adopt stacked metal to realize the central patch or mesh ground of the CCS TL. In Table I, all the designs of the CCS TL with below 50.0 , designed by following the second approach mentioned above, have SWF values exceeding the theoretical limit of the quasi-TEM TL. Meanwhile, these CCS TLs meet the 30.0% metal density requirement for all metal layers and need no additional chip area for filling dummy metal, attaining true miniaturization. To the best of the authors’ knowledge, the proposed design is the first to comply with CMOS metal density rules for designing the CMOS TL. To discuss these two design approaches in details, Fig. 4 plots the SWF at 10.0 GHz against average metal density for all the CCS TL designs in Table I. The average metal density (AMD), as follows in (1), indicates the average value of the metal densities for all six metal layers: Total metal area at layer Area of transmission line

% (1)

As shown in Fig. 4, the CCS TLs, which are marked by symbols in hollow squares , represent the special limiting designs presented in Section III-B. Additionally, the filled circles ( ) indicate that the metal densities of all six metal layers in the CCS TL design are higher than or equal to 30.0%. The half-filled circles ( ) mean at least one metal layer with metal density less than 30.0%. The quantity adjacent to the symbols represents the values of the CCS TL design listed in Table I. As shown in Fig. 4, the CCS TLs, which are designed by decreasing the ratio of / , show that the SWF increases from 0.97 to 2.03, while the AMD falls from 12.2% to 1.7%. The designs based on the first approach do not easily meet the required 30.0% metal density. Conversely, in the designs following the

second approach, the corresponding SWF increases from 1.97 to 4.79, and meanwhile the AMD also increases from 36.3% to 82.5%. This trend indicates that the second approach can realize a CCS TL with high SWF, and that the CCS TL design can easily meet the metal density requirement, enabling successful circuit miniaturization. The design approaches for the CCS TL, which can synthesize a TL with various structure parameters, reveal the fundamental modifications to the design of the CMOS TL. Furthermore, a glance back at Fig. 1 reveals that the CCS TLs can be realized in different areas for the same , thus attaining different factors. Hence, Section III-D is devoted to the discussion of the CCS TL designs with different area. D. Area-Influence Loss (AL) of CCS TL versus the characterFig. 5 plots the normalized area istic impedance for the CCS TL designs listed in Table I. , defined by (2) as follows, represents the ratio of The term the total occupying area of the meandered CCS TL with a fixed length of 270.0 m to the square of guided wavelength in free space at 10.0 GHz: Area

(2)

where denotes the velocity of light in free space and is the operating frequency. As shown in Fig. 5, the quantity adjacent to the symbols is the factor of the CCS TL listed in Table I. The symbols in Fig. 5 are identical to those shown in Fig. 4. As reported in Section III-B, the designs for the special limiting case of the CCS TL, which are meandered by following two basic rules defined in Section I, set the minimum line space at 2.0 m to achieve the smallest compact layout area. As reported in Section III-B, these special limiting cases of the CCS TL are designed according to the closed form in [28]. Therefore, as shown in Fig. 5, the limiting designs of the CCS TL de, which can form a noted by the symbols in hollow squares inversely virtually continuing curve, show the increase of proportional to the increase of . Such limiting designs are , and can be scaled down well controlled by only linewidth with the continuing improvement of semiconductor technology.

CHIANG et al.: DESIGN OF SYNTHETIC QUASI-TEM TL FOR CMOS COMPACT INTEGRATED CIRCUIT

Conversely, the design approaches presented in Section III-C, than that of the dewhich can synthesize a wider range of sign approaches in Section III-B and provide multiple designs distributions. As shown for one specific , lead to different between 88.1–104.0 , the CCS TL in Fig. 5, to synthesize value than the one predicted by the limrequires a higher iting case designs. This trend shows a reduction of the ratio of / in the CCS TL. Additionally, Fig. 5 shows two group for realizing a 35- CCS TL. The designs with different m, which achieve first design is the CCS TL with approaching that of the limiting case. The second dethe m, and results in below the predicted sign has factor of the CCS TL with value of the limiting case. The m is approximately 4.87, which is 39.14% higher m. The factor of the than that of the CCS TL with CCS TL is relatively proportional to the period of the unit cell. This observation reflects the fundamental physical phenomenon of the CCS TL design, which is studied in Fig. 6. The derivations in [29, Ch. 7.4] for a rectangular cavity in dominate-mode operation indicate that the conductor loss of the cavity is inversely proportional to its volume [29, Ch. 7.4, pp. 503, eq. (7.48)]. If the width, length, and height of the rectangular waveguide cavity are all identical, then the cavity is regarded as a cubic resonator, and the conductor loss in the resonator is related only to the quantity of the length since all the CCS TLs presented in Table I are designed on the silicon substrate with a fixed thickness, and meandered in a nearly square area, as shown in Fig. 1. Following the concept of the conductor loss in the cubic resonator, this study indicates that the loss of the meandered CCS TL with fixed length is exactly the same as the AL with the ratio of the square root of the normalized area to the factor. Additionally, shown in (3) as follows, the and the AL also can be represented by a function of factor after some algebraic manipulation:

(3) where , which denotes the total occupying area of the meandered CCS TL with a fixed length, is identical to that in (2), represents the operating frequency of the TL, and is the speed of light in free space. As defined in Section I, the length of all the CCS TLs in Table I is set to 270.0 m. Therefore, Fig. 6 plots the AL versus characteristic impedance for the CCS TLs in Table I at 5.0, 10.0, 20.0, and 30.0 GHz. The values of the parameters at 10.0 GHz are listed in Table I, and the values for 5.0, 20.0, and 30.0 GHz are obtained by following the same analytical procedures reported in Section II, except for the operating frequency. The definitions of the symbols, which represent various CCS TL designs in Fig. 6, are identical to those in Fig. 4. Due to the skin effect, the factor of the CCS TL is proportional to the square root of the frequency. Thus, the AL of the 50.7- CCS TL at 30.0 GHz is 0.89 10 , which is 6 3 and 1.5 times those at 5.0, 10.0, and 20.0 GHz, respectively. Such physical trends also can be observed at different CCS TL from 22.7 to 88.1 . designs with characteristic impedance

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Fig. 6. Characteristic impedance (Z ) against the AL for CCS TLs with a length of 270.0 m. The symbols in Fig. 6 are identical to those shown in Fig. 4.

TABLE II DESIGN GUIDELINES OF CMOS CCS TL

Furthermore, a close observation of the multiple CCS TL defrom 22.7 to 88.1 indicates that the signs with a specific corresponding AL value are nearly identical to each other at the same operating frequency, showing a constant ratio between the square root of the normalized area to the factor in different designs. This result confirms the observation of the two design approaches for the 35- CCS TL in Fig. 5. The CCS TL to raise the can be designed with a relatively high value of corresponding factor. Consequently, as shown in Fig. 5, the for a 22.7- CCS TL can be designed with factor of 4.03, or designed with for a factor of 4.42. Similarly, the 69.2- CCS TL can be designed for a factor of 2.27, or designed with with for a factor of 2.93. As reported here, the proposed CCS TL provides a high flexibility for synthesizing the desired guiding characteristics. By summarizing the design guidelines reported here, Table II shows the universal trends of synthesizing the desired guiding characteristics by adjusting the structural parameters of the CCS TL. Section IV applies the proposed CCS TL to designing a prac-band rat-race hybrid by following the tical example of the design guidelines reported here, demonstrating the capability of the CCS TL. IV. DESIGN EXAMPLE:

-BAND RAT-RACE HYBRID

Fig. 7(a) shows a 34.3-GHz CMOS rat-race hybrid design incorporating the proposed CCS TLs designed according to the guidelines reported in Section III. The operations of the rat-race

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TABLE III

Ka-BAND CMOS RAT-RACE DESIGNS

Ka

Fig. 7. -band CMOS rat-race hybrid. (a) Chip photograph of the prototype. (b) Simulated and measured results of the prototype.

and its equivalent TL network are well documented [29]. As shown in Fig. 7(a), the electrical length between Port 2 and Port 4 is three times as great as the quarter-wavelength and the remaining adjacent ports has the length of one quarter-wavelength. The reference impedance of all four ports is 50.0 , and the characteristic impedance of the TLs in the entire rat-race to establish the equal power split and is designed as 70.7 is applied to power combination. The CCS TL with the rat-race realization winding course shown in Fig. 1, and the design is fabricated by standard 0.18- m 1P6M CMOS technology. The chip area of the prototype shown in Fig. 7(a) is 420.0 m 540.0 m without the contact pads. The on-chip performances of the prototype are characterized by conducting the same experimental procedures reported in [23] and are compared to the theoretical data, which is computed by the full-wave HFSS simulations with the circuit layout in Fig. 7(a). Fig. 7(b) shows the composite plots, revealing good agreements between the measurements and HFSS simulations. The transmission coefficients, which are shown in Fig. 7(b) in

detail, are less than 4.0 dB from 34.0 to 38.0 GHz, indicating the intrinsic loss of less than 1.0 dB. Additionally, the two transmission coefficients in Fig. 7(b) are 3.94 and 3.75 dB at 34.3 GHz, showing an amplitude in balance of 0.19 dB. Such results show a nearly equal power distribution at two output ports of the prototype. The measured input return loss from 32.8 to 38.1 GHz, illustrated by the curve with hollow squares, is higher than 20.0 dB. The measured isolation from 31.5 to 40.0 GHz, illustrated by the curve with hollow triangles, is higher than 20.0 dB and has a maximum value of 58.0 dB at 36.3 GHz. From 31.0 to 37.0 GHz, the phase difference between two output ports is approximately 180 5 . By referring to the calculation of the area reduction factor (ARF) reported in [21, eq. (4)], the prototype in Fig. 7(a), which consisted of 14 18 unit cells with m, achieved an ARF of 96.9% at 34.3 GHz. -band rat-race, realized by incorA similar design of the porating CCS TLs on the standard 0.18- m 1P6M CMOS technology, was also recently reported in [23]. Table III summarizes these two rat-race hybrid designs for the following discussions. The ARF values shown in Table III are calculated by following the definition of Chen and Tzuang [21]. Two rat-race hybrids are designed at different operating frequencies, and implemented by different CCS TLs. The rat-race design in [23] based on a smaller period of unit cells reveals higher ARF values than those obtained by the design reported herein. This observation confirms the prediction reported in [21, Sec. II-B]. The smaller resulted in a higher ARF. Furthervalue of the periodicity more, the ARF is 97.1%, which is only 0.5% lower than that of [23], when the operating frequency of the rat-race design in Fig. 7(a) are scaled down from 34.3 to 30.0 GHz. Moreover, the scaling design, which is theoretically analyzed by performing the full-wave HFSS simulations mentioned above, achieved a transmission loss at 30.0 GHz of 4.2 dB, which is 22.2% lower than that of the design in [23]. Two different approaches are applied to designing CCS TLs, producing two rat-race hybrids with different performances in terms of loss, area, and ARF. The comparisons summarized above lead to the following observations. Increasing of the CCS TL enhances the factor of the CCS TL. As shown in Fig. 1, since is the main factor managing the occupying area of the CCS TL, increasing simultato increase and ARF to decrease. These obneously causes servations validate the design guidelines in Table II and the predictions of [21]. Moreover, the rat-race hybrid designs presented here demonstrate that CCS TL can systematically miniaturize the TL-based circuit with predictable electrical performances. The loss of the CCS TL-based circuit is confined to the desired area, and the CCS TL circuit designed by different approaches produced different ARF values for circuit miniaturization.

CHIANG et al.: DESIGN OF SYNTHETIC QUASI-TEM TL FOR CMOS COMPACT INTEGRATED CIRCUIT

V. CONCLUSION The CMOS CCS TL has been reported in detail. By following the universal design guidelines, the CCS TL can be designed with a wide range of characteristic impedance, high SWF, and the satisfaction of the metal density requirement. Additionally, when the physical length is fixed, the ratio of the CCS TL area factor approaches a constant and can to its corresponding be applied to estimating the cost of loss for the CMOS circuit miniaturization.

REFERENCES [1] C. Cojocaru, T. Pamir, F. Balteanu, A. Namdar, D. Payer, I. Gheorghe, T. Lipan, K. Sheikh, J. Pingot, H. Paananen, M. Littow, M. Cloutier, and E. MacRobbie, “A 43 mW Bluetooth transceiver with 91 dBm sensitivity,” in Proc. IEEE Solid-State Circuits Conf., 2003, vol. 1, 90–480. [2] M. Zargari, M. Terrovitis, S. Jen, B. Kaczynski, M. Lee, M. Mack, S. Mehta, S. Mendis, K. Onodera, H. Samavati, W. Si, K. Singh, A. Tabatabaei, D. Weber, D. Su, and B. Wooley, “A single-chip dual-band tri-mode transceiver for IEEE 802.11a/b/g wireless LAN,” IEEE J. Solid-State Circuits, vol. 39, no. 12, pp. 2239–2249, Dec. 2004. [3] P. Zhang, L. Der, D. Guo, I. Sever, T. Bourdi, C. Lam, A. Zolfaghari, J. Chen, D. Gambetta, B. Cheng, S. Gowder, S. Hart, L. Huynh, T. Nguyen, and B. Razavi, “A single-chip dual-band direct-conversion IEEE 802.11a/b/g WLAN transceiver in 0.18 m CMOS,” IEEE J. Solid-State Circuits, vol. 40, no. 9, pp. 1932–1939, Sep. 2005. [4] T. Maeda, H. Yano, S. Hori, N. Matsuno, T. Yamase, T. Tokairin, R. Walkington, N. Yoshida, K. Numata, K. Yanagisawa, Y. Takahashi, M. Fujii, and H. Hida, “Low-power consumption direction-conversion CMOS transceiver for multistandard 5-GHz wireless LAN systems with channel bandwidths of 5–20 MHz,” IEEE J. Solid-State Circuits, vol. 41, no. 2, pp. 357–383, Feb. 2006. [5] Y. Palaskas, A. Ravi, S. Pellerano, B. R. Carlton, M. A. Elmala, R. Bishop, G. Banerjee, R. B. Nicholls, S. K. Ling, N. Dinur, S. S. Taylor, 2 MIMO transceiver and K. Soumyanath, “A 5-GHz 108-Mb/s 2 RFIC with fully integrated 20.5-dBm P power amplifiers in 90-nm CMOS,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2746–2756, Dec. 2006. [6] C. Doan, S. Emami, A. Niknejad, and R. Brodersen, “Millimeter-wave CMOS design,” IEEE J. Solid-State Circuits, vol. 40, no. 1, pp. 144–155, Jan. 2005. [7] H. Shigematsu, T. Hirose, F. Brewer, and M. Rodwell, “Millimeterwave CMOS circuit design,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 472–477, Feb. 2005. [8] P.-S. Wu, H.-Y. Chang, M.-D. Tsai, T.-W. Huang, and H. Wang, “New miniature 15–20-GHz continuous-phase amplitude control MMICs using 0.18- m CMOS technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 10–19, Jan. 2006. [9] C.-H. Wang, Y.-H. Cho, C.-S. Lin, H. Wang, C.-H. Chen, D.-C. Niu, J. Yeh, C.-Y. Lee, and J. Chern, “A 60 GHz transmitter with integrated antenna in 0.18 m SiGe BiCMOS technology,” in Proc. IEEE SolidState Circuits Conf., 2006, pp. 659–668. [10] C.-M. Lo, C.-S. Lin, and H. Wang, “A Miniature V -band 3-stage cascode LNA in 0.13 m CMOS,” in IEEE Solid-State Circuits Conf., 2006, pp. 1254–1263. [11] G. Six, G. Prigent, E. Rius, G. Dambrine, and H. Happy, “Fabrication and characterization of low-loss TFMS on silicon substrate up to 220 GHz,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 301–305, Jan. 2005. [12] L. F. Tiemeijer, R. M. T. Pijper, R. J. Havens, and O. Hubert, “Low-loss patterned ground shield interconnect transmission lines in advanced IC processes,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 3, pp. 561–570, Mar. 2007. [13] I. Toyoda, T. Tokumitsu, and M. Aikawa, “Highly integrated threedimensional MMIC single-chip receiver and transmitter,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2340–2346, Dec. 1996.

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[14] T. Tokumitsu, M. Hirano, K. Yamasaki, C. Yamaguchi, K. Nichikawa, and M. Aikawa, “Highly integrated three-dimensional MMIC technology applied to novel masterslice GaAs- and Si-MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 32, no. 9, pp. 1334–1341, Sep. 1997. [15] I. Toyoda, K. Nichikawa, T. Tokumitsu, K. Kamogawa, C. Yamaguchi, M. Hirano, and M. Aikawa, “Three-dimensional masterslice MMIC on Si substrate,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2524–2530, Dec. 1997. [16] K. Nichikawa, K. Kamogawa, K. Inoue, K. Onodera, T. Tokumitsu, M. Tanaka, I. Toyoda, and M. Hirano, “Miniaturized millimeter-wave masterslice 3-D MMIC amplifier and mixer,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1856–1862, Sep. 1999. [17] C. Warns, W. Menzel, and H. Schumacher, “Transmission lines and passive elements for multilayer coplanar circuits on silicon,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 616–622, May 1998. [18] K. Hettak, G. Morin, and M. Stubbs, “The integration of thin-film microstrip and coplanar technologies for reduced-size MMICs,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 283–291, Jan. 2005. [19] M. Chirala and C. Nguyen, “Multilayer design techniques for extremely miniaturized CMOS microwave and millimeter-wave distributed passive circuits,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4218–4224, Dec. 2006. [20] C.-K. C. Tzuang, C.-H. Chang, H.-S. Wu, S. Wang, S.-X. Lee, C.-C. Chen, C.-Y. Hsu, K.-H. Tsai, and J. Chen, “An X -band CMOS multifunction-chip FMCW radar,” in IEEE MTT-S Int. Microw. Symp. Dig., 2006, pp. 2011–2014. [21] C.-C. Chen and C.-K. C. Tzuang, “Synthetic quasi-TEM meandered transmission lines for compacted microwave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 6, pp. 1637–1647, Jun. 2004. [22] C.-K. C. Tzuang, H.-H. Wu, H.-S. Wu, and J. Chen, “CMOS active bandpass filter using compacted synthetic quasi-TEM lines at C -band,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4548–4555, Dec. 2006. [23] S. Wang and C.-K. C. Tzuang, “Compacted Ka-band CMOS rat-race hybrid using synthesized transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., 2007, pp. 1023–1026. [24] A. B. Kahng, G. Robins, A. Singh, and A. Zelikovsky, “New and exact filling algorithms for layout density control,” in Proc. 12th Int. VLSI Design Conf., Jan. 1999, pp. 106–110. [25] T. Itoh, “Generalized spectral domain method for multiconductor printed lines and its applications to tunable suspended microstrips,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 12, pp. 983–987, Dec. 1978. [26] H. Kamitsuna, “A very small, low-loss MMIC rat-race hybrid using elevated coplanar waveguides,” IEEE Microw. Guided Wave Lett., vol. 2, no. 8, pp. 337–339, Aug. 1992. [27] G. E. Ponchak and A. N. Downey, “Characterization of thin film microstrip lines on polyimide,” IEEE Trans. Compon., Packag., Manuf. Technol. B, vol. 21, no. 2, pp. 171–176, May 1998. [28] F. Schnieder and W. Heinrich, “Model of thin-film microstrip line for circuit design,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 104–110, Jan. 2004. [29] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992.

Meng-Ju Chiang (S’05) received the B.S. degree in electrical engineering from Fong-Chia University, Taichung, Taiwan, R.O.C., in 2002, the M.S. degree in electronic engineering from the Chung-Cheng Institute of Technology, National Defense University, Taoyuang, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design of microwave/millimeter-wave passive circuits and antenna designs, CMOS RF integrated circuits, and development of advanced guiding structures for CMOS RF SOC.

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Hsien-Shun Wu (S’97–M’05) received the B.S. degree in electronic engineering from the National Taipei University of Technology, Taipei, Taiwan, R.O.C. in 1999, and the M.S. and Ph.D. degrees in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C. in 2001, and 2005, respectively. He is currently a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design of wireless system modules and the development of synthetic waveguides for RF circuits.

Ching-Kuang C. Tzuang (S’80–M’80–SM’92– F’99) received the B.S. degree in electronic engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1977, the M.S. degree from the University of California at Los Angeles (UCLA), in 1980, and the Ph.D. degree in electrical engineering from the University of Texas at Austin, in 1986. From 1981 to 1984, he was with TRW, Redondo Beach, CA, where he was involved with analog and digital MMICs. From 1986 to 2004, he was with

the Institute of Communication Engineering, National Chiao Tung University. In February 2004, he joined the Graduate Institute of Communication Engineering, Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., where he conducts research on advanced guiding structures for research and development of RF SOC and integrating active and passive microwave/millimeter-wave RF signal-processing components into a single chip. He has consulted the Wireless Communications Engineering Center (WiCE) on RF SOC/system-in-package (SIP). His research activities also involve the design and development of millimeter-wave and microwave active and passive circuits and the field theory analysis and design of various complex waveguide structures and large-array antennas. He has supervised 66 M.S. students and 23 Ph.D. students. Dr. Tzuang helped formed the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Taipei Chapter, and served as secretary, vice chairman, and chairman in 1988, 1989, and 1990, respectively. Since 2004, he has assisted the Taiwan Electrical and Electronic Manufacturers’ Association (TEEMA), promoting standardization and application of millimeter-wave technology.

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Analysis and Design of Monolithic Rectangular Coaxial Lines for Minimum Coupling Yuya Saito, Student Member, IEEE, and Dejan S. Filipovic´, Member, IEEE

Abstract—Coupling through holes etched in a common wall between the neighboring surface micromachined rectangular coaxial lines is studied in this paper. Analytical models for singleand multiple-hole structures are developed using Bethe’s small aperture theory, and excellent agreement with finite-element and finite-integration simulations is obtained. These models are used in a circuit-level simulator for efficient optimization of the selected multihole structure. An isolation test structure composed of 200- m center-to-center separated 300- m-tall 50-W rectangular coaxial lines is designed, built, and measured. Coupling levels below 55 dB over 1-cm-long coupling lengths are achieved -band. It is found that the probe radiation is throughout the more than 10 dB higher than the coupling through the common wall holes. This radiation severely affects the accuracy of the measurements and has to be considered when the coupling through the common walls is analyzed. Index Terms—Coupling analysis, small aperture theory.

Fig. 1. SEM image of an array of 50- -recta-coax lines fabricated on a 6-in Si wafer with the PolyStrata process (courtesy of Rohm and Haas Electronic Materials, Blacksburg, VA). Clearly depicted are the release holes needed for the removal of sacrificial photo resist.

-coaxial line, probe effects,

I. INTRODUCTION ARIOUS advances in microelectromechanical system (MEMS) fabrication techniques [1] including surface micromachining have contributed to several successful demonstrations of a low-loss miniature rectangular coaxial line, also known as a recta-coax. For example, a microfabrication process capable of depositing 40 layers of nickel for forming the 3-D structures including recta-coax lines and related components is demonstrated in [2]. All-copper recta-coax lines are also reported, however, their bandwidth is either narrow [3] or they are difficult to integrate with other components [4]. The recently introduced PolyStrata process [5], [6] overcomes most of the issues associated with the 3-D recta-coax -band cavity resonators based millimeter-wave circuits. [7], [8], hybrids [9], and antennas [10] fabricated with this technology have all shown excellent performance. A scanning electron microscope (SEM) photograph of an array of 50- -coaxial lines fabricated with the PolyStrata process is shown in Fig. 1. At least five sequentially deposited horizontal copper layers are needed to form a single recta-coax, while a minimum of nine layers are required for signal routing on

V

Manuscript received May 21, 2007; revised August 22, 2007. This work was supported by the Defense Advanced Research Projects Agency–Microsystems Technology Office under the 3-D Micro Electromagnetics Radio Frequency Systems (3d MERFS) Program. The authors are with the Department of Electrical and Computer Engineering, University of Colorado at Boulder, Boulder, CO 80309-0425 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.2007.910092

two stacked levels. In the process, the inner conductor is held by thin dielectric straps firmly attached to the vertical walls of the coaxial shield. These straps are either embedded in, or they are positioned underneath, the inner conductor. Once the recta-coax structure is formed, the sacrificial photo-resist is removed through holes located in the walls of the outer conductor, as depicted in Fig. 1. These holes are commonly referred to as release, etch, or weep holes. A common theme for surface micromachining recta-coax line processes [1], [2], [4]–[10] is the necessity for using the release holes. For higher package densities, it is desired that the neighboring lines are separated by a single wall. The thickness of this (common) wall should be at least several skin depths at the lowest operating frequency of the line. To facilitate easier fabrication, the release holes need to be integrated in the common wall. From the fabrication perspective, these holes should be as large as possible; however, this results in larger coupling through the holes. Although effects of the common wall holes can be directly accounted for in a computer-aided design, the detailed analysis pertaining to the coupling between the neighboring lines has not been demonstrated previously. This paper provides a comprehensive study of common wall hole effects on coupling between the surface micromachined recta-coax lines. Parameters such as hole location, dimension, and shape are addressed with respect to the aspect ratio of the photo-resist mold needed for the line formation. A test structure with four holes in a common wall between the recta-coax lines -band. The impact has been designed and measured over the of probe radiation on measurements is also considered. This paper is organized as follows. • Section II demonstrates some advantages of the recta-coax line and introduces an isolation test structure with holes in a common wall. Brief overviews of similar structures and relevant comparisons are also provided.

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Fig. 2. Simulated and measured normalized group velocity of a 50-

recta-coax line and comparison with simulated values for different microstrip lines. A cross section of the line is shown in the inset. Line parameters are fW ; W ; a; b; b ; g; tg = f130; 300; 37:6; 100; 50; 46:2; 18g m. The length of the layer 2 and 4 release holes is 300 m. 100-m-long 700-m separated dielectric straps have " = 2:85 and tan  = 0:045. All microstrip lines are printed on a 300-m-thick substrate, and their width is adjusted for 50 (quasi-static) characteristic impedance. Presented results are obtained with Ansoft Designer (AD) [13].

• Section III discusses the use of Bethe’s small aperture theory for determining coupling through a single hole. Effects of hole width and length are evaluated, and derived analytical models are favorably compared with the commercial finite-element method (FEM) software High Frequency Structure Simulator (HFSS) [11]. • Section IV gives the analysis, design, and optimization of a multiple-hole structure based on the single-hole analysis derived in Section III. Effects of dielectric straps and holes in the outer conductor are determined with the FEM and verified with the finite-integration technique (FIT) software CST Microwave Studio [12]. • Section V provides measured results and relevant comparisons with the models and discusses probe radiation effects on the accuracy of testing. II. RECTA-COAX ISOLATION TEST STRUCTURE When compared with various planar transmission lines at - and higher bands, a monolithic -recta-coax offers several advantages. These include lower loss, higher packaging densities, lower crosstalk, and much wider instantaneous single TEM mode bandwidth, just to mention a few. For example, plotted in Fig. 2 are normalized group velocities of guided waves by the recta-coax line (shown in the inset) and several 50- microstrip lines. The measured result for a fabricated recta-coax with structural parameters denoted in the caption is also given. As seen, the recta-coax maintains virtually unchanged group velocity (very close to 1) over more than two octaves. On the other hand, all microstrip lines undergo 7%–8% variation through the same bandwidth, indicating much stronger dispersion than the coax. The group velocity of the traveling TEM wave is below the speed of light due to the wave propagation inside copper, dielectric strap loading, and the presence of the release holes.

Specifically, at 50 GHz, skin effect, dielectric, and holes contribute to 13.6%, 72.8%, and 13.6% to the overall reduction of the group velocity, respectively. Note that for forming a 50recta-coax with the cross section of the outer conductor given in Fig. 2, the fabrication of layer 3 has to be performed by sequential deposition of two sub-layers, each 50- m tall. This way, the aspect ratio of the photoresist mold between the inner conductor and the shield is kept below 1.5 : 1. Low aspect ratios (as close to 1 as possible) are desired for maintaining good quality of line cross sections. As is well known, the coupling between transmission lines becomes more serious as the operation frequency increases. To address this issue, Ishikawa and Yamashita proposed the buried microstrip lines (BMSLs) [14]. They obtained over 30-dB isolation at -band. Lower couplings at higher frequencies can be obtained with multilayer polyamide structures [15], [16]; however, the dielectric losses are typically significant. Ponchak et al. utilized metal filled via fences and obtained reduced coupling between parallel microstrip lines for up to 5 dB [17]. In recent years, various MEMS foundries have been used for building low-loss high-isolation millimeter-wave lines. For example, Drayton and Katehi demonstrated bulk micromachined shielded lines with high isolation and low loss [18]. Surface micromachining was also used for creating the transmission lines with low levels of coupling. For example, Jeong et al. fabricated BMSLs and air-gap coaxial lines and reported isolation of above 40 and 52 dB, respectively [19]; however, the coupling length was only approximately 2 mm, and insertion loss remained relatively high. Table I gives the summary of isolation structures and reported performances for the above-mentioned technologies. For comparative purposes, our results are also included, though they are discussed later in this paper. Note that over 60-dB iso-band high-density transceiver lation is required for a typical network, and that even lower coupling is allowed for high precision/dynamic range measurement equipment. A drawing of a four-port recta-coax isolation test structure used here is shown in Fig. 3. Ports are denoted as: input (1), output (2), forward coupled (3), and backward coupled (4). The test structure is composed of a pair of 50- recta-coax lines from Fig. 2 separated by a common 70- m-thick wall. The release holes in the outer conductor are also depicted, while the common wall holes are not shown. The center-to-center distance between the two inner conductors is 200 m. In a five-layer surface micromachining recta-coax line process, the common wall release holes can be etched in layers 2, 3, or 4, as shown in Fig. 4. When the heights of layers 2 and 4 are the same, like here, effects of the holes in these two layers are the same. Notice that the layer 2 and 3 holes must be offset from the locations where the dielectric straps are placed. III. SINGLE-HOLE ANALYSIS A. Analysis Bethe’s small aperture theory [20] is applied to compute the scattered fields through a common wall release hole up to 50 GHz. In this frequency range, a hole is electrically small ). Consider two recta-coax lines with a single (dimensions hole in the common wall, as shown in Fig. 5. The release holes

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TABLE I SUMMARY OF THE PERFORMANCE AND CHARACTERISTIC FEATURES OF VARIOUS REPORTED ISOLATION STRUCTURES

Fig. 3. Sketch of a four-port test structure, including the dielectric straps and outer conductor holes.

Fig. 5. Recta-coax isolation test structure with a layer 3 hole. (a) Orientation of the electric and magnetic dipole moments. (b) Coordinate system and hole parameters.

fields produced by an equivalent electric dipole moment normal to the wall and an equivalent magnetic dipole moment tangential to the wall. These are shown in Fig. 5(a). The are given by expansion coefficients

Fig. 4. Cross-sectional view of the recta-coax isolation test structure. Shown are layer 3 hole (left) and a layer 2 hole (right).

(2)

(3) in the outer conductor and the dielectric straps are not included in the development of this model. The fields scattered through the hole into the second line can be found as

and are the free-space permittivity and permewhere and are the electric and magnetic polarizability ability, and are the incident electric and magnetic dyadics, and field. These incident fields are assumed to be uniform over the hole. Combining (1)–(3) yields

(1) where indicates forward or backward wave direction with respect to the -axis. The reference is located and are normalized electric at the center of the hole. and magnetic field vectors in the recta-coax line. From Bethe’s theory, the scattered field through a small hole is the sum of the

(4) where is the propagation constant, is the wave impedance is the component of the incident of the free space, and and electric field. In (4), it is assumed that the incident field

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Fig. 6. S -parameter couplings for three structures shown in the inset. 130 300 m recta-coax lines and 250 250 m SCL are used. Height of the hole is 100 m and aspect ratio AR is 1. Notice that the backward coupling is always larger.

2

2

the normalized electric field are the same since the two lines are identical. Electric and magnetic polarizabilities are denoted and , respectively. They account for an effect of the as finite wall thickness, and are computed by

(5) where and are the aspect ratios of the hole defined and , respectively [see Fig. 5(b)]. and are as dimensionless coefficients that depend on / [21]. Forward and backward couplings can now be computed as

(6) From (4)–(6), it is clear that , indicating that the waves are coupled asymmetrically to the second recta-coax line, with the power in the backward wave being larger than that in the forward wave [22]. B. Results and Discussion A comparison of coupling -parameters between analytical and numerical models of the recta-coax isolation test structure with layer 2 and 3 holes and the square coax line (SCL) with a layer 3 hole is shown in Fig. 6. The holes in the outer conductor and the dielectric straps are not included and their effects are dism, , cussed later. The hole dimensions are and . Inner cross section of a 50- SCL’s outer conductor is 250 250 m . Normalized electric field distribution, needed for Bethe’s small aperture model, is computed at the center of the hole using conformal mapping [23]. FIT simulations are virtually indistinguishable from the FEM and are not included for clarity. As seen, excellent agreement between the analytical and numerical results is obtained with less than 2-dB difference for all

2

Fig. 7. Effects of common wall width on coupling S -parameters for a 130 300 m recta-coax line shown in the inset. Height of the hole is 100 m and aspect ratio AR is 1.

structures. These results demonstrate that analytical implementation of Bethe’s theory can be used to accurately predict the coupling between the neighboring lines. Note that although the component of the incident electric field exists in the layer 2 hole, the effect of the component is very small, and the incident fields for the layer 2 hole are also assumed to be uniform. Also noticeable is that the hole in layer 3 contributes to the larger coupling than the hole in layer 2. This is expected, as the stronger electric field is incident upon the layer 3 hole. Thus, for the minimum coupling between the lines, layer 2 holes are desired and selected for later studies. The effect of the width of the common wall is demonstrated in Fig. 7. As expected, the longer the width, the smaller the coupling since the hole can be considered as a rectangular waveguide operating below the cutoff. Comparison between the analytical and FEM results for the coupling -parameters as a function of normalized length of is shown in Fig. 8. Relathe layer 2 hole at 50 GHz tive differences, including the case of layer 3 holes, are given in Table II. The analytical and FEM results agree well when the length of the hole is electrically small. It is interesting to see increases, the differences for start to increase that as around , while the differences for are less (for layer 2 holes). It is generthan 2 dB up to ally accepted that the maximum dimensions of a hole for which . These findings, Bethe’s theory still hold is approximately however, show that Bethe’s theory provides accurate results for the backward coupling when the length of the hole is larger than with approximately 2-dB difference when the hole size . To further probe the applicability range is approximately for Bethe’s theory, recall the comparison between computed and in [21] with experimental data from [24]. An excellent , while is approximately 4% agreement is obtained for lower than the measurements (for larger ). The forward and backward couplings corrected for the experimental are recalculated and results are also shown in Fig. 8. As seen, these , while little or no efcorrections improve the accuracy for . Note that from the fabrication perspecfect is observed on

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Fig. 8. Coupling S -parameters computed with analytical and FEM models versus normalized hole length at 50 GHz. A 130 300 m recta-coax line with a layer 2 hole is used. Aspect ratio AR is 100/70.

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Fig. 9. Coupling S -parameters for the isolation structure with 27 uniformly distributed holes in layer 2. Holes dimensions are h = 100 m, AR = 100=300, and, AR = 100=70. The inner cross section of the recta-coax line is 130 300 m .

2

TABLE II RELATIVE DIFFERENCE BETWEEN ANALYTICAL AND FEM RESULTS

tive, the length of the release holes should be as large as possible to allow for easier removal of sacrificial photo-resist. To accommodate this desired feature and still be able to use Bethe theory for multiple-hole structures, the 300- m-long holes (for ) are chosen. IV. MULTIPLE-HOLE ANALYSIS A. Analysis and Design The single-hole analysis presented in Section III is combined with transmission-line theory and utilized for the analysis of multiple-hole isolation structures. It is assumed that the scattered field at each hole is the same, as this field is relatively small compared to the incident field (port 1 in Fig. 3). Shown in Fig. 9 are forward and backward couplings of a 1-cm-long , aspect ratio of each isolation structure with 27 holes ( copper section between the two neighboring holes is ). These holes are opened in the common wall of layer 2 and are uniformly distributed along the axial direction. As seen, excellent agreement between analytical and FEM results is obtained with less than 1.9-dB maximum difference observed at the very small backward coupling locations. Notice that the forward coupling increases monotonically with frequency since all coupled waves are in phase. Backward coupling has pronounced

j j

Fig. 10. Analytical results for maximum S versus number of holes in layer 2. The height of the holes is 100 m and the aspect ratio is AR = 100=300.

local minima and maxima, and the period is directly related to overall length of the isolation structure. To assess the applicability of the analytical model when the distance between the release holes is reduced, two additional configurations with 30 and 33 release holes were created. The separations between the uniformly distributed holes were 34 and 3.1 m, respectively. Up to 1-dB difference for forward coupling and a small frequency shift for the backward coupling are obtained in comparison with the FEM. It is important to understand that these two cases (30 and 33 holes) require much larger aspect ratios than those currently ). permitted by the utilized technology ( The analytical results for maximum forward coupling for a different number of layer 2 holes of a 1-cm-long recta-coax line with different widths of the common wall are shown in Fig. 10. Distances between the holes are maintained uniform. The height of the hole and the inner cross section of the recta-coax line are is the same as in Fig. 9. As seen in Fig. 10, the maximum increased by 6 dB when the number of holes doubles. Notice that

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Fig. 11. Optimization example: four layer 2 holes in a 1-cm-long isolation test structure. Hole dimensions are h = 100 m, AR = 100=300, and AR = 100=70. (a) Sketch of a structure with uniformly distributed holes. (b) Schematic of a circuit model created in the Ansoft Designer circuit simulator.

the forward coupling only depends on the number of holes and their dimensions since, at port 3, the forward coupling waves add in phase. B. Circuit Optimization To demonstrate advantages of analytical modeling for fast and accurate (referred to here as efficient) optimization-driven design, a 1-cm-long line shown in Fig. 11(a) is used. The initial design has four layer 2 holes uniformly distributed in the common wall across the length of the test structure. The length of the holes is fixed, and the optimization goal is to determine the best distance between holes for minimum backward coupling. From the earlier discussion, it is clear that the forward coupling will remain unchanged. The optimization process is driven within the internal engine of the circuit simulator Ansoft Designer. The circuit model is developed as follows: first, the four-port -parameters of a short section of an isolation structure with a single hole are extracted with the FEM. As the optimization is intended only for coupled -parameters, a recta-coax is modeled as an ideal TEM transmission line element with 50- impedance. Each two-port line model is then connected between the four-ports developed in the first step. A schematic diagram of a circuit representation for the isolation structure is shown in Fig. 11(b). The total length of the line was fixed at 1 cm, while the lengths of ideal transmission line sections are optimized. As seen in Fig. 12(a), approximately 12-dB is obtained with the optimized improvement at 45 GHz for and are not significantly structure. At the same time, changed [see Fig. 12(b)]. Note that the optimized circuit simulator results for the return and insertion losses are lower than those computed with the FEM due to the use of an ideal transmission line model in Fig. 11. Optimized values for all distances between the holes are shown in Table III. It took less than 1 s for the entire optimization to complete. When compared with approximately 160 s per frequency point and unchanged mesh (no optimization) in a full-wave simulator, significant savings

Fig. 12. (a) Isolation and (b) return and insertion loss for the optimized structure. Its full-wave simulation is also included.

TABLE III DISTANCES BETWEEN THE HOLES FOR UNIFORM AND OPTIMAL DISTRIBUTIONS

2

Fig. 13. Top view drawing of isolation structure for the 130 300 m RCLs = with four layer 2 holes. The dimensions are d ; d ; L ; L ; W ; L 3:26; 3:36; 0:3; 0:15; 0:116; 0:1 mm.

f

g

f

g

in computational resources are achieved without any noticeable sacrifices in accuracy. C. Effects of Straps and Outer Conductor Holes To evaluate effects of dielectric straps and holes in the outer conductor, a 1-cm-long isolation structure with four layer 2 holes in the common wall, shown in Fig. 13, is simulated with the FEM and FIT. 18- m-thick and 100- m-long dielectric straps are periodically distributed between layers 2 and 3. Their

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Fig. 14. Comparison between the analytical and numerical coupling S -parameters for: (a) the designed isolation structure and (b) its return and insertion losses.

dielectric constant and are 2.85 and 0.045, respectively. A comparison between the analytical and full-wave results for coupling -parameters is shown in Fig. 14(a). The cross section of the isolation structure is shown in the inset. As seen, excellent agreement is obtained. The analytical model did not include dielectric straps and outer conductor holes, which implies their irrelevance to the overall coupling. Our simulations have shown the validity of this observation even for the isolation test structures with up to 27 holes in the common wall. For completeness, shown in Fig. 14(b) are the computed return and insertion losses of an isolation structure from Fig. 13. with a loss of 0.3 dB/cm at As seen, excellent agreement for 30 GHz is obtained. The FEM and FIT simulations show a very frequency begood agreement and the discrepancy in the havior is due to different wave port implementations in the two codes and the fact that the reflected power levels at the ports are dB). very small ( V. MEASUREMENTS AND DISCUSSION A photograph of the fabricated isolation test structure and a 3-D model of the probe-to-recta-coax transition are shown

Fig. 15. (a) Fabricated isolation test structure. (b) 3-D model of a probe to rectacoax transition. (c) Four-port measurement setup (courtesy of BAE Systems, Nashua, NH).

in Fig. 15(a) and (b), respectively. Measurements of similar two-port structures have shown that this transition has return -band [25]. The test structure loss better than 20 dB at the consists of 1-cm-long lines; other structural parameters are given in the caption of Fig. 13. -parameters are measured using an HP-8510C network analyzer with GGB Industries’ 150- m-pitch microwave probes. Measurements are calibrated using external short-open-load-thru (SOLT) calibration on alumina substrate. The utilized measurement setup is shown in Fig. 15(c). Measured results for four test structures are overlaid in Fig. 16 and are compared with the full-wave simulations labeled as “FEM wave port” and “FIT wave port.” Although FEM and FIT results agree very well, there is a significant difference when these are compared with measurements. To research the reasons for this discrepancy, a more realistic computational model with 3-D models of probes is developed within the FEM (see Fig. 17). Since the isolation structure is a symmetric four-port network, even- and odd-mode analysis with magnetic

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Fig. 18. FEM simulations for 27-hole isolation structure excited with ideal wave ports and models of realistic probes. Hole dimensions are h = 300 m, AR = 1, and AR = 300=70.

Fig. 16. Coupling S -parameters of the fabricated isolation test structure shown in Fig. 15. Four devices are fabricated and their measurements are overlaid. (a) Forward and (b) backward couplings.

Fig. 17. Sketch of the probe model with shielding PEC walls.

and electric walls along the symmetry planes is used to reduce the size of the computational model [26]. This way, the same convergence tolerance used in the “wave port” numerical experiment can be set. Simulation results, labeled “FEM with probes” in Fig. 16, show significant increase in coupling when compared with previous FEM and FIT computations. More importantly, the levels and trends match the measured data quite well.

To further investigate the probe effects, three computational experiments are designed. First, two vertical walls set to be perfect electric conductors (PECs) are created to emulate the radiation shield between the probes (see Fig. 17). These walls do not cut through the test structure, they are instead erected from the outer conductor upwards into the four orthogonal sides of the radiating box. As shown in Fig. 16, the results of this computational experiment, labeled “FEM with shield,” agree very well with the original model without the probes. This experiment clearly demonstrates that the radiation coupling between probes is much higher than the actual coupling through the common wall holes, thus significantly deteriorating (up to 30 dB) the accuracy of measurements. In the second computational experiment, an isolation test and structure with 27 holes, m, is created. As shown in Fig. 18, the effects of radiation coupling between probes are now less important, as the neighboring lines significantly couple through the common wall holes. Compared to Fig. 10, the forward coupling is over 40 dB increased since the height of the common wall holes is three times larger. Note that the commercial probes are typically coated with absorbing material to reduce these radiation effects. Nevertheless, the results presented here clearly demonstrate that the probe radiation should be considered when the measured isolation structures have a low level of coupling. To the best of our knowledge, a calibration technique able to fully eliminate the effects of probe radiation on the accuracy of the line isolation measurements has not been reported yet. A full-wave FEM experiment with the thru-reflect-line standards has shown that unless the probes in the thru and line setups are completely shielded, the deembedding does not work. Insertion losses and forward and backward couplings of a 1-cm isolation structure with deembedded probes and probe-to-recta-coax transitions are virtually indistinguishable from the raw data. The third computational experiment is designed to demonstrate two ways for characterizing the coupling between the probes. In the first setup, referred to as Model 1, the four probe to recta-coax transitions [see Fig. 15(b)] are placed at the same

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becomes feasible, the test structure used in the Model 1 computational experiment could also be fabricated and used in exclusive probe-to-probe coupling measurements. VI. CONCLUSION

Fig. 19. Near- and far-port probe-to-probe coupling levels for the two computational models obtained with the FEM. Model 1 determines exclusive probe-toprobe couplings. Model 2 takes into account effects of secondary scattering arising from the high power levels, i.e., low insertion losses, at port 2.

positions as in Fig. 15(a). Instead of a 1-cm-long coupling structure, these transitions are terminated with 50- loads. The termination loads are realized by thin-film resistors connected between the inner conductor and vertical walls shorting the outer walls of the recta-coax. To retain as many geometrical features of the isolation test structure, the 1-cm-long coupling section is replaced by a solid copper bar. For the port notation from Fig. 15(a), and assuming for example that port 1 is excited, this setup models only the direct coupling between the four probes connected to the test-structure ports. As seen from Fig. 19, the ) and near port ( ) are coupling levels to the far port ( below 70 and 64 dB, respectively. These low coupling levels verify the above claim that the only important coupling is direct probe-to-probe and that any scattered fields from other probes can be neglected. The second model, referred to as Model 2, is developed to determine the probe coupling in the realistic four-port measurement setup, as shown in Fig. 15(c). The common wall release holes present in the isolation structure from Fig. 15(a) are now closed, thus leaving the probe-to-probe coupling as the only coupling mechanism. The main difference with Model 1 is in the follows: in probe 1 to probe 3 (or 4) coupling, the scattering term from the (terminated) probe on port 2 now becomes important (due to the line’s low insertion loss) and has to be taken into account. This is best illustrated in Fig. 19 where consistently higher, up to 10 dB, increased coupling levels for Model 2 are computed. The increased couplings at approximately 33 GHz . As exare directly correlated with the probe heights pected, the coupling levels obtained with Model 2 are almost the same as those labeled as “FEM with probes” in Fig. 16. This is yet another proof that the couplings are dominated by the undesired probe-to-probe radiation. Note that in the practical realization for Model 2, one needs to build only the modified isolation test structure without any common wall release holes and use the 50- loads to terminate the two unused probes (in a two-port measurement setup). Once the integration of thin-film resistors

Effects of the release holes in a common wall between the surface micromachined recta-coax lines have been analyzed using Bethe’s small aperture theory. Excellent agreement between analytical and numerical results for the forward and backward couplings has been obtained. A single-hole analysis has been applied to the design of multiple-hole structures, and efficient circuit simulator driven optimization has been demonstrated. A recta-coax isolation test structure with four common wall holes has been designed, fabricated, and characterized. Obtained measured coupling levels are below 55 dB/cm -band. These results agree well with a comthroughout the putational model that includes probes. It is, however, shown that the probe radiation is significantly higher than the common wall coupling and that it must be considered when low-coupling lines are designed. ACKNOWLEDGMENT The authors would like to thank D. Fontaine and G. Potvin, both with BAE Systems, Nashua, NH, Dr. C. Nichols and his Microfabrication Team, Rohm and Haas Electronics Materials, Blacksburg, VA, Dr. J. Evans, Defense Advanced Research Projects Agency (DARPA)/Microsystems Technology Office (MTO), Arlington, VA, E. Adler, Army Research Laboratory (ARL), Adelphi, MD, and Dr. W. Wilkins and Dr. V. Sokolov, both with the Mayo Foundation, Rochester, MN. The authors would also like to thank M. Lukic´ , Dr. K. Vanhille, Dr. S. Rondineau, and Prof. Z. Popovic´ , all with the University of Colorado at Boulder. The PolyStrata process is a trademark of Rohm and Haas Electronic Materials. REFERENCES [1] R. Chen, “Micro-fabrication techniques,” Wireless Design Develop., pp. 16–20, Dec. 2004. [2] E. R. Brown, A. L. Cohen, C. A. Bang, M. S. Lockard, B. W. Byrne, N. M. Vandelli, D. S. McPherson, and G. Zang, “Characteristics of microfabricated rectangular coax in Ka band,” Microw. Opt. Technol. Lett., vol. 40, pp. 365–368, Mar. 2004. [3] I. Llamas-Garro and P. Hall, “A low loss wideband suspended coaxial transmission line,” Microw. Opt. Technol. Lett., vol. 43, pp. 93–95, Oct. 2004. [4] J. B. Yoon, B. I. Kim, Y. S. Choi, and E. Yoon, “3-D construction of monolithic passive components for RF and microwave ICs using thick-metal surface micromachining technology,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 279–288, Jan. 2003. [5] D. Sherrer and J. Fisher, “Coaxial waveguide microstructures and the method of formation thereof,” U.S. Patent Applicat. 2004/0 263 290A1, Dec. 30, 2004. [6] D. S. Filipovic, Z. Popovic´ , K. Vanhille, M. Lukic, S. Rondineau, M. Buck, G. Potvin, D. Fontaine, C. Nichols, D. Sherrer, S. Zhou, W. Houck, D. Fleming, E. Daniel, W. Wilkins, V. Sokolov, and J. Evans, “Modeling, design, fabrication, and performance of rectangular -coaxial lines and components,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 1393–1396. [7] K. J. Vanhille, D. L. Fontaine, C. Nichols, Z. Popovic´ , and D. S. Filipovic, “Ka-band miniaturized quasi-planar high-Q resonators,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 6, pp. 1272–1279, Jun. 2007. [8] K. J. Vanhille, D. L. Fontaine, C. Nichols, D. S. Filipovic, and Z. Popovic´ , “Quasi-planar high-Q millimeter-wave resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2439–2446, Jun. 2006.

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[9] K. J. Vanhille, D. S. Filipovic, C. Nichols, D. Fontaine, W. Wilkins, E. Daniel, and Z. Popovic´ , “Balanced low-loss Ka-band coaxial hybrid,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 3-8, 2007, pp. 1157–1160. [10] M. Lukic and D. S. Filipovic, “Surface-micromachined dual Ka-band cavity backed patch antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2107–2110, Jul. 2007. [11] “HFSS v10.0 User Manual,” Ansoft Corporation, Pittsburgh, PA, 2005. [12] “CST Microwave Studio User Manual,” ver. 2006.0.0, CST GmbH, Darmstadt, Germany. [13] “Designer v2.0 User Manual,” Ansoft Corporation, Pittsburgh, PA, 2005. [14] T. Ishikawa and E. Yamashita, “Low crosstalk characteristic of buried microstrip lines,” in IEEE MTT-S Int. Microw. Symp. Dig., May 16-20, 1995, pp. 853–856. [15] K. Kim, Q. Yongxi, F. Guojin, M. Pingxi, J. Judy, M. F. Chang, and T. Itoh, “A novel low-loss low-crosstalk interconnect for broadband mixed-signal silicon MMICs,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1830–1835, Sep. 1999. [16] G. E. Ponchak, E. M. Tentzeris, and J. Papapolymerou, “Coupling between microstrip lines embedded in polyimide layers for 3D-MMICs on Si,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag,, vol. 150, no. 5, pp. 344–50, Oct. 10, 2003. [17] G. E. Ponchak, D. Chun, J. G. Yook, and L. P. B. Katehi, “Experimental verification of the use of metal filled via hole fences for crosstalk control of microstrip lines in LTCC packages,” IEEE Trans. Adv. Packag., vol. 24, no. 1, pp. 76–80, Feb. 2001. [18] R. F. Drayton and L. P. B. Katehi, “Development of self-packaged high frequency circuits using micromachining techniques,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 9, pp. 2073–2080, Sep. 1995. [19] J. Inho, S. Seong-Ho, G. Ju-Hyun, L. Joong-Soo, N. Choong-Mo, K. Dong-Wook, and K. Young-Se, “High-performance air-gap transmission lines and inductors for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2850–2855, Dec. 2002. [20] H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev., vol. 66, pp. 163–182, Oct. 1944. [21] N. A. McDonald, “Electric and magnetic coupling through small apertures in shield walls of any thickness,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 10, pp. 689–695, Oct. 1972. [22] P. F. Wilson and M. T. Ma, “Shielding-effectiveness measurements with a dual TEM cell,” IEEE Trans. Electromagn. Compat., vol. EMC-27, no. 3, pp. 137–142, Aug. 1985.

[23] M. Lukic, S. Rondineau, Z. Popovic´ , and D. S. Filipovic, “Modeling of realistic rectangular -coaxial lines,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2068–2076, May 2007. [24] S. B. Cohn, “Determination of aperture parameters by electrolytic-tank measurements,” Proc. IRE, vol. 39, no. 11, pp. 1416–1421, Nov. 1951. [25] K. J. Vanhille, “Design and characterization of microfabricated threedimensional millimeter-wave components,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Colorado at Boulder, Boulder, CO, 2007. [26] J. Reed, “A method of analysis of symmetrical four-port networks,” IRE Trans. Microw. Theory Tech., vol. MTT-4, no. 10, pp. 246–252, Oct. 1956.

Yuya Saito (S’07) received the B.S. degree in electrical and electronic engineering from Hosei University, Tokyo, Japan, in 2002, the M.S. degree from the Tokyo Institute of Technology, Tokyo, Japan, in 2004, and is currently working toward the Ph.D. degree at the University of Colorado at Boulder. His research interests include analytical and numerical modeling and design of 3-D millimeter-wave components and subsystems.

Dejan S. Filipovic´ (S’97–M’02) received the Dipl. Eng. degree in electrical engineering from the University of Nis, Nis, Serbia, in 1994, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1999 and 2002, respectively. From 1994 to 1997, he was a Research Assistant with the University of Nis. From 1997 to 2002, he was a graduate student with The University of Michigan at Ann Arbor. He is currently and Assistant Professor with the University of Colorado at Boulder. His research interests are the development of millimeter-wave components and systems, multiphysics modeling, and antenna theory and design, as well as computational and applied electromagnetics.

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Calculation of the Properties of Reentrant Cylindrical Cavity Resonators Richard G. Carter, Senior Member, IEEE, Jinjun Feng, Member, IEEE, and Ulrich Becker

Abstract—The lowest resonant frequencies of reentrant cylindrical cavity resonators are calculated using the method of moments to obtain upper and lower bounds. The accuracy and converfactors and shunt gence of the results are investigated and the impedances calculated. A simple empirical assumption about the choice of basis functions leads to results that are of good accuracy and readily computed. The results obtained are compared with those of experiment and from calculations using MAFIA and Microwave Studio. Index Terms—Cavity resonators, frequency, method of moments, shunt impedance, factor, upper and lower bounds.

I. INTRODUCTION HE PROPERTIES of cavity resonators (resonant frequency, factor, and shunt impedance) are commonly calculated using finite-difference time-domain, finite-element, and other similar methods [1]. A variety of computer codes are available for this purpose and the power of modern computers means that the properties of many cavities can be calculated quickly and with good accuracy. However, some commonly encountered geometries, such as the cylindrical reentrant cavity shown in Fig. 1, include sharp edges, which are difficult to model accurately using these methods. It is usually necessary to use very large numbers of mesh cells to obtain good accuracy and the computational time becomes much longer. In an alternative approach, the electric and magnetic fields in each region of the cavity are represented by infinite series of basis functions, which are then matched by the imposition of . This method was proposed a continuity condition at for the reentrant cavity by Hansen [2] who assumed an approximate variation of the electric field on the boundary between them. The resonant frequency of the lowest TM mode was de. termined by matching the magnetic fields at the point In a further development of this method, Chu and Hansen [3] showed that upper and lower bounds to the resonant frequency could be obtained from assumed variations of the electric and magnetic fields on the boundary by requiring that the flux of the . This reactive Poynting vector should be continuous at

T

Manuscript received May 31, 2007; revised September 3, 2007. This work was supported in part by the Particle Physics and Astronomy Research Council, U.K., under Grant PPA/G/S/2000/00055. R. G. Carter is with the Engineering Department, Lancaster University, Lancaster LA1 4YR, U.K. (e-mail: [email protected]). J. Feng is with the Vacuum Electronics National Laboratory, Beijing Vacuum Electronics Research Institute, Beijing 100016, China (e-mail: [email protected]). U. Becker is with the Technical Support Group, CST GmbH, D-64289 Darmstadt, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.909750

Fig. 1. Cross section of a reentrant cylindrical cavity.

method is closely related to that used by Schwinger to determine upper and lower bounds for the admittances of discontinuities in waveguides [4], [5]. A further important step was taken by Taylor [6] who showed that it is not necessary to assume the forms of the electric and magnetic fields. Instead, the field expansions in the two regions can be linked by the continuity equations to give a determinant whose value is zero at the resonant frequency. It has been shown [7], [8] that this process can lead to upper and lower bounds to the frequency and is effectively a variational method. Since the method reduces the problem to a matrix equation, it falls within the general class of moment methods [9]. The problem of computing the resonant frequency of the cavity shown in Fig. 1 has been addressed by many other authors, see, e.g., [10]–[14] and the references therein. However, Taylor’s method is the best one available for this problem because the computation of upper and lower bounds to the resonant frequency means that the accuracy of the solution is always known. This method is, therefore, valuable for the rapid and accurate computation of the properties of reentrant cavities and for providing a method by which the accuracy of results obtained by other methods can be checked. It can also be used to compute the properties of any resonant structure, which can be divided into two or more simple regions in which the fields can be expressed in terms of basis functions, which satisfy Maxwell’s equations and the external boundary conditions [7], [8]. The purpose of this paper is to examine the convergence of Taylor’s method and to show how it can be used to obtain factors, and accurate values for the resonant frequencies, shunt impedances of a wide range of cavities. We shall see that good results can be obtained by retaining only a few terms in

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the series expansions of the fields when an empirical rule is applied.

is defined in the Appendix. Similarly the where the matrix at is continuity condition for

II. THEORY

when

The theory of the method presented here is based on that described by Taylor, but uses matrix algebra. This makes it easier to understand and apply. In region I, the axial component of and the azimuthal component of , which satisfy the boundary conditions on the axis, may be written as [6] (1) and (2)

(11) and inteWhen both sides of (11) are multiplied by to , we obtain the matrix equation grated from (12) is defined in the Appendix. It should be where the matrix when is autonoted that the boundary condition for matically satisfied by the distribution of the surface current. It is now a simple matter to eliminate all the column vectors from (7), (8), (10), and (12) to of the coefficients, except obtain

where the functions are defined in the Appendix, the prime denotes differentiation with respect to the argument (3)

(13) so that the eigenvalue

is the solution of (14)

and (4) where

. Now, when

, let (5)

is the unit matrix. This is equivalent to the solution where obtained by Taylor [6] and Jaworski [14] (but note that the factor in [14, eq. (17)] is incorrect and should be replaced by of 2 2). If, alternatively, the column vector is retained, we find that (15)

(6) where and are constant coefficients. Eliminating the coefficients between (1), (2), (5), and (6), we obtain (7) where and are column vectors of the coefficients in (5) and (6) and is a diagonal matrix whose elements are defined in the Appendix. A similar equation can be derived for region II as (8) where the elements of are defined in the Appendix and are column vectors of the coefficients of the and and in region II at analoFourier expansions of gous to those in (5) and (6). is continuous at for The condition that and if is if if (9) . When both sides of (9) are multiplied by and integrated from to , we obtain the matrix equation where

(10)

The derivations of (13)–(15) assume that both series of basis functions contain an infinite number of terms. In practice, the series must be terminated at and in regions I and II, respectively. Taylor [6] states that when the inner coefficients are eliminated, there is a discontinuity in at , leading to an upper bound for , and that the elimination of leads to a discontinuity in and to a the outer coefficients lower bound for . The existence of upper and lower bounds is said to be a consequence of regarding the truncated series of basis functions as a trial function in the variational sense. However, this explanation cannot be correct because when the is used in both (13) and (15), the same same choice of and value of is computed in each case. It is, therefore, necessary to examine more closely the relationship between the method of moments and the variational method implied by the truncation of the series. III. VARIATIONAL METHOD The variational method for this problem can be established in a simpler manner than in [7] and [8] by applying Poynting’s theorem [15] separately to the two regions. If the electric field , then in region I, is taken to vary with time as (16)

where is the instantaneous stored energy in region and the integral is taken over its surface. However, since we have

CARTER et al.: CALCULATION OF PROPERTIES OF REENTRANT CYLINDRICAL CAVITY RESONATORS

defined and in terms of basis functions, which satisfy the external boundary conditions, the integral must be 0, except on . A similar equation can be written for region II. These equations may be added together to give the rate of change with time of the total stored energy in the cavity as

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We use (10) to express (20) in terms of

as

(21)

(17) . The where the integration is taken over the surface at negative sign in the integrand is necessary because the flux of the Poynting vector in (16) is defined in terms of the outward normal for the region. Now the left-hand side of (17) must be 0 at all times for the oscillations in a lossless cavity. When and are exact solutions to the problem, their tangential components are and the integral in (17) is identically 0. continuous when Let us now assume that the fields in the two regions are defined by two series of basis functions as above and choose to . If we retruncate the series in the inner region when quire that the continuity condition (9) is satisfied, then an infinite number of terms is required in the outer region. The coefficients of the magnetic fields in the two regions are given by (7) and (8), and we note that these depend upon . There is now a discontinuity in when and the continuity condition (11) cannot be satisfied. We may, however, impose some other, approximate, continuity condition and an approximate value of is then determined. Following Schwinger and Saxon [4], let us choose that the flux of the reactive Poynting vector is continuous . Equation (17) then shows that the total stored energy at in the cavity is constant as, physically, it must be. Other possible choices of approximate continuity condition do not guarantee this. The flux of the reactive Poynting vector out of region I is given by (18)

We now make use of (5) and (6) and note that the basis functions are mutually orthogonal so that

so that the approximate continuity condition is (22) Note that this expression is quadratic in the coefficients , which are, as yet, undetermined. Their values can be found by treating them as variational parameters and requiring that the value of should be stationary for small variations of them. Thus, (23) The first term is 0 when is stationary and the values of and are then the solution of the set of equations obtained by equating the second term in (23) to 0. It can be shown, after some manipulation, that this condition, which is a linear function of the coefficients, is identical to (13). Thus, the truncation of the series in , while retaining an infinite number of terms in the series in , leads to a stationary value of . The same procedure may be carried out when the series for in the outer region is truncated at , while retaining an infinite number of terms in the inner region. The condition that should be stationary is then (15). Chu [8] has shown that the two values of determined in this way must be different and are upper and lower bounds to the exact solution. Thus, the method of moments naturally leads to upper and lower bounds for through the truncation of the series of basis functions in either region. We see that the explanation offered by Taylor [6] is misleading and that it is the method of truncation, and not the choice of coefficients to be retained, which leads to upper and lower bounds. These bounds are absolute limits and make it possible to determine the resonant frequency of the cavity with known accuracy. IV. IMPLEMENTATION

(19) where the expansion has been terminated at II, the series is infinite and we have

. In region

(20)

The method described in Section III was implemented using Mathcad 13.1 The solutions to (14) were found using the secant method, which was terminated when the fractional error in was less than 10 . Once had been determined, the eigenvector was obtained from (13) using the Mathcad function eigenvec with an eigenvalue of unity. The eigenvector was norat . The other column vectors malized to a gap voltage of coefficients for the electric and magnetic fields were obtained from (7), (8), and (10) so that the electric and magnetic fields 1Mathcad 13 is a product of the Parametric Technology Corporation, Needham, MA. [Online]. Available: http://www.ptc.com

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Fig. 3. Convergence of the upper and lower bound frequencies for the same cavity as Fig. 2 with results from Jaworski [14]. Fig. 2. Convergence of frequencies computed for a cavity for which 6:004 mm, r = 42:29 mm, z = 7:958 mm, and z = 22:792 mm.

r

=

throughout the cavity could be computed to the level of approximation determined by the choices of and . The stored enand the energy loss per cycle were obtained by inteergy grations over the volume of the cavity and over its surface so that the factor could be calculated. The results presented below assumed that the cavities were made of copper with conductivity 5.7 10 S m . Finally, the ratio of the shunt impedance to was calculated using [16] (24) where is the resonant frequency. The calculations usually take a few seconds on a PC with a 3-GHz Pentium 4 processor and 1 Gb of RAM. V. CONVERGENCE OF THE METHOD The convergence of the method outlined above was studied using a cavity for which the frequency had been computed by Jaworski [14]. Fig. 2 shows that the resonant frequencies comdecreased smoothly to a lower bound as puted for constant was increased. Similarly, the resonant frequencies computed for constant increased smoothly to an upper bound as was increased. It can also be seen that the bounds obtained are nested within one another so that they are truly upper and lower bounds, as expected. In each case, the calculations were taken to (or ) and the infinite limits found by linear extrapola, the frequency is always tion. It may be noted that when close to the upper bound. Fig. 3 shows the convergence of the infinite limits to the upper and lower bounds as the number of terms in the truncated series was progressively increased. When (or ) the bounds were 2.12588 and 2.12591 GHz giving a frequency of 2.1259 GHz to an accuracy of four decimal places. Fig. 3 also shows the lower bound results obtained by Jaworski [14] who was apparently unaware of the possibility of obtaining an upper bound. In his results, the maximum value of was 10 and linear extrapolation was used to obtain a resonant frequency of 2.1276 GHz, which is in error by approximately 0.08%. The difference between this result and those ob-

Fig. 4. Convergence of the upper and lower bounds for N = 8 and M = 24.

tained here was probably caused by not using a large enough (the value used was not stated). Fig. 2 shows that value of this would lead to an overestimate of the frequency, as observed. The method described here can give as high an accuracy as may be desired by suitable choices of and , but care is needed to ensure that proper convergence has been obtained. VI. APPROXIMATE METHOD Fig. 2 shows that the frequencies computed using constant values of , while tending to a lower bound, give frequencies that lie above the true frequency when is not much greater for which the frethan . In every case, there is a value of quency is close to the true frequency. Examination of this point on each curve showed that a very good estimate of the final freis taken to be the closest integer quency can be obtained if ]. This choice makes the length of the smallest to [ wavelength of the basis functions approximately the same in both regions. While this is intuitively attractive, we have not been able to find any theoretical reason why it should be so. The approximate frequency calculated using this relationship has the advantage that good accuracy can be obtained without the need for extrapolation of the results or for the use of large numbers of terms. Error limits can be obtained as before, if necessary, by

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Fig. 7. Difference between the resonant frequencies of the test cavities computed with Microwave Studio and those obtained when N = 8. Fig. 5. Convergence of the approximate frequency.

Fig. 8. Difference between the Q factors of the test cavities computed with Microwave Studio and those obtained when N = 8.

Fig. 6. Shapes of the test cavities.

fixing

and increasing , as shown in Fig. 4. When and , the frequency is 2.1258 GHz to an accuracy of 1.4 MHz and 0.9 MHz. In practice, the accuracy of the result is greater than this because it lies almost midway between the bounds. Fig. 5 shows the results obtained using the . The frequency was approximate method for values of and computed using in this way up to , the result is 2.1259 GHz to an it was found that, for accuracy of four decimal places. Thus, the frequency computed is actually accurate to 0.004%. when To evaluate the usefulness of the approximate method, nine very different cavities were selected from the set described by Hamilton et al. [17], which were also used by Fujisawa [18]. Since the shape of a cavity can be described by three normalized dimensions, the objective was to investigate all combinations of the extreme values of these together with one case in the center of the range. The shapes of the cavities are shown in Fig. 6 where the numbering corresponds to that used by Fujiof each cavity were sawa. The resonant frequency and . computed to eight significant figures with In every case, the results were found to have converged to at . A study of the difleast five significant figures when and those for ferences between the results for showed that, in almost every case, the difference was less than 0.005%. The exceptions were cavities 4 and 52 in which is

Fig. 9. Difference between the R=Q of the test cavities computed with Microwave Studio and those obtained when N = 8.

small compared with and is appreciably less than . For those cavities, the differences were less than 0.01%, except for : 0.011%). Thus, the properties of cavity 52 ( : 0.016%, any cavity whose shape lies within the range of those computed can be calculated with an error of less than 0.01% using the ap. proximate method and VII. COMPARISON WITH OTHER METHODS The properties of the cavities in Fig. 6 were computed using Microwave Studio (MWS)2 with the default setting of automatic mesh refinement, which seeks the frequency to an accuracy of better than 1%. The differences between these results and those are shown obtained with the approximate method in Figs. 7–9. The differences are less than 1.0% for the fre. The frequenquency, 2.2% for the factor, and 5.7% for factors show signs of systematic errors of around cies and 0.5% and 1.4%, respectively. To investigate the comparison between the methods in more detail, results were computed for 2Microwave Studio and MAFIA are products of CST GmbH, Darmstadt, Germany. [Online]. Available: http://www.cst.com

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the change in frequency is 0.03%, which is also negligible. If (the most sensitive dimension) is in error it is assumed that by 10 m, the change in the frequency is of the order of 0.1%. It, therefore, seems probable that the differences shown in Fig. 10 are caused by experimental errors.

TABLE I COMPARISON BETWEEN RESULTS FROM MAFIA-2D AND FROM THE APPROXIMATE METHOD

VIII. CONCLUSION

Fig. 10. Difference between measured and calculated frequencies.

four cavities using MAFIA-2D, which employs the same algorithm as Microwave Studio, but which assumes cylindrical symmetry. The first three of these were selected because the differences between the results obtained by the two methods had been particularly large in previous tests. The results in Table I show excellent agreement. The resonant frequency of cavity 60 calculated using MAFIA-2D and extrapolated to an infinite number of mesh nodes was 9.5339 GHz. Calculations using MWS and more than 2 million mesh points gave a frequency of 9.53032 GHz. Extrapolation of the MWS results to an infinite number of mesh points gave 9.5341 GHz. The total CPU times using a PC with a 2.8-GHz Xeon processor and 2-GB RAM were 2 h for MAFIA-2D and 6 h for MWS. The comparison shows that very close agreement can be obtained between the results from the different methods if sufficient care is taken. The resonant frequencies of cavities for which experimental data is available [11], [12], [14] were computed using the ap. The upper and lower bounds were proximate method and , respectively, by a further computed by increasing factor of eight. The differences between the measured frequencies and those computed, shown in Fig. 10, are generally less than 0.5%. The upper and lower bounds lie within 0.1% of the approximate frequency with a single exception (cavity 7, upper bound, 0.12%). The correction for the difference between a cavity with walls of infinite and finite conductivity was computed using [19]

A detailed study has been carried out of the method proposed by Taylor [6] for computing the properties of cylindrical reentrant cavity resonators. This method is superior to those described by later authors and seems to have been overlooked by them. The theory of the method has been presented in terms of matrix algebra, which makes it simple to implement. The upper and lower bounds to the resonant frequencies of cavities can be computed by suitable choices of the numbers of basis and outer regions. It has been functions in the inner shown that this is equivalent to the variational method proposed by Schwinger for calculating the properties of discontinuities in waveguides. The method is capable of any desired accuracy if sufficiently large numbers of basis functions are used. It has been shown that, when the numbers of basis functions in the two regions are linked by a simple empirical formula, the frequency computed lies very close to the exact value. Tests with cavities of widely differing shapes showed that this result is valid for all cavities of this type. For practical purposes, it is suffi; the resonant frequency, factor, and cient to choose are then generally accurate to better than 0.01%, which is more than adequate for most purposes. It has been shown that the results agree with those extrapolated from computations using Microwave Studio and MAFIA-2D to an accuracy of better than 0.002%. Comparisons with the results of experiment and of computations using Microwave Studio, with automatic mesh refinement set to 1% accuracy in the frequency, showed differences in the resonant frequencies of less than 1%. This method provides a fast and accurate way of computing the properties of reentrant cavity resonators, disc-loaded waveguides, and other similar microwave structures. It is valuable for benchmarking results obtained by other methods. APPENDIX The radial variations of the fields in (1) are given by

(A1) otherwise where and are Bessel functions using the usual notation. The corresponding expression in region II is

(25) and found to be less than 0.02% in every case. This is much smaller than the differences observed so that the error involved in assuming that the boundaries are perfect conductors is negligible. If the relative permittivity of air is taken to be 1.00054,

otherwise. (A2)

CARTER et al.: CALCULATION OF PROPERTIES OF REENTRANT CYLINDRICAL CAVITY RESONATORS

The working equations for calculating the elements of the diagonal matrices in (7) and (8) are

(A3) otherwise

otherwise (A4) where

and . It was found that large argument approximations [20] to these expressions were required to avoid floating point overflow errors when the numerical values of the arguments of the Bessel functions exceeded 700. The definitions of the elements of the matrices in (10) and (12) are

(A5)

(A6)

ACKNOWLEDGMENT The authors wish to acknowledge the helpful comments of a referee on a previous version of this paper, which led us to study the method of upper and lower bounds in greater depth. The authors are grateful to Prof. R. Tucker, Physics Department, Lancaster University, Lancaster, U.K., for many helpful discussions. REFERENCES [1] M. N. O. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. Boca Raton, FL: CRC, 2000.

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[2] W. W. Hansen, “On the resonant frequencies of closed concentric lines,” J. Appl. Phys., vol. 10, pp. 38–45, 1939. [3] E. L. Chu and W. W. Hansen, “Disk-loaded waveguides,” J. Appl. Phys., vol. 20, pp. 280–285, Mar. 1949. [4] J. Schwinger and D. S. Saxon, Discontinuities in Waveguides. New York: Gordon and Breach, 1968. [5] R. E. Collin, Field Theory of Guided Waves. New York: McGrawHill, 1960. [6] R. Taylor, “Calculation of resonant frequencies of re-entrant cylindrical electromagnetic cavities,” J. Nucl. Energy C, Plasma Phys., vol. 3, pp. 129–134, 1961. [7] J. S. Bell, “A variational approach to disc-loaded waveguides,” Atom. Energy Res. Establishment, Harwell, U.K., AERE Rep. G/R 680, 1951. [8] E. L. Chu, “Upper and lower bounds for composite-type regions,” J. Appl. Phys., vol. 21, pp. 454–467, May 1950. [9] R. F. Harrington, Field Computation by Moment Methods. Malabar, FL: Krieger, 1982. [10] D. M. Bolle, “Eigenvalues for a centrally loaded circular cylindrical cavity,” IRE Trans. Microw. Theory Tech., vol. MTT-10, pp. 133–138, Mar. 1962. [11] K. Uenakada, “LCR equivalent circuit of re-entrant cavity resonator,” (in Japanese) Trans. Inst. Electron. Commun. Eng. Jpn., vol. 53-B, pp. 51–58, 1970. [12] K. Uenakada, “Equivalent circuit of re-entrant cavity,” IEEE Trans. Microw. Theory Tech., vol. MTT-21, pp. 48–51, Jan. 1973. [13] A. G. Williamson, “The resonant frequency and tuning characteristics of a narrow-gap re-entrant cavity,” IEEE Trans. Microw. Theory Tech., vol. MTT-24, pp. 182–187, Apr. 1976. [14] M. Jaworski, “On the resonant frequency of a re-entrant cylindrical cavity,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 4, pp. 256–260, Apr. 1978. [15] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics. New York: Wiley, 1965. [16] R. G. Carter, Electromagnetic Waves: Microwave Components and Devices. London, U.K.: Chapman & Hall, 1990. [17] D. R. Hamilton, J. K. Knipp, and J. B. K. Kuper, Klystrons and Microwave Triodes. New York: McGraw-Hill, 1948, pp. 77–79. [18] K. Fujisawa, “General treatment of klystron resonant cavities,” IRE Trans. Microw. Theory Tech., vol. MTT-6, no. 10, pp. 344–358, Oct. 1958. [19] J. C. Slater, Microwave Electronics. New York: Van Nostrand, 1950. [20] N. W. McLachlan, Bessel Functions for Engineers. Oxford, U.K.: Clarendon, 1955. Richard G. Carter (M’97–SM’01) received the B.A. degree in physics from the University of Cambridge, Cambridge, U.K., in 1965, and the Ph.D. degree in electronic engineering from the University of Wales, Wales, U.K., in 1968. From 1968 to 1972, he was involved with highpower traveling-wave tubes as a Development Engineer with the English Electric Valve Company Ltd. In 1972, he joined the Engineering Department, University of Lancaster, initially as a Lecturer, then as a Senior Lecturer in 1986, and then as a Professor of electronic engineering in 1996. His research interests include electromagnetics and microwave engineering with particular reference to the theory, design, and computer modeling of microwave tubes and particle accelerators. Prof. Carter is a Fellow of the Institution of Engineering and Technology (IET). He is a member of the Vacuum Electronics and Compact Modeling Technical Committees of the IEEE Electron Devices Society

Jinjun Feng (M’94) received the B.Sc. degree in electronic engineering from Tsinghua University, Beijing, China, in 1988, and the M.Sc. and Ph.D. degrees in physical electronics from the Beijing Vacuum Electronics Research Institute (BVERI), Beijing, China, in 1990 and 2001, respectively. His doctoral thesis concerned field emitter array cathodes and their application in microwave devices. In 1990, he was involved with vacuum microelectronics and microwave tubes with the BVERI. In 1997, he became a Senior Engineer. In 2000, he became a Research Professor. From 1999 to 2001, he was Head of the

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Device Computer-Aided Design (CAD) Division, and since 2003, has been Vice Director of the Vacuum Electronics National Laboratory, BVERI. In 1997, he was a Visiting Research Fellow with Lancaster University, and in 2001 was a Post-Doctoral Research Associate. His research interests include field emitter array cathode technology and microwave vacuum devices using microfabrication technology. Dr. Feng is a Senior Member of the Chinese Institute of Electronics (CIE). He is a member of the Chinese Vacuum Society (CIV). He has been the treasurer of the IEEE Electron Devices Society Beijing Chapter since 1998.

Ulrich Becker was born in 1967. He received the Engineering degree in electronics and Ph.D. degree in electromagnetic simulation in interaction with free moving charges from the Technical University of Darmstadt, Darmstadt, Germany, in 1992 and 1997, respectively. Since 1998, he has been with the Technical Support Group, CST GmbH, Darmstadt, Germany, where he provides general tools for electromagnetic simulation in different areas.

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A Space-Mapping Approach to Microwave Device Modeling Exploiting Fuzzy Systems Slawomir Koziel, Senior Member, IEEE, and John W. Bandler, Life Fellow, IEEE

Abstract—We present a novel surrogate modeling methodology based on a combination of space mapping and fuzzy systems. Fine model data, the so-called base set, is assumed available in the region of interest. Although we do not assume any particular location of the base points, it is preferable that they form a uniform mesh. The standard space-mapping surrogate is established using available fine model data. The fuzzy system is then set up to interpolate the differences between the space-mapping surrogate and the fine model at all base points. Our new methodology offers significant advantages with respect to some of the previous space-mapping approaches to modeling, which are: 1) it handles any base set and 2) the number of space-mapping parameters does not limit the accuracy of the surrogate. Moreover, it exhibits comparable or better accuracy than the recently published modeling technique utilizing space mapping and radial basis functions. We also consider a hierarchical fuzzy space-mapping modeling, which relies on a fuzzy interpolation of space-mapping parameters and subsequent fuzzy interpolation of the residuals between the fine and surrogate model. Examples demonstrate the robustness of our approach and give a comparison with other space-mapping-based modeling techniques. Index Terms—Computer-aided design (CAD), electromagnetic (EM) modeling, fuzzy systems, microwave circuits, space mapping, surrogate modeling.

I. INTRODUCTION

S

TATISTICAL analysis and yield optimization are crucial to manufacturability-driven designs in a time-to-market development environment and demand fast accurate device and component models. Full-wave electromagnetic (EM) simulations of microwave structures offer accuracy at the cost of CPU effort. High CPU cost is undesirable from the point of view of direct statistical analysis and design. The space-mapping concept [1]–[8] addresses this issue. Space mapping assumes the existence of “fine” and “coarse” models. The “fine” model may be a high fidelity CPU-intensive EM simulator, undesirable for direct statistical analysis and design. The “coarse” model can be a simplified representation such as an equivalent circuit with empirical formulas. Space-mapping modeling [9]–[16] and neuro-space-mapping Manuscript received May 25, 2007; revised September 13, 2007. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007239 and Grant STPGP336760, and by Bandler Corporation. S. Koziel is with the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1 (e-mail: [email protected]). J. W. Bandler is with the Simulation Optimization Systems Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1, and also with Bandler Corporation, Dundas, ON, Canada L9H 5E7 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.909605

modeling [17]–[19] exploit the speed of the coarse model and the accuracy of the fine model to develop fast accurate enhanced models (surrogates) valid over a wide range of parameter values. The main factor that distinguishes space mapping from many other surrogate-based modeling methodologies (e.g., [20]–[26]) is the use of physics-based coarse models, which allows good modeling accuracy with a small amount of fine model data. The standard space-mapping modeling methodology [11], [12] is based on setting up the surrogate model using a small points, where amount of fine-model data (usually, is the number of design variables). Extraction of the model parameters is performed over the whole set of this data. This methodology is simple and gives reasonable accuracy, which, however, may not be sufficient for some applications. To improve modeling performance, additional fine model information needs to be involved. Unfortunately, this approach to space mapping is not able to effectively harness a large amount of data, i.e., increasing the number of base points does not help if the number of space-mapping parameters (model flexibility) remains unchanged [15]. Space-mapping modeling with variable weight coefficients [14], [15] is aimed at overcoming these limitations. It indeed provides much better accuracy than the standard method, however, at the expense of significant increase of the evaluation time, which is due to a separate parameter extraction required for each evaluation of the surrogate model. This limits potential applications of the method. A recently published modeling technique utilizing space mapping and radial basis function interpolation [16] gives modeling accuracy comparable with [14] without compromising computational cost. Moreover, because of the fact that the surrogate is based on the underlying coarse model, modeling accuracy is substantially better than for radial basis function interpolation used directly. Unfortunately, the problem of determining interpolation coefficients may be ill conditioned and the method may be very sensitive to some control parameters. In this paper, we present another approach that combines standard space-mapping modeling with fuzzy system interpolation. This technique has the same advantages as the methodology [16], however, it is simpler to implement. Moreover, in some cases, it allows us to improve modeling accuracy even further. II. FUZZY SPACE-MAPPING SURROGATE MODELING Let : , and : denote the fine and coarse model response vectors. For and may represent the magnitude of example,

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a transfer function at chosen frequencies. We denote by the region of interest in which we want an enhanced matching between the surrogate and fine model. We assume is an -dimensional interval in with the center at that reference point

[

(1) determines the size of . We use to denote the region of interest defined by and . Suppose we have the base set , where is the number of base points, such that the . fine model response is known at all points is defined as in [16] A generic surrogate model where

(2) Matrices

and are obtained by the parameter extraction

(3) Apart from model (2) and (3), optional frequency scaling can be implemented that works in such a way that the coarse model is evaluated at a different frequency than the fine model using [11]. More general spacethe transformation mapping models can be found, e.g., in [3] and [11]. Let us introduce the so-called characteristic distance of the base set depending on the size of the region of interest and the number of base points, defined as (4) If the base points are uniformly distributed in is just an average distance between neighboring points. We will use parameter to characterize and compare different . base sets On top of the standard space-mapping surrogate, we use fuzzy and interpolation of the difference between the fine model standard surrogate. Fuzzy systems have been successfully used in the microwave area by other authors (e.g., [27]–[29]). In this study, we use a fuzzy system with triangle membership functions and centroid defuzzification [30]. We assume that we have , where and data pairs . Membership functions for the th variable are defined as shown in Fig. 1. Each interval , is divided into subintervals (fuzzy regions). The number corresponds to the number of . In base points and is given by the formula consists of base points uniformly distributed particular, if , then is exactly the number in the region of interest of points of this uniform grid along any of the design variable axes. In general, is chosen in such a way that the number of

0

Fig. 1. Division of the input interval x  ;x and the corresponding membership functions.

+  ] into fuzzy regions

-dimensional subintervals (and, consequently, the maximum number of rules) is not larger than the number of base points. into subintervals creThe division of values . In the case of a uniform ates cobase set, points incide with the base points. Value corresponds to the fuzzy for for region , and for ). We also use the symbol to denote the -dimensional fuzzy region . For any given , the value of membership function de. In this paper, termines the degree of in the fuzzy region we only use triangular membership functions; one vertex lies at the center of the region and has membership value unity; the other two vertices lie at the centers of the two neighboring regions, respectively, and have membership values equal to zero. Having defined the membership functions, we need to generate the fuzzy rules from given data pairs. We use if–then rules is in , THEN , where is the reof the form IF sponse of the rule. At the level of vector components it means IF

is in AND THEN

AND is in

is in

AND (5)

are components of vector . In genwhere eral, it may happen that there are some conflicting rules, i.e., rules that have the same IF part, but a different THEN part. We resolve such conflicts by assigning a degree to each rule and accepting only the rule from a conflict group that has a maximum degree. A degree is assigned to a rule in the following way. For is in AND is in AND AND the rule “IF is in THEN ,” the degree of this rule, denoted , is defined as by (6) Having resolved the conflicts we have a set of nonconflicting . We denote by rules, which we denote as the output of our fuzzy system, which is determined using a centroid defuzzification (7) where is an -dimensional fuzzy region corresponding to the th rule, and is the output of the th rule.

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Fig. 3. Coarse model of the second-order dual-behavior resonator filter (Agilent ADS). Fig. 2. Geometry of the second-order dual-behavior resonator filter [31].

Our surrogate model combining the standard space-mapand the fuzzy system is defined as ping surrogate (8) It should be noted that although the idea of combining space mapping with fuzzy systems is similar to the idea of combining space mapping with radial basis function interpolation described in [16], the latter technique is more difficult to implement. In particular, the problem of determining interpolation coefficients may be ill conditioned, especially if the number of base points is large. Moreover, the radial basis function interpolation may be very sensitive to control parameters, and typically, some sort of adjusting algorithm is necessary in order to find the proper values of these parameters. Fuzzy systems are free of these problems.

Fig. 4. Geometry of the dual-band microstrip bandpass filter [34].

III. EXAMPLES Here we compare the modeling accuracy for the standard space-mapping modeling methodology [11], space-mapping modeling with variable weight coefficients [14], space mapping with radial basis function interpolation [16], and the space mapping with fuzzy system interpolation described in Section II. In our comparison, we also include direct interpolation of the fine model data using a fuzzy system. A. Test Problem Description Problem 1: Second-order dual-behavior resonator microstrip filter [31] shown in Fig. 2. The fine model is simulated in FEKO [32]. The coarse model (see Fig. 3) is the circuit model implemented in Agilent ADS [33]. The design parameters are . The response vector consists of transmission coeffiin the frequency band GHz with samples cient taken every 200 MHz. The region of interest is defined by the mm and the deviation reference point mm. Problem 2: Dual-band microstrip bandpass filter [34] shown in Fig. 4. The fine model is simulated in FEKO [32]. The coarse model (see Fig. 5) is the simplified equivalent-circuit model implemented in Agilent ADS [33]. The design parameters are

Fig. 5. Coarse model of the dual-band microstrip bandpass filter (Agilent ADS).

. Parameter is set to 0.47 mm. The rein the sponse vector consists of transmission coefficient frequency range from 1.5 to 5.5 GHz. The reference point is mm and the region size is mm. Problem 3: Double-folded stub filter [1] shown in Fig. 6. The fine model is simulated with Sonnet’s em [35] using a high-resolution grid with a 0.0254 mm 0.0254 mm cell size.

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TABLE I BASE SET DATA FOR TEST PROBLEMS 1–3

Fig. 6. Double-folded stub filter [1].

TABLE II MODELING RESULTS FOR TEST PROBLEM 1. VERIFICATION FOR 30 RANDOM TEST POINTS

Fig. 7. Coarse model of the double-folded stub filter (Agilent ADS).

The coarse model (see Fig. 7) is the equivalent-circuit model implemented in Agilent ADS [33]. The design parameters . Parameter is set to 0.254 mm. The are response vector consists of transmission coefficient in the frequency range from 6 to 20 GHz. The reference mm and the region size is point is mm.

TABLE III MODELING RESULTS FOR TEST PROBLEM 2. VERIFICATION FOR 30 RANDOM TEST POINTS

B. Experimental Setup For each of the test problems, we performed a number of numerical experiments using the standard space-mapping surrogate model, space mapping with variable-weight coefficients, space mapping with radial basis functions, space mapping with fuzzy systems, as well as direct fuzzy interpolation (i.e., the fuzzy system directly interpolating the fine model data). Table I shows details of the base sets used in our simulations. The base sets have growing numbers of points (and decreasing characteristic distance ) in order to examine the dependence of the modeling error on the amount of fine model data used to create the model. Accuracy was tested using 30 test points randomly distributed in the region of interest. The error measure used was the norm of the difference between the fine model response and the corresponding surrogate model response.

C. Numerical Results and Discussion Tables II–IV show numerical results (error statistics) for the considered models with all the base sets considered. Figs. 8–13 show error plots, i.e., the modulus of the difference between the fine model and the corresponding surrogate model response

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TABLE IV MODELING RESULTS FOR TEST PROBLEM 3. VERIFICATION FOR 30 RANDOM TEST POINTS

Fig. 10. Test problem 2: error plots for the standard space-mapping model.

Fig. 11. Test problem 2: error plots for space mapping combined with the fuzzy system.

Fig. 8. Test problem 1: error plots for the standard space-mapping model.

Fig. 12. Test problem 3: error plots for the standard space-mapping model.

Fig. 9. Test problem 1: error plots for space mapping combined with the fuzzy system.

versus frequency, for the standard space-mapping model and the space mapping with fuzzy interpolation; data obtained for was used in all cases. Figs. 14–16 show the the base set average modeling error versus the characteristic distance . It follows from the results that the modeling accuracy provided by the new model (2)–(8) is comparable with or better than the accuracy of space mapping enhanced by radial basis function interpolation. All the other space-mapping approaches, as well as the direct fuzzy interpolation, are clearly outperformed by these two techniques. As expected, the accuracy of the standard space-mapping model is almost independent of the density

of the base set. Other approaches exhibit improvement of the modeling quality with decrease of the characteristic distance of the base set. It should be mentioned that the computational cost of the model (2)–(8) is almost the same as the cost of the coarse model because once the parameters are established (including fuzzy rules), evaluation of formula (7) (defuzzification) is very fast. Space mapping combined with radial basis function interpolation exhibits similar advantages with respect to computational efficiency, although, as mentioned in Section I, it has some inherent problems such as sensitivity to the control parameters and the possibility of the parameter calculation being ill conditioned. On the other hand, space mapping with variable weight coefficients, which performs well in terms of accuracy, suffers

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Fig. 13. Test problem 3: error plots for space mapping combined with the fuzzy system.

Fig. 16. Test problem 3: average modeling error versus characteristic distance . Data for the standard space-mapping model (o), space mapping with variable weight coefficients ( ), space mapping with radial basis functions ( ), space mapping with the fuzzy system ( ), and direct fuzzy interpolation (+).

2

3

D. Fuzzy Systems With Alternative Membership Functions

Fig. 14. Test problem 1: average modeling error versus characteristic distance . Data for the standard space-mapping model (o), space mapping with variable weight coefficients ( ), space mapping with radial basis functions ( ), space mapping with the fuzzy system ( ), and direct fuzzy interpolation (+).

2

3

Fig. 15. Test problem 2: average modeling error versus characteristic distance . Data for the standard space-mapping model (o), space mapping with variable weight coefficients ( ), space mapping with radial basis functions ( ), space mapping with the fuzzy system ( ), and direct fuzzy interpolation (+).

2

3

from computational overhead related to separate parameter extractions required for each evaluation of the model. Overall, the presented combination of space mapping and fuzzy systems seems to be a robust alternative to the existing modeling techniques based on space mapping.

The fuzzy system described in Section II uses triangle membership functions. This kind of model has an interpolation property provided that the base set is a uniform mesh and that base points are located at the centers of the membership functions. Here, we compare the accuracy of the space-mapping surrogate model (2)–(8) with the space-mapping model using the fuzzy system with Gaussian and Z-shaped membership functions. These functions can be beneficial in some cases, especially if the fine model response exhibits highly nonlinear behavior. Also, the resulting surrogate model is smooth, which is not the case for triangular functions. The Gaussian membership , with function is given by being the control parameter, which must be optimized in general in order to obtain the best performance. The Z-shaped function , as is defined, for if , and if , where . Definitions for other intervals are similar. Table V presents a comparison of the average modeling error and . It is for Problems 1–3 with the base sets seen that the performance of the surrogate model is very similar for all membership functions considered. It should be noted that the model using the fuzzy system with unoptimized Gaussian membership functions exhibits the worst performance, which is most likely because this model does not exhibit an interpolation property. IV. POSSIBLE EXTENSIONS OF FUZZY SPACE MAPPING Apart from the fuzzy space-mapping surrogate model (8) described in Section II, it is possible to employ fuzzy systems to approximate the space-mapping parameters in a regular . In order to discuss this concept, space-mapping model we will use the following notation. Let be a compact way of denoting the space-mapping surrogate is a vector of the model parameters. Let model, where denote a fuzzy system that approximates variable and is built using data

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TABLE V MODELING ACCURACY FOR SPACE-MAPPING SURROGATE WITH FUZZY SYSTEM USING DIFFERENT MEMBERSHIP FUNCTIONS

Fig. 17. (a) Fine and (b) coarse model, two-section capacitively loaded impedance transformer [36]. TABLE VI BASE SET DATA FOR TWO-SECTION TRANSFORMER EXAMPLE

pairs with from a given base set denotes the response of the fuzzy system at point . Using this notation, fuzzy space-mapping model (8) can be . Thus, we can written as call it the fuzzy output space-mapping model. As mentioned before, one of the possible extensions of the model (8) is to use a fuzzy system not only to approximate the differences between the fine model and the regular space-mapping model, but also to approximate space-mapping parameters in the regular spacemapping model. Using our notation, the extended model has the following form:

TABLE VII MODELING RESULTS FOR TWO-SECTION TRANSFORMER EXAMPLE. AVERAGE ERROR FOR 30 RANDOM TEST POINTS

(9) where, in general, both base sets and can be different, although in practice they should be the same in order to efficiently use the available fine model data. Another possibility is to use a fuzzy system only to approximate the space-mapping parameters, which would give the following model: (10) Using space-mapping terminology, model (10) is the fuzzy input space-mapping surrogate, while model (9) is the fuzzy input and output space-mapping surrogate. It should be noted that model (10) does not fully use available fine model information, and, therefore, model (9) is expected to outperform model (10) when using the same base set . An extended fuzzy space-mapping model (9) may seem attractive, however, there is an issue that makes its actual usefulness questionable. In order to model the space-mapping parameters with a fuzzy system, one has to extract the optimal set of paand then hope to rerameters for each of base points from trieve the optimal set of parameters for any other point from the region of interest using a fuzzy system. The problem is that, typically, the optimal space-mapping parameter set corresponding to a given design variable vector is nonlinearly dependent on the vector and it might be very difficult to model this dependency with a fuzzy system (or, more generally, with any other approach).

For illustrative purposes, consider a capacitively loaded 10 : 1 two-section impedance transformer example [36]. The “coarse” model and “fine” model, both implemented in MATLAB, are an ideal two-section transmission line and a capacitively loaded transmission line, as shown in Fig. 17. The electrical and are chosen as design parameters. The lengths response vector consists of reflection coefficient in the frequency range from 0.5 to 1.5 GHz. The reference point is . We consider the region of interest defined by a 10% deviation from . The base sets considered for this problem are shown in Table VI. In Table VII, we compare modeling accuracy for the space-mapping model (8), as well as fuzzy space-mapping models (9) and (10). Space-mapping parameters for models (9) and (10) were obtained using space mapping with the variable weight coefficient modeling technique [14]. It follows from the results that although models (9) and (10) retain the pattern of improving accuracy with growing density of the base set, their performance is not as good as for model (8). V. CONCLUSION A modeling methodology has been presented that combines the standard space-mapping technique and fuzzy interpolation. The new methodology can efficiently handle any base set. As

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with most of the recent space-mapping-based surrogate modeling techniques, it has the property of increasing modeling accuracy when the number of base points increases. Examples have demonstrated the robustness of our method. It follows that the new approach provides modeling accuracy comparable with or better than the recently published space mapping enhanced by radial basis function interpolation, and outperforms any other space-mapping approach. The new technique is easy to implement and computationally efficient. ACKNOWLEDGMENT The authors thank Sonnet Software Inc., Syracuse, NY, for em and Agilent Technologies, Santa Rosa, CA, for making ADS available. REFERENCES [1] J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R. H. Hemmers, “Space mapping technique for electromagnetic optimization,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 12, pp. 536–544, Dec. 1994. [2] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 337–361, Jan. 2004. [3] S. Koziel, J. W. Bandler, and K. Madsen, “A space mapping framework for engineering optimization: Theory and implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3721–3730, Oct. 2006. [4] D. Echeverria and P. W. Hemker, “Space mapping and defect correction,” Int. Math. J. Comput. Methods Appl. Math., vol. 5, no. 2, pp. 107–136, 2005. [5] H.-S. Choi, D. H. Kim, I. H. Park, and S. Y. Hahn, “A new design technique of magnetic systems using space mapping algorithm,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3627–3630, Sep. 2001. [6] M. A. Ismail, D. Smith, A. Panariello, Y. Wang, and M. Yu, “EMbased design of large-scale dielectric-resonator filters and multiplexers by space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 386–392, Jan. 2004. [7] S. Amari, C. LeDrew, and W. Menzel, “Space-mapping optimization of planar coupled-resonator microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2153–2159, May 2006. [8] J. E. Rayas-Sánchez and V. Gutiérrez-Ayala, “EM-based Monte Carlo analysis and yield prediction of microwave circuits using linear-input neural-output space mapping,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4528–4537, Dec. 2006. [9] M. H. Bakr, J. W. Bandler, and N. Georgieva, “Modeling of microwave circuits exploiting space derivative mapping,” in IEEE MTT-S Int. Microw. Symp. Dig., Anaheim, CA, Jun. 1999, pp. 715–718. [10] J. W. Bandler, N. Georgieva, M. A. Ismail, J. E. Rayas-Sánchez, and Q. J. Zhang, “A generalized space mapping tableau approach to device modeling,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 67–79, Jan. 2001. [11] S. Koziel, J. W. Bandler, A. S. Mohamed, and K. Madsen, “Enhanced surrogate models for statistical design exploiting space mapping technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 1609–1612. [12] J. W. Bandler, Q. S. Cheng, and S. Koziel, “Simplified space mapping approach to enhancement of microwave device models,” Int. J. RF Microw. Comput.-Aided Eng., vol. 16, no. 5, pp. 518–535, 2006. [13] J. C. Rautio, “A space mapped model of thick, tightly coupled conductors for planar electromagnetic analysis,” IEEE Micro, vol. 5, no. 3, pp. 62–72, Sep. 2004. [14] S. Koziel and J. W. Bandler, “Space-mapping-based modeling utilizing parameter extraction with variable weight coefficients and a data base,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 1763–1766.

[15] S. Koziel, J. W. Bandler, and K. Madsen, “Theoretical justification of space-mapping-based modeling utilizing a data base and on-demand parameter extraction,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 12, pp. 4316–4322, Dec. 2006. [16] S. Koziel and J. W. Bandler, “Microwave device modeling using space-mapping and radial basis functions,” in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, 2007, pp. 799–802. [17] L. Zhang, J. J. Xu, M. Yagoub, R. T. Ding, and Q. J. Zhang, “Neuro-space mapping technique for nonlinear device modeling and large signal simulation,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 173–176. [18] V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, and Q.-J. Zhang, “Advanced microwave modeling framework exploiting automatic model generation, knowledge neural networks, and space mapping,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1822–1833, Jul. 2003. [19] L. Zhang, J. Xu, M. C. E. Yagoub, R. Ding, and Q.-J. Zhang, “Efficient analytical formulation and sensitivity analysis of neuro-space mapping for nonlinear microwave device modeling,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2752–2767, Sep. 2005. [20] P. Burrascano, M. Dionigi, C. Fancelli, and M. Mongiardo, “A neural network model for CAD and optimization of microwave filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, 1998, pp. 13–16. [21] S. F. Peik, R. R. Mansour, and Y. L. Chow, “Multidimensional Cauchy method and adaptive sampling for an accurate microwave circuit modeling,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2364–2371, Dec. 1998. [22] X. Ding, V. K. Devabhaktuni, B. Chattaraj, M. C. E. Yagoub, M. Doe, J. J. Xu, and Q. J. Zhang, “Neural network approaches to electromagnetic based modeling of passive components and their applications to high-frequency and high-speed nonlinear circuit optimization,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 436–449, Jan. 2004. [23] T. W. Simpson, J. Peplinski, P. N. Koch, and J. K. Allen, “Metamodels for computer-based engineering design: Survey and recommendations,” Eng. Comput., vol. 17, no. 2, pp. 129–150, Jul. 2001. [24] N. V. Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidynathan, and P. K. Tucker, “Surrogate-based analysis and optimization,” Progress Aerosp. Sci., vol. 41, no. 1, pp. 1–28, Jan. 2005. [25] A. A. Mullur and A. Messac, “Metamodeling using extended radial basis functions: A comparative approach,” Eng. Comput., vol. 21, no. 3, pp. 203–217, Apr. 2006. [26] T. W. Simpson, T. M. Maurey, J. J. Korte, and F. Mistree, “Kriging models for global approximation in simulation-based multidisciplinary design optimization,” AIAA J., vol. 39, no. 12, pp. 2233–2241, Dec. 2001. [27] V. Miraftab and R. R. Mansour, “Computer-aided tuning of microwave filters using fuzzy logic,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2781–2788, Dec. 2002. [28] V. Miraftab and R. R. Mansour, “A robust fuzzy-logic technique for computer-aided diagnosis of microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 450–456, Jan. 2004. [29] V. Miraftab and R. R. Mansour, “EM-based microwave circuit design using fuzzy logic techniques,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 153, no. 6, pp. 495–501, Dec. 2006. [30] L.-X. Wang and J. M. Mendel, “Generating fuzzy rules by learning from examples,” IEEE Trans. Syst., Man, Cybern., vol. 22, no. 6, pp. 1414–1427, Nov./Dec. 1992. [31] A. Manchec, C. Quendo, J. F. Favennec, E. Rius, and C. Person, “Synthesis of capacitive-coupled dual-behavior resonator (CCDBR) filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2346–2355, Jun. 2006. [32] “FEKO® User’s Manual, Suite 4.2,” EM Softw. Syst. S.A. (Pty) Ltd., Stellenbosch, South Africa, Jun. 2004. [Online]. Available: http://www.feko.info [33] Agilent ADS. ver. 2003C, Agilent Technol., Santa Rosa, CA, 2003. [34] Y. P. Zhang and M. Sun, “Dual-band microstrip bandpass filter using stepped-impedance resonators with new coupling schemes,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3779–3785, Oct. 2006. [35] em. ver. 11.52, Sonnet Software Inc., North Syracuse, NY, 2007. [36] M. H. Bakr, J. W. Bandler, K. Madsen, and J. Søndergaard, “An introduction to the space mapping technique,” Optim. Eng., vol. 2, no. 4, pp. 369–384, Dec. 2001.

KOZIEL AND BANDLER: SPACE-MAPPING APPROACH TO MICROWAVE DEVICE MODELING EXPLOITING FUZZY SYSTEMS

Slawomir Koziel (M’03–SM’07) received the M.Sc. and Ph.D. degrees in electronic engineering, M.Sc. degrees in theoretical physics and in mathematics, and Ph.D. degree in mathematics from the Gdansk University of Technology, Gdansk, Poland, in 1995, 2000, 2000 and 2002, and 2003, respectively. He is currently a Research Associate with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada. He has authored or coauthored over 100 papers. His research interests include space mapping, circuit theory, analog signal processing, evolutionary computation, and numerical analysis.

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John W. Bandler (S’66–M’66–SM’74–F’78– LF’06) studied at Imperial College. He received the B.Sc. (Eng.), Ph.D., and D.Sc. (Eng.) degrees from the University of London, London, U.K., in 1963, 1967, and 1976, respectively. In 1969, he joined McMaster University, Hamilton, ON, Canada. He is a Professor Emeritus. He was President of Optimization Systems Associates Inc., which he founded in 1983, until November 20, 1997, the date of acquisition by Hewlett-Packard Company. He is President of Bandler Corporation, Dundas, ON, Canada, which he founded in 1997. Dr. Bandler is Fellow of several societies, including the Royal Society of Canada. He was the recipient of the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Microwave Application Award.

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Synthesis of Compact Lumped Models From Electromagnetic Analysis Results James C. Rautio, Fellow, IEEE

Abstract—Synthesis of compact lumped ( ) models from the results of an electromagnetic (EM) analysis of planar circuits is described. The technique requires precise EM analysis -parameter data at two or more frequencies. Data at up to five frequencies allows synthesis of more complicated models. For each port-to-port (i.e., “branch”) connection, 662 potential branch models are first synthesized and tested. Branch models that best match the EM results at all nonsynthesis frequencies are selected. For structures for which a compact lumped model is appropriate, this technique yields models that often provide direct physical insight into the electrical nature of the modeled structure. For electrically large structures, the technique is extended by the use of supplemental internal EM analysis ports. The technique is closed form; iteration is not used. Index Terms—Compact models, electromagnetic (EM) analysis, lumped models, method of moments, model extraction, model-order reduction, model synthesis, reduced-order systems.

I. INTRODUCTION

T

HERE HAS been significant work over several decades [1]–[23] with the goal of extracting a lumped model from numerical response data. Some of this work [2]–[9] involves various techniques that extract a pole-zero description from time and/or frequency domain data, sometimes requiring some form of iteration, optimization, or fitting. In addition, due to the distributed nature of the circuits being modeled, the model for even a simple circuit can become complicated with the resulting model bearing no physical resemblance to the circuit being modeled. The model is then effectively an abstract curve fitting with no direct correspondence to the physical structure being modeled. The advantage of these models is that extremely arbitrary circuits can be modeled over wide frequency ranges. In other cases, a specialized model topology is selected and element values are extracted from measured or electromagnetic (EM) analysis data [1], [10]–[23]. In some cases, the element values give insight into the physical circuit being modeled. A generalization of this second approach is described in this paper. While it is possible, using this generalized approach, to model a wide variety of circuits, it cannot be applied to every possible microwave circuit. Rather, it can be applied only to circuits that can be modeled within a specific (but large) set of physically appropriate lumped elements. The allowed circuit size can be made arbitrarily large by adding supplemental inManuscript received June 15, 2007; revised August 19, 2007. The author is with Sonnet Software Inc., North Syracuse, NY 13212 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.909141

Fig. 1. Example branch models. Branch model IX-4 was used in my original lumped model synthesis effort. It is equivalent to XI-19, as can be seen after noting that the III-1 and IV-1 branch models are equivalent, as is discussed in the text.

ternal EM analysis ports (that facilitate the modeling process); however, the circuits selected for the model between each port (including the supplemental internal ports) must still come from within the allowed solution space. The advantages of this approach are that the lumped models often correspond, element by element, to the circuit being modeled and that the resulting models tend to be especially compact. The approach involves no iteration of any kind. Typical synthesis time for a two-port circuit is less than 1 s using prototype software written in Visual BASIC macros on Microsoft Excel. Synthesis time increases with the number of ports squared. Pro. duction code will be compiled C This paper begins with a detailed description of the technique, followed by a discussion of the effect of EM analysis error, an illustration of the need for negative valued elements, and closes with an example. In this study, I use the term “element” to denote an individual resistor, inductor, or capacitor. A “branch” or “branch model” is a two-terminal collection of elements for connection between two nodes. Ground is considered to be a node. A “model” is a collection of branch models forming a nodal network. II. DESCRIPTION OF THE TECHNIQUE The synthesis technique described here is a generalization of [1], where an -port circuit is analyzed electromagnetically at two frequencies. The result is converted to -parameters from which the admittance connecting each node (i.e., port) is calculated. Next, the IX-4 branch model in Fig. 1 is synthesized from the resulting admittance data. There are four unknown elements , , and ). We write the expression for the admittance ( , of this branch model, and given the real and imaginary admittance data at two frequencies, we generate four simultaneous equations. The equations are nonlinear, but they do posses a so-

0018-9480/$25.00 © 2007 IEEE

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lution. In [1], an approximate solution was presented. The exact synthesis is (1) (2) (3) Numeric subscripts indicate data from one of the two synthesis frequencies with the node-to-node admittance being , and being the radian frequency. The model is pasand . It is stable if has the sive if , and has the same sign as , and same sign as . Note that negative elements are allowed in a branch model that is both passive and stable. If one or more of the four elements are close to 0, then the and are close to branch is simplified. For example, if . This process is repeated for all pos0, we have a series sible node-to-node (and node-to-ground) connections. For each node-to-node admittance, a lumped equivalent circuit is synthesized and the model supplemented. The problem is that there are numerous branch models that might be needed, but cannot be represented as a special case of the IX-4 branch model. For example, two two-element branch and series ), and numerous three-elemodels (series ment branch models are left unconsidered. In this new approach, every possible zero-, one-, two-, three-, and four-element branch model is synthesized. Many five-, six-, seven-, and eight-element branch models are also synthesized. Based on comparison with EM analysis data not used for synthesis, the best branch model to represent the EM analysis result is then selected. We presently synthesize 662 candidate branch models (this includes different ways of synthesizing the same branch model) for each node-to-node connection in a given -port. Thus, for a two-port, which has three node-to-node connections (ground is a node), we consider 662 potential models, and of these we select the model that best matches the EM analysis results at the nonsynthesis frequencies. Thus, we select the best possible model from a comprehensive solution space of 290 117 528 potential models for a two-port. This new approach allows an efficient automation of compact model generation. Performed manually, a designer proposes a lumped model. By some combination of calculation, guessing, and optimization, the designer determines if that model can then adequately represent the EM results. If not, then the designer speculatively selects another compact lumped model and repeats the tedious extraction and evaluation. With existing model extraction techniques, the best compact model is highly unlikely to be found and compact model generation cannot be automated because of the high degree of skill required, a computer cannot replace the engineer. In this approach, a new technique synthesizes and evaluates hundreds of millions of models with selection of the best model taking place in real time. III. CANDIDATE BRANCH TOPOLOGIES There is one possible zero-element topology, a single one-element topology, two two-element topologies, four three-element

Fig. 2. There are four possible three-element branch topologies and eight possible four-element topologies. The elements are ordered as indicated.

topologies, and eight four-element topologies. All possible three- and four-element topologies are shown in Fig. 2. Each topology has a final element connection. The final connection for all the odd-numbered (Roman numeral) topologies is in parallel and it is thus most convenient to write an equation for admittance. The final connection for all even-numbered topologies is in series. Now we most conveniently write an equation for impedance. Note that the next higher even-numbered (Roman numeral) topology represents the dual (i.e., swap and , and swap parallel and series connections) for each odd-numbered topology. Thus, when synthesis equations are derived for a type-III topology, we immediately have the solution for a type-IV topology with the changes: 1) swap and ; 2) invert ; and 3) swap impedance and admittance. Thus, synthesis equations need be derived for only half of the branch models. Table I shows how each of the three- and four-element topologies are populated. For example, a type III-1 branch model (Fig. 1) is a resistor in parallel with a series connected ” resistor and inductor. This is indicated by the entry “1 under the heading “III, IV.” Branch model elements are listed and are arbitrarily considered to in a specific order: be in smallest to largest order (as in “1, 2, 3”). The element populations are ordered in Table I as though they are in numeric immediately follows , etc. order. Thus, The first (most significant) element is assigned to element 1 (always an element involved in the final connection) in Fig. 2, . Thus, element 1 etc. For example, branch model IV-7 is a (in topology IV of Fig. 2) is a capacitor, element 2 is a resistor, and element 3 is an inductor. The zero-, one-, and two-element topologies are not assigned numerical designations; rather we simply use their common ,” etc. names, e.g., “Series There are two zero-element branch models (an open and short circuit), three one-element branch models, six two-element branch models, 20 three-element branch models, and 90

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TABLE I BRANCH MODEL SPECIFICATION

TABLE II BRANCH MODEL EQUIVALENCES

four-element branch models, for a total of 121 possible branch models of up to four elements. IV. EQUIVALENCES BETWEEN BRANCH MODELS A number of the branch models are equivalent. This means that given the values for either branch model, the other branch model can be uniquely determined. The impedance of both branch models is identical at all frequencies. For example, Fig. 1 shows branch models III-1 and IV-1. That equivalence exists is seen by inspection. The only difference between the two branch models is the location where the parallel connected resistor is tapped into the series-connected resistor. The resistor and inductor values can be selected so that both branches have exactly the same impedance at all frequencies. The three-element equivalences are useful in identifying fourelement equivalences. For example Fig. 1 also shows the XI-19 branch model. Noting that this is IV-1 with a capacitor connected in parallel, we have IX-4 of Fig. 1 by changing the IV-1 portion to the equivalent III-1 branch model. It is instructive to fill in a similar table with the schematics of each equivalent branch model. Patterns quickly emerge, such as described above. Space limits do not permit presentation of such a table here. In this approach, we synthesize all possible branch models, even if there are equivalences, because one branch model might have more desirable element values or yield a better fit to actual data. In fact, in some cases, one branch model might have all positive element values while the equivalent branch model has some negative values. Take, for example, the first line of Table II. An III-1 is equivalent to an IV-1, as described above. In addition, it is also equivalent to an III-2. This seems counter-intuitive because the III-1 is an inductor embedded in a resistive network, while III-2 is a capacitor similarly embedded. Note that III-1 is not listed in boldface, while III-2 is bold. This indicates that if one of the branch models has all positive elements, the other must have

Branch models in each row are equivalent. Bold face indicates some elements are negative when the nonbold-faced branch models are all positive. The XI and XII branch models in the first five rows are not equivalences, rather, they reduce to the three-element branch models on the same row.

some negative elements. Indeed, the equivalence from III-1 to III-2 is

(4) where the prime indicates the III-2 element value. Equivalence equations have been derived for many, but not all, of the listed equivalences. We determine equivalence by noting that the impedance or admittance equations have the same form and by comparing numerical results for a specific case of one branch model synthesized from data generated by the other branch model. We do not synthesize 12 of the four-element branch models because they reduce to three-element branch models. These are the XI and XII branch models listed in the first five lines of Table II. Thus, 108 branch models of up to four elements are synthesized. Strictly speaking, these four-element branch models are not equivalent to the three-element branch models even though element values can be selected that yield identical impedance at all frequencies. The reason is that given a specific three-element branch model, one cannot uniquely determine the corresponding four-element branch model; there are an infinite number of solutions. There are no equivalences for branch models with fewer than three elements. All three- and four-element equivalences are summarized in Table II. All the lossless equivalences have been previously reported [24]. We expect these equivalences to be useful for filter and matching network synthesis. Numerous equivalences between branch models of up to eight elements have been observed and are not reported here.

RAUTIO: SYNTHESIS OF COMPACT LUMPED MODELS FROM EM ANALYSIS RESULTS

V. SYNTHESIS STRATEGIES Each synthesis derivation starts by writing an expression for the admittance (parallel final connection) or impedance (series final connection). Sometimes both admittance and impedance expressions can be used to give additional synthesis opportunities. The resulting expression is split into real and imaginary parts. If one is writing the expression for the admittance of a branch model, it is split into two expressions, one for conducand another for susceptance . tance In the first strategy, one takes EM data at a sufficient number of frequencies and writes an equation for both the real and imaginary parts at each frequency. For example, when synthesizing a three-element branch model, we require three equations. With data at two frequencies, we have two equations for conductance and two equations for susceptance. This forms a set of four nonlinear simultaneous equations. We select three equations (the fourth equation is satisfied only if we happen to synthesize the correct model for the data) and find an algebraic solution. This is the strategy used for the synthesis of IX-4 above. In some cases, the solution involves multiple roots. Every root must be evaluated and the resulting branch model considered. In the second strategy, only the imaginary part of the admittance or impedance is used. A three-element branch model now requires data from three frequencies. In the third strategy, only the real part is used. The second strategy cannot be used for any branch model whose susceptance (or reactance) is unmodified by one or more elements. Likewise, the third strategy cannot be used for any branch model whose conductance (or resistance) is unmodified by one or more elements. For example, a parallel RC cannot be synthesized based only on the branch model’s conductance or only on its susceptance, but it can be synthesized based only on its resistance or only on its reactance. If the second strategy is used, additional branch models can be synthesized. For example, note that a resistor added in parallel to a branch model has no effect on the resulting susceptance. Thus, if a branch model is synthesized based only on its susceptance, we may now use the conductance to determine a resistor connected in parallel and add one more element to the synthesized branch model. Likewise, if reactance is used to synthesize a branch model, a resistor can be added in series with the branch model. Thus, for each branch model that can be synthesized using the second strategy, one more branch model with an additional resistor can also be synthesized. If the third strategy is used, we can increase the complexity of the synthesized branch model by adding any lossless branch model. For example, if the synthesis is based on conductance only, we can add any lossless branch model in parallel and leave the conductance unchanged. The added lossless branch model is synthesized based on the unused susceptance information. The dual situation benefits a synthesis based only on resistance. The branch model complexity can be increased by synthesizing a lossless branch model based on the unused reactance information. There are 16 lossless branch models of up to four elements that can be added in this manner. Thus, for every branch model that can be synthesized using the third strategy, another 16 branch models can be synthesized of up to eight elements.

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Branch models should be synthesized in all ways possible. This is because one synthesis strategy might be sensitive to difficulties that are transparent to another synthesis. For example, synthesis of a low-loss resonant branch model might have difficulties if based only on the real part of the impedance or admittance data. The combination of all possible synthesis strategies applied to all possible zero- through four-element branch models (yielding branch models of up to eight elements) provides 662 candidate branch models for every node-to-node connection. As a practical matter, minimum and maximum limits should be set on the allowed values of all lumped elements. For example, a capacitor whose value is below a specified minimum limit should be treated as an open circuit. This significantly reduces the complexity of the resulting model in that high impedance shunt and low impedance series elements do not clutter the model. In addition, the limits can be used to adjust the model complexity and accuracy as desired. VI. SYNTHESIS ERROR CONSIDERATIONS A great deal of effort has been expended over the years to extract lumped models that are causal, passive, and stable. If one were to have exact EM data for a causal, passive, stable structure and an extracted model that exactly matches the EM data, then the model must be causal, passive, and stable. Thus, models that are not causal, stable, or passive must be the result of some combination of EM analysis error and extraction/synthesis error. The technique reported here can result in nonpassive unstable models when the structure is unsuitable for a compact lumped model within the technique’s solution space at the selected synthesis frequencies. Typically, this is the result when a structure has a large distributed component. When this technique fails to provide an adequate model, any of the more general model extraction techniques, mentioned in Section I, should be used. A model that is inadequate for the structure being modeled is a failure due to extraction/synthesis error. It has been my observation that even minor changes in EM data can result in large differences in the quality of the synthesized model. Thus, this technique is not especially useful for measured data due to measurement noise. All of my EM synthesis efforts use -parameters evaluated to full double precision. If the same -parameter data is truncated to the reduced precision common in -parameter data files (the old-style “Touchstone” format), then substantially degraded models can result. The EM analysis used in this study, i.e., [25] and [26], typically has over 100 dB of dynamic range [27], [28]. In addition, the EM analysis uses an exact deembedding algorithm [29], [30]. By “exact” we mean that provided the deembedding assumptions are not violated (i.e., no over-moded port connecting lines), then the port discontinuities are removed to within the numerical precision of the underlying EM analysis. There is no deembedding error added on top of the already existing EM analysis error. EM deembedding algorithms that require knowledge of the characteristic impedance of the port connecting lines are typically not exact for inhomogeneous or lossy geometries due to

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Fig. 3. Negative valued element situations. The circuits for a; c; and d require use of negative valued elements in their  -models.

ambiguity in characteristic impedance definitions. We suggest that nonpassive and unstable models extracted from such EM data might be the direct result of the very small, but nonphysical error introduced by the deembedding due to lack of knowledge of the exact characteristic impedance. This problem is especially clear for causality. For lossy transmission lines, characteristic impedance is complex (as in real and imaginary). In order for a system to be causal, the magnitude and phase of the characteristic impedance must be related by a Hilbert transform [31]. Use of any characteristic impedance that is not causal (e.g., a pure real characteristic impedance for a lossy line) results in noncausal -parameters. Even if a model extraction forces causality, then the noncausality of the -parameter data could make itself known in some other way, perhaps as an unstable or nonpassive model. In our approach, all characteristic impedances used are causal [30], the deembedding error is 0, and the dominant source of EM analysis error is error due to finite subsection size [27], [28]. It is our experience that error due to subsection size does not impact the stability or passivity of the resulting models. Rather, the percent error translates directly into percent error of the value of the dominant elements in a model. Typically, the subsection error (i.e., error due to meshing the circuit for EM analysis) is around 1% or less and is easily quantified. VII. NEGATIVE ELEMENT CONSIDERATIONS When a nonpassive unstable model, for whatever cause, is a common extraction result, one can compensate by requiring all element values to be positive. This is a common, but unacceptable model restriction because certain situations require negative elements. Fig. 3(a) shows coupled inductors and Fig. 3(b) shows the is negative, a negative inductor equivalent -model. When (between the two nodes) is required. The negative inductor can be hidden by use of a phase reversing transformer or a controlled source, but the negative inductor is still there. In a similar example, Fig. 3(c) shows a coupled line. At low frequency (i.e., the length of the line is short compared to

wavelength), there is a mutual inductance between the two lines. is inversely proportional to . The circuit theory small), we have the Using the small angle approximation ( impedance between ports 1 and 2 proportional to with the sign resulting from conversion of the -parameter to the node-to-node connecting admittance. Notice that for a , the node-to-node impedance, , lossless situation requires a negative inductor (a positive capacitor has the wrong frequency variation). For the lossy situation, the node-to-node . Thus, in impedance also includes a negative resistance order to model forward coupled crosstalk, negative elements are required. Another situation is the interface between two circuits that do not share a perfect common ground. This could be the case with two connected circuits that simply have different grounds, or a circuit with a single resistive ground (e.g., silicon). A model is shown in Fig. 3(d). Ports 1 and 2 each have different ground references. Voltage placed across port 1 results in current flowing out of the ground terminal on port 2. The -parameters for this circuit are (5) This yields the -model of Fig. 3(e), including a negative resistor. The stability and passivity of individual branches are not required for a complete model to be passive and stable. This situation also arises when modeling the interaction (perhaps for signal integrity purposes) between two different circuits, each using its own ground, and a signal on one circuit can induce current in the ground of the other circuit. Notice that while the circuits of Fig. 3(d) and (e) are exactly equivalent no matter what termination is attached to ports 1 and 2, the same is not true if one were to add circuitry connecting nodes 1 and 2 together. For this reason, a model that contains multiple grounds that is later used within a larger circuit, nodal connections should be made only between nodes that are all referenced to exactly the same ground. This rule is sometimes violated when RF circuits on silicon are simulated by connecting together individual components available in a process design kit (PDK). The ports of the components in the PDK might have different ground references. Thus, inclusion of negative valued elements in a lumped model is required for modeling a wide range of circuits, whether it is done on a general and explicit basis, as described here, or on a hidden basis that is limited to special situations, e.g., by including phase-reversing transformers. Since negative elements are required for maximum generality, passivity and stability cannot be enforced by restricting the model to positive elements. The only alternative is to use high-quality noise-free EM data. VIII. EXAMPLE SYNTHESIS To illustrate this model synthesis approach, a six-turn circular spiral inductor on silicon has been selected, and a scale image is inset into Fig. 4. The linewidth is 10 m and the gap dielecis 2 m. It is on top of 10 m of lossless , tric, which itself is on top of 100 m of silicon ( conductivity S/m). The inside end of the spiral passes

RAUTIO: SYNTHESIS OF COMPACT LUMPED MODELS FROM EM ANALYSIS RESULTS

Fig. 4. Modeled data from the circuit in Fig. 5. The model was synthesized from the EM analysis data at the indicated frequencies.

Fig. 5. Synthesized compact model for a spiral inductor on silicon shows the physical inductor (7.96 nH) and associated circuitry modeling the multiple loss mechanisms. Units are ohms ( ), nanohenrys (nH), and picofarads (pF).

out 4 m underneath the spiral. Analysis reference planes are located at the start of the vertical segment of feed line. The cell size is 2- m square and the box size is 500- m square. Conformal meshing [32] is used for the circular portion of the spiral. The small cell size allows high edge current. Metal thickness is not included, as it is not needed for validation of this synthesis. The inductor was analyzed electromagnetically from 0.1 to 10 GHz. The model was synthesized from the EM data at five evenly spaced frequencies from 2.0 to 2.8 GHz. The inductance of the spiral itself is the 7.96-nH inductor in the schematic of Fig. 5. Much of the additional complexity of the model is due to skin-effect resistance (present in both the metal of the spiral and the silicon substrate). Lumped resistors do not vary with the square root of frequency. In addition, the metal loss has a very complicated frequency dependence [33] that is also not modeled by frequency-independent resistors. The model includes two negative elements. A pole-zero SPICE analysis1 shows all poles and zeros are stable and a transient analysis does, in fact, converge nicely. In addition, note that the series arm is a X-2 branch model, equivalent (Table II) to, among others, the X-1 with all positive element values, and thus, the model is passive and stable. In [13]–[16], various combinations of a pair of coupled inductors with a resistor across the terminals of one inductor are used 1[Online].

Available: http://www.aimspice.com

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to model the series arm of a spiral inductor. That four-parameter branch model reduces to the three-element IV-4 branch model. The idea is to increase resistance as the frequency increases. Note that a portion of the series arm branch model synthesized here (Fig. 5) resembles an IV-4 only with a negative parallel RC instead of a positive parallel . This configuration effectively decreases resistance as frequency decreases. The inductors in the shunt branches have been seen in the synthesis of a wide variety of spiral inductors using this technique. We suggest that these inductances might actually be physical, as opposed to an abstract curve-fitting artifact, and are perhaps due to the inductance of the ground return current flowing in the surface of the silicon. The shunt arms of the model include the IV-8 branch model often used for this purpose in spiral inductor modeling. The parallel RC models the resistance and capacitance of the silicon substrate, while the series capacitor models the capacitance of the lossless dielectric between the spiral and silicon. When this study was initially submitted, [23] reported use of the X-2 branch model (which includes a series inductor) to model the shunt arms of a spiral inductor model, and the X-2 branch model in parallel with a series RL is to model the series arm. The resistor in parallel with the capacitor in [23] is sometimes negative, and [23] reported that conservation of energy is not violated, however, stability is not discussed. Use of the III-1 branch model (Fig. 1) for spiral inductor compact modeling is reported in [10]–[12]. The III-7 branch model combined with a parallel RC is used in [17]–[19]. The X-2 branch model, combined with coupled inductors, is used to build general models in [9]. Modeled versus EM analysis results are shown in Fig. 4 with synthesis frequencies indicated. The inductor resonance and high-frequency are well modeled even though the synthesis frequencies are far below resonance. Models for numerous circuits have been synthesized. We have successfully synthesized lossless and metal loss only spiral inductors based on EM data at very low frequencies with results valid far above resonance. Lengths of transmission line have in some cases been successfully synthesized well above one half-wavelength in length yielding simple pure lumped models, a most unexpected result. Simple pure lumped broadband models for antennas have been synthesized. In research to be reported in the future, we can now synthesize ideal transmission lines, multiple coupled lines, and -port tee networks. Compact model extraction of coupled lines from EM data is described in [20], however, they “calibrate” the EM analysis to match measurements by significant adjustments to the EM analysis substrate conductivity, making the analyzed substrate conductivity significantly different from that measured at dc. This approach is dangerous because a common measurement flaw can masquerade as modified substrate conductivity [34]. A patent on this compact model synthesis technique has been submitted. IX. CONCLUSION A technique to synthesize a lumped model based on highquality EM analysis data has been described. All possible combinations of resistors, inductors, and capacitors up to a certain

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level of complexity have been synthesized from the EM data. The branch models that best match the EM analysis data at all nonsynthesis frequencies have then been selected for the model. Using this approach, the best of literally hundreds of millions of possible models have been selected. If there exists a compact lumped model within the synthesis solution space that can model a given structure, then this technique finds that model.

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[1] J. C. Rautio, “Synthesis of lumped models from -port scattering parameter data,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 3, pp. 535–537, Mar. 1994. [2] A. Kogure, “Automatic SPICE models and -parameters analysis,” (in Japanese) Design Wave Mag., no. 20, pp. 145–151, Mar. 1999. [3] I. Timmins and K. L. Wu, “An efficient systematic approach to model extraction for passive microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1565–1573, Sep. 2000. [4] S. D. Corey and A. T. Yang, “Automatic netlist extraction for measurement-based characterization of off-chip interconnect,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 1934–1940, Oct. 1997. [5] R. Araneo, “Extraction of broadband passive lumped equivalent circuits of microwave discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 393–401, Jan. 2006. [6] R. Neumayer, A. Stelzer, F. Haslinger, and R. Weigel, “On the synthesis of equivalent-circuit models for multiports characterized by frequency-dependent parameters,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2789–2796, Dec. 2002. [7] L. F. Tiemeijer, R. J. Havens, R. De Kort, Y. Bouttement, P. Deixler, and M. Ryczek, “Predictive spiral inductor compact model for frequency and time domain,” in IEEE Int. Electron Devices Meeting Tech. Dig., Dec. 2003, pp. 36.4.1–36.4.4. [8] S. Sercu and L. Martens, “A new algorithm for experimental circuit modeling of interconnection structures based on causality,” IEEE Trans. Compon., Packag., Manuf. Technol. B, vol. 19, no. 2, pp. 289–295, May 1996. [9] T. Mangold and P. Russer, “Full-wave modeling and automatic equivalent circuit generation of millimeter-wave planar and multilayer structures,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 851–858, Jun. 1999. [10] T. Kamgaing, M. Petras, and M. Miller, “Broadband compact models for transformers on conductive silicon substrates,” in IEEE Radio Freq. Integrated Circuits Symp., 2004, pp. 457–460, Paper TU1D-4. [11] T. Kamgaing, T. Meyers, M. Petras, and M. Miller, “Modeling of frequency dependent losses in two-port and three-port inductors on silicon,” in IEEE Radio Freq. Integrated Circuits Symp., Jun. 2002, pp. 307–310, Paper TU3A-2. [12] T. Kamgaing, T. Meyers, M. Petras, and M. Miller, “Modeling of frequency dependent losses in two-port and three-port inductors on silicon,” in IEEE MTT-S Int. Microw. Symp. Dig., 2002, pp. 153–156, Paper TU3A-2. [13] D. Melendy, P. Francis, C. Pichler, K. Hwang, G. Srinivasan, and A. Weisshaar, “A new wideband compact model for spiral inductors in RFICs,” IEEE Electron Device Lett., vol. 23, no. 5, pp. 273–275, May 2002. [14] A. Watson, Y. Mayevskiy, P. Francis, K. Hwang, G. Srinivasan, and A. Weisshaar, “Compact modeling of differential spiral inductors in Si-based RFICs,” in IEEE MTT-S Int. Microw. Symp. Dig., 2004, pp. 1053–1056, Paper WEIF-7. [15] Y. Mayevskiy, A. Watson, P. Francis, K. Hwang, and A. Weisshaar, “A new compact model for monolithic transformers in silicon-based RFICs,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 6, pp. 419–421, Jun. 2005. [16] A. Watson, D. Melendy, P. Francis, K. Hwang, and A. Weisshaar, “A comprehensive compact-modeling methodology for spiral inductors in silicon-based RFICs,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 849–857, Mar. 2004. [17] K. Naishadham, “Experimental equivalent-circuit modeling of SMD inductors for printed circuit applications,” IEEE Trans. Electromagn. Compat., vol. 43, no. 4, pp. 557–56, Nov. 2001.

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[18] K. Naishadham, “Measurement-based closed-form modeling of surface-mounted RF components,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2276–2286, Oct. 2002. [19] J. Sieiro, J. M. López, J. Cabanillas, J. A. Osorio, and J. Samitier, “A physical frequency-dependent compact model for RF integrated inductors,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 384–392, Jan. 2002. [20] T. Kim, X. Li, and D. J. Allstot, “Compact model generation for on-chip transmission lines,” IEEE Trans. Circuits Syst. 1, Reg. Papers, vol. 51, no. 3, pp. 459–470, Mar. 2004. [21] A. Lauer and I. Wolff, “A conducting sheet model for efficient wide band FDTD analysis of planar waveguides and circuits,” in IEEE MTT-S Int. Microw. Dig., Jun. 2000, vol. 1, pp. 255–258. [22] M. Ingalls and G. Kent, “Measurement of the characteristics of highceramic capacitors,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. CHMT-12, no. 4, pp. 487–495, Dec. 1987. [23] K. Y. Lee, S. Mohammadi, P. K. Bhattacharya, and L. P. B. Katehi, “A wideband compact model for integrated inductors,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 9, pp. 490–492, Sep. 2006. [24] A. I. Zverev, Handbook of Filter Synthesis. New York: Wiley, 1967, pp. 522–527. [25] J. C. Rautio and R. F. Harrington, “An electromagnetic time–harmonic analysis of shielded microstrip circuits,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 8, pp. 726–730, Aug. 1987. [26] J. C. Rautio, “A time–harmonic electromagnetic analysis of shielded microstrip circuits,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Syracuse Univ., Syracuse, NY, 1986. [27] J. C. Rautio, “An ultrahigh precision benchmark for validation of planar electromagnetic analyses,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 11, pp. 2046–2050, Nov. 1994. [28] J. C. Rautio, “Testing limits of algorithms associated with high frequency planar electromagnetic analysis,” in Proc. 33rd Annu. Eur. Microw. Conf., Munich, Germany, 2003, pp. 463–466. [29] J. C. Rautio, “A de-embedding algorithm for electromagnetics,” Int. J. Microw. Millimeter-Wave Comput.-Aided Eng., vol. 1, no. 3, pp. 282–287, Jul. 1991. [30] J. C. Rautio and V. I. Okhmatovski, “Unification of double-delay and SOC electromagnetic deembedding,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2892–2898, Sep. 2005. [31] D. F. Williams, B. K. Alpert, U. Arz, D. K. Walker, and H. Grabinski, “Causal characteristic impedance of planar transmission lines,” IEEE Trans. Adv. Packag., vol. 26, no. 2, pp. 165–171, May 2003. [32] J. C. Rautio, “A conformal mesh for efficient planar electromagnetic analysis,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 257–264, Jan. 2004. [33] J. C. Rautio and V. Demir, “Microstrip conductor loss models for electromagnetic analysis,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 915–921, Mar. 2003. [34] J. C. Rautio and R. Groves, “A potentially significant on-wafer highfrequency measurement calibration error,” IEEE Micro, pp. 94–100, Dec. 2005.

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James C. Rautio (S’77–M’78–SM’91–F’00) received the B.S.E.E. degree from Cornell University, Ithaca, NY, in 1978, the M.S. degree in systems engineering from the University of Pennsylvania, Philadelphia, in 1982, and the Ph.D. degree in electrical engineering from Syracuse University, Syracuse, NY, in 1986. From 1978 to 1986, he was with General Electric, initially with the Valley Forge Space Division, then with the Syracuse Electronics Laboratory. During this time, he developed microwave design and measurement software and designed microwave circuits on alumina and on GaAs. From 1986 to 1988, he was a Visiting Professor with Syracuse University and Cornell University. In 1988, he joined Sonnet Software, Liverpool, NY, full time, a company he had founded in 1983. In 1995, Sonnet Software was listed on the Inc. 500 list of the fastest growing privately held U.S. companies, the first microwave software company ever to be so listed. Today, Sonnet Software is the leading vendor of 3-D planar high-frequency EM analysis software. Dr. Rautio was the recipient of the 2001 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Microwave Application Award. He was appointed an MTT-S Distinguished Microwave Lecturer for 2005–2007 lecturing on the life of James Clerk Maxwell.

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Design of a High-Power Superconducting Filter Using Resonators With Different Linewidths Xubo Guo, Bin Wei, Xiaoping Zhang, Bisong Cao, Shichao Jin, Huili Peng, Longma Gao, and Baoxin Gao

Abstract—This paper presents a design method to develop a high-power superconducting microstrip filter with different linewidth resonators. The linewidth of every resonator is optimized using this method to meet the same maximum current density. Thus, the power-handling capability of the five-pole filter on a limited substrate size can reach a maximum value. The design method is verified by the electromagnetic simulation of the current density distribution. The high-power superconducting filter is designed and fabricated on a 38 mm 30 mm 0.505 mm LaAlO3 substrate. The maximum resonator linewidth is 7 mm, and the minimum one is 1.6 mm. The filter has a center frequency of 2006 MHz and a narrow bandwidth of 1.0%. The measured power level is up to 35 dBm at 65 K without an evident change in insertion loss. Index Terms—Bandpass filters, high-power filters, high-temperature superconductors.

I. INTRODUCTION N RECENT years, high-temperature superconducting (HTS) filters with excellent performance have been developed for various applications [1]–[3]. Compact planar HTS filters have the advantages of extremely low insertion loss, high out-of-band rejection, and steep band edges because the surface resistance of superconducting materials at microwave frequencies is two orders of magnitude smaller than that of normal conductors [4]. The field test results of HTS receiver filters show that they can greatly improve the receiving sensitivity and selectivity of base stations, leading to an increased performance of the mobile communication network [5], [6]. However, most filters made with HTS thin films have a limited power level of milliwatts, which restricts their practical applications. Generally, the power-handling capability of an HTS filter is limited when the current density in the filter circuit exceeds some critical value. The current density at any point in the filter circuit is proportional to the square root of the input power. The power-handling capability of an HTS filter can be increased by using thin films with a high RF critical current density, or by reducing the maximum current density in the filter at a given input power. In this paper, we focus primarily on the latter approach. - or dual-mode Several high-power HTS filters with a -mode resonator have been reported thus far [7]–[11]. A

I

Manuscript received April 11, 2007; revised August 27, 2007. This work was supported by the National Natural Science Foundation of China under Grant 60471001. X. Guo, B. Wei, X. Zhang, B. Cao, S. Jin, H. Peng, and L. Gao are with the Department of Physics, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). B. Gao is with the Department of Electronics Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.909604

ring resonator carries only the radial component of the current density, and there is current free parallel to the edge of the disk patch. A dual-mode rectangular resonator has large sizes in both dimensions, and the distribution of current density is more uniform. Generally, these two types of resonators require a larger size substrate. Recently, we have reported a high-power narrowband stripline filter [12]. The power-handling capability of the stripline resonator was studied, and it increases linearly with the resonator linewidth. A two-pole HTS stripline filter with equal width resonators was designed. As is already known, it is hard to design a narrowband filter because the required weak coupling implies widely spaced resonators. The stripline structure can greatly reduce the spacing between resonators and can be used to realize a high-power filter. However, there is great difficulty in its assembly and tuning. In this paper, we choose the TEM mode microstrip structure to design a high-power filter. Similar to [12], the power-handling capability of the microstrip resonator is also studied. Furthermore, the power level of each resonator in a five-pole filter is simulated, and a quantitative method is obtained to determine the linewidth of every resonator in the microstrip filter. Therefore, the substrate area can be effectively used to maximize the power-handling capability. The measured power level of the HTS filter is also presented and discussed. II. POWER DESIGN METHOD A. Simulation Approach The experimental maximum power can be predicted by [13]

handled by the filter (1)

where is the RF critical current density of the HTS film, and is the maximum current density in the filter simulated by Sonnet Software Inc.’s EM software [14]. The software assumes a 1-V sinusoidal source at port 1 in 50- impedance; therefore, the resulting current density data are normalized to 5-mW input power. The following approximations are made in the simulations. 1) A zero thickness ideal conductor model is used for the superconducting circuit because there is currently no available commercial software providing an HTS material model to effectively perform an electromagnetic analysis of planar RF circuits. Although the properties of HTS materials are different from those of ideal conductors, our experience show that Sonnet Software Inc.’s EM using ideal conductor models can accurately predict the electrical performance of HTS thin films [3], [12], [15]. 2) Sonnet Software Inc.’s EM subdivides a filter circuit into finite size “cells,” which is an important factor in the simulation. Higher accuracy and finer current density distribu-

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Fig. 1. Lumped-element equivalent circuit of a five-pole bandpass filter. Each resonator is represented by an LC parallel circuit. The resonators are coupled together with J-inverters.

tion can be achieved by choosing a smaller cell size, while the computation time is significantly increased. A cell size of 0.02 mm 0.02 mm is used in the following simulations as a compromise between the computation time and accuracy. The prediction of power-handling capability using (1) is validated by the simulated and tested results of a number of HTS planar filters and resonators in [13]. Thus, we put an emphasis on the filter design process. As in (1), the key issue is to reduce in a filter at the given input the maximum current density power. B. Discussion of Power-Handling Capability We begin with a lumped-element equivalent circuit of a fivepole bandpass filter to investigate the power level within every resonator, as shown in Fig. 1. The filter has a center frequency of 2006 MHz with a 1% bandwidth. Each resonator is represented by an LC parallel circuit, and these resonators are coupled together with J-inverters. From the source of the filter, the five resonators are sequentially called Resonators 1–5. The circuit parameters are calculated by the general procedures of coupled-resonator filters [16]. We simulate the lumped-element circuit using SPICE software. The input power of the filter is 5 mW. The solid lines in Fig. 2 show the power level within the capacitance of every resonator as a function of frequency. The power levels of the five resonators are quite different from one another [17]. For example, the maximum power of Resonator 2 is approximately two times that of Resonator 4 and five times that of Resonator 5. If microstrip resonators with an equal linewidth were used in the HTS filter design, as in [18], Resonator 2 will have a higher current density than the other four resonators. As a result, it will limit the overall power-handling capability of the filter. We propose a possible solution to resolve this issue by equalizing the current density, i.e., the higher power the resonator carries, the greater the linewidth that should be used. Therefore, although these resonators have different power levels, the maximum current density of every resonator reaches the same value. Thus, the substrate area is most effectively used to maximize the power-handling capability of the HTS filter. C. Determination of Resonators’ Linewidths The relationship between the resonator linewidth and the power-handling capability is calculated first. The maximum flowing in one direction within the instantaneous power half-wavelength resonator is [18] (2)

Fig. 2. Power level (solid lines) within every resonator of the five-pole lumpedelement filter simulated by SPICE software. The results (circles and squares) for the resonators of the microstrip filter in Fig. 5 are simulated by Sonnet Software Inc.’s EM software and will be discussed later. Resonator 1 is close to the source, and Resonator 5 is close to the load.

where is the maximum current at its middle point, and is the characteristic impedance of the resonator, which is the

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TABLE II PARAMETERS OF EVERY RESONATOR IN THE FIVE-POLE MICROSTRIP FILTER

Fig. 3. Sketch of the microstrip structure. TABLE I VARIABLE FOR RESONATORS WITH DIFFERENT LINEWIDTHS

function of the resonator linewidth , thickness , and relative dielectric constant of the substrate (Fig. 3). occurs at the edges of the The maximum current density middle point of the resonator. Defining variable as the ratio of to the average one , the maximum current density (3) the maximum instantaneous power written as

in the resonator can be

(4) The variable in (4) cannot be expressed in a simple form. We simulate the current density distribution of resonators using Sonnet Software Inc.’s EM software and obtain the numeric values of for different linewidths, as shown in Table I. An LaAlO substrate with a thickness of 0.505 mm and a relative dielectric constant of 23.75 is used in the simulations. The resonant frequency of the resonators is 2006 MHz. The relationship between the resonator linewidth and the power-handling capability in (4) applies only at the resonant frequency. However, we assume that the relationship also exists at frequencies in the passband. Hence, the linewidth of each resonator in the five-pole microstrip filter can be determined by the following steps. of each resonator in the Step 1) The maximum power lumped-element equivalent circuit is obtained from Fig. 2. It is related with the maximum instantaneous in the half-wavelength microstrip respower onator as

Fig. 4. Power-handling capability versus the resonator linewidth of the microstrip structure. The power increases linearly with the resonator linewidth.

Step 3) Given the maximum instantaneous power of Resonator 2 in Table II and the linewidth of 7 mm, we calculate, using (4), the maximum current den, which is 690 A/m. sity Step 4) Substituting this current density value into (4), we obtain the relationship between the resonator linewidth and the power-handling capability. As plotted in Fig. 4, the power-handling capability of the microstrip resonator increases linearly with the linewidth. Step 5) With the relationship in Fig. 4, the linewidths of the other four resonators are chosen to afford their maximum instantaneous power in Table II. The last column of Table II presents the initial values of the resonator linewidths. III. FILTER DESIGN PROCEDURE

(5) where is the maximum energy stored within the lumped-element resonator or the half-wavelength microstrip resonator. Therefore, we get . The values of these power levels are listed in Table II. Step 2) We use a 7-mm linewidth for Resonator 2, which carries the highest power level in the filter.

The HTS microstrip filter is designed by the well-known procedures of coupled-resonator filters [16]. Fig. 5 shows the filter layout. The initial values of the resonator linewidths are determined in Section II. The length and width of each resonator are adjusted slightly so that the five resonators have the same resonant frequency. The linewidth of Resonator 5 is increased to 1.6 mm for facilitating the fabrication. Parallel-coupled feed lines, instead of tap-coupled ones, are chosen to avoid currents crowding at the tap points. The linewidth of the feed lines is

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Fig. 5. Layout of the five-pole microstrip bandpass filter. The width of every resonator is optimized to meet the same maximum current density. The filter is designed for a 2006-MHz center frequency and a 1% bandwidth using a 38 mm 30 mm 0.505 mm LaAlO substrate.

2

2

0.16 mm with a 50- characteristic impedance. The five resonators are staggered to decrease the coupling and reduce the spacings between the resonators. We use an LaAlO substrate with a thickness of 0.505 mm and a relative dielectric constant of 23.75. The total area of the five-pole filter is 38 mm 30 mm. The aforementioned power design method is verified by the current density distribution of the filter circuit simulated by Sonnet Software Inc.’s EM software [14]. The maximum instantaneous power of the microstrip resonator is compared with the result of the lumped-element circuit (Fig. 2). They are well matched. In addition, Fig. 6 presents the maximum current densities at the edges of the resonators and feed lines as a function of frequency. As we use a zero thickness lossless model for the superconducting circuit, the simulated sheet current densities are in the unit of amperes per meter. The maximum current density of every resonator has the same peak value of approximately 600 A/m (except for Resonator 5, whose linewidth has been increased from 0.4 to 1.6 mm), showing that the power-handling capability of the filter is optimized and is not limited by one single resonator. However, the simulated maximum current densities in the resonators are not symmetrical around the center frequency, which are different from the simulation of the lumped-element circuit. The maximum current densities in the 0.16-mm feed lines are less than 400 A/m because the power flowing in the thin feed lines are far less than the resonant power in the resonators. IV. FABRICATION AND MEASUREMENTS We fabricated the five-pole microstrip filter using a doublesided YBCO film deposited on an LaAlO substrate and then packed it in a copper shield box. The filter was mounted onto a cryogenic platform, which was connected to the cooled finger of a Stirling cooler and enclosed in a vacuum chamber. Fig. 7 shows the frequency response of the microstrip filter simulated by Sonnet Software Inc.’s EM software and that measured by the Agilent 8720/ES network analyzer at 65 K. No tuning was performed in the measurement. The filter has a center frequency

Fig. 6. Simulated maximum current densities at the edges of the resonators and feed lines in the five-pole microstrip filter as a function of frequency.

of 2006 MHz and a narrow bandwidth of 20 MHz. The experimental insertion loss of the device is 0.6 dB, and the return loss is better than 13.6 dB. There is a transmission zero at approximately 2020 MHz in Fig. 7, which mainly results from the nonadjacent couplings between Resonators 1 and 3, 2 and 4, and 3 and 5. To find out the origin of the transmission zero, we simulate the adjacent and nonadjacent couplings of the five-pole microstrip filter using Sonnet software. The obtained coupling matrix is presented in Table III. The nonadjacent couplings are approximately 25%–50% of the adjacent ones and cannot be neglected. and the reThe calculated response of the coupling matrix sponse of Chebyshev filter without nonadjacent couplings are shown in Fig. 8. A transmission zero appears on the high side of the passband because of the couplings between the nonadjacent resonators. The power-handling capability of the HTS filter was measured by an HP 438A power meter. The filter structure in Fig. 5

GUO et al.: DESIGN OF HIGH-POWER SUPERCONDUCTING FILTER USING RESONATORS WITH DIFFERENT LINEWIDTHS

Fig. 7. Simulated and measured frequency response of the HTS filter.

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Fig. 9. Measured insertion-loss variations versus the input power of the HTS filter. The insertion loss increases less than 0.2 dB as the input power increases from 10 to approximately 35 dBm. The superconducting state then breaks down after the power increases even a little.

0

COUPLING MATRIX

M

TABLE III FOR THE FIVE-POLE MICROSTRIP FILTER

TABLE IV TEST RESULTS OF THE HTS FILTER AT DIFFERENT FREQUENCIES

M

Fig. 8. Comparison between the Chebyshev filter response and the calculated response of the coupling matrix .

is not symmetrical and the filter is not reciprocal from a high power operation point of view. Therefore, the input and output of the filter cannot be switched in the measurement. Fig. 9 shows the insertion loss versus the input power of the HTS filter at three in-band frequencies. The insertion loss of the HTS filter changes slightly as the input power increases. For example, the insertion loss at 2008 MHz increases less than 0.2 dB as the input power

increases from 10 to 35 dBm. The superconducting state then apparently breaks down after the power increases to even a little more than 35 dBm. As the input power decreases, the filter performance is fully recovered, and no electrical damage to the HTS filter is observed. The predictive power-handling capability of HTS circuits using (1) was investigated by a number of microwave filters and resonators with various topologies [13]. We extend this investigation by examining the same filter at different frequencies. Given the simulated maximum current density in Fig. 6 and the measured maximum power at several frequencies, we deduce, using (1), the RF critical current density of the HTS film, as shown in Table IV. As the unit of current density in (1) is amperes per meter, we make the conversion by assuming that the current is evenly distributed throughout the entire has an average value 500-nm-thick HTS film. The deduced of 1.84 10 A/cm . These values are self-consistent and consistent with the results in [13], adding evidence to the assertion that the power-handling capability of the HTS filter design may be predicted by the simulated current density distribution. V. CONCLUSION We have proposed a power design method to optimize the linewidth of every resonator in the HTS filter so that the sub-

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strate area can be best utilized to maximize the power-handling capability. The current density distribution in the filter circuit is simulated by Sonnet Software Inc.’s EM software. The maximum current density of every resonator reaches the same peak value, verifying our design method. We also deduced the RF of the HTS film from the measured critical current density maximum power of the HTS filter and the simulated current density data. The values of are self-consistent and also consistent with the results reported by other researchers. ACKNOWLEDGMENT The authors wish to thank the National Laboratory for Superconductivity, Institute of Physics, Chinese Academy of Sciences (CAS), Beijing, China, for providing the HTS thin films. REFERENCES [1] R. R. Mansour, “Microwave superconductivity,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 750–759, Mar. 2002. [2] G. Zhang, M. Zhu, B. Cao, W. He, X. Guo, S. He, B. Liu, Y. Wang, X. Zhang, Y. Zhao, M. Liu, Z. Han, and B. Gao, “Design and performance of a compact forward-coupled HTS microstrip filter for a GSM system,” IEEE Trans. Appl. Supercond., vol. 12, no. 4, pp. 1897–1901, Dec. 2002. [3] X. Guo, B. Cao, B. Wei, X. Zhang, Z. Yin, W. He, S. He, L. Gao, M. Zhu, B. Gao, B. Aminov, M. Getta, S. Kolesov, H. Piel, N. Pupeter, and D. Wehler, “A high performance HTS filter subsystem for CDMA mobile communications,” Chin. J. Phys., vol. 42, no. 4, pp. 463–467, Aug. 2004. [4] W. Rauch, E. Gornik, G. Solkner, A. A. Valenzuela, F. Fox, and H. thin films studied Behner, “Microwave properties of YBa Cu O with coplanar transmission line resonators,” J. Appl. Phys., vol. 73, no. 4, pp. 1866–1872, Feb. 1993. [5] M. I. Salkola and D. J. Scalapino, “Benefits of superconducting technology to wireless CDMA networks,” IEEE Trans. Veh. Technol., vol. 55, no. 3, pp. 943–955, May 2006. [6] Z. Yin, B. Wei, B. Cao, X. Wang, X. Guo, X. Zhang, L. Gao, Y. Piao, M. Zhu, Y. Liang, F. Wang, H. Piel, B. Aminov, F. Aminova, M. Getta, S. Kolesov, A. Knack, N. Pupeter, and D. Wehler, “Field trial of an HTS filter system on a CDMA base station,” Chin. Sci. Bull., vol. 52, no. 2, pp. 171–174, Jan. 2007. [7] A. Flores, C. Collado, C. Sans, J. O’Callaghan, R. Pous, and J. Fontcuberta, “Full-wave modeling of HTS dual-mode patch filters and staggered coupled-line filters,” IEEE Trans. Appl. Supercond., vol. 7, no. 2, pp. 2351–2354, Jun. 1997. [8] A. Baumfalk, H. Chaloupka, S. Kolesov, M. Klauda, and C. Neumann, “HTS power filters for output multiplexers in satellite communications,” IEEE Trans. Appl. Supercond., vol. 9, no. 2, pp. 2857–2861, Jun. 1999. [9] G. Bertin, B. Piovano, L. Accatino, G. Dai, R. Tebano, and F. Ricci, “A novel rounded-patch dual-mode HTS microstrip filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, vol. 2, no. 6–11, pp. 1113–1116. [10] H. F. Huang, J. F. Mao, and Z. H. Luo, “Novel miniature high power ring filter,” Phys. C, vol. 420, no. 3–4, pp. 125–129, Apr. 2005. [11] H. Chaloupka, M. Jeck, B. Gurzinski, and S. Kolesov, “Superconducting planar disk resonators and filters with high power handling capability,” Electron. Lett., vol. 32, no. 18, pp. 1735–1737, Aug. 1996. [12] X. Guo, X. Zhang, B. Cao, B. Wei, L. Mu, Y. Lang, L. Gao, and B. Gao, “HTS narrowband stripline filter at 2.1 GHz with high power handling capability,” Microw. Opt. Technol. Lett., vol. 49, no. 2, pp. 254–257, Feb. 2007. [13] F. S. Thomson, R. R. Mansour, S. Ye, and W. Jolley, “Current density and power handling of high-temperature superconductive thin film resonators and filters,” IEEE Trans. Appl. Supercond., vol. 8, no. 2, pp. 84–93, Jun. 1998. [14] “EM User’s Manual, Version 8.52,” Sonnet Softw. Inc., North Syracuse, NY, 2002. [15] X. Guo, X. Zhang, B. Cao, B. Wei, L. Mu, J. Tian, L. Gao, Y. Liang, and B. Gao, “A novel HTS shuttle-shape resonator for high-power application,” J. Supercond. Novel Magn., vol. 20, no. 1, pp. 37–41, Jan. 2007.

[16] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters Impedance-Matching Networks and Coupling Structures. Norwood, MA: Artech House, 1980. [17] R. R. Mansour, B. Jolley, S. Ye, F. S. Thomson, and V. Dokas, “On the power handling capability of high temperature superconductive filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 7, pp. 1322–1338, Jul. 1996. [18] G. C. Liang, D. Zhang, C. F. Shih, M. E. Johansson, R. S. Withers, D. E. Oates, A. C. Anderson, P. Polakos, P. Mankiewich, E. Obaldia, and R. E. Miller, “High-power HTS microstrip filters for wireless communication,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 3020–3029, Dec. 1995. Xubo Guo was born in Shanxi Province, China, on May 7, 1981. He received the B.S. and M.S. degrees in physics from Tsinghua University, Beijing, China, in 2001 and 2004, respectively, and is currently working toward the Ph.D. degree at Tsinghua University, Beijing, China. From 2004 to 2005, he was engaged in research on technical barriers to trade (TBT) as a Research Assistant with the Shenzhen Institute of Standard and Technology, Guangdong, China. Since July 2005, he has been with the Department of Physics, Tsinghua University. His research interests include design and fabrication of HTS filters for both receiving and transmitting applications.

Bin Wei was born in Zhejiang Province, China, on January 22, 1974. He received the B.S. and Ph.D. degrees in physics from Tsinghua University, Beijing, China, in 1996 and 2003, respectively. Since 2003, he has been an Assistant Researcher with the Department of Physics, Tsinghua University. His current research interests include HTS microwave devices and their applications in wireless communication systems.

Xiaoping Zhang was born in Liaoning Province, China, on November 26, 1957. She received the B.S. degree in physics from Liaoning University, Shenyang City, China, in 1982. From 1982 to 2000, she was with the Fundamental Department, Televise University, Shenyang China. In 2000, she joined the Physics Department, Tsinghua University, Beijing China, as a Senior Engineer. Her current research interests include HTS materials and HTS microwave devices.

Bisong Cao was born in Jiangsu Province, China, on October 26, 1946. He received the B.S. and M.S. degrees in physics from Tsinghua University, Beijing, China, in 1970 and 1981, respectively, and the Ph.D. degree in material science from Tokyo University, Tokyo, Japan, in 1989. Since 1980, he has been with the Department of Physics, Tsinghua University, Beijing, China, where he is currently a Professor. His current research interests include HTS physics and HTS microwave devices and their applications. He has authored or coauthored over 100 papers on materials and HTS microwave devices in journals and international conferences. Prof. Cao is a committee member of the Superconductor Electronics Branch, Chinese Society for Electronics. He is also a committee member of the Low Temperature Physics Branch, Chinese Society for Physics.

Shichao Jin was born in Hebei Province, China, on November 8, 1980. He received the B.S. degree in physics from Lanzhou University, Gansu, China, in 2004, and is currently working toward the Ph.D. degree at Tsinghua University, Beijing, China. In September 2004, he joined the Department of Physics, Tsinghua University. His research interest is RF and microwave application of high-temperature superconductors, especially the design and fabrication of HTS filters and subsystems.

GUO et al.: DESIGN OF HIGH-POWER SUPERCONDUCTING FILTER USING RESONATORS WITH DIFFERENT LINEWIDTHS

Huili Peng was born in Shandong Province, China, on June 12, 1982. She received the B.S. degree in physics from Xi’an Jiaotong University, Shannxi, China in 2004, and is currently working toward the Ph.D. degree at Tsinghua University, Beijing, China. In September 2004, she joined the Department of Physics, Tsinghua University. Her research interest includes RF and microwave application of HTSs, especially the design and tuning of HTS filters and subsystems.

Longma Gao was born in Heilongjia Province, China, on December 28, 1978. He received the B.S. degree in physics from Tsinghua University, Beijing, China, in 2001. Since August 2001, he has been a Research Assistant with the Department of Physics, Tsinghua University. His research interests include development of HTS subsystems and their applications mobile communication systems.

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Baoxin Gao was born in Beijing, China, in 1934. He is currently a Professor with the Department of Electronics Engineering, Tsinghua University, China. He has authored or coauthored approximately 200 technical papers and has authored 11 scientific books. His research areas include microwave active and passive circuits, microwave integrated circuits (MICs), microwave computer-aided design (CAD) software techniques, RF systems of wireless communication, photonic bandgap (PBG) circuits, photoelectronic integrated circuits, and microelectromechanical systems (MEMS). Mr. Gao is a Fellow of the Chinese Institute of Electronics (CIE). He is chairman of the MIC and the Technology Society of China.

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Input and Output Cross-Coupled Wideband Bandpass Filter Hussein Shaman, Student Member, IEEE, and Jia-Sheng Hong, Senior Member, IEEE

Abstract—A new wideband bandpass filter configuration with input–output cross-coupling is investigated. The new filter is based on a transmission line filter with short-circuited stubs. In order to allow the filter to exhibit high selectivity filtering characteristic and to enhance the group delay with only a few resonators, cross-coupling between the input and output feed lines is introduced. As a result, new symmetrical pairs of transmission zeros are generated at the lower and upper stopbands, leading to a quasi-elliptic function response that improves the passband and out-of-band performances. A general circuit model for the proposed filter is presented and a demonstrator with approximately 48% ripple bandwidth at a midband frequency of 3 GHz is developed. The design is successfully realized in theory and verified by full-wave electromagnetic simulation and the experiment, where excellent agreement is obtained. Index Terms—Cross-coupling, microstrip filters, parallel-coupled lines, transmission lines filters, wideband bandpass filters.

I. INTRODUCTION

R

ECENTLY, there has been increased interest in wideband bandpass filters for use in modern wireless communication systems. This is due to today’s growing consumer demand for the convenience and flexibility of wireless connectivity, which can be achieved through ultra-wideband (UWB) technology. Various wideband bandpass filters have been developed [1]–[5]. Since lumped elements are difficult to implement at high frequencies, distributed elements are commonly used to design bandpass filters with very wideband. These distributed elements can be obtained by using Richard’s transformation [6]. In addition, Kuroda’s identities can be implemented to simplify the filter realization by adding redundant transmission line sections, which do not affect the filter response [7]. This type of filter is commonly known as a conventional stub filter [8]. The connecting lines of the conventional stub filter may be modified to have twice the length of the stubs in order to achieve a so-called optimum performance, for which the connecting lines also contribute to the selectivity of filter [9]. In general, a standard short-circuited stub filter, either conventional or optimum, has all its attenuation poles at dc and . at even multiples of a fundamental midband frequency Therefore, it requires more sections to improve its performance. Manuscript received June 13, 2007; revised September 9, 2007. This work was supported in part by the U.K. Engineering and Physical Science Research Council. The work of H. Shaman was supported by the Saudi Interior Ministry. The authors are with the Department of Electrical, Electronic, and Computer Engineering, Heriot-Watt University, Edinburgh EH14 4AS, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910066

Fig. 1. Configuration of the proposed cross-coupled wideband bandpass filter.

Alternatively, each short-circuited stub can be replaced with a two-section open-circuited stub in order to obtain pairs of transmission zeros in both sides of the passband, which enhance the filter selectivity [10]. However, this increases the filter size by approximately 50%. Transmission zeros may also be implemented with input and output or source–load coupling, which has been reported mostly for narrowband filter applications [11]–[13]. Recently, we have reported a preliminary investigation of an UWB bandpass filter with input/output (I/O) cross-coupling [14]. The same principle is applied to UWB stepped-impedance resonator filters to enhance the out-of-band performance [15]. As a result, new pairs of transmission zeros or attenuation poles were generated at each side of the desired passband. However, the attenuation level in the upper stopband was poor. It was later found that the poor attenuation level in the upper stopband was due to unequal modal phase velocities of coupled microstrip lines. Therefore, this issue will be addressed in this paper. More recently, we have also applied the I/O cross-coupling to a wideband bandstop filter, consisting of open-circuited stubs, to introduce two new attenuation poles in the stopband [16]. Nevertheless, the I/O cross-coupling in [14]–[16] does not add any new transmission poles in the passbands. In this paper, a more comprehensive study is carried out for a general configuration of the wideband transmission line filter in Fig. 1, which is developed from a standard short-circuited stub filter with additional coupled lines. It will be shown that the resultant filter can contribute not only a new pair of attenuation poles at each side of the passband, but also a new pair of transmission poles in the passband. Furthermore, it will be demonstrated that the new filter has an additional capability for group-delay equalization. Thus, based on the proposed configuration, a selective and linear phase wideband filter can be designed. In Section II, a comparison study is performed to show the interesting and advantageous characteristics of the proposed filter. In Section III, an improved microstrip filter of this type is demonstrated on a low-cost GML 1000 substrate with a relative

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For the five-pole conventional bandpass filter of Fig. 2(a), (3a) (3b) (3c) with (4a) (4b) (4c)

(4d) where is the terminal admittance and and are the oddand even-mode input admittances of the conventional filter. For the proposed cross-coupled filter of Fig. 2(b), (5a) (5b) (5c)

Fig. 2. Wideband filter circuit models. (a) Conventional bandpass filter with five short-circuited stubs. (b) Proposed cross-coupled wideband bandpass filter based on a conventional bandpass filter with three short-circuited stubs.

where and are the odd- and even-mode input admittances of the proposed filter, and are defined by dielectric constant of 3.2 and a thickness of 0.762 mm. The theoretical, simulated, and measured results are presented. This is followed by a conclusion in Section IV.

(6a) (6b)

II. COMPARISON STUDY FOR THE PROPOSED FILTER For the comparison study, Fig. 2(a) shows a conventional filter consisting of five short-circuited stubs separated by four connecting lines. The characteristic impedances of the , while the short-circuited stubs are defined by characteristic impedances for the connecting lines are defined . The electrical length of the connecting by lines and the short-circuited stubs is 90 at a fundamental midband frequency . Following the general configuration of Fig. 1, a proposed cross-coupled wideband bandpass filter is shown in Fig. 2(b). The I/O cross-coupling is implemented by replacing the first and last stubs of the conventional filter with a quarter-wavelength parallel coupled line section that has a pair of even- and and , respectively. odd-mode impedances defined by Assuming that both filters in Fig. 2 are symmetric with respect to the I/O ports, which is the case for most practical realizations, an odd- and even-mode analysis [17] can be applied. Hence, and the reflection coefficient the transmission coefficient for both the filters can be defined by (1) (2)

with (7a) (7b) and have the same expressions as (4c) and (4d). in which The electrical lengths of the even- and odd-mode for the coupled and , which are equal only for pure lines are defined by TEM transmission lines such as a stripline. Since all the transat mission line elements have an electrical length of the midband frequency , the frequency-dependent characteristics can be calculated for . The design equations used to determine the characteristic impedances of the circuit model of Fig. 2(a) are described in [8]. For example, the calculated characteristic impedances for are a ripple fractional bandwidth of approximately 48% at is the terminal impedance. listed in Table I, where Although similar design equations are not available for the proposed cross-coupled wideband filter, an optimization procedure can be used to design this type of filter. To this end, a computer-aided design (CAD) program can be developed based on the formulations given above or a commercially available

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TABLE I CIRCUIT PARAMETERS FOR A CONVENTIONAL BANDPASS FILTER WITH FIVE SHORT-CIRCUITED STUBS ( = 90 AT f )

TABLE II CIRCUIT PARAMETERS FOR A PROPOSED CROSS-COUPLED FILTER WITH THREE SHORT-CIRCUITED STUBS ( = 90 AT f )

design tool such as Microwave Office [18] can be used. Table II shows a set of circuit parameters optimized for the proposed cross-coupled filter with three short-circuited stubs in Fig. 2(b). Using the formulations given above, the frequency responses of these two filters can be computed and the results are plotted in Fig. 3 for comparison. As demonstrated in Fig. 3(a), both filters have the same ripple bandwidth for a return loss of 20 dB. On both sides of the primary passband, the conventional filter of Fig. 2(a) has its transmission zeros or attenuation poles at , as expected. However, for the proposed filter of dc and at Fig. 2(b), more attenuation poles appear on both sides of the passband, leading to an improved selectivity and a better stopband performance. The I/O cross-coupling, which is introduced, allows the attenuation poles to be split and allocated at different frequencies. Both the conventional and proposed filters exhibit five reflection coefficient zeros in the passband due to the fact that both filters has four nonredundant unit elements and one short-circuited stub in a canonical form [19]. In addition, the proposed filter has two more unit elements, i.e., a coupled line section, but these additional unit elements are redundant and do not add any reflection zero in the passband. Even with a higher selectivity, it is very interesting to see from Fig. 3(b) that the proposed filter exhibits a better group-delay performance than the conventional filter does. This indicates the proposed filter also has a capability of self-equalizing group delay or of producing a linear phase response. In light of all this, the magnitude and group-delay responses shown in Fig. 3 suggest that the proposed filter exhibits two new imaginary axis zeros and one real axis zero when referencing to a complex frequency plane of a low-pass prototype. In principle, the zeros may be found when expressed in terms of the low-pass from the roots of complex frequency variable. Nevertheless, as matter of fact, a small group-delay ripple introduced at the middle band for the proposed filter has shown a typical effect resulting from the real axis zero in order to reduce the overall delay variation over a larger part of the passband, as documented in [17]. We can also discuss how the proposed filter generates the new transmission zeros and poles in light of the above formulations.

Fig. 3. Frequency responses of the filters in Fig. 2, using the circuit parameters displayed in Table I for Fig. 2(a) and Table II for Fig. 2(b), respectively. (a) Magnitude. (b) Group delay.

It can be seen from (1) and (5) that the new transmission zeros . Figs. 4 and 5 depict the typical reoccur when and at the low and high frequencies. It can sponses of be seen in Figs. 4 and 5 that at two new different frequencies in the lower stopband and at two other new frequencies in the upper stopband. Therefore, new pairs of attenuation poles can be generated at different frequencies on each side of the passband when applying cross-coupling between the I/O feed lines. The numerator of (2) is 0 at four different frequencies inside the passband for both the conventional filter and cross-coupled filter, as illustrated in Fig. 6. In addition, the denominator of (2) is infinity, while the numerator is finite at the center frequency for both filters. Hence, the new filter with only three stubs is a 5 filter. It can also be seen from Fig. 6(b) that the function for the proposed filter exhibits two singularities at approximately 0.8 and 1.2 GHz. This is interesting as the allocation of these two singularities seems to coincide with the two lowest ripples in the group delay response of Fig. 3(b). For the proposed filter, a flatter group-delay response may be obtained by slightly tuning the I/O cross-coupling, i.e., the and , while keeping the other parameters unvalues of changed, though this can affect the ripples in the stopband. For

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TABLE III CIRCUIT PARAMETERS OF THE PROPOSED CROSS-COUPLED FILTER FOR DIFFERENT COUPLINGS

Y

Y

of the proposed cross-coupled wideband filter at the low

Y

Y

of the proposed cross-coupled wideband filter at the high

Fig. 4. and frequencies.

Fig. 5. and frequencies.

Fig. 7. (a) Magnitude responses and (b) group delays of the proposed structure with varying values of and .

Z

Z

(8)

S

N

Fig. 6. and for the: (a) five-pole conventional filter and (b) proposed cross-coupled filter.

comparison, Table III tabulates the circuit parameters for three cases, i.e., three different couplings. Note that Case A is, in fact, the one we have discussed thus far. For our discussion, we may define a cross-coupling coefficient as

From Table III, it can be found that the coupling coefficient is equal to 0.083, 0.063, and 0.049 for Case A, Case B, and Case C, respectively. Fig. 7 displays the magnitude responses and group delays of the cross-coupled filter using the circuit parameters shown in Table III. As can be seen from Fig. 7(b), Case C, which has the smallest coupling among the three cases, has the flattest groupdelay response over the central 83% of the passband. A tradeoff is a slight reduction in the selectivity and a result of the unequalripple stopbands, which can be observed from Fig. 7(a).

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Fig. 9. Magnitude responses of the new filter using stripline ( and microstrip coupled lines ( 6  at f ).

=

=

at f )

Fig. 8. Comparison between the proposed structures with three short-circuited stubs for Case C and a conventional filter with five short-circuited stubs. (a) Magnitude responses. (b) Group delays.

In order to obtain a sufficient improvement over the five-stub conventional filter, Case C is chosen for demonstration and the magnitude responses and group delays of both filters are depicted in Fig. 8. It is demonstrated that the proposed structure with only three stubs can exhibit a better filtering selectivity characteristic than that of a conventional filter with five stubs. For example, the selectivity of the desired passband at 30 dB is improved by approximately 15%, and the stopband is widened by approximately 10% over the five-stub conventional filter, as shown in Fig. 8(a). Therefore, high selectivity is obtained with only a few via-hole groundings. In addition, the group delay of the proposed filter is much more enhanced than that of a conventional filter, as displayed in Fig. 8(b). III. IMPLEMENTATION AND EXPERIMENTAL PERFORMANCE For the experimental demonstration, Case C is chosen for implementation. The proposed cross-coupled wideband bandpass filter is implemented on a low-cost GML 1000 substrate with a relative dielectric constant of 3.2 and a thickness of 0.762 mm. Due to the inhomogeneous dielectric material in microstrip line, the even- and odd-mode propagation velocities for microstrip coupled lines are not equal and the match between them becomes frequency dependent. This results in poor directivity as

Fig. 10. (a) Microstrip layout (unit: millimeters) and (b) a photograph of the fabricated cross-coupled wideband bandpass filter.

the frequency increases. A stripline structure is one solution for the coupled lines to have identical even- and odd-mode propagation constants. To see the effect of modal phase velocities of the

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the dielectric overlay to equalize the modal phase velocities of the microstrip coupled-line section for the I/O cross coupling, it would not be possible to realize the new transmission zeros, as displayed in Fig. 11(a). Although the proposed filter has only three stubs, it exhibits better frequency selectivity than that of a conventional filter with five short-circuited stubs. Furthermore, the fabricated filter confirms the better group-delay performance of the proposed filter, as shown in Fig. 11(b). Due to the fabrication tolerance, there is a slight deviation between the simulated and measured return losses of the proposed filter in the frequency range from 3 to 4 GHz. This also causes the measured return loss of proposed filter to show only four reflection poles in the desired passband. IV. CONCLUSION

Fig. 11. Magnitude responses of predicted and measured results. (a) Magnitude responses. (b) Group delays.

coupled-line section on the performance of the proposed filter, Fig. 9 shows the circuit-simulated magnitude responses for the and new filter using microstrip coupled lines ( at ) and stripline coupled lines ( at ). It is clearly demonstrated that the filter using the stripline coupled lines shows a much better stopband performance than the filter using microstrip coupled lines does. However, the stripline is difficult to implement, especially when via-hole groundings are used. Alternatively, a dielectric overlay [20]–[22] can be implemented on top of each line of the microstrip parallel-coupled lines to equalize the modal phase velocities in the coupled microstrip lines. We will adopt this approach for our demonstrator. The filter is fabricated using print circuit board (PCB) technology and Fig. 10 depicts the final layout of the microstrip design and the photograph of the fabricated filter with attached overlays and subminiature A (SMA) connectors. Fig. 11 demonstrates the frequency responses and group delays of the full-wave electromagnetic (EM) simulation and the experiment where excellent agreement is obtained. The EM simulation is done using a commercially available tool [23]. The filter exhibits a highly selective wideband bandpass performance with a ripple bandwidth of approximately 48% at a center frequency of 3 GHz and shows an enhanced group delay. The measured insertion and return losses are found to be less than 1 dB at the center frequency and better than 12 dB over the entire passband, respectively. Meanwhile, the measured shows a new pair of attenuation poles at each of the lower and upper stopbands, which improves the frequency selectivity and the out-of-band performance. Without the implementation of

In this paper, we have proposed a novel cross-coupled wideband bandpass filter, which is developed from a standard shortcircuited stub filter with additional coupled lines. The coupledline section facilitates the I/O cross-coupling. A general circuit model for the proposed filter has been presented and a comparison study has been carried out to demonstrate the interesting characteristics of the proposed filter. It has been shown that new symmetrical pairs of transmission zeros can be generated at the lower and upper stopbands, leading to a quasi-elliptic function response that improved the passband and out-of-band performances. Furthermore, the proposed filter exhibits a linear phase response, leading to a self-equalization of the group delay. These attractive characteristics have been verified experimentally by a fabricated microstrip filter of this type. The experimental microstrip filter also demonstrates the importance of equalizing the modal phase velocities of the coupled line section for achieving a desired performance in practice. REFERENCES [1] A. Saito, H. Harada, and A. Nishikata, “Development of bandpass filter for ultra wideband (UWB) communication systems,” in Proc. IEEE Ultra Wideband Syst. Technol. Conf., 2003, pp. 76–80. [2] H. Ishida and K. Araki, “Design and analysis of UWB bandpass filter with ring filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 1307–1310. [3] K. Li, D. Kurita, and T. Matsui, “An ultra-wideband bandpass filter using broadside-coupled microstrip-coplanar waveguide structure,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 675–678. [4] C.-L. Hsu, F.-C. Hsu, and J.-T. Kuo, “Microstrip bandpass filters for ultra-wideband (UWB) wireless communications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 679–682. [5] H. Wang, L. Zhu, and W. Menzel, “Ultra-wideband (UWB) bandpass filters with hybrid microstrip/CPW structure,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 844–846, Dec. 2005. [6] P. I. Richards, “Resistor-transmission-line circuits,” Proc. IRE, vol. 36, no. 2, pp. 217–220, Feb. 1948. [7] H. Ozaki and J. Ishii, “Synthesis of a class of strip-line filters,” IRE Trans. Circuit Theory, vol. CT-5, no. 6, pp. 104–109, Jun. 1958. [8] G. Mattaei, L. Young, and E. M. T. Jones, Microwave filters, Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980. [9] R. Levy, “A new class of distributed prototype filters with applications to mixed lumped/distributed component design,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 12, pp. 1064–1071, Dec. 1970. [10] P. Cai, Z. Ma, X. Guan, X. Yang, Y. Kobayashi, T. Anada, and G. Hagiwara, “A compact UWB bandpass filter using two section opencircuited stubs to realize transmission zeros,” in Asia–Pacific Microw. Conf., Dec. 2005, vol. 5, p. 14. [11] S. Amari, “Direct synthesis of folded symmetric resonator filters with source-load coupling,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 6, pp. 264–266, Jun. 2001.

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[12] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 1–10, Jan. 2003. [13] K. T. Jokela, “Narrow-band stripline or microstrip filters with transmission zeros at real and imaginary frequency,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 6, pp. 542–547, Jun. 1980. [14] H. Shaman and J.-S. Hong, “A novel ultra-wideband (UWB) bandpass filter (BPF) with pairs of transmission zeros,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 121–123, Feb. 2007. [15] M. Mokhtaari, J. Bornemann, and S. Amari, “Folded compact ultrawideband stepped-impedance resonator filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 747–750. [16] H. Shaman and J.-S. Hong, “Wideband bandstop filter with cross-coupling,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 8, pp. 1780–1785, Aug. 2007. [17] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [18] Microwave Office. Appl. Wave Res., El Segundo, CA, 2005. [19] H. J. Carlin and W. Kohler, “Direct synthesis of bandpass transmission line structures,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 5, pp. 283–297, May 1965. [20] B. Sheleg and B. E. Spielman, “Broadband directional couplers using microstrip with dielectric overlays,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 12, pp. 1216–1220, Dec. 1974. [21] D. D. Paolino, “MIC overlay coupler design using spectral domain techniques,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 9, pp. 646–649, Sep. 1978. [22] J. L. Klein and K. Chang, “Optimum dielectric overlay thickness for equal even- and odd- mode phase velocities in coupled microstrip circuits,” Electron. Lett., vol. 26, pp. 274–276, 1990. [23] “EM User’s Manual,” ver. 10, Sonnet Software Inc., North Syracuse, NY, 2005.

Hussein Shaman (S’05) was born in Najran, Saudi Arabia, in 1973. He received the Bachelor of Engineering (B.Eng.) degree in electrical and electronic engineering from Heriot-Watt University, Edinburgh, U.K., in 2005, and is currently working toward the Ph.D. in electrical engineering at Heriot-Watt University. His research concerns UWB microwave filters for radar and wireless communications.

Jia-Sheng Hong (M’94–SM’05) received the D.Phil. degree in engineering science from the University of Oxford, Oxford, U.K., in 1994. His doctoral dissertation concerned EM theory and applications. In 1994, he joined the University of Birmingham, Edgbaston, Birmingham, U.K., where he was involved with microwave applications of high-temperature superconductors, EM modeling, and circuit optimization. In 2001, he joined the Department of Electrical, Electronic, and Computer Engineering, Heriot-Watt University, Edinburgh, U.K., where he is currently a faculty member leading a team for research into advanced RF/microwave device technologies. He has authored or coauthored over 130 journal and conference papers. He authored Microstrip Filters for RF/Microwave Applications (New York: Wiley, 2001) and RF and Microwave Coupled-Line Circuits, Second Edition (Boston: Artech House, 2007). His current interests involve RF/microwave devices such as antennas and filters for wireless communications and radar systems, as well as novel material and device technologies including RF microelectromechanical systems (MEMS) and ferroelectric and high-temperature superconducting devices.

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Dual-Band and Triple-Band Substrate Integrated Waveguide Filters With Chebyshev and Quasi-Elliptic Responses Xiao-Ping Chen, Ke Wu, Fellow, IEEE, and Zhao-Long Li, Student Member, IEEE

Abstract—In this paper, synthesis and design techniques of dualand triple-passband filters with Chebyshev and quasi-elliptic symmetric frequency responses are proposed and demonstrated for the first time on the basis of substrate integrated waveguide technology. The inverter coupled resonator section is first investigated, and then a dual-passband Chebyshev filter, a triple-passband Chebyshev filter, and a dual-passband quasi-elliptic filter, which consist of the inverter coupled resonator sections, are synthesized from the generalized low-pass prototypes having Chebyshev or quasi-elliptic responses, respectively. Subsequently, theses filters with a symmetric response are designed and implemented using -band the substrate integrated waveguide scheme over the frequency range. The inverter coupled resonator sections composed of side-by-side horizontally oriented substrate integrated waveguide cavities are coupled, in turn, by post-wall irises. 50microstrip lines are used to directly excite the filters. Measured results are presented and compared to those simulated by Ansoft’s High Frequency Structure Simulator (HFSS) software package. A good agreement between the simulated and measured results is observed, which has also validated the proposed concept of design and synthesis with the substrate integration technology.



Index Terms—Dual-passband filter, inverter coupled resonator section, quasi-elliptic, substrate integrated waveguide, triple-passband filter.

I. INTRODUCTION

R

ECENT advances in the development of microwave and millimeter-wave communication systems has greatly stimulated the demand on multifunctional and frequency-agile transceiver systems for an efficient utilization of more and more frequency channels. Under this trend, the development of highly integrated multifunctional and multiband components such as antennas and filters plays a pivotal role for more convenient and compact products [1]. A number of topologies have been studied and developed to realize high-quality filters with dual-passband or multipassband responses, which hold the promise for multichannel system design. They usually fall into the following three different categories. Manuscript received June 19, 2007; revised August 21, 2007. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The authors are with Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montreal, Montreal, QC, Canada H3C 3A7 (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.909603

1) A dual-passband filter can be realized by cascading a broadband bandpass and a bandstop filter; however, resulting in a larger circuit size. The bandpass and bandstop filters are analytically synthesized and separately designed, and then cascaded to fulfill the required dual-passband characteristics based on a numerical optimization [2]. 2) Novel resonant structures with a great degree of design freedom, e.g., a stepped-impedance resonator, dual-behavior resonator, etc., are used for designing dual-passband filters because their two dominant resonances coincide with the two center frequencies of two designated passbands by adjusting their geometric parameters. The resonators are properly placed so that appropriate coupling in the structure can be established. Similarly, this topology can also be analytically synthesized, but an optimization scheme is needed to design these filters [3]–[12]. 3) Transmission zeros produced by cross-coupling or bandstop resonators are used for splitting single passband into dual passbands or multipassbands based on a single filter circuit [13]–[15]. For the topology with cross-coupling, a single wideband filter whose bandwidth covers the entire bandwidth of the dual-passband or multipassband filter can be constructed as the initial filter for the synthesis. This topology can then be numerically synthesized by optimizing coefficients of the numerator and denominator of the filtering function of the single wideband filter to determine the prototype function and the positions or locations of the reflection and transmission zeros of the dual-passband or multipassband filters [16]. Alternatively, the prototype function of this topology can also be obtained from the piecing functions of multiindividual filters, each of which is analytically synthesized according to the single passband filter approach. At last, a generalized coupling matrix can be extracted by optimization and the design parameters of , and coupling coefficients , external quality factor can be obtained by de-normalresonant frequencies izing the coupling matrix [17]. It is very easy to control the attenuation of the inner stopband by changing the position of transmission zeros. Every band is also characterized by different in-band and out-of-band responses. The topology with the transmission zeros generated by the bandstop resonators can be analytically synthesized from the generalized low-pass prototype with Chebyshev or quasi-elliptic responses. The design parameters of coupling coefficients , external quality factor , and resonant frequencies can then be obtained. In this topology, a bandpass resonator

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and all the bandstop resonators that are properly coupled to this bandpass resonator comprise the inverter coupled resonator section. This section can be used as the basic unit for the dual-passband or multipassband filters in which every band has the same in-band and out-of-band responses [16]. Recently, the substrate integrated waveguide, which is synthesized on a planar substrate with linear periodic arrays of metallic vias or metallic slots by standard printed circuit board (PCB) or other planar circuit processes, has provided a very attractive platform to design low-cost and highly integrated waveguide filters [18], [19]. In this paper, a dual-passband Chebyshev filter, triple-passband Chebyshev filter, and dual-passband quasi-elliptic filter with inverter coupled resonator sections are proposed and realized on the basis of the substrate integrated waveguide technology for the first time. This paper is organized as follows. In Section II, the inverter coupled resonator sections are presented for the dual- and triple-passband filters. Sections III–V provide the synthesis and realization by the substrate integrated waveguide technology for the proposed dual- and triple-passband substrate integrated waveguide filters with Chebyshev or quasi-elliptic response. Measured results are presented and compared with simulated counterparts. Finally, Section VI presents conclusions. II. INVERTER COUPLED RESONATOR SECTIONS Fig. 1 shows the topologies of inverter coupled resonator sections consisting of one bandpass resonator and one or two bandstop resonators. The bandstop resonators are coupled to the bandpass resonator through the admittance inverters in order to achieve dual- and triple-passband filtering responses. For the inverter coupled resonator section in Fig. 1(a), the admittance viewed from the input/output is given by

Fig. 1. Topologies of inverter coupled resonator sections for: (a) dual- and (b) triple-passband filter.

(1)

where

(2) When approaches infinity, a transmission zero at will appear. If approaches the zero, two reflection zeros at ) will be produced. Similarly, for the inverter coupled resonator section, as shown in Fig. 1(b), the admittance viewed from the input/output is determined by

Fig. 2. Transmission responses of the inverter coupled resonator sections for dual- and triple-band filters.

where

(4)

(3)

Two transmission zeros at and three reflection zeros at may be produced. Fig. 2 shows transmission responses of the inverter coupled resonator sections for the dual- and the triple-passband filters, where the frequency is normalized to that

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Fig. 4. Synthesized responses of a dual-passband Chebyshev filter.

Fig. 3. (a) Topology and (b) low-pass prototype of the proposed dual-passband substrate integrated waveguide filter with Chebyshev response consisting of the inverter coupled resonator sections.

of transmission zero for the dual-passband filter, and the frequency is normalized to that of the middle reflection zero for the triple-passband filter. These two types of sections can be used as basic building blocks for the construction of dual- and the triple-passband filters, respectively. If the sections are adequately connected with reference to the constructive recombination criteria, dual- or triple-passband Chebyshev or quasi-elliptic responses will appear. In a similar way, the inverter coupled resonator section is also applicable in the design of multipassband filters. III. DUAL-PASSBAND SUBSTRATE INTEGRATED WAVEGUIDE FILTER WITH CHEBYSHEV RESPONSE

Assuming the filter has two passbands of and of the low-pass prototype circuit should be mapped into the passband right edge angular frequencies and . Similarly, should be mapped and . into the left edge angular frequencies Therefore, susceptance parameters and resonant frequenof the bandpass and bandstop resonators, and cies can be calculated from the last equation in (5) admittance and . Alternatively, analytical by using and can be obtained by the expressions for relationship between the roots and the coefficients of the last is an odd function and , equation in (5) because , and are roots of the last equation in (5) [16]. The coupling coefficients between the bandpass resonators can then be obtained by

(6) The coupling coefficient between the bandstop and bandpass resonators is given by

A. Circuit Synthesis Fig. 3 depicts the topology of the proposed dual-passband substrate integrated waveguide filter with a Chebyshev response, and its low-pass prototype with one type of reactive elements and admittance inverters. In order to map the low-pass prototype circuit onto its practical bandpass counterpart, the resonator admittance and inverter parameters of both circuits should be equalized, respectively. Since the inverter parameters in a practical filter are always assumed to be frequency independent, then

(7) The external quality factor of the first/last bandpass resonators is formulated as follows:

(8)

(5)

The above procedure is used to synthesize a dual-passband Chebyshev filter with the two passbands of 19.85–20.15 and 20.85–21.15 GHz and the return loss of 20 dB and three reflection zeros in each band. The design parameters are GHz, and GHz. Fig. 4 shows the synthesized responses of the dual-band bandpass filter.

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waveguide cavity bandpass–bandstop resonators with input and output, respectively. From the coupling curve, two split resonant-mode frequencies can easily be identified by two resonance peaks. The coupling coefficient between the two bandpass resonators can then be extracted by using the following expression [21]:

(11) where and are the higher and lower resonant frequencies, respectively. The coupling coefficient between the bandpass and bandstop resonators should be extracted by using the following relationship [21]: Fig. 5. Geometrical structure of the proposed dual-passband substrate integrated waveguide filter with Chebyshev response.

(12) B. Design and Realization by the Substrate Integrated Waveguide Technology Fig. 5 depicts a geometrical structure of the proposed dualpassband substrate integrated waveguide filter having a Chebysubstrate integrated waveguide cavity shev response. A bandstop resonator is coupled with a substrate integrated waveguide cavity bandpass resonator through a post-wall iris in the common post wall, which consists of the inverter cousubpled resonator section. The coupling between two strate integrated waveguide cavity bandpass resonators are also realized by the post-wall iris in the common post wall. The substrate integrated waveguide cavity bandpass first/last resonators are directly excited by 50- microstrip lines with coupling slots. The design procedure can be summarized as follows. First, substrate integrated waveguide bandthe initial sizes of stop and bandpass cavity resonators are determined by setting the resonant frequency to their design values by using the following formula [20]:

(9) where

where and are the higher and lower resonant frequencies, and are resonant frequencies of the bandpass and and bandstop resonators in the absence of coupling, respectively. In order to determine the external quality factor, numerical analysis with HFSS is also carried out with respect to the substrate integrated waveguide cavity bandpass resonator connected to 50- microstrip lines as its input and output. The coupling is controlled by changing the length of coupling slot with a fixed coupling slot width of 0.26 mm and a fixed post-wall iris is calculated width of 3.1 mm. The external quality factor by [21] (13) where is the frequency at which reaches its maximum value and is the bandwidth for which the attenuation of is 3 dB from its maximum value. Finally, the design parameters are used to estimate initial sizes of the entire filter. A fine tuning procedure is needed to optimize the entire filter in the design. Table I shows dimensions of the proposed dual-pass substrate integrated waveguide filter with a Chebyshev response whose photograph of fabrication is shown in Fig. 6. The full-wave simulation response is also shown in Fig. 7. C. Fabrication and Measurements

(10) and are the width and length of the substrate integrated waveguide cavity, respectively. and are the diameter of metallized via-holes and center-to-center pitch between two adjacent via-holes. is the light velocity in vacuum, and is the dielectric constant of substrate. To determine the internal coupling coefficients, a commercial full-wave electromagnetic simulator [Ansoft’s High Frequency Structure Simulator (HFSS)] is used to simulate a pair of coupled substrate integrated waveguide cavity bandpass resonators and a pair of coupled substrate integrated

The proposed dual-passband substrate integrated waveguide filter with a Chebyshev response was implemented on a substrate of Rogers RT/Duroid 5880, which has a height of 0.508 mm by our standard PCB process. The metallized via-holes of linear arrays have a diameter of 0.5 mm and a center-to-center pitch of 1 mm. An Anritsu 37397C vector network analyzer and Anritsu Wiltron 3680 K test fixture are used to measure the filter. A thru-reflect-line (TRL) calibration is performed in order to remove effects of the test fixture. Fig. 7 shows simulated and measured frequency responses. The measured passband return losses for both of the passbands are below 18 dB, while the minimum insertion losses are

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TABLE I DIMENSIONS OF THE PROPOSED DUAL-PASSBAND SUBSTRATE INTEGRATED WAVEGUIDE FILTER WITH CHEBYSHEV RESPONSE

Fig. 8. Synthesized responses of the triple-passband Chebyshev filter.

loss if the diameters of via-holes and the center-to-center pitch between the via-holes are correctly chosen [22]–[24]. There is a slight frequency shift by approximately 0.3% due to the tolerance of fabrication and parametric uncertainty of substrate. IV. TRIPLE-PASSBAND SUBSTRATE INTEGRATED WAVEGUIDE FILTER WITH CHEBYSHEV RESPONSE A. Circuit Synthesis

Fig. 6. Proposed dual-passband substrate integrated waveguide filter with Chebyshev response.

Except that the admittance in Fig. 3(a) should be replaced with that given by (3), the topology and low-pass prototype of the proposed triple-passband substrate integrated waveguide Chebyshev filter with three passbands and are similar to of those in Fig. 3. The same procedure is used to obtain the and resonant frequencies susceptance parameters of the bandpass and bandstop resonators, and from the last equation in (5) by using the admittances . Equation (6) is then used to between the bandpass calculate the coupling coefficients and between resonators, (7) for the coupling coefficients the bandpass and bandstop resonators, and (8) for the external quality factor. The design parameters of a triple-band Chebyshev filter with selected passbands of 19.90–20.15, 20.70–21.05, and 21.70–22.05 GHz, and a return loss of 20 dB and three transmission poles in each band are as follows: GHz, GHz, and GHz. Fig. 8 shows the synthesized responses of the triple-passband Chebyshev filter.

Fig. 7. Full-wave simulation and measurement responses of the proposed dualpassband substrate integrated waveguide filter with Chebyshev response.

approximately 1.37 and 1.1 dB over the first and second passbands, respectively. The insertion losses are attributed mainly to conductor and dielectric losses because the substrate integrated waveguide becomes radiationless or free from leakage

B. Design and Realization by the Substrate Integrated Waveguide Technology The geometrical structure of the proposed triple-passband substrate integrated waveguide filter with a Chebyshev response substrate integrated waveis depicted in Fig. 9. Two guide cavity bandstop resonators are simultaneously coupling substrate integrated waveguide cavity bandpass with a resonator by the post-wall irises in the common post walls,

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TABLE II DIMENSIONS OF THE PROPOSED TRIPLE-PASSBAND SUBSTRATE INTEGRATED WAVEGUIDE FILTER WITH CHEBYSHEV RESPONSE

Fig. 9. Geometrical structure of the proposed triple-passband substrate integrated waveguide filter with Chebyshev response.

which consist of the inverter coupled resonator section. The coupling between two substrate integrated waveguide cavity bandpass resonators are also realized by the post-wall iris in substrate integrated the common post wall. The first/last waveguide cavity bandpass resonators are directly excited by 50- microstip lines with coupling slots. The filter can be designed in the same manner as in substrate integrated Section III. A pair of coupled waveguide cavity resonators with input and output is used to determine the coupling coefficients by using (11) for synchronously tuned bandpass resonators and (12) for asynchronously substrate tuned bandpass and bandstop resonators. A integrated waveguide cavity bandpass resonator connected with 50- microstip lines as the input and output is used to determine the external quality factor by using (13). Table II gives dimensions of the entire filter whose photograph of fabrication and full-wave simulation response are described in Figs. 10 and 11, respectively. C. Fabrication and Measurements The proposed triple-passband substrate integrated waveguide filter with a Chebyshev response was also fabricated on the same substrate of Rogers RT/Duroid 5880 that has a height of 0.508 mm by our standard PCB process. The metallized viaholes of linear arrays were made with a diameter of 0.5 mm and a center-to-center pitch of 1 mm. The filter was measured by our Anritsu 37397C vector network analyzer and Anritsu Wiltron 3680K test fixture whose effects were removed by the TRL calibration. Fig. 11 depicts full-wave simulation and measurement responses. The measured passband return losses are all below 19 dB, while the minimum insertion losses are approximately 1.6, 0.9, and 0.85 dB over the first, second, and third passbands, respectively. Again, the insertion losses are attributed mainly to the conductor and dielectric losses because the leakage loss from the apertures between via-holes can be ignored [22]–[24]. The third passband has a slight frequency shift by approximately 0.3% and the bandwidths of the first and the second passbands have an offset by approximately 0.2%.

Fig. 10. Propose triple-passband substrate integrated waveguide filter with Chebyshev response.

V. DUAL-PASSBAND SUBSTRATE INTEGRATED WAVEGUIDE FILTER WITH QUASI-ELLIPTIC RESPONSE A. Circuit Synthesis In order to improve the selectivity of the dual-band filter, a dual-passband substrate integrated waveguide filter with a quasi-elliptic response that consists of the inverter coupled resonator section has been proposed. Fig. 12 shows the topology of the filter and its low-pass prototype with one type of reactive elements and admittance inverters. The negative cross-coupling between resonators 1 and 4 is introduced for generating a quasi-elliptic response. The filter synthesis follows the subsequent steps. First, the transfer function of the low-pass prototype can be calculated according to the specifications of each passband that has the same specifications and then generalized element values can be obtained by using the poles

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Fig. 13. Synthesized response of the dual-passband quasi-elliptic filter. Fig. 11. Full-wave simulation and measurement responses of the proposed triple-passband substrate integrated waveguide filter with Chebyshev response.

Fig. 14. Geometrical structure of the proposed dual-passband substrate integrated waveguide filter with quasi-elliptic response.

For our selected dual-passband quasi-elliptic filter with two passbands of 19.825–20.175 and 20.825–21.175 GHz and return loss of 25 dB and four reflection zeros in each band, the generalized element values of the low-pass prototype are given by . The design parameters can be obtained as follows: GHz, and shows the synthesized responses.

GHz. Fig. 13

Fig. 12. (a) Topology and (b) low-pass prototype of the proposed dual-passband substrate integrated waveguide filter with quasi-elliptic response consisting of the inverter coupled resonator sections.

B. Design and Realization by the Substrate Integrated Waveguide Technology

and zeros of the transfer function [25]. Next, the low-pass prototype can be mapped into its practical counterpart by the same procedure as discussed in Section III. Finally, the design parameters can be obtained.

Fig. 14 depicts the geometrical structure of the proposed dualpassband substrate integrated waveguide filter with a quasi-elliptic response. The filter has the same inverter coupled resonator section as that in Section III. A structure including a magnetic coupling iris and a balanced microstrip line with a pair

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TABLE III DIMENSIONS OF THE PROPOSED TRIPLE-PASSBAND SUBSTRATE INTEGRATED WAVEGUIDE FILTER WITH CHEBYSHEV RESPONSE

Fig. 15. Proposed dual-passband substrate integrated waveguide filter with quasi-elliptic response.

Fig. 16. Full-wave simulation and measurement responses of the proposed dual-passband substrate integrated waveguide filter with quasi-elliptic response.

of metallized via-holes is placed between substrate integrated waveguide cavity resonators 2 and 3 to invert the phase of the signal. Therefore, a mixed coupling including both positive and negative coupling that cancel with each other is produced. The structure can be optimized to ensure that the negative coupling is stronger than the positive coupling and a small amount of the negative coupling can be canceled by tuning the width of the magnetic post-wall iris. This structure fits well with the substrate integrated waveguide technology and provides an accurate negative coupling with an inductive post-wall iris [26], substrate integrated waveguide cavity [27]. The first/last bandpass resonators are directly excited by 50- microstip lines with coupling slots. The procedure given in Section III can be used to design the substrate integrated waveguide filter. A pair of coupled cavity resonators with input and output is used to determine the coupling coefficients by using (11) for synchronously tuned bandpass resonators and (12) for asynchronously tuned bandpass and bandstop resonators. A substrate integrated

waveguide cavity bandpass resonator connected with 50- microstip lines as the input and output is used to determine the external quality factor by using (13). Table III gives dimensions of the entire filter whose photograph of fabrication and full-wave simulation response are described in Figs. 15 and 16, respectively. C. Fabrication and Measurements The proposed dual-passband substrate integrated waveguide filter with a quasi-elliptic response was also realized on the same substrate of Rogers RT/Duroid 5880 that has a height of 0.508 mm by our standard PCB process. The metallized viaholes of linear arrays of were made with a diameter of 0.5 mm and a center-to-center pitch of 1 mm. The filter is measured by our Anritsu 37397C vector network analyzer and Anritsu Wiltron 3680K test fixture whose effects can be effectively removed by the TRL calibration. Fig. 16 presents full-wave simulation and measurement responses. The measured passband return losses are all below 16 dB, while the minimum insertion losses are approximately 2.2 and 1.2 dB in the first and second passbands, respectively. Again, the conductor and dielectric losses are responsible for the insertion losses because the leakage loss from the apertures between the via-holes can be ignored [22]–[24]. There is a frequency shift by approximately

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0.4 % because the cross-coupled filter is more sensitive to the fabrication tolerance. VI. CONCLUSION In this paper, the inverter coupled resonator section has been studied and discussed in detail, which has demonstrated that such a section can be used as the basic building block in the design of dual-passband or multipassband filters that are proposed and developed in this study. Those novel filters are easily realized by the substrate integrated waveguide technology for better device and system integration. The synthesis and design techniques of the proposed dual- and triple-passband filters with Chebyshev and quasi-elliptic responses have been given in detail. A number of practical examples including dual-passband Chebyshev, triple-passband Chebyshev, and dual-passband quasi-elliptic filters over -band have been investigated, designed, and implemented on a Rogers/RT Duroid 5880 substrate through our standard PCB process. The measured results, which agree with the simulated results, suggest that the proposed filters present attractive performances and compact size at a low cost. With the use of the substrate integrated waveguide scheme, those proposed filter structures can directly be integrated with other planar circuits. ACKNOWLEDGMENT The authors express their gratitude to J. Gauthier, R. Brassard, and S. Dube, all with the Poly-Grames Research Center, École Polytechnique de Montreal, Montreal, QC, Canada, for their technical assistance in the fabrication of the experimental prototypes. REFERENCES [1] M.-I. Lai and S.-K. Jeng, “Compact microstrip dual-band bandpass filters design using genetic-algorithm techniques,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 160–168, Jan. 2006. [2] L.-C. Tsai and C.-W. Hsue, “Dual-band bandpass filters using equallength coupled-serial-shunted lines and -transform technique,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1111–1117, Apr. 2006. [3] C. Quendo, E. Rius, and C. Person, “An original topology of dual-band filter with transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 1093–1096. [4] C. Quendo, E. Rius, A. Manchec, Y. Clavet, B. Potelon, J.-F. Favennec, and C. Person, “Planar tri-band filter based on dual behaviour resonator (DBR),” in Proc. Eur. Microw. Conf., Oct. 1995, pp. 269–272. [5] H. Y. A. Yim and K.-K. M. Cheng, “Novel dual-band planar resonator and admittance inverter for filter design and applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 2187–2190. [6] M.-H. Hsu and J.-F. Huang, “Annealing algorithm applied in optimum design of 2.4 GHz and 5.2 GHz dual-wideband microstrip line filters,” IEICE Trans. Electron., vol. E88-C, pp. 47–56, Jan. 2005. [7] A. A. A. Apriyana and Y. Zhang, “A dual-band BPF for concurrent dual-band wireless,” in Proc. Electron. Packag. Technol., Oct. 2003, pp. 145–149. [8] J.-T. Kuo and H.-S. Cheng, “Design of quasi-elliptic function filters with a dual-passband response,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 10, pp. 472–474, Oct. 2004. [9] J.-T. Kuo, T. H. Yeh, and C. C. Yeh, “Design of microstrip bandpass filters with a dual-passband response,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1331–1336, Apr. 2005. [10] J.-F. Huang and M.-H. Hsu, “Design and implementation of high performance and miniaturization of SIR microstrip multi-band filters,” IEICE Trans. Electron., vol. E88-C, pp. 1420–1428, Jul. 2005. [11] S. Sun and L. Zhu, “Compact dual-band microstrip bandpass filter without external feeds,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 10, pp. 644–646, Oct. 2005.

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[12] C.-F. Chen, T.-Y. Huang, and R.-B. Wu, “Design of dual- and triple-passband filters using alternately cascaded multiband resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 9, pp. 3550–3558, Sep. 2006. [13] J. Lee, M. S. Uhm, and I.-B. Yom, “A dual-passband filter of canonical structure for satellite applications,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 271–273, Jun. 2004. [14] J. Lee, M. S. Uhm, and J. H. Park, “Synthesis of a self-equalized dualpassband filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 256–258, Apr. 2005. [15] P. Lenoir, S. Bila, F. Seyfert, D. Baillargeat, and S. Verdeyme, “Synthesis and design of asymmetrical dual-band bandpass filters based on equivalent network simplification,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 3090–3097, Jul. 2006. [16] G. Macchiarella and S. Tamiazzo, “Design techniques for dual-passband filters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3265–3271, Nov. 2005. [17] M. Mokhtaari, J. Bornemana, K. Rambabu, and S. Amari, “Coupling matrix design of dual and triple passband filters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3940–3945, Nov. 2006. [18] K. Wu, “Integration and interconnect techniques of planar and nonplanar structures for microwave and millimeter-wave circuits-current status and future trend,” in Proc. Asia–Pacific Microw. Conf., Taiwan, R.O.C., Dec. 2001, pp. 411–416. [19] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [20] Y. Cassivi, L. Perregrini, P. Arcioni, M. Bressan, K. Wu, and G. Conciauro, “Dispersion characteristics of substrate integrated rectangular waveguide,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 2, pp. 333–335, Feb. 2002. [21] J.-S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Applications. New York: Wiley, 2001, pp. 257–258. [22] D. Deslandes and K. Wu, “Accurate modeling, wave mechanisms, and design considerations of a substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2516–2526, Jun. 2006. [23] Y.-L. Zhang, W. Hong, K. Wu, J.-X. Chen, and H.-J. Tang, “Novel substrate integrated waveguide cavity filter with defected ground structure,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1280–1287, Apr. 2005. [24] Y. Cassivi, L. Perregrini, K. Wu, and G. Conciauro, “Low-cost and high- millimeter-wave resonator using substrate integrated waveguide technique,” in Proc. Eur. Microw. Conf., Milan, Italy, Sep. 2002, pp. 1–4. [25] S. Amari, “Synthesis of cross-coupled resonator filters using an analytical gradient-based optimization technique,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1559–1564, Sep. 2000. [26] C.-Y. Chang and W.-C. Hsu, “Novel planar, square-shaped, dielectric-waveguide, single-, and dual-mode filters,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2527–2536, Nov. 2002. [27] X.-P. Chen and K. Wu, “Substrate integrated waveguide cross-coupled filter with negative coupling structure,” IEEE Trans. Microw. Theory Tech., to be published.

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Xiao-Ping Chen was born in Hubei Province, China, in 1974. He received the Ph.D. degree in electrical engineering from Huazhong University of Science and Technology, Wuhan, China, in 2003. From 2003 to 2006, he was a Post-Doctoral Researcher with the State Key Laboratory of Millimeter-Waves, Radio Engineering Department, Southeast University, Nanjing, China, where he was involved with the advanced microwave and millimeter-wave circuit for communication system applications under the financial support of the China Post-Doctoral Fund. He is currently a Post-Doctoral Fellow with the Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montreal, Montreal, QC, Canada. He has authored or coauthored over 20 papers in referred journals and conference proceedings. His research interests are dielectric resonator filters and antennas, microwave measurement, and substrate integrated circuits (SICs) for microwave and millimeter-waves applications.

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Ke Wu (M’87–SM’92–F’01) is Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering with the École Polytechnique de Montréal, Montréal, QC, Canada. He also holds a Cheung Kong endowed chair professorship (visiting) with Southeast University, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China, and the City University of Hong Kong. He has been the Director of the Poly-Grames Research Center. He has authored or coauthored over 515 referred papers and several books/book chapters. He has served on the Editorial/Review Boards of numerous technical journals, transactions, and letters, including being an Editor and Guest Editor. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors. He is also interested in the modeling and design of microwave photonic circuits and systems. Dr. Wu is a Fellow of the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is a member of the Electromagnetics Academy, Sigma Xi, and URSI. He has held key positions in and has served on various panels and international committees including the chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. He

is currently the chair of the joint IEEE Chapters of the Microwave Theory and Techniques Society (MTT-S)/Antennas and Propagation Society (AP-S)/Lasers and Electro-Optics Society (LEOS), Montréal, QC, Canada. He is an elected IEEE MTT-S Administrative Committee (AdCom) member for 2006–2009 and serves as the chair of the IEEE MTT-S Transnational Committee. He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award.

Zhao-Long Li (S’01) was born in Zhenjiang, China. He received the B.S. degree in radio science from Lanzhou University, Lanzhou, China, in 1998, and the M.S. degree in radio engineering from Southeast University, Nanjing, China, in 2002. Since 2002, he has been with Département de Génie Électrique, Poly-Grames Research Center, École Polytechnique de Montreal, Montreal, QC, Canada. His current research interests involve integrated radar front-end design, antenna arrays, microwave/millimeter-wave circuit design, and monolithic microwave integrated circuit design.

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A Physical Model of Solenoid Inductors on Silicon Substrates Chih-Ming Tai and Chien-Neng Liao

Abstract—In this paper, a physical model is presented to predict the frequency-dependent characteristics of solenoid-type inductors on standard silicon substrates. The model considers the skin-depth effect in the conductor, interwinding capacitance, parasitic capacitance between the conductor and substrate, substrate resistance, and substrate capacitance of a solenoid inductor on the silicon substrate, which are all computed based on the inductor’s geometric dimensions and related material properties. Surface-micromachined inductors of various geometries have been tested to validate the physical model and found a satisfactory consistency between the measured results and the theoretical predictions in the multigigahertz frequency range. It is also suggested that the increase of the solenoid aspect ratio is beneficial in enhancing quality factors of solenoid inductors on the silicon substrate at a high frequency. Index Terms—Inductor model, quality factor, self-resonance, silicon, solenoid inductor, substrate loss, surface micromachining.

I. INTRODUCTION HE INDUCTOR is one of the key components in many RF devices such as bandpass filters, voltage-controlled oscillators, and amplifier matching networks [1]–[3]. A growing demand for compact mobile communication applications motivates the need for miniaturized inductors, which, in general, take the largest system space as compared to other passive components. Indeed, direct integration of inductors with RF integrated circuits (RFICs) is an even more favorable option in terms of system size, power loss, and reliability considerations [4]. Although modern silicon technology offers the advantage of mature processing techniques, silicon-based on-chip inductors usually suffer serious quality factor degradation at high frequency due to capacitive coupling and ohmic loss in silicon substrates [5]. In recent years, several approaches have been proposed to enhance quality factors of silicon-based inductors. Yue and Wong have inserted an additional ground shielding layer in between the inductor and the substrate to minimize the substrate loss [6]. Huo et al. have used benzocyclobutene (BCB) as an inter-layer dielectric to reduced capacitive coupling [7]. Moreover, a suspended inductor structure has also been proposed via bulk micromachining to minimize the substrate loss [8]. Nevertheless, these methods either have complex process sequences

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Manuscript received January 17, 2007; revised June 13, 2007. This work was supported by the Ministry of Economic Affairs, Technology Development Program for Academia under Grant 91-EC-17-A-08-S1-003 and by National Nano Device Laboratories under Grant NDL-94S-C096. The authors are with the Department of Materials Science and Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.910069

Fig. 1. (a) Scanning electron microscope micrograph. (b) Process sequences of the solenoid inductors on silicon substrates.

Fig. 2. Schematic diagram of an on-chip solenoid inductor.

or are uncommon options for current silicon process technology, and hence, are not widely adopted in RFIC manufacturing. For conventional planar spiral inductors, the way to raise the inductance-to-area ratio is to increase the number of metal turns at the cost of increasing parasitic capacitance and decreasing quality factor [9]. Besides, the magnetic field induced current crowding effect in the inner coils will raise the inductor resistance markedly, leading to serious degradation of the quality factor at high frequency [10]. As compared to conventional spiral inductors, solenoid inductors have small parasitic coupling with the substrate because they only have a

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TABLE I CALCULATION FORMULAS OF THE INDUCTIVE, RESISTIVE, AND CAPACITIVE COMPONENTS IN THE EQUIVALENT CIRCUIT OF AN ON-CHIP SOLENOID INDUCTOR

portion of metal wires in contact to the silicon substrate [11]. A small parasitic coupling will minimize degradation of the quality factor and raise the self-resonance frequency of on-chip inductors. Therefore, solenoid inductors are expected to have larger quality factors than planar spiral inductors within the same chip area at high frequency [12], [13]. Precise modeling of the on-chip inductor’s high-frequency properties is not a trivial task owing to many parasitic contributions involved in the RF frequency domain. A nine-element single- circuit is the most commonly used equivalent-circuit model for on-chip inductors, which consists of various inductive, resistive, and capacitive components [14]. A number of modifications have been made on the lumped-element model to accurately simulate the high-frequency characteristics of on-chip inductors [15]. Although the full-wave electromagnetic (EM) analysis techniques can be used to model the high-frequency properties of on-chip inductors with reasonable accuracy, it is very time consuming and lacking in insight into the relation between inductor physical sizes and device

Fig. 3. Single- equivalent circuit for the solenoid inductor on silicon substrate.

performance. In our previous study [16], these equivalent-circuit parameters of inductors were obtained by using Agilent’s IC-CAP device modeling software. The extracted parameters may not be directly correlated with the dimensions and material properties of the inductor. Therefore, it would be beneficial for the designers if there is a physical model available for solenoid inductors. Yue et al. presented a physical model for planar

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TABLE II NUMERICAL RESULTS OF THE CALCULATED CIRCUIT COMPONENTS IN A SOLENOID INDUCTOR AND THE MATERIAL PROPERTIES AND PHYSICAL DIMENSIONS EMPLOYED

spiral inductors, which correlates the inductor physical sizes directly to each circuit element, and finds satisfactory consistency between the experimental data and the modeled results in the multigigahertz range [5]. It is noted that the major differences between the planar spiral inductor and the solenoid inductor are the inductive, resistive, and capacitive contributions of the “via” segments, and the unequal conductor-substrate capacitance for the upper and bottom levels of the conductors in the inductor structure. In this paper, we have developed a physical model for on-chip solenoid inductors, which provides a feasible means of predicting the performance of the solenoid inductor simply based on their physical dimensions and material properties. The solenoid inductors with different geometries were fabricated using surface micromaching techniques and measured up to 40 GHz to validate the model experimentally. In addition, the geometric dependence on the performance of on-chip solenoid inductors was also investigated. II. EXPERIMENT Solenoid inductors were fabricated on standard silicon wafers by surface micromachining techniques. Multiple lithographic processes were performed to make photoresist molds followed by copper electroplating in forming a 3-D solenoid inductor structure. Fig. 1 shows a scanning electron microscope image and the process sequences of the solenoid inductors. Firstly, a orientation standard 4–in p-type 1- cm silicon wafer of was spin coated with a layer of polyimide (dielectric constant ) for electrical isolation, whose thickness was measured to be 15 m after the polyimide was thermally cured at 350 C for 1 h. A 100-nm-thick copper film was sputtered on the wafer, serving as a seeding layer for subsequent electroplating process. The first photo mask was used to pattern the photoresist mold of the bottom copper wires. A copper electroplating process was then applied to form the 10- m-thick patterned copper structure. Without removal of the first photoresist layer, the second and third photolithography and electroplating processes were carried out in sequence for making the 40- m-thick copper vias and the 10- m-thick top copper wires. Finally, the solenoid structure was successfully realized on the silicon wafer by stripping off all three layers of photoresist and etching away the exposed copper seed layer. All the processes have been completed below 350 C without substrate etch, making the technology fully CMOS compatible. The aspect ratio (AR) of the

solenoid inductor is defined as H/W, as shown in Fig. 1. The solenoid inductors fabricated were measured from 0.2 to 40 GHz using a vector network analyzer (HP 8510C) and two coplanar probes. A short-open-load-thru (SOLT) calibration procedure has been performed up to the probe tips prior to the inductor measurements. An open structure that has the signal and ground pads only was measured to deduct the contribution of the probe pads. The inductance and quality factor were then obtained from the deembedded -parameters as follows [17]: (1) (2) and where of the admittance

denote the imaginary and real parts , respectively.

III. MODELING OF ON-CHIP SOLENOID INDUCTORS Fig 2 shows a schematic of a solenoid inductor on the silicon substrate, illustrating the inductor’s physical parameters and all the inductive, resistive, and capacitive components. All these components can be calculated from the material properties and geometric dimensions of the solenoid inductor according is to the formulas listed in Table I. The series inductance computed by adapting the Greenhouse and Grover algorithm of each [18]–[20], which accounts for the self-inductance between line segment and the mutual inductance each pair of line segments in the coil, where denotes the length, width, and thickness of the line segment , respecrepresents the mutual inductance of the two parallel tively. denotes the mutual inductance of the upper and segments. bottom segments at an angle of . The series resistance of the inductor is strongly dependent on frequency due to the skin and proximity effects [21]. The skin effect forces the electric current to flow within the region near the conductor surface at high frequency. We can assume the electric current flowing in the periphery of the conductor with a characteristic skin depth [22]. On the other hand, the magnetic field induced by the current in the neighboring conductor may result in current crowding phenomena in the metal line, which is called the proximity effect [10]. The series resistance can be equated to be

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where is a parameter in the order of unity [18]. We found that the value of 1 would best fit the experimental data at low frequencies for the solenoid inductors investigated. The interis calculated by summing the capacwinding capacitance itance of each pair of line segments and vias. The capacitive coupling between the inductor and silicon substrate is attributed and , which denote the couto two components, i.e., plings of the upper conductor—silicon substrate and the bottom conductor—silicon substrate, respectively. Finally, the substrate and , resistance and capacitance are also modeled by respectively, which depend on silicon substrate resistivity, dielectric constant of insulating layer, and area of the bottom conductor in contact to the silicon substrate. The equivalent circuit of the solenoid inductor is described by a single circuit shown and are further in Fig. 3. The parasitic components reduced into and based on fundamental circuit theory

(3) Finally, the quality factor and inductance of the solenoid inductor are equated according to the above equivalent-circuit model [5], shown in (4) and (5) at the bottom of this page. IV. RESULTS AND DISCUSSION Solenoid inductors with various geometric sizes were fabricated and measured to verify the physical model in the frequency range from 0.2 to 40 GHz. The inductor’s physical dimensions, related material properties, and all the calculated inductive, resistive, and capacitive components were listed in Table II. Based on the AR definition in Fig. 1, the AR values of the solenoid inductors can be approximately calculated to be (6) Fig 4 shows the quality factor and inductance of these inductors as functions of frequency. The experimental results were found to reasonably agree with the theoretical predictions, except the inductance results in the frequency regime close to the self-resonance point. The deviation of the measured inductance is possibly attributed to the capacitive effect induced in-

substrate loss factor

Fig. 4. Plots of: (a) quality factor and (b) inductance with respect to frequency for the solenoid inductors with different AR values.

ductance peaking near resonance. Since the measurements appear to be more peaky than the modeled ones for the and data, it is reasonably expected that the substrate losses are less than predicted for the high AR solenoid inductors. Indeed, the calculated inductance in the physical model only accounts for the self- and mutual-inductance of the metal wires, and the eddy-current effect in the silicon substrate has been neglected. The approximation may be justified if the inductors are not applied in a very high-frequency

self-resonance factor (4)

(5)

TAI AND LIAO: PHYSICAL MODEL OF SOLENOID INDUCTORS ON SILICON SUBSTRATES

Fig. 5. Plots of: (a) substrate loss factor and self-resonance factor and (b) distributed components R and C with respect to frequency for the solenoid inductors with increasing AR values from 0.1 to 0.5.

range. Therefore, it is not surprising that the modeled inductance is somewhat higher than the experimental data around the self-resonance frequency, as shown in Fig. 4(b). The inmultiplied by subductor quality factor is equal to strate loss and self-resonance factors according to (4). Both factors are mainly dependent on the inductor’s distributed paramand . Fig. 5 shows the frequency dependence of the eters substrate loss factor, self-resonance factor, and distributed paand for the solenoid inductors with increasing rameters AR values from 0.1 to 0.5. The substrate loss factor is equal to and has the reciprocal is apdependence of the fourth power of frequency since proximately proportional to . Therefore, the substrate loss factor decreased drastically with increasing frequency. However, the inductors with higher AR values appeared to have less degradation of the quality factor at the same frequency. Due to the fixed via height, the solenoid inductors with higher AR value means the shorter solenoid width, and in turn, the smaller area, increases and are in contact with the silicon substrate. Thus, decreases with increasing AR values from 0.1 to 0.5 owing

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Fig. 6. Plots of the modeled quality factors of the AR = 0:1 and AR = 10 solenoid inductors with respect to frequency. (a) Physical model. (b) EM simulations.

to the enlarged and the reduced and , as shown in Fig. 5(b). , Since the quality factor is proportional to the term it is rather difficult to differentiate the AR effect on the properand ties of the solenoid inductors if they have different values. Thus, we compute the quality factors of the and inductors with increasing frequency according to the physical model, as shown in Fig. 6(a). Both the solenoid inductors have the same physical dimensions, and hence, the simand values. Due to the lack of experimental results of ilar the high-AR solenoid inductor, an EM simulation analysis has also been performed on the solenoid inductors using a 2.5-D EM solver (Sonnet), as shown in Fig. 6(b). It is found that the EM simulation results are slightly lower than the predictions from the physical model, but the trend remains the same in the frequency range of interest. The superior RF performance of the high-AR inductor should be attributed to the reduced parasitic coupling between the conductor and silicon substrate, which are reflected by a lesser degree of quality degradation caused by the substrate loss and self-resonance effects, as shown in Fig. 7. In addition, the only way to increase inductance without occupying

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), which is mainly the corresponding low-AR one ( attributed to the reduced coupling between the inductor and siland small . It has also been sugicon substrate, i.e., large gested that the high-AR inductors can be encapsulated using a low- dielectric for supporting and protecting the inductor without substantial degradation of the quality factor.

Fig. 7. Plots of the substrate loss factor and self-resonance factor with respect to frequency for the inductors with AR = 0:1 and 10.

much precious chip area is to increase the via height of the solenoid inductors. Hence, we may conclude that the solenoid inductor with large AR value is a preferred option in terms of the quality factor and inductance density considerations. Although the high-AR solenoid inductor has the desired high-frequency performance, there are still some technical problems associated with the mechanical support and protection of the high-AR structure. One feasible solution is to encapsulate the entire structure with spin-on dielectric. A low-dielectric-constant polyimide, for example, is one of the possible candidates that are currently used in integrated circuit (IC) packaging. However, the parasitic capacitance induced by encapsulation and its influence on the inductor characteristics is the should be evaluated. The capacitive component most likely affected parameter in the lumped-element circuit model of the encapsulated inductor. The immediate impact on inductor performance is the decline of of the increased the self-resonance factor according to (4). Nevertheless, the self-resonance factor is less significant than the substrate loss factor, except at the frequency near the self-resonance regime, as shown in Fig. 5(a). Therefore, it is reasonably expected that the encapsulation material may have little impact on the inductor characteristics in a multigigahertz frequency range. Choi et al. have reported on the effect of encapsulation on the suspended spiral inductor using a BCB spin-on dielectric and found only a slight degradation of the quality factor [23]. Therefore, it is a technically feasible option to preserve the integrity of the high-AR solenoid inductors without sacrificing inductor performance by adapting the encapsulating techniques. This is an essential requirement for the proposed solenoid inductors to be implemented on RFICs. V. CONCLUSION In this study, a physical model for solenoid-type inductors on silicon substrates has been presented to predict the inductor performance in the multigigahertz frequency range. The modeled results agree reasonably well with the measured data for the solenoid inductors with various AR values. By assuming the same physical dimensions of the solenoid inductors, the high-AR one ) demonstrates a 25% higher peak quality factor than (

REFERENCES [1] M. J. Yu, H. H. Wu, and Y. J. Chan, “900 MHz/1.8 GHz thin-film microwave bandpass filter,” in Proc. Asia–Pacific Microw. Conf., Dec. 1999, pp. 686–689. [2] E. C. Park, S. H. Baek, T. S. Song, J. B. Yoon, and E. Yoon, “Performance comparison of 5 GHz VCOs integrated by CMOS compatible high Q MEMS inductors,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 721–724. [3] Y. E. Chen, Y. K. Yoon, J. Laskar, and M. Allen, “A 2.4 GHz integrated CMOS power amplifier with micromachined inductors,” in IEEE MTT-S Int. Microw. Symp. Dig, May 2001, pp. 523–526. [4] J. W. M. Rogers, V. Levenets, C. A. Pawlowicz, N. G. Tarr, T. J. Smy, and C. Plett, “Post-processed Cu inductors with application to a completely integrated 2-GHz VCO,” IEEE Trans. Electron Devices, vol. 48, no. 6, pp. 1284–1287, Jun. 2001. [5] C. P. Yue, C. Ryu, J. Lau, T. H. Lee, and S. S. Wong, “A physical model for planar spiral inductors on silicon,” in Int. Electron Devices Meeting Tech. Dig, Dec. 1996, pp. 155–158. [6] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Si-based RF ICs,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, May 1998. [7] X. Huo, K. J. Chen, and P. C. H. Chan, “Silicon-based high-Q inductors incorporating electroplated copper and low-K BCB dielectric,” IEEE Electron Device Lett., vol. 23, no. 6, pp. 520–522, Sep. 2002. [8] H. Jiang, Y. Wang, J. L. A. Yeh, and N. C. Tien, “On chip spiral inductors suspended over deep copper-lined cavities,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2415–2423, Dec. 2000. [9] C. B. Sia, B. H. Ong, K. W. Chan, K. S. Yeo, J. G. Ma, and M. A. Do, “Physical layout design optimization of integrated spiral inductors for silicon-based RFIC applications,” IEEE Trans. Electron Devices, vol. 52, no. 12, pp. 2559–2567, Dec. 2005. [10] W. B. Kuhn and N. M. Ibrahim, “Analysis of current crowding effects in multiturn spiral inductors,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 31–39, Jan. 2001. [11] J. B. Yoon, B. K. Kim, C. H. Han, E. Yoon, and C. K. Kim, “Surface micromachined solenoid on-Si and on-glass inductors for RF applications,” IEEE Electron Device Lett., vol. 20, no. 9, pp. 487–489, Sep. 1999. [12] C. S. Lin, Y. K. fang, S. F. Chen, C. Y. Lin, M. C. Hsieh, C. M. Lai, T. H. Chou, and C. H. Chen, “A deep submicrometer CMOS process compatible high-Q air-gap solenoid inductor with laterally laid structure,” IEEE Electron Device Lett., vol. 26, no. 3, pp. 160–162, Mar. 2005. [13] C. H. Chen, Y. K. Fang, C. W. Yang, and C. S. Tang, “A deep submicron CMOS process compatible suspending high-Q inductor,” IEEE Electron Device Lett, vol. 22, no. 11, pp. 522–523, Nov. 2001. [14] H. Ronkainen, H. Kattelus, E. Tarvainen, T. Riihisaari, M. Andersson, and P. Kuivalainen, “IC compatible planar inductors on silicon,” Proc. Inst. Elect. Eng.—Circuits, Devices, Syst., vol. 144, no. 1, pp. 29–35, Feb. 1997. [15] A. C. Watson, D. Melendy, P. Francis, K. Hwang, and A. Weisshaar, “A comprehensive compact-modeling methodology for spiral inductors in silicon-based RFICs,” IEEE Trans. Microw. Theory Tech, vol. 52, no. 3, pp. 849–857, Mar. 2004. [16] C. M. Tai and C. N. Liao, “High-quality solenoid inductors on silicon wafers,” in Proc. Int. Commun., Circuits, Syst. Conf., Jun. 2006, pp. 866–869. [17] J. B. Yoon, Y. S. Choi, B. I. Kim, Y. Eo, and E. Yoon, “CMOS-compatible surface-micromachined suspended-spiral inductors for multigigahertz silicon RF ICs,” IEEE Electron Device Lett., vol. 23, no. 10, pp. 591–593, Oct. 2002. [18] H. M. Greenhouse, “Design of planar rectangular microelectronic inductors,” IEEE Trans. Parts, Hybrids, Packag., vol. PHP-10, no. 2, pp. 101–109, Jun. 1974. [19] Y. K. Koutsoyannopoulos and Y. Papananos, “Systematic analysis and modeling of integrated inductors and transformers in RF IC design,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 8, pp. 699–713, Aug. 2000.

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[20] F. W. Grover, Inductance Calculations. New York: Van Nostrand, 1946. [21] Y. S. Choi and J. B. Yoon, “Experimental analysis of the effect of metal Thickness on the quality factor in integrated spiral inductors for RF ICs,” IEEE Electron Device Lett., vol. 25, no. 2, pp. 76–79, Feb. 2004. [22] R. F. Dana and Y. L. Chow, “The current distribution and AC resistance of a microstrip structure,” IEEE Trans. Microw. Theory Tech, vol. 38, no. 9, pp. 1268–1277, Sep. 1990. [23] Y. S. Choi, E. Yoon, and J. B. Yoon, “Encapsulation of the micromachined air-suspended inductors,” in IEEE MTT-S Int. Microw. Symp. Dig, Jun. 2003, pp. 1637–1640. Chih-Ming Tai received the B.S. and M.S. degrees in mechanical engineering from National Central University, Chung-Li, Taiwan, R.O.C., in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree at National Tsing-Hua University, Hsinchu, Taiwan, R.O.C.

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Chien-Neng Liao received the B.S. degree from National Tsing-Hua University, Hsinchu, Taiwan, R.O.C., in 1990, the M.S. degree from the University of Texas at Austin, in 1995, and the Ph.D. degree from the University of California at Los Angeles (UCLA), in 1999, all in materials science and engineering. In 2001, he joined the faculty of the Department of Materials Science and Engineering, National Tsing-Hua University, Hsinchu, Taiwan, R.O.C., as an Assistant Professor, and became an Associate Professor in 2006. His research interests include the area of passive RF microelectronics.

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Half Mode Substrate Integrated Waveguide 180 3-dB Directional Couplers Bing Liu, Wei Hong, Member, IEEE, Yan Zhang, Hong Jun Tang, Xiaoxin Yin, and Ke Wu, Fellow, IEEE

Abstract—In this paper, two types of half mode substrate integrated waveguide 180 couplers are proposed, the phase-shifting structures of which are composed of metallic via fins or etched air slots. Each coupler maintains the good performance of the substrate integrated waveguide coupler with a half reduction in size and a 180 phase difference between through and coupling -, and -bands ports. Prototypes working in the -, -, are designed and fabricated using the standard printed circuit board process. The consistency between the measured data and simulated results shows that the proposed couplers have the merits of low profile and good performance. Index Terms—Half mode substrate integrated waveguide, 180 directional coupler, substrate integrated waveguide.

I. INTRODUCTION Fig. 1.

T

HE 180 couplers with input “sum” and “difference” ports are required for many applications where the corresponding incident signals need to be divided equally between the output ports, being in-phase and out-of-phase, respectively, whereas the conventional rectangular waveguide 180 couplers are highly costly and very difficult to integrate into planar circuits [1], [2]. Recently some new planar waveguide structures, called substrate integrated waveguide, as well as laminated waveguide or post-wall waveguide, were proposed and applied to the designs of high-quality microwave and millimeter-wave components [3]–[18]. However, the sizes of substrate integrated waveguide couplers may conflict with the integrations of low-profile components. Lately, a new guided wave structure called the half model substrate integrated waveguide was proposed [19], which can reduce the size of the substrate integrated waveguide device nearly in half without deteriorating the performance [20]. Based on the half mode substrate integrated waveguide technology and narrow-wall coupling mechanism, two types of compact 180 couplers are proposed in this paper. Manuscript received October 24, 2006; revised July 30, 2007. This work was supported by the National Natural Science Foundation of China under Grant 60621002. B. Liu is with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). W. Hong, Y. Zhang, H. J. Tang, and X. Yin are with the State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]). K. Wu is with the Department of Electrical Engineering, Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC, Canada H3V 1A2 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.909749

E -field distributions of dominant mode in both structures. II. PRINCIPLE

The advantages of the half mode substrate integrated waveguide over the substrate integrated waveguide are mainly smaller size and lower insertion loss. The reduction in size may lead to a lower dielectric and metal loss. Meanwhile, less vias can avoid unwanted discontinuity. When a substrate integrated waveguide works only in the dominant mode, the tangent electric field is equal to the maximum value and normal magnetic field is equal to zero at the symmetrical plane along the propagation direction so the symmetrical plane can be equivalent to a magnetic wall. With this character, the substrate integrated waveguide can be bisected with a fictitious magnetic wall and each half after cutting can almost keep the half field mode, actually a half-width mode. distribution of the Generally, the width-to-height ratio of a half mode substrate integrated waveguide may exceed 10 so the power is nearly fully constrained inside the half-width mode structure and the power leakage from the cutting side aperture is neglectable. As for a conventional rectangular metallic waveguide, if it experiences the same operation, most of power will escape from the cutting side for its small width-to-height ratio and lack of dielectric. A typical -field distribution of the dominant mode in the half mode substrate integrated waveguide is illustrated in Fig. 1 in comparison with that in the substrate integrated waveguide. III. DESIGN CONSIDERATION In this study, two types of half mode substrate integrated -, and waveguide 180 couplers working in the -, -, -bands are investigated and the prototypes are shown in Fig. 2(a)–(c), where the areas enclosed by the white dashed

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LIU et al.: HALF MODE SUBSTRATE INTEGRATED WAVEGUIDE 180 3-dB DIRECTIONAL COUPLERS

Fig. 3.

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E -field distributions in the proposed X - and Ku-band couplers.

Fig. 4. Configurations of the proposed X - and Ku-band couplers.

TABLE I OPTIMIZED COUPLER PARAMETERS

Fig. 2. Proposed C -, X -, Ku-, and Ka-band couplers.

line are the proposed 180 couplers. Each of them consists of two half mode substrate integrated waveguides with a common narrow wall consisting of metallic vias. The phase-shifting structure is implemented by either inserting metallic via fins in the through branch or etching slots on the top metal plate of the coupling branch, , just as shown in Fig. 3, which also illus-band trated the simulated -field distributions in - and couplers, and ports 1–4 are the input, through, coupling, and isolating ports, respectively. Fig. 4 shows the configurations of the proposed - and -band couplers, and the optimized geometry parameters are listed in Table I. As discussed above, the width of a half mode substrate integrated waveguide is half that of the corresponding substrate integrated waveguide so the width of the substrate integrated waveguide working in the same frequency range should be determined at first according to the design consideration discussed in [18]. The width of the transition between the half mode substrate integrated waveguide and the and , as shown in Fig. 4) is also microstrip (

TABLE II THICKNESS OF SUBSTRATE FOR COUPLERS

nearly one-half that of the substrate integrated waveguide so the design of the transitions is similar to the discussions in [9]. and , as As for the lengths of coupling slots ( shown in Fig. 4), the initial values for optimization can be set like those of the rectangular waveguide narrow-wall couplers, which have the same coupling ratio and working frequency band. The design procedure of the phase-shifting structures will be discussed in Section IV.

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TABLE III MEASURED PERFORMANCES OF THE PROPOSED COUPLERS

IV. SIMULATED AND EXPERIMENT RESULTS A. Measured and Simulated Results The printed circuit board (PCB) layouts of the proposed couplers shown in Fig. 2 are fabricated on the substrates with and , except for the -band case, of 2.65. The corresponding thicknesses of the which has a proposed couplers are listed in Table II. All the proposed structures were simulated with Ansoft’s High Frequency Structure Simulator (HFSS), and a vector network analyzer was used for the measurements. In order to obtain two-port measurement, the unconnected ports are loaded with matched loads. The measured data shown below exclude the insertion losses of the half mode substrate integrated waveguide-to-microstrip transitions, the 2.4–3.5-mm adapters, and the subminiature A (SMA) connectors. All these extra insertion losses for each proposed coupler are calibrated with a segment of through microstrip, parts of which are shown on the right of Fig. 2(a) and (c). The length of the through microstrip is equal to the summation of the corresponding microstrip lengths between SMA connectors and the couplers. The analysis of the measured data is listed in Table III, followed by the performances of -band couplers, plotted in detail in Figs. 5–10 for illustration. B. Discussions About the Results When analyzing the -parameters performance of the -band coupler (Type II), we found some deviations between the measured data and simulated results, which, in our opinion, should be caused by the variation of the dielectric constant so further simulations were run to find out the truth. The corresponding simulated and measured -parameters and phase difference are compared in Figs. 11–13, where we can find that when the is changed to 2.11 in the simulation, the simulated results matched the measured data much better than the original so the actual value of the of the 1-mm-thick substrate is probably much closer to 2.11 rather than 2.2. The couplers working in other bands are fabricated on PCBs of Rogers 5880, which have good qualities.

Fig. 5.

S

and

S

performance (

Ka-band type I).

Fig. 6.

S

and

S

performance (

Ka-band type I).

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Fig. 7. Phase difference (P ) performance (Ka-band type I).

Fig. 10. Phase difference (P ) performance (Ka-band type II).

Fig. 8. S

and S

performance (Ka-band type II).

Fig. 11. Effects of different " on S

and S .

Fig. 9. S

and S

performance (Ka-band type II).

Fig. 12. Effects of different " on S

and S .

In this study, we achieved 180 phase differences between the coupled and through ports by using the phase-shifting structure mentioned above. Actually, the configurations of the fins

and slots will greatly affect the performances of the couplers and isolation , as well as the on the return loss output phase differences, especially in the -band. Fig. 14

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Fig. 13. Effects of different " on phase difference.

Fig. 15. Simulated phase difference corresponding to different H

.

Fig. 14. Configuration of the fins of type-I Ka-band coupler.

shows the configuration of the fins used in the -band coupler (type I) and the parameters to be optimized in the design. -band, if the element of the fins is still composed by In the two vias or more, just as in the cases in the - or -bands, the size of the fins will conflict with the coupler size. Since the radius of vias is limited by the PCB process and cannot be reduced unrestrictedly, the original fins are replaced by a single row of metallic vias with effective shifting-phase ability. and , both of The two parts shown in Fig. 14, i.e., which composed of two metallic vias, have deep impacts on the ( and ) are coupler performance. If all the vias of of the same radius and at a same high horizontal level ( and ), the return loss and isolation will be worsened, and if we reduce the values by a same level, the performance will be improved at the expense of the increase of phase-shifter length, which will increase the total size of the coupler. Thus, a and to improve the tradeoff is made by introducing performance of return loss. ( ), but Further research proves that it is not the the that plays a major part in the phase shifting, just as shown in Figs. 15 and 16. In this study, the measured data of -band coupler has an of 0.85 mm with the type-I vias of 0.25-mm radius. The corresponding phase differences are shown in Fig. 15, where we can find that the shifting phase is strengthened as the value of increases. ability of The phase difference is also sensitive to the number of vias . The phase differences corresponding to consisting within more cases shown in Fig. 16 illustrate that when

Fig. 16. Simulated S -parameters corresponding to different N

.

decreases, the amplitude of the shifted phase mainly caused by will decrease nearly 11 at each drop. As a conclusion, when we design the structures of fins or etched slots, we can take the following steps (the design procedure of the etched slot is similar). and to make sure the phase Step 1) Determine difference is approximately 180 . and to improve Step 2) Adjust the parameters of the performances of return loss and isolation. Step 3) Consider the additional phase shifter caused in Step ( and 2) and adjust the parameters of ) slightly to optimize the entire performance of the coupler. V. CONCLUSION Two types of a half mode substrate integrated waveguide 180 coupler have been proposed in this paper. The advantages of these half mode substrate integrated waveguide couplers are compact size, low insertion loss, and easy embedment in planar circuits. All the couplers are designed and fabricated using a standard PCB process and are measured. The measured data

LIU et al.: HALF MODE SUBSTRATE INTEGRATED WAVEGUIDE 180 3-dB DIRECTIONAL COUPLERS

and simulated results are in good agreement. With the promising technology of the half mode substrate integrated waveguide, the proposed couplers can serve as building blocks in the design of highly integrated microwave and millimeter-wave circuits and systems. REFERENCES [1] F. Arndt, T. Sieverding, and P. Anders, “Optimum field theory design of broadband E -plane branch guide phase shifters and 180 couplers,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 12, pp. 1854–1861, Dec. 1990. [2] F. Arndt, T. Sieverding, and P. Anders, “Rigorous modal-S -matrix design of a new class of broadband 180-branch guide couplers,” in IEEE MTT-S Int. Microw. Symp. Dig., May 8–10, 2003, vol. 1, pp. 577–580. [3] K. Wu, D. Deslandes, and Y. Cassivi, “The substrate integrated circuits—A new concept for high-frequency electronics and optoelectronics,” in 6th Int. Telecommun. Modern Satellite, Cable, Broadcast. Service. Conf., 2003, pp. P-III–P-X. [4] Y. L. Zhang, W. Hong, F. Xu, K. Wu, and T. J. Cui, “Analysis of guided wave problems in SIW—Numerical simulations and experiment results,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, vol. 3, pp. 2049–2052. [5] Y. Sakakibara, A. Kimura, J. Akiyama, M. A. Hirokawa, and N. Goto, “Alternating phase-fed waveguide slot arrays with a single-layer multiple-way power divider,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 144, pp. 425–430, 1997. [6] A. Zeid and H. Baudrand, “Electromagnetic scattering by metallic holes and its applications in microwave circuit design,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1198–1206, Apr. 2002. [7] W. Hong, “Development of antennas, components and subsystems based on SIW technology,” in IEEE Int. Microw., Antennas, Propag. and EMC Technol. Wireless Commun. Conf., Aug. 8–12, 2005, pp. 14–17, keynote speech. [8] Y. Li, W. Hong, G. Hua, J. X. Chen, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 446–448, Sep. 2004. [9] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Guided Wave Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [10] J. Hirokawa and M. Ando, “Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates,” IEEE Trans Antennas Propag., vol. 46, no. 5, pp. 625–630, May 1998. [11] A. Piloto, K. Leahy, B. Flanick, and K. A. Zaki, “Waveguide filters having a layered dielectric structures,” U.S. Patent 5 382 931, Jan. 17, 1995. [12] J. X. Chen, W. Hong, Z. C. Hao, H. Li, and K. Wu, “Development of a low cost microwave mixer using a broadband substrate integrated waveguide (SIW) Coupler,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 2, pp. 84–86, Feb. 2006. [13] Z. C. Hao et al., “Multilayered substrate integrated waveguide (MSIW) elliptic filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 95–97, Feb. 2005. [14] Z. C. Hao, W. Hong, J. X. Chen, X. P. Chen, and K. Wu, “Compact super-wide bandpass substrate integrated waveguide (SIW) filters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2968–2977, Sep. 2005. [15] Y. Zhang, W. Hong, K. Wu, J. Chen, and H. Tang, “Novel substrate integrated waveguide cavity filter with defected ground structure,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1280–1287, Apr. 2005. [16] G. Q. Luo et al., “Theory and experiment of novel frequency selective surface based on substrate integrated waveguide technology,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4035–4043, Dec. 2005. [17] H. J. Tang, W. Hong, Z. C. Hao, J. X. Chen, and K. Wu, “Optimal design of compact millimeter wave SIW circular cavity filters,” Electron. Lett., vol. 41, no. 19, pp. 1068–1069, 2005. [18] L. Yan and W. Hong, “Investigations on the propagation characteristics of the Substrate Integrated Waveguide based on the Method of Lines,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 152, no. 1, pt. H, pp. 35–42, 2005. [19] W. Hong, B. Liu, Y. Q. Wang, Q. H. Lai, and H. J. Tang, “Half mode substrate integrated waveguide: A new guided wave structure for microwave and millimeter wave application,” in Joint 31st Int. Infrared Millimeter Waves Conf./14th Int. Terahertz Electron. Conf., Shanghai, China, Sep. 18–22, 2007, keynote talk. [20] B. Liu, W. Hong, Y. Q. Wang, Q. H. Lai, and K. Wu, “Half mode substrate integrated waveguide (HMSIW) 3 dB coupler,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 1, pp. 22–24, Jan. 2007.

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Bing Liu was born in Jiangsu Province, China, on October 18, 1976. He received the B.S., and M.S. degrees in radio engineering from Southeast University, Nanjing, China, in 1998 and 2003, respectively, and is currently working toward the Ph.D. degree at Southeast University. From 1998 to 2001, he was a Communication Engineer with Jinxin Communication Technology Ltd., Nanjing, China. His current research interests include microwave and millimeter-wave components, circuits, and antennas.

Wei Hong (M’92) was born in Hebei Province, China, on October 24, 1962. He received the B.S. degree from the Zhenzhou Institute of Technology, Zhenzhou, China, in 1982, and the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1985 and 1988, respectively, all in radio engineering. Since 1988, he has been with the State Key Laboratory of Millimeter Waves, Southeast University, where he is currently a Professor with the Department of Radio Engineering. In 1993 and 1995–1998, he was a short-term Visiting Scholar with the University of California at Berkeley and University of California at Santa Cruz, respectively. He has been engaged in numerical methods for electromagnetic problems, millimeter-wave theory and technology, antennas, electromagnetic scattering, inverse scattering and propagation, RF front-ends for mobile communications, and the parameters extraction of interconnects in very large scale integration (VLSI) circuits, etc. He has authored or coauthored over 200 technical publications. He authored Principle and Application of the Method of Lines (Southeast Univ. Press, 1993, in Chinese). He has served as a reviewer for many technique journals such as Proceedings of the IEE (Part H) and Electronics Letters. Dr. Hong is a Senior Member of the China Institute of Electronics (CIE). He has served as the reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the two-time recipient of the First-Class Science and Technology Progress Prize presented by the State Education Commission in 1992 and 1994, respectively. He was the recipient of the Fourth-Class National Natural Science Prize in 1991, and the Third-Class Science and Technology Progress Prize of Jiangsu Province. He was also the recipient of the Foundation for China Distinguished Young Investigators presented by the National Science Foundation (NSF) of China.

Yan Zhang was born in Hei Bei Province, China, in 1983. He received the B.S. degree in radio engineering from Southeast University, Nanjing, China, in 2006, and is currently working toward the M.S. degree at Southeast University. His current research concerns microwave components and circuits.

Hong Jun Tang was born in Sichuan Province, China, on February 15, 1971. He received the B.S. degree in radio engineering from the Sichuan Institute of Light Industry and Chemical Technology, Zigong, China, in 1992, the M.S. degree in circuits and system from the University of Electronic Science and Technology of China, Chendu, China, in 2000, and is currently working toward the Ph.D. degree at Southeast Univesity, Nanjing, China. From 1992 to 2002, he was with the Sichuan Institute of Light Industry and Chemical Technology. His research interests include microwave and millimeter-wave circuits.

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Xiaoxin Yin received the M.S. and Ph.D. degrees from Southeast University, Nanjing, China, in 1989 and 2001, respectively, both in electromagnetic field and microwave technology. From 1983 to 1986 and 1989 to 1998, he was with the Department of Physics, University of Petroleum, Dongying, China, as a Lecturer. Since 2001, he has been with the Department of Radio Engineering, Southeast University. His current research interests include computational electromagnetics and antennas.

Ke Wu (M’87–SM’92–F’01) is Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering with the École Polytechnique (University of Montréal), Montréal, QC, Canada. He also holds a Cheung Kong endowed chair professorship (visiting) with Southeast University, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China, and the City University of Hong Kong, Hong Kong. He has been the Director of the Poly-Grames Research Center as well as the Founding Director of the Canadian Facility for Advanced Millimeter-Wave Engineering (FAME). He has authored or coauthored over 530 referred

papers and several books/book chapters. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers and sensors. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of Microwave Journal, Microwave and Optical Technology Letters, and Wiley’s Encyclopedia of RF and Microwave Engineering. He is an Associate Editor for the International Journal of RF and Microwave Computer-Aided Engineering. Dr. Wu is a Fellow of the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities). He is a member of Electromagnetics Academy, Sigma Xi, and the URSI. He has held key positions in and has served on various panels and international committees including the chair of Technical Program Committees, International Steering Committees, and international conferences/symposia. He has served on the Editorial/Review Boards of many technical journals, transactions, and letters including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He is currently the chair of the joint IEEE Chapters of Microwave Theory and Techniques Society (MTT-S)/Antennas and Propagation Society (AP-S)/Lasers and Electro-Optics Society (LEOS), Montréal, QC, Canada. He is an elected IEEE MTT-S Administrative Committee (AdCom) member for 2006–2009 and serves as the chair of the IEEE MTT-S Transnational Committee. He was the recipient of many awards and prizes including the first IEEE MTT-S Outstanding Young Engineer Award.

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Reduced-Length Rat-Race Couplers Mrinal Kanti Mandal, Student Member, IEEE, and Subrata Sanyal, Member, IEEE

Abstract—In this paper, analysis of a rat-race coupler having three equal arms of arbitrary electrical length is presented. All of these couplers have lower characteristic impedances than the conventional 1.5- coupler and some of them have smaller electrical lengths. Bandwidths in terms of isolation, matching, amplitude, and phase imbalances are also discussed. A reduced-size rat-race coupler of length 19 18 at the operating frequency of 0.9 GHz has been fabricated in the microstrip line. Experimental results agree well with the theoretical predictions. Index Terms—Rat-race coupler, size reduction.

I. INTRODUCTION HE 180 rat-race coupler is one of the most fundamental components in microwave and millimeter-wave frequency range. It is often used in various circuit applications such as mixers, multipliers, amplifiers, beam formers, etc. The conventional ring coupler has an electrical length of 1.5 at the operating frequency and, hence, occupies a large circuit area [1]. Several approaches were reported to address this problem. One efficient approach of miniaturization is to increase effective electrical length by using a slow wave effect. Slow wave effects are realized usually by increasing effective series inductance or shunt capacitance of the transmission line. In [2], a slotted ground plane was used to increase series inductance, while in [3] and [4], shunt open stubs and artificial transmission lines were used, respectively, to increase shunt capacitance. Another way of increasing capacitance is to use a spiral compact microstrip resonant cell [5]. All of these design approaches were based on a conventional rat-race coupler of electrical length as a basic section. 1.5 having For the capacitive loading used in [3]–[5], it is necessary to increase the series inductance in order to keep the line charwith being the port acteristic impedance fixed at impedance. Series inductance was increased by decreasing the linewidth of the main line. Therefore, the upper limit of miniaturization using these procedures is given by the fabrication limit of a high impedance line. The slotted ground plane approach does not have this high impedance line requirement. However, due to the etched slots in the ground plane, the structure should be suspended for proper functioning. This structure also has the problem of radiation losses, especially at higher frequencies. Reference [6] describes the design of two unconventional ratbased on the race couplers of electrical lengths 1.25 and

T

Manuscript received March 13, 2007; revised June 6, 2007. The authors are with the Electronics and Electrical Communication Engineering Department, Indian Institute of Technology, Kharagpur 721302, India (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910058

Fig. 1. Rat-race coupler configuration.

and sections, respectively. Although the 1.25- coupler , it has a lower requires a higher than the conventional value for the second one. Both of these couplers have reduced sizes. In this paper, rat-race couplers having electrical lengths other than those reported in [1] and [6] are also investigated. It is shown that an infinite number of solutions exist for coupler design at a given frequency. For all the solutions, characteristic im. In some cases, pedances are less than the conventional coupler electrical lengths are less than 1.5 . The procedures of [3]–[5] by using capacitive loading can be used for further miniaturization. Due to the lower values, the slow wave effect can be used for more effective miniaturization than a 1.5coupler under the same fabrication limit of a high impedance line. Bandwidth variations of these unconventional couplers are also investigated. II. ANALYSIS AND DESIGN A. Coupler Analysis Fig. 1 shows the coupler configuration to be investigated. Electrical lengths and corresponding characteristic impedances of each arm of the coupler is shown in this figure. Ports 1 and 4 are considered as the input ports and ports 2 and 3 are considered as the output ports. The coupler will be a rat-race coupler if the following conditions are simultaneously satisfied. 1) A signal applied to port 1 will be evenly split into two in-phase components at ports 2 and 3, and port 4 will be isolated. 2) If the input is applied to port 4, it will be equally split into two components with a 180 phase difference at ports 2 and 3, and port 1 will be isolated. 3) When operated as a combiner, with input signals applied at ports 2 and 3, the sum of the inputs will be formed at port 1, while the difference will be formed at port 4. To fulfill the above conditions, solutions for the electrical lengths and characteristic impedances are obtained using evenand odd-mode analysis [1]. Even- and odd-mode decomposed circuits of the current coupler for port 1 excitation are shown in Fig. 2(a) and (b), respectively. The transmission parameters for even-mode excitation at port 1 are

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(3d)

Respective parameters for odd-mode excitation are (4a) Fig. 2. Even- and odd-mode decomposition of the rat-race hybrid when port 1 is excited. (a) Even mode. (b) Odd mode.

(1a)

(4b)

(4c)

(1b)

(4d)

(1c)

Coupler in-band, as well as out-of-band responses can be predicted by using (1a)–(4d), [1]. and isolation At operating frequency, matching conditions for port 1 excitation yield

(1d)

(5a)

Respective parameters for odd-mode excitation are (2a) (2b) (5b)

(2c)

(2d)

is the port impedance. Here, The same results are obtained when the matching and isolation conditions are applied at port 4. , we get Assuming (6a) (6b)

Next, consider the port 4 excitation. Corresponding even- and odd-mode decomposed circuits are the same as those in Fig. 2 with the exception that ports 1 and 2 are replaced by ports 3 and 4, respectively. The transmission parameters for even-mode excitation at port 4 are then (3a) (3b)

(3c)

Equation (6a) and (6b) imply an infinite number of solution sets for and . Each solution set is repeated with a period . These solutions for the first two periods are of 180 for . In Fig. 3, the shown in Fig. 3 for port impedance maximum value of is 70.71 . It is obtained for and , which is the conventional case. The conditions in (6a) and (6b) simplify the scattering parameter expressions for the output ports as (7a) (7b) (7c) (7d)

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Fig. 4. Comparison of computed S -parameters of the coupler with  = 60 (solid lines) and 90 (dashed lines) for port 1 excitation.

Fig. 3. Solutions for Z;  and  . (a) First and (b) second periods.

6

Fig. 5. Amplitude ( 0.5 dB) and phase imbalance (0 tions with  for port 1 excitation.

65

) bandwidth varia-

where

It is evident from (7a)–(7d) that the phase conditions of 0 and 180 , respectively, for the sum and difference port excitations and 3-dB power coupling at the output ports are automatically . Therefore, satisfied by our initial assumption of ports 1 and 4 are, respectively, the sum and difference port of the coupler. B. Bandwidth Considerations In the following, coupler bandwidth variations with are considered. The various coupler upper and lower cutoff frequencies are defined for 0.5-dB amplitude imbalance, 0 5 and 180 5 phase imbalances between the outputs, respectively, for the sum and difference port excitations and for 20-dB matching and isolation. It is observed that amplitude and phase responses are symmetrical about the operating frequency

at and its odd multiples. The coupler band of operation shifts towards higher frequencies if decreases below 90 and vice versa. As an example, the transmission line -paand 60 are shown in Fig. 4. Due to rameters for and lower this asymmetry in the -parameters, upper cutoff frequencies are not symmetrical about the operating frequency. In bandwidth variation plots, the fractional bandwidths . are defined as For port 1 excitation, variations of the amplitude and 0 5 phase imbalance bandwidths with are shown in Fig. 5. For a conventional rat-race coupler, respective imbalance bandwidths are 16.07% and 22.67%. For port 4 excitation, variations of amplitude and 180 5 phase imbalance bandwidths are shown in Fig. 6. Corresponding imbalance bandwidths for a conventional coupler are 15.78% and 22.46%, respectively. The jumps in Figs. 5 and 6 arise due to the asymmetric humps in imbalance bandwidth variation plots. As an example, the 0 5 phase imbalance plots for sum port excitation are shown for and and , respectively, in Fig. 7. For all

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6

Fig. 6. Amplitude ( 0.5 dB) and phase imbalance (180  for port 4 excitation.

65

) variations with

Fig. 7. Phase imbalance between the output ports for port 1 excitation.

of these , upper cutoff frequency remains almost unchanged. However, if is decreased below 65.4 , the lower cutoff frequency jumps to a lower value. Therefore, a jump appears at in the phase imbalance plot in Fig. 5. Figs. 8 and 9 shows the variations of the matching and isolation bandwidths for sum and difference ports excitations, respectively. For a conventional coupler, these bandwidths are, respectively, 27.84% and 31.35% for excitation at port 1, and 32.22% and 31.35% for excitation at port 4. For port 1 excitation, the amplitude and 0 5 phase imbalance bandwidths range from 9.5% to 33.47% and 16.06% to 36.21%, range from 50 to 130 . Corresponding respectively, for a matching and isolation bandwidths vary from 3.26% to 27.83% and 23.18% to 31.35%, respectively. For port 4 excitation, amplitude and 180 5 phase imbalance bandwidths range from 8.85% to 33.43% and 15.78% to 29.31%, respectively. Corresponding matching and isolation bandwidth ranges are from 3.26% to 32.22% and 23.18% to 31.35%, respectively. The conventional coupler has the least bandwidths if only phase imbalances are considered. The plots in Figs. 5, 6, 8, and 9 can be used to optimize the coupler size depending upon the phase,

Fig. 8. Matching and isolation fractional bandwidth variations with  for port 1 excitation.

Fig. 9. Matching and isolation fractional bandwidth variations with  for port 4 excitation.

amplitude, isolation and matching requirements. Coupler deand periods are not considered here as sign with higher they show no improvement from bandwidth point-of-view and also since the resulting circuit area becomes larger. III. FABRICATION AND MEASUREMENTS and operAs an example, a rat-race coupler with GHz has been fabricated on a 1.58-mm-thick ating at and a loss FR4 substrate with dielectric constant tangent of 0.022. Corresponding is 38.46 . Coupler physical dimensions are obtained by using the full-wave simulator IE3D. Measured, IE3D simulated and transmission line computed -parameters are shown in Fig. 10, and corresponding output amplitude imbalances are shown in Fig. 11. The phase responses are shown in Fig. 12. Measurements have been carried out using an Agilent Technologies 8510 C vector network analyzer. Measurements show that matching and isolation are better than 20 dB from 872 to 918 and 778 to 1023 MHz, respectively. Within the matching bandwidth, amplitude variations at the two output ports are from 3.3 to 3.5 and 3.3 to 3.6 dB, respectively. Output phase variations, when the sum and the difference ports

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Fig. 12. Phase imbalance between the output ports (ports 2 and 3) when the sum and difference ports are excited.

Fig. 13. Fabricated rat-race coupler. Inner and outer radii of the ring are 28.7 and 33.4 mm, respectively.

Fig. 10. S -parameter variations of the coupler. Plot of: (a) S and S and (b) S and S . Substrate thickness = 1:58 mm, " = 4:3. Coupler dimensions: inner radius of the ring = 28:7 and outer radius = 33:4 mm.

IV. CONCLUSION In this paper, analysis and design of an unconventional ratrace coupler has been presented. Analysis is carried out using even- and odd-mode excitations. It is shown that a rat-race hybrid can be designed with lower electrical length and lower characteristic impedance than a conventional 1.5- coupler. Bandwidth variations in terms of matching, isolation, amplitude, and phase imbalance are also described. It is observed that these reduced-length couplers are mainly limited by matching bandwidth. Amplitude and phase imbalance bandwidths are better than a conventional coupler. Theoretical predictions are verified in microstrip technology. The fabricated coupler has 50.7% area of a conventional one. REFERENCES

Fig. 11. Amplitude imbalance between the output ports for port 1 excitation.

are excited, are from 0.31 to 2.08 and 179.6 to 182.6 , respectively. A photograph of the fabricated coupler is shown in Fig. 13.

[1] D. M. Pozar, Microwave Engineering. New York: Wiley, 1988. [2] Y. J. Sung, C. S. Ahn, and Y.-S. Kim, “Size reduction and harmonic suppression of rat-race hybrid coupler using defected ground structure,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 7–9, Jan. 2004. [3] M.-L. Chuang, “Miniaturized ring coupler of arbitrary reduced size,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 1, pp. 16–18, Jan. 2005. [4] K. W. Eccleston and S. H. M. Ong, “Compact planar microstripline branch-line and rat-race couplers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 10, pp. 2119–2125, Oct. 2003.

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[5] J. Gu and X. Sun, “Miniaturization and harmonic suppression of branch-line and rat-race hybrid coupler using compensated spiral compact microstrip resonant cell,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 1211–1214. [6] D. Kim and G.-S. Yang, “Design of new hybrid-ring directional coupler using =8 or =6 sections,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 10, pp. 1779–1784, Oct. 1991.

Mrinal Kanti Mandal (S’06) was born in West Bengal, India, in 1977. He received the B.Sc. degree (with honors) in physics and B. Tech. and M. Tech. degrees in radiophysics and electronics from the University of Calcutta, Calcutta, India, in 1998, 2001 and 2003, respectively, and is currently working toward the Ph.D. degree in microwave engineering at the Indian Institute of Technology Kharagpur, Kharagpur, India. He has authored or coauthored over 25 journal and conference papers. His research interests include the

design and performance improvement of passive RF and microwave components.

Subrata Sanyal (M’86) was born in 1952. He received the B.E. degree from the Indian Institute of Science (IISc) Bangalore, Banglaore, India, in 1975 and 1977, respectively, and the M.Tech. and Ph.D. degrees from the Indian Institute of Technology (IIT) Kharagpur, Kharagpur, India, in 1977 and 1987, respectively. From 1977 to 1980, he was an Electronics Engineer with the Radio Astronomy Group, Tatsa Institute of Fundamental Research (TIFR), Ooty, India. From 1990 to 1992, he was a Post-Doctoral Research Fellow with Queen Mary and Westfield College. Since 1984, he has been a faculty member with IIT Kharagpur, where, in 2007, he became a Professor. His research interest has been in the area of electromagnetic scattering and passive RF components.

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Time-Varying Matching Networks for Signal-Centric Systems Xin Wang, Student Member, IEEE, Linda P. B. Katehi, Fellow, IEEE, and Dimitrios Peroulis, Member, IEEE

Abstract—A time-varying matching network is proposed for signal-centric systems. This matching network achieves significantly higher bandwidth than conventional designs with the same circuit topology and complexity. This is achieved by exploiting information about the incident pulse. Time-varying elements control the initial conditions of the reactive components so that excellent matching is obtained over a very wide frequency range. In particular, experimental and theoretical results show that a single-stage ladder with one single-pole double-through switch achieves a nearly perfect match with negligible dispersion for a monocycle with a 10 : 1 10-dB bandwidth. Unlike conventional matching networks, time-varying designs are not limited by Bode–Fano gain–bandwidth restrictions, but only by the nonidealities of their circuit components. Moreover, it is experimentally shown that tunable time-varying designs reduce the reflected energy by 40%–45% and improve the transmitted energy efficiency of the same monocycles by 20%–25% compared to conventional tunable designs of the same complexity. Index Terms—Impedance matching, signal-centric systems, time-varying circuits.

I. INTRODUCTION MPEDANCE matching plays an important role in improving the efficiency of power transfer for most microwave and RF systems. The most common design of an impedance-matching circuit is a linear low-loss two-port network that minimizes reflections and maximizes power transfer between the source and load over a prescribed bandwidth. It is well known that perfect match (zero reflection and total transmission) can be realized by linear time-invariant networks only at a finite number of discrete frequencies. The design of a wideband impedance matching is generally a network synthesis problem subject to the gain–bandwidth restrictions derived by Bode and Fano [1]–[3]. In general, two major pathways are available to synthesize a conventional matching network that achieves a desired gain–bandwidth performance, which are: 1) utilize a well-known approximation (e.g., Butterworth, Chebyshev, elliptic, etc.) and the respective ladder or 2) calculate (analytically or numerically) the required positive real impedance

I

Manuscript received March 11, 2007; revised August 11, 2007. This work was supported by the National Science Foundation under Award ECCS-0638531. X. Wang and D. Peroulis are with the School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]). L. P. B. Katehi is with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61820 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910082

and synthesize it as a lossless two-port network based on Darlington’s theorem [4]. The first technique is usually preferred in practical applications due to its simplicity and its ability to yield, in general, good results. For more demanding applications that require optimal performance, it may be necessary to follow the second technique by employing one of the available methodologies based on either analytical (e.g., [5] and [6]) or numerical techniques. For example, the real frequency technique yields nearly optimal results with reasonable computational cost [4]. Regardless of the specific technique followed, conventional matching networks do not utilize any available information of the incident signal. The proposed matching networks vary in real time to adapt to the incoming pulse and, as a result, they provide maximum transmission and minimum reflection over a very wide frequency range. A pulse-based transmitter knows the starting time, duration, data rate, and amplitude of the transmitted pulses. Receiving systems may also require knowledge of the incoming signal, namely, the shape of the pulse and the time of arrival. If the system operates in a communication environment where the signals are known, the time of arrival may be detected through a power detecting circuit. A number of techniques exist today focused on estimating the time of arrival. These include, for example, threshold/leading edge detection using a tunnel diode or avalanche transistor and correlation detection using a matched filter or a bank of correlators. In a complex environment where the incoming pulses are arbitrarily generated or are dynamically changing due to multipath, a more advanced sensor that can provide the appropriate information about the time of arrival is needed. In this case, a wide variety of algorithms with various complexities implementing the timing acquisition and estimation have been proposed such as high-resolution methods [7], direct path signal detection algorithms [8], and the cyclostationarity-based timing estimation [9]. An overview of the different available techniques is provided in [10]–[12]. Fig. 1 schematically shows the application of the proposed matching network in a cognitive radio receiver with information of the pulse determined by using one of the above approaches. Due to the ability of the proposed matching network to vary in real time, an appropriate dynamic feedback loop can change the state of the network to provide matching to the incoming pulse or trail of pulses in real time. Assuming information of the incident pulse or trail of pulses is known, it may be possible to significantly improve the achieved loss/bandwidth for the same network complexity. Alternatively, it may be possible to considerably reduce the network complexity for the same loss/bandwidth performance.

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Fig. 1. Cognitive matching network (CMN) utilized in a receiver.

In this paper, we demonstrate a design methodology to achieve the above goals by imposing nonzero initial conditions on the network’s reactive components. By introducing time-varying elements that provide controlled nonzero initial conditions of the system, we develop intriguing designs with performance not possible under conventional design rules. The implementation of a matching network with nonzero initial conditions for an and load has been presented in [13] and [14] by the same authors. The two conference presentations focus primarily on the early stages of the study and demonstrate the possibility of widening the operating bandwidth by pre-storing energy in the capacitive element of the matching network. Building on the two conference presentations, this paper provides an extensive description of the design process and the method by which the initial pre-stored energy at each storage circuit element of the impedance matching network is determined in order to dynamically adapt the incoming signal to the load. In Section I, we define the problem, while in Section II, we introduce the appropriate time-domain parameters utilized in the design. In Section III, we introduce the proposed time-varying matching network design via a second-order circuit and present a frequency-domain analysis of the design (Section IV). In Section V, we discuss the extension of the time-varying design for a variety of pulse types. Performance limitations due to nonideal circuit components are discussed in Section VI. Experimental verifications are given in Section VII. In the presented study and for the sake of simplicity, it is assumed that the source and load impedances are purely resistive. However, the proposed technique may also be extended to complex source and load impedances. In this case, it may be necessary to have pre-stored dc energy in the reactive components of the source and the load. Consequently, the source and load – characteristic should be known over a broadband range in the frequency domain, including dc. Most practical circuits and devices (e.g., amplifiers and antennas) have input or output imnetwork, pedances that can be approximated by a simple which leads to simpler time-domain designs and provides explicit solutions. The design of the time-varying matching network varies with the topology of the load and, as a result, a general analytical solution for an arbitrary complex impedance is difficult to find. Numerical approaches may be preferred in such cases. In Section VIII, two examples of and loads are provided to demonstrate the implementation of the method to complex loads with excellent results in terms of achieving broadband matching.

Fig. 2. Pulse signal traveling along a transmission line to the load with a matching network.

II. PROBLEM SETUP AND DEFINITIONS The impedance-matching problem considered in this paper is is incident to illustrated in Fig. 2. A wideband pulse signal the input port with characteristic impedance . and are the total voltage and current at the input port of the matching netand are the voltage and current on the load . work, while Although the proposed design is applicable to complex load imand are assumed pedances, for the sake of simplicity, both to be purely resistive. Section VIII discusses complex loads. Under the above assumptions, the incident and reflected voltages at the input port can be expressed as (1) (2) Since the signals of interest are short pulses, it becomes appropriate to characterize matching in terms of energy rather than and transpower ratios. In this paper, we define reflection energy coefficients as mission (3) (4) where subscript denotes energy. In (3) and (4), the total incident energy , reflected energy , and transmitted energy are given by (5) (6) (7) In (5)–(7), the integrations are performed over the duration of the pulse signal. When the pulsewidth spreads due to dispersion, the range of integration is extended to cover the entire time inis a monochromatic signal, terval of the pulse. Notice that if and become equivalent to the return and insertion losses of the matching network at this frequency, provided that the integrations in (5)–(7) are performed over one period of the signal. and . From (2), (3), A perfect match requires : and (6), the following expressions are equivalent to (8) (9)

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Fig. 3. (a) Sinusoidal monocycle. (b) Switch-capacitor network.

Fig. 4. (a) Incident pulse. (b) Switch-capacitor network.

the latter of which means that the effective impedance seen at is . the input port by the input voltage III. DESIGN PRINCIPLE IN THE TIME DOMAIN When we design a network to match a resistive load to a sinusoidal signal propagating on a transmission line of characteristic impedance , we introduce a sequence of discontinuities physically separated so that the generated multiple reflections cancel to produce a near-zero reflected signal. Common networks include quarter-wavelength transformers or lumped elements. When the incoming signal is a pulse, the above design provides a zero reflection only at discrete frequencies, while it introduces high reflections at all other frequencies. To eliminate these unwanted reflections, a nonlinear component can be added to the network to delay the unwanted reflections in a way that they too cancel effectively. Such a nonlinear component can be a switch–capacitor combination with the capacitor(s) precharged appropriately. For pulses such as the sinusoidal monocycle, which is the simple truncation of a sinusoidal signal [see Fig. 3(a)], the switch–capacitor network can be very simple, as shown in Fig. 3(b). However, as the incident pulse becomes more complicated, the nonlinear component of the matching network becomes more complex [Fig. 4(a) and (b)]. Optimal matching can be accomplished via an appropriate selection of the dc source(s) and the time-varying on–off activation of the switches. In the following, we describe the design approach for the case of a monocycle (Fig. 3). A similar approach may be followed for other pulse shapes. The design approach consists of the following three main steps. Step 1) Choose circuit topology and component values. Step 2) Choose appropriate pre-stored energy levels that define the necessary initial conditions. Step 3) Design switching network to implement Step 2). To demonstrate these steps, in Sections III-A–C, we present the design process for a simple second-order circuit with a . More complex loads are shown in resistive load Section VIII.

Fig. 5. Conventional single-stage LC ladder as a matching network between the transmission line Z and the load Z . (a) Circuit schematic. (b) Typical input and output voltage waveforms when perfect match is achieved at one single frequency. (c) Typical input and output voltage waveforms when a monocycle signal is fed to the circuit.

A. Step 1: Circuit Topology and Component Values The design of matching networks for monochromatic signals is well established and is the basis for the design of the time varying matching network of Fig. 3. If the monocycle is extended to a periodic signal, it becomes a sinusoid whose spectrum is a single frequency , where is the duration of the pulse. Equivalently, the monocycle is a sinusoid truncated (or, generally, , with to the interval ). The performance of the system to the monocycle differs from that of the single-frequency signal due to the frequencies introduced by the truncation. Consequently, the performance of the system to the monocycle can become theoretically identical to that of the single-frequency signal if the voltage values at the leading and trailing edges of the monocycle are adjusted to represent those of the single-frequency signal. This is accomplished as shown in Section III-B. circuit As is well known [3], a simple second-order (Fig. 5) can easily match a single-frequency incident signal given by (10)

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when

where

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and

become

where

. In this case, the resulting

is 0, and

(11)

and

(19)

(12)

and are 0, as is the case in most conIf both ventional applications, the reflection (17) becomes

is (20) (13)

is the phase difference between where and . Typical waveforms of and are shown in Fig. 5(b). Obviously this is not a unique circuit topology and many different solutions exist. As is the case for conventional matching network design, the preferred solution depends on the application at hand. B. Step 2: Pre-Stored Energy Levels—Initial Conditions As shown in Appendix I, the reflected voltage and the in Fig. 5(a) satisfy the following two transmitted voltage second-order differential equations:

increases. Similarly, with which increases as set to 0 in (18), becomes

and

(21) . whose waveform is shown in Fig. 5(c) for in (17) suggests that The expression of the solution of the reflection can be eliminated by imposing appropriate initial conditions. Specifically, if and

(22)

becomes identical to 0. This requires pre-stored then energy in the capacitor of the matching network. More complex circuits may require pre-stored energies in several of their reactive components. C. Step 3: Switching Network for Initial Conditions (14) and

(15) Assuming an incident pulse given by otherwise

Based on the aforementioned analysis, we propose a timevarying matching network design, as shown in Fig. 6(a), which provides nonzero initial conditions using a switched capacitor with an output that is precharged by a dc voltage source resistance . At time , the switch disconnects the capacitor from the dc source and connects it to the load, whereas at time , the capacitor is detached from the load and connected back to the dc source via the switch. With the dc source chosen to be

(16) (23)

and inductor and capacitor values, as calculated in (11) and (12), the solutions to the previous equations between and are (Appendix II)

the initial conditions in (22) are satisfied. As a result, becomes identical to (13) for duces to 0, and Moreover, the conditions at time are found as

re.

(24) (25)

and

(17)

(18)

which are the same as (22). , remains at the constant voltage level , For as shown in Fig. 6(b), and is isolated by the switch from the rest of the circuit. This results in a first-order system that can be described by the following differential equation: (26)

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which is exactly one cycle of the load voltage of (13). Therefore, the response of the time-varying design (Fig. 6) to a sinusoidal monocycle incident signal is identical to one cycle of the response of the conventional design (Fig. 5) to a continuous sinusoidal signal. This can also be seen by comparing the waveforms in Figs. 6(b) and 5(b). With the above and solutions, we have achieved and . Here, it is worth mentioning that the response of the circuit of Fig. 6(a) can be seen as the superposition of two separate responses: one due to the monocycle input signal and the second one due to the pre-stored capacitor energy. Consider for instance for . When the capacitor is not the solution of precharged, the solution is given by (21). When the capacitor is precharged, but no signal is present at the input, the solution of can be found by letting and in (18), leading to (29) Adding (21) and (29) yields (28). Fig. 6(c) shows the energy stored in the capacitor and the and ) as a function of time. These inductor (denoted as energies can be calculated from (30) (31) The total stored energy normalized to the incident pulse energy is (32)

Fig. 6. Proposed time-varying matching network for sinusoidal monocycle signal. (a) Switched capacitor design of the matching network. (b) Input and output voltage waveforms. (c) E (the energy stored in the capacitor C ) and E (the energy stored in the inductor L) as a function of time.

which is homogeneous because solution of (26) can be easily found to be

when

. The

(27) in which the latter equality is the result of applying the initial condition (25). Consequently, and for . Now we have the complete solution of , which is idencan be written tical to 0 for all . The complete solution of as

otherwise (28)

It is interesting to note that the initial and final energies of both reactive components are identical. Therefore, from the energy conservation point of view, the energy delivered to the load is solely due to the energy of the incident pulse. This can also be seen from the fact that , which seems to imply that there is no need to recharge the capacitor after each pulse and, hence, the dc voltage source can theoretically be removed. Practically, however, the capacitor voltage may not exactly end up with its initial voltage level due to nonideal factors, as discussed in Section VI (e.g., leakage through the switch). Therefore, the dc voltage source is indispensable for restoring and maintaining the necessary voltage level on the capacitor during the intermission between consecutive pulse signals. of the dc voltage source needs to be suffiThe resistance ciently low so that the overall charging time is shorter than the time between two consecutive pulses. For example, we could en, where is the on resistance of sure that the switch and is the minimum time between two consequential pulses. This guarantees that reaches more than after the capacitor is charged for a period of time of . It needs to be pointed out that no pre-stored energy in the inductor is needed, as shown by (22). This might have not been the case should a different topology had been chosen. Having zero pre-stored energy in the inductors of a circuit is a good design principle since it eliminates undesired dc currents. This may be a useful guideline for practical designs.

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which is generally dependent on the complex frequency . This dependence can be eliminated by properly choosing the and values as follows. First, with , and reduces to (36)

We then match the coefficients of the and constant terms of the numerator to those in the denominator, i.e., choose the and values such that (37) (38) Fig. 7. Equivalent circuit of the time-varying matching network in Fig. 6(a) with t = 0 and t = T in the: (a) time domain and (b) Laplace transform domain.

In this case, the impedance becomes a constant constant can be chosen to be , i.e., let

. This

(39)

IV. FREQUENCY DOMAIN POINT OF VIEW Strictly speaking, the proposed design, as shown in Fig. 6(a), is a nonlinear time-varying system. Its frequency-domain behavior is dependent on the input signal itself. Consequently, it is not possible to directly obtain a frequency-domain representation of this circuit. Nevertheless, as shown here, it is possible to obtain a time-invariant circuit that is equivalent to the original time-varying one under proper conditions. This new circuit has the advantage of providing an intuitive design technique in the frequency domain that yields the same results with Section III. The new time-invariant circuit is shown in Fig. 7(a). In this circuit, the switch effect is represented by: 1) an initial voltage across the capacitor and 2) a delta-function current source with an unknown amplitude . By appropriately choosing and for the given input of (16), it is possible to match the response of this circuit with the one of Fig. 6(a) for all times. Here we show the method to accomplish this, as well as the complete frequency-domain response. and Without loss of generality, it is assumed that so that the initial condition at is automatically included in the standard Laplace transform analysis. Fig. 7(b) shows the Laplace transformation of the circuit in of is given by Fig. 7(a). The Laplace transform (33) , therefore, the input impedance For perfect match, can be readily calculated to be

then (40) which means the impedance of the load is transformed from to over the entire frequency range defined by specific input voltage . By solving the equations of (37)–(39), the necessary and values are found to be (41) (42) (43) and, hence, (44) Notice that and in (41)–(43) have exactly the same values as in (11), (12), and (23). Fig. 8 shows an example of the input impedance of the time-varying design compared to that of the conventional design as a function of frequency. As seen in this figure, the proposed technique maintains constant impedance over a very wide frequency range for the given input signal. and load With (40), the expression of the input current voltage can be derived as

(34) where

(45) (35) (46)

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Fig. 8. Effective impedance Z = R + jX seen at the input port of the proposed matching network of Fig. 6(a) for a 1-ns sinusoidal monocycle, compared to that of the conventional matching network in Fig. 5(a) designed at 1 GHz. In both cases, Z = 50 ; Z = 800 ; L = 30:8 nH, and C = 0:77 pF. The initial capacitor voltage V in Fig. 6(a) is given by (43).

Correspondingly, in the time domain, otherwise

(47)

which is the same as the previously obtained expression with given by (43). At jumps from to 0 due to the impulse current source , which is also equivalent to the effect of the switch disconnecting the capacitor from the . load at in the proFig. 9 shows typical spectra of load voltages posed design and the conventional design, both fed with the in the same incident sinusoidal monocycle. It is seen that time-varying design has a much wider bandwidth compared to the conventional design. As a result, it also prevents pulse dispersion. V. EXTENSION TO OTHER PULSE SIGNALS In the above analysis, the pulse signal is assumed to be a sinusoidal monocycle so that analytical solutions of the time-varying design with perfect match can be found. The sinusoidal monocycle has the advantage of being short in duration and relatively easy to generate. For this reason, it is considered for ultra-wideband communications. On the other hand, it has relatively large spectral sidelobes in the frequency domain [15]–[17]. Other wideband signals may require a different time-varying network. Fortunately, Gaussian pulses and their derivatives can be closely approximated by sinusoidal pulses with one or more cycles. In these cases, we can follow a similar design with very good results for practical cases. As an example, Fig. 10 shows a 1.2-ns Gaussian monocycle (first derivative of a Gaussian pulse) with an amplitude of 50 mV. This pulse can be expressed as (48)

V

V

V

Fig. 9. Spectra of: (a) , (b) in the proposed design, and (c) in the conventional design. Each graph is normalized to its own maximum. Both designs have the same component values as described in Fig. 8.

where is a time constant that controls the pulse duration, is the pulse amplitude, and is the position of the center of ns, mV, and the pulse. For this example, ns. A sinusoidal monocycle with the same amplitude can be fit to this Gaussian monocycle if and pulsewidth of (49) As a result, the matching network of Fig. 6(a) can be employed to approximately match the Gaussian monocycle. We have experimentally found that optimum matching is achieved if (50) For this example, Fig. 11 shows the reflected and transmitted voltage waveforms when this time-varying matching network is used for the aforementioned Gaussian monocycle pulse. For comparison, the dashed lines show the same waveforms when a conventional matching network [see Fig. 5(a)] with the same and components is used for the same incident Gaussian monocycle. The time-varying design has a significantly improved performance. In particular, its reflection and transmission coeffi-

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Fig. 10. Comparison between a Gaussian monocycle and a sinusoidal monocycle in the: (a) time domain and (b) frequency domain.

Fig. 11. Simulated waveforms when the time-varying matching network of Fig. 6(a) and conventional matching network of Fig. 5(a) are used for a Gaussian monocycle signal with time constant a = 0:303 ns. In both cases, Z = 50 ; Z = 800 , and the L; C; and V values are the same as those in the example in Fig. 8 to achieve matching for a 1-ns sinusoidal monocycle.

cients are % and design, the same coefficients are

%. For the conventional % and %.

VI. NONIDEALITY FACTORS In conventional matching networks, nonideal components primarily lead to increased insertion loss. For the proposed time-varying networks, they also lead to leakage of the pre-stored energy. This can be observed by comparing the and across the capacitor of voltage values

Fig. 12. Simulation of the capacitor voltage of Fig. 6(a) when an 800- load is matched to Z = 50 for a 1-ns 50-mV sinusoidal monocycle. Nonideal components with the shown equivalent circuits are assumed for the simulation. Proper choice of the dc voltage ensures the initial and final voltages of the capacitor are equal.

Fig. 6(a) at the beginning and ending of the incident pulse. (Section III), While under ideal conditions the loss of the reactive and switching components will cause , which translates to pre-stored energy leakage. This is always the case if is chosen according to (43). As an example, Fig. 12 shows the simulated time-domain capacitor voltage for a 1-ns 50-mV incident sinusoidal pulse with nonideal circuit components. The assumed models for all circuit components are shown in Fig. 12(a)–(c). These represent a 30-nH inductor with of 50 at 1 GHz, a 0.77-pF capacitor with of 200 at 1 GHz and a solid state switch with is chosen on state resistance of 3 . As Fig. 12(d) shows, if according to (43), which requires 0.194 V at ns, (solid line) ends at only 0.185 V at ns, which leads to approximately 10% of dc energy loss. This dc energy leakage can be compensated by choosing a so that ends at the same voltage level as its inismaller tial value. This happens at the cost of inferior matching since part of the incident energy is utilized to recharge the capacitor. The dashed line in Fig. 12(d) shows that, when a smaller initial V is chosen, the capacitor voltage starts voltage and ends at the same voltage level. Fig. 13 shows the dc energy leakage, as well as the transmission and reflection coefficients of the circuit of Fig. 6(a) for and inductor values. The dc energy leakage different is calculated as the difference of the energy stored in the capacitor between the times and . Obviously, as the inductor decreases, decreases, and increases. Depending on the application, it may be necessary to trade off between the matching performance and the maximum allowable dc energy leakage. If zero leakage is desired, it is necessary to choose a . In Section VII, the experimental results have suboptimal

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slow speed of the commercial off-the-shelf switches ( 4 ns). However, the design is frequency scalable. Every monocycle has a 10 : 1 10-dB bandwidth regardless of its center frequency. Therefore, application of the design methodology to a higher frequency range can be achieved by scaling down the inductor and capacitor values, as well as adopting faster switches with switching time approximately one-tenth of the pulsewidth or shorter. A. Single-Switch

Matching Network

Fig. 14 shows an example where a matching network is designed to match a 460- load to a 50- source. The following three experiments are conducted to evaluate the performance of the time-varying network: • matching of a 5-MHz sinusoid with a conventional matching network; • matching of a 200-ns (5-MHz center frequency) sinusoidal matching netmonocycle with the same conventional work; • matching of a 200-ns (5-MHz center frequency) sinusoidal monocycle with the proposed time-varying network and values as the conventional one. the same

Fig. 13. Simulation of the matching performances of the circuit of Fig. 6(a) for different V and inductor Q values with: (a) Z : Z = 16 : 1 and (b) Z : Z = 100 : 1. The switch is assumed to have an on resistance of 3 , and the capacitor is assumed to have a Q of 200. E is the dc energy leakage.

been achieved with energy leakage.

values chosen to result in zero pre-stored

VII. EXPERIMENTAL RESULTS AND DISCUSSION This section provides experimental verifications of the proposed matching network design. The experiments were conducted in a low-frequency range (1–10 MHz) because of the

A M/A-COM SW-239 GaAs monolithic microwave integrated circuit (MMIC) single-pole double-throw (SPDT) switch with 4-ns switching time, a 4.7- H Coilcraft 1008LS surface MHz, and a 200-pF ESS mount inductor with CDC ceramic disk capacitor are employed in the circuits. The sinusoidal monocycle and switch control signals are generated and synchronized by four Agilent 33250A arbitrary waveform generators. An HP 54602B 150-MHz oscilloscope is used for the measurements. Simulations for all cases are performed using Agilent ADS and the equivalent-circuit components provided by the manufacturers. Due to lack of an appropriate switch model, the on-state is represented with a 3- resistor that produces an insertion loss equivalent to that of the switch, as given in the datasheet. Fig. 14(c)–(f) illustrates the reflected and transmitted voltage waveforms of the conventional and time-varying circuits when fed by a 1-V peak-to-peak 200-ns sinusoidal monocycle. All measurements are in excellent agreement with the simulation results. The reflection and transmission energy coefficients are listed in Table I. The major sources of loss for all cases are listed in Table II. It is seen that the conventional matching network has a poor matching performance for the wideband sinusoidal monocycle: 5.4-dB (28.8%) reflection and 2.7-dB (53.7%) transmission coefficients are measured. Moreover, the pulsewidth is widened from 200 ns to more than 300 ns. On the other hand, the time-varying network achieves a 13-dB (5%) reflection and 1.3-dB (74.1%) transmission coefficients for the same pulse signal. Furthermore, nearly no dispersion is observed. In this case, the loss is primarily due to the low of the inductor and capacitor, as well as the nonideality of the switch. For comparison, the conventional matching network was measured to have a return loss of approximately 20 dB (1%) and an insertion loss of 0.8 dB (83.2%) at 5 MHz. Therefore, the additional 0.5-dB (9.1%) transmission loss in the time-varying design is a result of the switch loss.

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Fig. 14. Matching of a 460- load to a 50- source for a sinusoidal monocycle signal by the conventional circuit and the proposed time-varying circuit. (a) Conventional design. (b) Time-varying design. (c) Reflection waveforms and conventional design. (d) Reflection waveforms and time-varying design. (e) Load waveforms and conventional design. (f) Load waveforms and time-varying design.

TABLE I MEASURED (MEAS.) AND SIMULATED (SIM.) REFLECTION AND TRANSMISSION COEFFICIENTS OF THE EXAMPLE IN FIG. 14

TABLE II MAJOR SOURCES OF LOSS OF THE MATCHING NETWORK DESIGNS IN THE EXAMPLE OF FIG. 14

As shown in Fig. 14(d) and (f), unwanted spikes with relatively large magnitude appear in this particular implementation due to clock feedthrough when the switch changes states.

This noise needs to be carefully considered if such networks are employed in receiver architectures. The clock-feedthrough can be minimized by choosing higher performance switches and by compensation schemes that have already been proposed (e.g.,

WANG et al.: TIME-VARYING MATCHING NETWORKS FOR SIGNAL-CENTRIC SYSTEMS

Fig. 15. (a) Conventional and (b) time-varying tunable matching network designs.

[18]–[21]). Additionally, the spikes may be readily filtered out by the communication system itself. B. Tunable Two-Switch

Matching Network

Time-varying designs can be extended to tunable matching networks. To demonstrate this concept, we consider the conventional tunable matching network of Fig. 15(a) that is designed to achieve matching in the frequency range of 5–10 MHz and load range of 200–1000 . In this circuit, is chosen to be a fixed 8.2- H chip inductor with MHz. The two varactors and are implemented with ZEEX ZC836 varactor diodes MHz, minimum tuning ratio 1 : 5). The (minimum and via circuit is tuned by controlling the capacitances and separately. Specifically, for the frequency range of ranges from 5–10 MHz with a fixed 400- load, the required ranges from 100 to 210 pF. For the load 40 to 250 pF, while ranges range of 200–1000 with fixed frequency of 6 MHz, from 120 to 290 pF and from 110 to 230 pF. Fig. 15(b) shows the time-varying counterpart with the same inductors and varactors that is designed to achieve matching for sinusoidal monocycles with pulsewidth ranging from 100 to 200 ns and loads ranging from 200 to 1000 . The additional and are used to provide the initwo dc voltage sources and , respectively. The switches are tial voltages across realized as explained in Section VII-A. Both switches connect nodes 1 to 3 in the beginning of the pulse signal and connect nodes 1 to 2 in the end of the pulse. Fig. 16 compares the performance of the two tunable matching networks fed with the same monocycle pulses. For each load and each pulsewidth, the transmission and reflection coefficients are measured for the following three cases.

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Case 1) Time-varying matching network. Case 2) Conventional matching network with the same and values as in Case 1). Case 3) Conventional matching network with and tuned to achieve maximum . The measurements show that the time-varying network outperforms the conventional ones in all cases. In particular, 1.4–2 dB and 6–12 dB (40–45%) reduc(20–25%) improvement in is observed. Fig. 16(a) also show the matching pertion in formances for loads larger than 1000 , which are beyond the decreases rapidly for designed tuning range. As expected, these loads. To estimate the loss added by the switches, we compared the transmission and reflection coefficients of the following: • time-varying matching network when fed with a sinusoidal pulse of duration ; • conventional matching network when fed with a single. frequency signal at The measured results are shown in Fig. 17. For all cases, of the time-varying design with a pulse input is approximately 0.6 dB (9%) lower than the conventional design with a singlefrequency input. The measurements and simulation results agree with each other in all cases, except for the largest loads and narrowest pulsewidths. This is due to the higher loss of the varactors near the edge of their tuning range that is not predicted by their models.

VIII. EXTENSION TO

AND

LOADS

In the following, for the sake of completeness, we discuss and the design methodology as it applies to more complex loads. The design proceeds as previously discussed. First, a single-frequency matching network is designed at the center frequency of the sinusoidal monocycle. Second, the required pre-stored energy is calculated. Third, the switching network is implemented to provide the needed pre-stored energy. Since no major design differences exist, we simply summarize the results of the following examples. A. Capacitive

Load

Fig. 18 shows the matching network design for a complex pF and a resistor load consisting of a capacitor assuming a 1-ns incident sinusoidal monocycle. The conventional matching network designed at single frequency GHz consists of a 54.6-nH inductor and a 0.314-pF capacitor. The time-varying design requires in addition a dc source and a switch to provide pre-stored energy for the matching net. As already discussed work capacitor and load capacitor in Section III, the dc source value can be found from the voltat time when the input signal is the periages on and odic extension of the pulse signal. The exact solution requires pre-stored energy in the inductor as well. However, its value is 2.3% of the incident energy and, therefore, it can be ignored. Fig. 19 shows the matching performance of the time-varying design compared to that of the conventional matching network for the 1-ns sinusoidal monocycle in both the time and frequency

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Fig. 17. Measured and simulated matching performances of the time-varying matching network fed with a sinusoidal monocycle compared with those of the conventional matching network fed with a single-frequency signal: (a) for different loads and a fixed 167-ns width (period) of the pulse (single- frequency) signal and (b) for different width (period) of the pulse (single- frequency) signal and a fixed 400- load.

B. Inductive

Fig. 16. Measured performances of the time-varying matching network (circles) in Fig. 15 compared to those of the conventional matching network with two different designs: conventional design 1 (down triangle) has the same C and C values as in the time-varying design; conventional design 2 (up triangle) has the C and C values tuned to produce maximum T . All circuits are fed with the same sinusoidal monocycles. In (a), the pulsewidth is fixed at 167 ns. In (b), the load is a fixed 400- resistor.

Load

Fig. 20 shows the matching network design for an inductive load consisting of an inductor nH and a resistor . For a 1-ns incident sinusoidal monocycle, the nH and matching network component values are pF. Pre-stored energy in the capacitor is provided by the dc source in the time-varying design. The performances of the time-varying and conventional designs are compared in Fig. 21. Similar to the capacitive load, the time-varying design exhibits superior performance with negligible pulse distortion and a minor reflection coefficient. The required pre-stored energy in the inductor is 6.3% of the incident energy and has again been neglected. IX. CONCLUSION

domains. As expected, benefits similar to the ones presented for the simple resistive loads are also observed here.

A time-varying matching network with nonzero initial conditions is shown to significantly improve the gain–bandwidth

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Fig. 18. (a) Conventional and (b) time-varying matching network designs for a complex RC load.

Fig. 20. (a) Conventional and (b) time-varying matching network designs for a complex RL load.

Fig. 19. Matching performance of the time-varying design compared with that of the conventional matching network for complex RC load.

Fig. 21. Matching performance of the time-varying design compared to that of the conventional matching network for complex RL load.

performance for monocycles. The proposed design pre-stores energy in critical reactive components so that proper initial conditions are set that facilitate matching of the incoming pulse. Theory and experiments show that nearly perfect match can be networks over a 10 : 1 obtained from simple single-ladder bandwidth. This idea is also extended to tunable time-varying networks with the additional feature of adapting to the duration

of the incoming pulse and the value of the load. The proposed designs are only limited by the quality of their reactive components, the speed and loss of their switches, and by the need to know the incident pulse shape. Future research needs to focus on extracting the required pulse information, as well as on scaling the design to higher frequencies and embedding it in practical RF front-ends.

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APPENDIX I DERIVATION OF THE DIFFERENTIAL EQUATIONS AND GOVERNING

Equation (57) is homogeneous, whereas (58) is nonhomogeneous with a particular solution already given by (13). Therefore, the general form of the solution can be written as

Here we derive the differential equations (14) and (15) that and transmitted signal govern the reflected signal in Fig. 5(a). and , we have With the definition of (60)

(51) (52) According to Kirchhoff’s voltage and current law, (53)

(61)

(54)

and Next, we determine the unknown coefficients by matching the initial conditions of the circuit at time . and are continuous at time , and we Notice that have

Substitute (51) and (52) into (53), (55)

(62) (63)

Substitute (52) and (55) into (54), we obtain the differential . equation (14) that governs the reflected signal and obtain With (51) and (52), we can eliminate (56)

(64) (65) Substitute (62) and (64) into (52) and set to

Substitute (56) into (53) to eliminate and then use (54) to , we obtain the differential equation (15) that goveliminate erns the transmitted signal .

, then (66)

Substitute (62), (63), (65),, and (66) into (55) with set to we obtain

,

APPENDIX II AND SOLUTION OF The solutions of and in (17) and (18) are derived here. Substitute (11), (12), and (16) into (14) and (15). The differand become ential equations governing

(67) Substitute (64) and (65) into (54) with set to

, we obtain

(57) (68) On the other hand, from (60) and (61), we have (58) (69) Both (57) and (58) are second-order linear differential equations with the same constant coefficients and, hence, have the same characteristic equation whose roots are

(70)

(59)

(72)

(71)

WANG et al.: TIME-VARYING MATCHING NETWORKS FOR SIGNAL-CENTRIC SYSTEMS

With (65)–(72),

and

can be solved as (73) (74) (75) (76)

Finally, we substitute (73)–(76) into (60) and (61) to obtain the and , as given in (17) and complete solution of (18). REFERENCES [1] H. W. Bode, Network Analysis and Feedback Amplifier Design. New York: Van Nostrand, 1945. [2] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” J. Franklin Inst., vol. 249, pp. 57–83, Jan. 1950. [3] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1997. [4] H. J. Carlin and P. P. Civalleri, Wideband Circuit Design. Boca Raton, FL: CRC, 1998. [5] D. C. Youla, “A new theory of broadband matching,” IEEE Trans. Circuit Theory, vol. CT-11, no. 1, pp. 30–50, Mar. 1964. [6] W.-K. Chen, “Explicit formulas for the synthesis of optimum broadband impedance-matching networks,” IEEE Trans. Circuits Syst., vol. CAS-24, no. 4, pp. 157–169, Apr. 1977. [7] M. A. Pallas and G. Jourdain, “Active high resolution time delay estimation for large BT signals,” IEEE Trans. Signal Process., vol. 39, no. 4, pp. 781–188, Apr. 1991. [8] J.-Y. Lee and R. A. Scholtz, “Ranging in a dense multipath environment using an UWB radio link,” IEEE J. Sel. Areas Commun., vol. 20, no. 12, pp. 1677–1683, Dec. 2002. [9] Z. Tian, L. Yang, and G. Giannakis, “Symbol timing estimation in ultrawideband communications,” in Proc. Asilomar Signals, Syst., Comput. Conf., 2002, pp. 1924–1928. [10] S. Gezici, Z. Tian, G. Giannakis, H. Kobayashi, A. Molisch, H. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios,” IEEE Signal Process. Mag., vol. 22, pp. 70–84, Jul. 2005. [11] S. Tekinay, “Wireless geolocation systems and services,” IEEE Commun. Mag., vol. 36, pp. 28–28, Apr. 1998. [12] J. H. Reed, Ed., An Introduction to Ultra Wideband Communication Systems. Upper Saddle River, NJ: Prentice-Hall, 2005. [13] X. Wang, L. P. B. Katehi, and D. Peroulis, “Time-domain impedance RC loads,” in IEEE adaptors for pulse-based systems with high MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1741–1744. [14] X. Wang, L. P. B. Katehi, and D. Peroulis, “Time-varying matching network for antennas in pulse-based systems,” in IEEE Antennas Propag. Symp. Dig., Jun. 2007, pp. 89–92. [15] A. Azakkour, M. Regis, F. Pourchet, and G. Alquie, “A new integrated monocycle generator and transmitter for ultra-wideband (UWB) communications,” in Proc. IEEE RF IC Symp., Jun. 2005, pp. 79–82. [16] B. Godara, G. Blamon, and A. Fabre, “UWB: A new efficient pulse shape and its corresponding simple transceiver,” in 2nd Int. Wireless Commun. Syst. Symp., 2005, pp. 365–369. [17] H. Schantz, The Art and Science of Ultrawideband Antennas. Norwood, MA: Artech House, 2005. [18] C. Eichenberger and W. Guggenbuhl, “Dummy transistor compensation of analog MOS switches,” IEEE J. Solid-State Circuits, vol. 24, no. 8, pp. 1143–1146, Aug. 1989. [19] S. D. Willingham and K. W. Martin, “Effective clock-feedthrough reduction in switched capacitor circuits,” in Proc. IEEE Int. Circuits Syst. Symp., May 1990, vol. 3, pp. 2821–2824. [20] M. Song, Y. Lee, and W. Kim, “A clock feedthrough reduction circuit for switched-current systems,” IEEE J. Solid-State Circuits, vol. 28, no. 2, pp. 133–137, Feb. 1993. [21] R.-M. Weng and C.-W. Lai, “Clock-feedthrough cancellation with compensation circuit for switched-current systems,” in IEEE Electron Devices and Solid-State Circuits Conf., Dec. 2005, pp. 593–595.

Q

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Xin Wang (S’07) was born in Jiangsu Province, China, in 1976. He received the B.S. and M.S. degrees in electrical engineering from Tsinghua University, Beijing, China, in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree in electrical and computer engineering at Purdue University. His current research interests include design and application of time-varying matching networks for signal-centric systems.

Linda P. B. Katehi (S’81–M’84–SM’89–F’95) received the Diploma degree from the School of Mechanical and Electrical Engineering, National Technical University of Athens, Athens, Greece, in 1977, and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1981 and 1984, respectively. Following her undergraduate studies, she worked for two years as a Senior Engineer with the Naval Research Laboratory. She joined UCLA in Fall 1979. From 1984 to 2002, she was a faculty member with the Electrical Engineering and Computer Science Department, The University of Michigan at Ann Arbor, where she was the Associate Dean for Academic Affairs from 1998 to 2002. From 2002 to 2004, she was the Dean of Engineering and as faculty member of the Electrical and Computer Engineering, Purdue University, West Lafayette, IN. She is currently the Provost and Vice Chancellor for Academic Affairs and Professor of Electrical and Computer Engineering with the University of Illinois at Urbana-Champaign. As a faculty member, she has focused her research on the development and characterization of 3-D integration and packaging of high-frequency circuits with particular emphasis on microelectromechanical systems (MEMS) devices, high- passives, and embedded filters. She pioneered the development of on-wafer packaging for high-density high-frequency monolithic Si-based circuit and antenna architectures that lead to low-cost high-performance integrated circuits for radar, satellite, and wireless applications. She has authored or coauthored over 500 papers published in refereed journals and symposia proceedings. She holds 13 U.S. patents. Prof. Katehi is a member of the National Academy of Engineering. She is a Fellow of American Association for the Advancement of Science (AAAS). She serves on numerous scientific committees including the Nominations Committee for the National Medal of Technology, the Board of the AAAS, the Kauffman National Panel for Entrepreneurship, the National Science Foundation (NSF) Advisory Committee to the Engineering, the NRC Telecommunications Board, the NRC Army Research Laboratory Advisory Committee on Sensors and Electronics Division, the NSF Advisory Committee to Computer and Information Science and Engineering (CISE), the National Aeronautics and Space Administration (NASA) Aeronautics Technical Advisory Committee (ARAC), and the Department of Defense (DoD) Advisory Group on Electron Devices. She has been the recipient of numerous national and international technical awards and to distinctions as an educator.

Q

Dimitrios Peroulis (S’99–M’04) received the Diploma degree in electrical and computer engineering from the National Technical University of Athens, Athens, Greece, in 1998, and the M.S.E. and Ph.D. degrees in electrical engineering from The University of Michigan at Ann Arbor, in 1999 and 2003, respectively. Since August 2003, he has been an Assistant Professor with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN. His current research is focused on MEMS for multifunctional communications systems and sensors. Dr. Peroulis was the recipient of several teaching awards presented by Purdue University, including the Teaching for Tomorrow Award (2006). He was also the recipient of three Student Paper Awards presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) (2001 and 2002) and the IEEE Antennas and Propagation Society (IEEE AP-S) (2001).

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Multigigahertz Causal Transmission Line Modeling Methodology Using a 3-D Hemispherical Surface Roughness Approach Stephen Hall, Steven G. Pytel, Member, IEEE, Paul G. Huray, Member, IEEE, Daniel Hua, Anusha Moonshiram, Member, IEEE, Gary A. Brist, and Edin Sijercic

Abstract—As computer clock speeds continue to increase at a rate dictated by Moore’s Law, the system buses must also scale in proportion to the processor speed. As data rates increase beyond 5 Gb/s, the historical methods used to model transmission lines start to break down and become inadequate for the proper prediction of signal integrity. Specifically, the traditional approximations made in transmission line models, while perfectly adequate for slower speeds, do not properly account for the extra losses caused by surface roughness and do not model the frequency dependence of the complex dielectric constant, producing incorrect loss and phase-delay responses, as well as noncausal waveforms in the time domain. This paper will discuss the problems associated with modeling transmission lines at high frequencies, and will provide a practical modeling methodology that accurately predicts responses for very high data rates. Index Terms—Causality, dielectric constant, interconnect, loss tangent, surface roughness, transmission line.

I. INTRODUCTION S SYSTEM speeds increase with Moore’s Law, the assumptions historically used to create transmission line models begin to break down. This is especially apparent when observing the system -parameters or analyzing eye diagrams constructed from pseudorandom bit streams. Most simulators employ transmission line models that are derived from quasi-static 2-D field solvers, which generally yield frequency-independent inductance and capacitance matrices, a conductance matrix that varies linearly with frequency, and a resistance matrix that varies with the square root of frequency. Although these approximations are valid for low frequencies, they do not account for frequency-dependent behavior that becomes significant above a few gigahertz, resulting in simulation errors for high data rates. Specifically, the assumption and loss of frequency-independent dielectric constant tangent (tan ) leads to noncausal time-domain responses, loss errors, and phase miscalculations. The problem of frequency invariant dielectric properties for transmission lines in high-speed digital designs was initially discussed by the

A

Manuscript received June 14, 2007; revised September 17, 2007. S. Hall, S. G. Pytel, D. Hua, A. Moonshiram, G. A. Brist, and E. Sijercic are with the Intel Corporation, Hillsboro, OR 97124 USA (e-mail: Stephen.h. [email protected]). P. G. Huray is with the Intel Corporation, Hillsboro, OR 97124 USA, and also with the Department of Electrical Engineering, University of South Carolina, Columbia, SC 29201 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910076

Fig. 1. Insertion loss and phase delay per inch for a 7-in transmission line showing errors induced by traditional models, " = tan  = 3:9=0:0073 at 1 GHz, roughness of copper foil h = 5:8 m modeled with (2) and (3).

authors in [1]–[3]. The modeling methodology discussed in this paper has been expanded to account for additional physical mechanisms that describe the behavior of dielectrics over a wide band of frequencies and has been validated against vector network analyzer (VNA) measurements. A detailed derivation of dielectric dispersion effects from induced molecular dipoles was discussed in [4], which are expanded upon in this paper to produce a practical transmission line methodology. Additionally, existing methods of calculating the extra losses due to conductor surface roughness work well for relatively smooth surfaces, but begin to break down for the roughest copper commonly used in the industry after 5 GHz. A detailed overview of nonclassical losses due to surface roughness was presented by the authors in [5], which described the physical aspects of copper foils used to construct printed circuit boards and explained the existing methods of predicting the extra losses. While the idea of using an array of interacting hemispheres to represent the surface roughness profile was described by the authors in [6], a new approach to the hemisphere method is presented in this paper, producing a model that is easier to implement, more computationally efficient, and proven accurate by comparison to laboratory measurements. The fundamental physics of this new surface roughness model was discussed by the authors in [7] and is further developed into a usable model in this paper. Fig. 1 depicts a comparison to a measurement of the insertion loss and phase delay per inch of a 7-in traditional transmission

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line model that assumes constant values for the inductance , , and , where the conductance and recapacitance sistance is calculated with (1) and (2), respectively, as follows: (1) (2) In (2), is the classic skin resistance for a smooth conis the Hammerstad coefficient (3), which is ductor and based on a corrugated 2-D surface, as calculated in [8], and is used to model the extra losses caused by the copper surfaces on a transmission line that are often purposely roughened to promote adhesion to the dielectric [9], [5] (3) where is the root mean square (rms) value of the surface is the skin depth, is the roughness height, is the permeability of free conductivity of the metal, and space. Fig. 1 illustrates some of the problems associated with the traditional transmission line modeling methodology by comparing a transmission line model to a measured response. In this exbegins ample, the magnitude of the modeled insertion loss to deviate from measured values after 5 GHz and the phase delay is constant over frequency with an error of 4.5 ps/in. These errors scale with longer line lengths and are magnified when calculating an eye diagram from a pseudorandom bit stream because the model will not properly predict loss, delay, or intersymbol interference. As data rates continue to increase, the errors introduced by the traditional modeling methodology become a significant roadblock to high-speed bus design for data rates above approximately 5 Gb/s. In order to correctly model the propagation of a signal on a transmission line, it is necessary that the complex impedance function be analytic [10]. Consequently, a change in the impedance must correspond to a specific change in the loss. A proper relationship between the impedance and loss is required to balance the amount of energy that is propagated down the transmission line versus the amount of energy that is dissipated through the conductor and dielectric losses. Assumptions that violate this principle effectively cause a portion of the signal to be propagated when it should have been attenuated or vice-versa. This paper presents a methodology for modeling transmission lines that has been proven accurate through correlation to VNA measurements up to 30 GHz. Section II describes a frequencydependent dielectric model that ensures causality enforcement and significantly reduces the phase-delay errors. Section III will present a new method for predicting the extra losses due to surface roughness. Finally, Section IV will discusses how this methodology can be implemented with HSPICE and other conventional simulators. II. DIELECTRIC MODEL The classical model for dielectric behavior is derived from four distinct mechanisms, as shown in Fig. 2 [4].

Fig. 2. Pictorial of how complex dielectric constant varies with frequency and the physical mechanism that dictates the behavior.

At low frequencies, conduction electrons within the dielectric provide a low-frequency loss mechanism, but since the conductivity for most dielectrics is very small, these losses for practical insulators are usually ignored. At intermediate frequencies (the value of which will vary for each dielectric), polar molecules in the dielectric form electric dipoles that partially align with the applied time-dependent electric field. As the frequency of the applied electric field increases, the molecular inertia makes it difficult for the molecular dipoles to follow the applied field, causing the losses to increase. Accordingly, the inability of the dipoles to follow the external fields results in a decrease in the along with an increase in the dielecdielectric permittivity . At very high frequencies, induced dipoles tric losses can be created in individual atoms or molecules when the applied electric field is oscillating near a natural frequency. This resonance induces large losses with a corresponding decrease in the dielectric constant. At even higher frequencies, plasma oscillations of conduction electron clouds cause an additional loss mechanism. These frequencies are rarely reached in practice and the effect is normally embedded in a high-frequency constant . Huray et al. described the dielectric behaviors above by analyzing an analogous oscillator in [4] and [1] in which they preserved causality. Consequently, the frequency-dependent dielectric constant of an insulating material is complex, shown in (4) as follows: (4) Furthermore, the complex dielectric permittivity must be an analytic function that retains a causal relation between the electric field in the transmission line material and the applied electric field, as shown by the Kramers and Krönig relations and demonstrated in [4]. To understand this concept, the most general, linear, and causal relationship between and in a homogeneous medium is shown in (5) as follows: (5) where is the electric displacement field and is a function that describes the response of an ordinary linear dielectric to the . Note, for (5) to remain causal, electric field at time must vanish for , meaning that the material cannot respond prior to the application of the electric field. If it assumed that the and can be written in the classical relationship between form , then the dependence on time can be derived

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by Fourier superposition, as done by Jackson in [11], yielding , as shown in (6) as follows: the form of

Assuming a time–harmonic field, substitution of (9) into Ampere’s equation gives

(6) is expressed in terms of a Fourier transform, the Since , inverse transform yields the dielectric constant in terms of shown as follows in (7):

(7) If the complex frequency is restricted to the upper half of the complex plane, then (7) can be shown to be analytic providing is always finite [12]. Therefore, the real and imaginary that parts of the permittivity are interdependent and must be related by the Cauchy–Riemann equations. Essentially, this means that a change in the real part of the permittivity must correspond to a specific change in the loss tangent. Since traditional transmission line models employ frequency, the required relationship independent capacitance matrices between the real and imaginary components of the dielectric permittivity (alternatively, the relationship between the loss tangent and capacitance) is not maintained, and nonphysical results are sometimes observed. Furthermore, if the frequency dependence of the dielectric permittivity and loss tangent is not correctly modeled, neither the losses, nor the phase velocity can be correctly predicted for a wideband transient signal because the harmonic content will be correct at no more than one frequency. Combining the four mechanisms of dielectric behavior to obtain a frequency-dependent model of the dielectric constant yields (8), as follows, which is guaranteed to yield a causal [4]: relationship between and (8)

where and are the frequencies where the dielectric variis the ations are occurring, is the operating frequency, variation of dielectric constant over the frequency of interest, is the conductivity of the dielectric material, is the dielectric constant value at very high frequencies, and is the permitaccounts for the damping tivity of free space. The term of the molecular dipoles in the mid-frequency ranges, the term accounts for resonance of induced atomic and molecular dipoles or plasma oscillations, and the final term accounts for the low-frequency loss of the dielectric, which is usually ignored. Note that the last term in (8) is singular at dc, which indicates that the equation breaks down at very low frequencies. This term describes the behavior of lossy dielectrics that have conduction electrons that are free to move according to Ohm’s law, shown in (9) as follows:

(9)

(10) Rearranging (10) yields the dependence of Ampere’s equation on a permittivity, as described by (4) and the conduction electrons. It should be noted that the pole created at is artificial based on the rearrangement of (10). Since due to being small in practical dielectrics, the authors have neglected the term when correlating models to measurements. Equation (8) is a general formula that describes the dielectric material and is suitable for calculating the frequency depenand . Since the particular implementation of dence of (8) is dependent on the characteristics of the dielectric, the most and straightforward usage requires a measured response of so the damping poles and resonant peaks can be identified. A more practical approach can be derived by ignoring the resonance term and integrating (8) to create an infinite pole model of the classic Debye model, as proposed by Djordjevic et al. [13], which is shown in (11). This approach has been shown to be accurate for FR4 and many other dielectrics used in the computer industry through comparisons to VNA measurements, as shown in [1] and [2],

(11) is the total variation between the lower frequency where and the upper limit limit of the model is the slope of the dielectric permittivity is the dielectric permittivity at very high per decade, and frequencies. In order to calculate the dielectric permittivity and loss tangent for a wide band, it is necessary to know the value and at one frequency. Since a specific relationship of and must be preserved, and can be between calculated if and are known at a single frequency, which allows the wideband response to be calculated. Expanding (11) into real and imaginary terms helps facilitate the calculation of and . Assuming , the real and imaginary parts of (11) can be approximated with (12) and (13) [13] as follows: (12)

(13) Fig. 3 depicts an example of the modeled and measured relationship between the loss tangent and dielectric permittivity, as calculated with (11), and measured with a Fabry–Perot open

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Fig. 3. Measured dielectric responses compared to models calculated with the generalized infinite pole model (11).

cavity resonator. The modeled response of Fig. 3 was calculated from a known data point at 1 GHz where , which was measured using a split-post resonator. Note the deviation between the measured data points and the model has a maximum error of only 0.0009 at 30 GHz with a maximum error less than .01 for at 20 GHz. and Additionally, since an explicit relationship between is required, if the real portion of the dielectric constant is carefully measured, then the loss tangent can be directly calculated and vice-versa using (12) and (13). To construct a transmission line model, the real portion of (11) is used to calculate the frequency-dependent capacitance and the ratio of the real and imaginary parts are used to , as shown in (14) calculate the loss tangent and subsequently and (15) as follows: (14) (15) and , are calculated by The seed values of a 2-D electromagnetic field solver at a reference frequency, , where and are known. Equations (14) and (15) simply scale the output of a standard frequency-inand calculated from variant field solver with (11) and (4) to achieve the proper frequency dependent properties. The seed dielectric permittivity and loss tangent are calculated in (16) and (17) as follows:

Fig. 4. Accuracy gained by implementing a frequency-dependent dielectric using (11), 7-in microstrip, " = tan  = 3:9=0:0073 at 1 GHz, rms rough= 5:8 m modeled with (2) and (3). ness of copper foil h

dielectric materials that are linear, meaning that the real and imaginary parts will not change with the applied field strength. In general, the dielectric materials used to construct printed circuit boards, multichip modules, and packages in the electronics industry are linear. Note that this methodology does not require the dielectric to be homogeneous as long as the spatial variations of the nonuniform regions along each axis are linear and small compared to the wavelength of interest, which allows the bulk properties of the permittivity to be used. Alternatively, for linear materials that are neither homogeneous, nor isotropic, this methodology remains valid if the permittivity is expressed ). Since many dielectric as a function of position (i.e., materials, such as FR4, used in the electronics industry are composites, and therefore, nonhomogeneous, the results in the paper assume the bulk properties of the permittivity. This methodology is not appropriate for nonlinear dielectrics such as metamaterials, which have a negative index of refraction or semiconductors. Fig. 4 depicts the increased accuracy when the infinite pole dielectric model (11) is used to calculate frequency-dependent and matrices with (14) and (15). Although the frequency-dependent dielectric model provides an accuracy improvement in the phase delay, the insertion loss error is still significant. This error will scale with line length and will dramatically affect the resultant eye created from a long pseudorandom bit pattern. To correct this error, the conductor model must be improved.

(16) III. CONDUCTOR MODEL (17) is the equivalent capacitance per unit length of the where transmission line if the dielectric was replaced with air, which is calculated from the speed of light in a vacuum, and the inductance matrix , resulting in the effective dielectric constant . and loss tangent at the reference (or seed) frequency The methodologies presented in this paper for modeling the complex permittivity as a function of frequency will hold for

At low frequencies, the current flows throughout the entire cross section of the conductor. As frequencies increase, the current begins to migrate towards the edge or skin of the conductor [14]. The current induces a magnetic field and subsequently an inductance, which is the circuit parameter used in the transmission line model to represent energy stored in the magnetic field. The total inductance is calculated in (18) as follows: (18)

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where is the portion of the inductance caused by currents flowing inside the conductor material (usually copper) and is caused by currents flowing on the surface of the conductor. As the current migrates to the periphery of the conductor at very high frequencies, the total inductance is decreased by because current ceases to flow in the interior of the conductor. Consequently, the total energy that can be stored in the magnetic field is reduced. Additionally, as frequencies increase, the skin effect will also cause an increase in the resistance because the currents are flowing in a smaller area. Subsequently, the skin effect simultaneously causes a decrease in the inductance and an increase in the resistive losses, implying a necessary relationship between the energy stored in the magnetic field and the energy dissipated by the conductor, which is classically described as follows in (19) [2]:

Fig. 5. Pictorial transmission line cross section showing rough copper and relation to skin depth.

(19) In simple terms, if the energy cannot be stored in the magnetic field, it must be dissipated by the skin-effect resistance. If the transmission line model does not provide for the necessary variations in the inductance and resistance, energy is not conserved and an erroneous waveform is anticipated because a portion of the signal is being transmitted when it should have been attenuated. The classical transfer function of a perfectly matched lossy transmission line is shown as follows in (20), where and length [15]:

(20) The Fourier transform of the transmission line must satisfy the following constraints [10]:

for

(21) (22)

where (21) ensures that the impulse response of the system is real and (22) ensures that it is causal. Djordjevic et al. [13] and Arabi et al. [15] show that (20) will not satisfy the above constraints unless the correct frequency dependencies of all transmission line parameters are included. and To properly account for the frequency variation of , the nonideal effects of the copper must be considered. The problem is that most (if not all) 2-D field solvers calculate the resistance and inductance assuming smooth conductors. Real copper surfaces, however, are purposely roughened to promote adhesion to the dielectric when manufacturing printed circuit boards. The resulting copper surfaces have a “tooth structure,” as depicted in Fig. 5. When the tooth height is comparable to the skin depth, the smooth copper assumptions break down. The rms tooth height of common copper foils used to manufacture printed circuit boards range from 0.3 to 5.8 m, with peak

Fig. 6. Measured results of identical 7-in transmission lines with relatively = 1:2 m) and rough (h = 5:8 m) copper showing how smooth (h surface roughness affects losses.

heights exceeding 11 m [5]. Note that the skin depth in copper at 1 GHz is 2 m, indicating that, for many copper foils, most of the current will be flowing in the tooth structure for multigigabit designs [14]. Since the rough copper surface affects current flow, it will also affect power dissipation, and thus, insertion loss. Fig. 6 shows the measured results of two identical transmission lines built with rough and smooth copper. The copper foil used to construct the test boards was characterized with an optical profilometer prior to lamination, yielding an rms tooth height of m for the smooth copper and m for the rough copper. Note the significant increase in insertion losses due to the roughness. The traditional way to account for surface roughness losses in a transmission line model is to use the Hammerstad equation, as shown in (2) and (3). This approach has been shown to be accurate by comparison to VNA measurements for copper surface roughness profiles less than approximately 2 m (rms). As roughness profiles are increased past 2 m, new models are required. Fig. 7 shows the accuracy of the Hammerstad model by comparing measured transmission line structures constructed with relatively smooth and a very rough copper to simulations using (3). Notice that the Hammerstad model is considerably less accurate for the rough copper case. To understand why the accuracy breaks down for some copper types, it is useful to explore the assumptions behind (3), which assumes a 2-D corrugated surface similar to that shown in Fig. 8 [7]. Additionally, other publications such as

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Fig. 10. Surface profile measurement of rough copper foil.

Fig. 7. Example of the Hammerstad model accuracy for: (a) relatively smooth and (b) rough copper foils, 7-in microstrip line.

Fig. 8. 2-D corrugated surface assumption behind the Hammerstad equation.

Fig. 9. Surface profile measurement of relatively smooth copper foil.

[16] make similar 2-D approximations of the surface roughness profile. By comparison, Fig. 9 depicts the measured surface of relatively smooth copper foil using an optical profilometer. Note the surface shown in Fig. 9 might be described as corrugated with sparse protrusions on the surface, suggesting that the Hammerstad (3) might be adequate for approximating the surface roughness losses. Conversely, Fig. 10 depicts an optical profilometer measurement of a very rough copper foil and Fig. 11 depicts the same surface at 5000 magnification using a scanning electron microscope (SEM). Note the significant difference between the corrugated surface assumed by (3) and the depicted surface, suggesting that the Hammerstad (3) will not work well for this type of 3-D surface profile. The surfaces depicted in Figs. 10 and 11 can be characterized as random protrusions sitting on a flat plane, which pre-

Fig. 11. SEM photograph of rough copper at 5000 angle.

2 magnification at a 30

cludes rigorous derivation of an analytical formula to calculate the extra losses due to current flowing in the tooth structure. Subsequently, for the rough copper, an approximation of the tooth structure is required so that an analytical solution can be derived. As a first approximation, a hemispherical boss sitting on a plane was chosen to represent the individual surface protrusions, as shown in Fig. 12. The complete surface is modeled using hemispheres randomly distributed on a flat plane. A TEM wave is assumed incident on the hemisphere at a grazing angle of 90 with respect to the flat plane and with the -field (the magnetic field intensity) tangential to the surrounding plane, as shown in Fig. 12. In order to find the power dissipated by the structure, the absorption and scattering of the incident TEM wave upon the hemisphere was calculated. The problem of scattering of a plane wave from the hemispherical protrusion on the flat surface is solved using the method of images. This method is based on the fact that the field computed with a charge configuration above an infinite perfect electrical conducting (PEC) plane is electrically equivalent to the combination of a charged hemisphere and its image (inverted hemisphere) below the interface with the PEC plane removed. In essence, the scattering of a TEM wave incident on a hemispherical protrusion is equivalent to the scattering of the same wave incident on the hemisphere and its image hemisphere with the plane removed.

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and is the magnitude of the magnetic where field. Note that reasonable accuracy (at least up to 30 GHz) can be in (23) is considered obtained when only the first term when calculating (27). Equation (27) calculates the power loss of a hemisphere sitting on a perfect conducting plane. However, since real planes are not perfect conductors, the good conductor losses must be accounted for. The time-averaged power absorbed by a flat conducting plane per unit area is calculated in (28) [11] as follows: (28) In order to approximate the losses of a single hemispherical boss sitting on a flat plane of finite conductivity, the power loss of the hemisphere is added to the loss of the plane less the base area of the hemisphere Fig. 12. Simplified hemispherical shape approximating a single surface protrusion, top and side views, current direction, and applied TEM field orientations shown.

For a good conducting protrusion (as opposed to a PEC protrusion), a plane electromagnetic wave incident on a conducting sphere will be partly scattered and partly absorbed. The total power scattered and absorbed from a sphere divided by the incident flux is known as the total cross section and is calculated by Jackson [11], shown as follows in (23), with units of square meters: (23) where is the speed of light and the scattering coefficients are approximated assuming that , where is the sphere radius, and are given by (24) and (25) [11] as follows: (24)

(25)

The Poynting vector gives the power of an electromagnetic wave in units of watts per square meter, shown as follows in (26): (26) Subsequently, the total power absorbed or scattered is calculated by multiplying (23) and (26) and dividing by 2 so the results are for that of a hemisphere

(27)

(29) where is the tile area of the plane surrounding the protruis the base area of the hemisphere. sion (see Fig. 12) and Note that (29) is an approximation because it assumes that the on the tile is not affected by the presence of magnetic field the hemisphere, and the loss of the surrounding plane is simply a function of the area. A complete description of how the magnetic field behaves in the presence of a protrusion is described by Huray et al. in [7]. To gain an intuitive understanding of how the current on a plane will flow in the presence of a protrusion, it is useful to solve for the surface currents on a PEC sphere, which is a good approximation of how the current will flow at very high frequencies when the skin depth is small compared to the sphere because the current will only flow in the surface. If the magnetic field is defined in terms of a magnetic scalar potential, the (in amps per surface current density on the hemisphere square meter) can be derived as follows in (30) [7], [17]: (30) where is the angle between the applied magnetic field and the current flow (see Fig. 12). If the uniform current streamlines on the plane are matched with those of the sphere, the influence of a hemisphere sitting on a plane can be observed. Fig. 13 depicts how current will flow in the presence of a spherical protrusion. Note that the current on the flat portion of the plane is drawn towards the protrusion with the highest density on the top and minimal current density on the side perpendicular to the current flow. Lines of constant magnetic field intensity, which are orthogonal to the lines of current flow, are not shown to reduce complexity. The current crowding at the top of the protrusion effectively decreases the area where current flows, increases the path length, and thus helps explain the physical mechanisms that cause extra losses on a rough surface.

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Fig. 13. Current streamlines of flowing over a single protrusion.

In order to calculate a new correction factor for use in (2), the ratio of the power absorbed with and without a good conducting protrusion present must be derived as shown in (31) at the bottom of this page. This can be simplified as

(32) is the tile area of the plane surrounding where is the base area of the hemispheres, as the protrusion and shown in Figs. 12 and 13. Note that (32) becomes invalid when the skin depth is greater than the surface protrusion height. At these frequencies, the power dissipated by a flat plane with an area equal to the base of the protrusion will be greater than the power dissipated by the protrusion. Subsequently, a knee frequency can be defined when , where the roughness begins to significantly affect the is losses. Below the knee frequency, the correction factor unity. Subsequently, implementation of this correction factor is shown in (33) as follows: when when

Fig. 14. Example of a profilometer measurement of a rough copper sample showing peak heights that range from 0.7 to 8.5 m. The flat surface is assumed to be at 0.5 m.

. To obtain these input parameters, the surface is measured using a profilometer, as shown in Fig. 14. To facilitate this volume equivalent model, the tooth shape is approximated as one-half of a spheroid instead of a hemisphere, which more closely resembles the shape of the surface protrusions. The spheroid is, in turn, equated to one-half the volume of a sphere to calculate the radius of a hemisphere with the same volume as the surface protrusion, as follows in (34):

(34) tooth base width tooth height, and where radius of a hemisphere with equivalent volume of the tooth and . Note that is the which is used to calculate rms value of the tooth heights. The base area of the hemisphere, , is then calculated with (35) as follows: i.e., (35)

(33)

The square tile area of the surrounding flat plane is calculated based on the distance between peaks leading to (36) as follows:

To implement (33) accurately, the surface shown in Figs. 10 and 11 must somehow be represented by equivalent hemispheres. We have found that the additional surface area caused by simple hemisphere bosses is insufficient to account for the measured surface roughness losses. This is not surprising when one compares the additional hemisphere model area to the 3-D SEM microphotograph areas shown in Fig. 11. To account for additional surface area the rms volume of the rough surface must be calculated, and the radius of volume equivalent hemispheres are determined to compute the scattering coefficients and from (24) and (25). The rms distance between peaks in the roughness profile is used to calculate the tile area

(36) If the rms values of and values are calculated, the surface shown in Fig. 10 and measured in Fig. 14 can be represented as the equivalent surface in Fig. 15. A comparison between the correction factor calculated from Hammerstad (3) and (33) with the modified equivalent volume is shown in Fig. 16. Note that (33) saturates at a much higher value than Hammerstad, which will always saturate at a value of 2. Note that the correction factor calculated in (33) is discontinuous at approximately 100 MHz, which is the point where the surface of the flat plane with an area equal to the base of

(31)

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Fig. 17. Accuracy gained by implementing a frequency-dependent dielectric using (11) and the surface roughness model (33), 7-in microstrip, " = tan  = 3:9=0:0073 at 1 GHz, rms roughness of copper foil = 5:8 m, d = 9:4 m. h Fig. 15. Equivalent surface represented by hemispheres with the same rms volume as the measured surface profile.

Fig. 16. Hammerstad correction factor (3) compared to (33) (rms roughness: h = 5:8 m, d = 9:4 m).

the hemisphere has more losses than the hemisphere. Initially, we worried that the discontinuity would induce nonphysical glitches into the time-domain responses. However, simulations of pulses as fast as 30 Gb/s were performed using this method in HSPICE with no apparent nonphysical aberrations observed. Implementation of this method could include a smoothing function to guarantee a continuous transition between plane losses and hemisphere losses. To implement (33) in a transmission line model, the freand are calculated by scaling the quency dependence of values output from the 2-D electromagnetic field solver (i.e., the seed values), shown as follows in (37) and (38): when

IV. IMPLEMENTATION (37)

when when (38) when

where and are the surface resistance and inductance output from the field solver for a smooth conductor at the reference (seed) frequency, is calculated with (33), is the conductor thickness, is the skin depth, and is the frequency where the skin depth equals the thickness of the conductor. Note that (38) assumes the field solver outputs only the external inductance value, or the reference (seed) frequency is high enough so that the internal inductance is small. Fig. 17 depicts the increased accuracy gained in both magnitude and phase when (11) is used to calculate frequency-depenand matrices with (14) and (15), the surface roughdent ness correction factor is calculated with (33), and and are calculated with (37) and (38). Note that this methodology ignores the interactions between protrusions, which would affect the losses at high frequencies. However, as Fig. 17 shows, it provides a significant increase in accuracy for rough copper. The small deviation that occurs between the model and the measurement in Fig. 17 is because (29) essentially uses superposition to combine the losses of the tile and protrusion and does not account for the interaction between the hemisphere and plane. Consequently, the formula is very accurate: 1) at low frequencies, where the skin depth is large compared to the protrusion and the loss is primarily due to the plane and 2) at high frequencies, where the skin depth is small compared to the protrusion and the loss is primarily due to the roughness. At intermediate frequencies, where the skin depth is on the same order as the roughness height, an error is introduced. Additionally, the roughness shape was approximated with an ellipsoid and the interactions between spheres was neglected. Nonetheless, Fig. 17 shows that the methodology produces very reasonable accuracy over a wide bandwidth with a simple easy-to-use formula.

Implementation of this method in HSPICE or similar simulators require the use of a tabular transmission line model that and values at each frequency. To calculates the do so, the seed values of the table must be calculated with a 2-D electromagnetic field simulator such as AnsoftQ2D or Linpar at a reference frequency where the loss tangent and dielectric

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OR. Without their engineering and measurement expertise, this study would not have been possible. REFERENCES

Fig. 18. Demonstration of time-domain accuracy for a 30-Gb/s pulse (33-ps wide). New model calculated with (11) and (33); Traditional model calculated with frequency invariant tan =" and (3), 5-in microstrip, " = tan  = 3:9=0:0073 at 1 GHz, rms roughness of copper foil h = 5:8 m, d = 9:4 m. Simulation performed in HSPICE using the tabular transmission line model.

constant are known. Preferably, the reference frequency should be high enough so that the skin depth is small so that internal inductance can be considered negligible. The frequency dependence is calculated by scaling the terms from the seed values calculated at the reference frequency, as shown in (14) and (15) for the dielectric and (37) and (38) for the conductor. However, note that the method of calculating the surface roughness should be chosen carefully. If relatively correction factor smooth copper is being used, with an rms value of the surface roughness less than 2 m, then Hammerstad’s formula (3) has been shown to adequately approximate the surface roughness losses. However, for very rough copper (which is often preferred by printed circuit board vendors due to its decreased tendency of delaminating), (33) will approximate the surface roughness losses with significantly more accuracy. V. CONCLUSION This paper has presented a methodology for modeling transmission lines that has been shown to be accurate to at least 30 GHz. Although equipment limitations did not allow the demonstration of higher frequency accuracy for this paper, indirect validation of pulse responses indicate that this methodology works well up to 30 Gb/s, as demonstrated in Fig. 18. Note that this methodology correctly predicts both the amplitude and delay of the pulse response. A specific modeling method for predicting the frequency-dependant properties of dielectric materials, which is necessary to generate real and causal transient responses, was presented. Furthermore, a brand new modeling methodology for calculating the extra losses from surface roughness were presented that facilitates the modeling of the industries roughest coppers.

[1] S. G. Pytel, G. Barnes, D. Hua, A. Moonshiram, G. Brist, R. Mellitz, S. Hall, and P. G. Huray, “Dielectric modeling, characterization, and validation up to 40 GHz,” presented at the 11th Signal Propag. on Interconnects Workshop, Genoa, Italy, May 13–16, 2007. [2] S. Hall, T. Liang, H. Heck, and D. Shykind, “Modeling requirements for transmission lines in multi-gigabit systems,” in IEEE Elect. Perform. Electron. Packag. Symp., Oct. 25–27, 2004, pp. 67–70. [3] T. Liang, S. Hall, H. Heck, and G. Brist, “A practical method for modeling PCB transmission lines with conductor surface roughness and wideband dielectric properties,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1780–1783. [4] P. G. Huray, S. G. Pytel, R. I. Mellitz, and S. H. Hall, “Dispersion effects from induced dipoles,” presented at the 10th Signal Propag. on Interconnects Workshop, Berlin, Germany, May 09–12, 2006. [5] G. Brist, S. Hall, S. Clouser, and T. Liang, “Non-classical conductor losses due to copper foil roughness and treatment,” in ECW 10 Conf. at IPC Circuits Designers Summit, May 2005, pp. S19-2-1–S19-2-1-11. [6] E. Sijercic, A. Moonshiram, O. Oluwafemi, and G. Brist, “Modeling surface roughness with an array of hemispheres,” IEEE Trans. Microw. Theory Tech., Dec. 2006, submitted for publication. [7] P. G. Huray, S. Hall, S. G. Pytel, F. Oluwafemi, R. Mellitz, D. Hua, and P. Ye, “Fundamentals of a 3-D “snowball” model for surface roughness power losses,” presented at the IEEE Signals Propag. on Interconnects Conf., Genoa, Italy, May 14, 2007. [8] S. P. Morgan, Jr., “Effects of surface roughness on eddy current losses at microwave frequencies,” J. Appl. Phys., vol. 20, pp. 352–362, 1949. [9] Hammerstad and Jensen, “Accurate models for microstrip computer aided design,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1980, pp. 407–409. [10] S. Ramo, Fields and Waves in Communication Electronics, 2nd ed. New York: Wiley, 1965. [11] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1999. [12] W. LePage, Complex Variables and the Laplace Transform for Engineers. New York: Dover, 1961. [13] A. Djordjevic, R. Biljic, V. Likar-Smiljanic, and T. Sarkar, “Wideband frequency domain characterization of FR4 and time domain causality,” IEEE Trans. Electromagn. Compat., vol. 43, no. 4, pp. ??662?–667, Nov. 2001. [14] S. Hall, G. Hall, and J. McCall, High-Speed Digital System Design. New York: Wiley, 2000. [15] T. Arabi, A. Murphy, T. Sarkar, R. Harrington, and A. Djordjevic, “On the modeling of conductor and substrate losses in multi-conductor transmission line systems,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1090–1097, Jul. 1991. [16] L. Proekt and A. Cangellaris, “Investigation of the impact of conductor surface roughness on interconnect frequency-dependent ohmic loss,” in Proc. 53rd Electron. Compon. Technol. Conf., May 27–30, 2003, pp. 1004–1010. [17] Orlando and Delin, Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991.

Stephen Hall joined the Special Purpose Processor Division, Mayo Foundation, in 1992, where he developed multigigabit modeling techniques for X -band digital radar and serial optical links. In 1996, he joined the Intel Corporation, Hillsboro, OR, where he was Lead Designer for buses on Pentium II–IV systems, coordinated research with universities, lead research teams in the area of high-speed modeling, and taught signal integrity courses. He authored the textbook High-Speed Digital System Design (Wiley 2000). He is currently involved in researching new modeling and measurement solutions for channel speeds as high as 30 Gb/s.

ACKNOWLEDGMENT The authors wish to acknowledge the assistance and support of O. Oluwafemi, Intel Corporation, DuPont, WA, D. Shykind, Intel Corporation, Hillsboro, OR, P. Hamilton, Intel Corporation, Hillsboro, OR, and G. Barnes, Intel Corporation, Hillsboro,

Steven G. Pytel (S’01–M’02) was born in Peru, IL, on April 19, 1974. He received the B.S.E.E. degree from Northern Illinois University, DeKalb, in 2002, and the M.S.E.E. and Ph.D. degrees from the University of South Carolina, Columbia, in 2004 and 2007, respectively.

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From 1992 to 2001, he served within the United States Navy as an Electrical Technician, followed by an RF internship with Northrop Grumman during Summer 2002. In January 2005, he joined the Intel Corporation, Hillsboro, OR, as a Signal Integrity Engineer. His current research interests include high-speed serial signaling, transmission line modeling (surface roughness and dielectric losses), and statistical analysis of digital circuits.

Paul G. Huray (M’01) received the B.S. degree in engineering and Ph.D. degree in physics from the University of Tennessee, Knoxville, in 1964 and 1968, respectively. He then performed post-doctoral work in physics with the University of North Carolina and in higher education management with Harvard University. He is currently a Professor of electrical engineering with the University of South Carolina, Columbia, where his research concerns signal integrity. Prior to 1988, he was a Senior Policy Analyst for the White House Office of Science and Technology Policy, and prior to 1985, he held a joint appointment as a Professor with the University of Tennessee and the Oak Ridge National Laboratory, where he conducted research in solid-state physics.

Daniel Hua received the A.B. degree in applied mathematics and Ph.D. degree in engineering sciences from the University of California at Berkeley, in 1985 and 1992, respectively. He has held several post-doctoral and lecturer appointments with the Centre National de la Recherche Scientifique, Orleans, France, the Lawrence Livermore National Laboratory, and the University of California at Berkeley. He is currently with the Intel Corporation, Hillsboro, OR.

Anusha Moonshiram (S’02–M’06) received the B.S.E.E. and M.S.E.E. degrees from Tufts University, Medford, MA, in 2003 and 2004, respectively. From May 2001 to 2004, she was a Research Assistant with the Millimeter Wave Laboratory, Tufts University, where she concentrated on error analysis of dielectric measurements of low-loss and lossy materials at millimeter-wave frequencies using the open resonator system, discrete Fourier transform spectroscopy method, network analyzer, and the -band high-power spectrometer. Since July 2004, she has been with the Intel Corporation, Hillsboro, OR, where she is focused on signal integrity topics including high-frequency measurement and 2-D/3-D modeling. She has authored ten IEEE publications Ms. Moonshiram is a Society of Women Engineers (SWE) member.

W

Gary A. Brist received the B.S. degree in electrical engineering from Montana State University, Bozeman, in 1990, and the M.S. degree in electrical engineering from Purdue University, West Lafayette, IN, in 1992. He is currently a Staff Printed Circuit Board (PCB) Technologist with the Intel Corporation, Hillsboro, OR, where he is involved with the System Materials Technology Development Group. Prior to joining the Intel Corporation, he held various engineering and engineering management positions within the PCB fabrication industry.

Edin Sijercic was born in Banja Luka, Bosnia and Hercegovina, in 1975. He received the M.S. degree in electric engineering and M.S. degree in physics from Portland State University, Portland, in 2002 and 2003, respectively. He is currently a Senior Signal Integrity Engineer with the Intel Corporation, Hillsboro, OR.

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A Novel 180 Hybrid Using Broadside-Coupled Asymmetric Coplanar Striplines Lap K. Yeung, Member, IEEE, and Yuanxun E. Wang, Member, IEEE

Abstract—In this paper, a new way to realize a 180 hybrid is presented. This technique utilizes a broadside-coupled asymmetric coplanar-stripline section to meet the tight coupling requirement that usually arises from the traditional reduced-size hybrid-ring configuration. In addition, this coplanar technique facilitates the implementation of a hybrid ring in a cascadable format. While providing a simple design procedure for the device, analytical details are also discussed. An experimental prototype, operating at 2.4 GHz, has been fabricated using conventional printed circuit board technology. Measured performances show good agreement with analytical and full-wave simulations, showing promising potential for a variety of applications such as balanced mixers, power amplifiers, and antenna feed networks. Index Terms—Coplanar stripline (CPS), coupled transmission lines, multilayer substrate, 180 hybrid.

I. INTRODUCTION HE 180 hybrid coupler is a fundamental component in microwave circuits and is used extensively in a variety of applications including balanced mixers, multipliers, power amplifiers, and antenna feed networks. However, the conventional transmishybrid ring, which consists of three sections of sion line and one section of transmission line, is inherently narrowband and large in size, as these shortcomings are mainly line section, March [1] proposed due to the presence of a coupled-line section having a method of replacing it with a short-circuited diametrically opposing ends to simultaneously improve the bandwidth and reduce the size of the ring. Similar line in the section with broadideas, by replacing the band phase inverters, have also been proposed [2]–[6]. Nonetheless, all of these methods do not lend themselves in an easy way to be realized in a microstrip format. Besides both bandwidth and size limitations, the conventional hybrid ring does not have a convenient layout for multiple cascading, which can be quite undesired since many applications, especially the antenna feed network discussed in [7], require the use of two or more 180 hybrids in a cascaded manner. In this study, a new small-size 180 hybrid based on the configuration suggested by March is proposed. This hybrid utilizes a pair of broadside-coupled asymmetric coplanar stripline cou(ACPS) to achieve strong coupling required by the pled-line section, and at the same time, to facilitate a cascadable

Fig. 1. Schematic for a reduced-size hybrid ring of 3-dB coupling.

realization. While providing a simple design procedure for the device, its analytical model is also discussed. In addition, some of the layout issues are also addressed. An experimental verification is provided to validate the proposed concept.

T

Manuscript received June 4, 2007; revised August 17, 2007. The authors are with the Electrical Engineering Department, University of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910067

II. THEORY A. Reduced-Size Hybrid Ring The hybrid ring proposed by March has a wider operating bandwidth and smaller size than its conventional counterpart. This is due to the fact that a short-circuited coupled-line line section, which is the major section is used instead of a limiting factor for the conventional ring. Fig. 1 illustrates the schematic layout for this reduced-size hybrid ring. According to the theory, the even- and odd-mode impedances and should be set to (1a) (1b) for a 3-dB power-split ratio. Hence, for a 50- transmission and should be 176.2 and 30.2 , respectively. system, As this coupled-line section can well approximate a phase-reversing network over a wide frequency range, a band-broadened hybrid ring has resulted. Moreover, the four arcs are now all a quarter-wave in length, and thus leads to an overall size reduction. B. Broadside-Coupled ACPSs According to the schematic-level design, even- and odd-mode impedances for the coupled-line section are 176.2 and 30.2 , respectively. These values represent a pair of strongly coupled lines that cannot be easily realized by a conventional edge-coupled microstrip pair. To overcome such a problem, a broadside-coupled ACPS section is used instead. Fig. 2 shows the configuration of such a coupled line pair.

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Fig. 2. Pair of broadside-coupled ACPSs with the corresponding even- and odd-mode equivalents.

Due to symmetry, fundamental properties of a broadside-coupled ACPS pair are completely characterized by its even- and odd-mode equivalents. Analytical expressions for these impedances can be obtained by the conformal mapping technique similar to the one for analyzing a broadside-coupled coplanar waveguide (CPW) [8]. Let the even- and odd-mode quasi-static capacitances per and , reunit length in the substrate region be spectively. By applying the logarithmic ( - to -plane) and Schwarz–Christoffel ( -plane to -plane) then transforms or

(2a) or

Fig. 3. Even- and odd-mode impedances for a pair of broadside-coupled ACPSs on a substrate of " = 10:2.

Using the same procedure, but no logarithmic transform, the capacitance per unit length in the air region can be obtained as

(7) where

(8) Notice that (7) and (8) are valid for both even- and odd-mode cases. Based on these expressions, the even- and odd-mode effective permittivities and characteristic impedances are, respectively,

(2b) (9a)

to those diagrams shown in Fig. 2. We can obtain analytical expressions to calculate these capacitances with

(9b) (3)

and (10a)

(4) (10b) where

is the complete elliptical integral of the first kind and

(5)

(6)

Fig. 3 shows the even- and odd-mode impedances for a pair of broadside-coupled ACPSs obtained from (9) and (10) based on . As seen from this figure, the odd-mode a substrate of impedance is almost constant for large values of gap-to-height ratio. For small values, it only increases slowly with the ratio. According to the calculations, and should be approximately and , respectively, in order to achieve the equal to desired impedances, as stated previously.

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Fig. 4. Equivalent model for the line section with a crossover junction.

Fig. 6. Metallization patterns on the top and bottom PCB layers of the proposed hybrid.

Fig. 5. (a) 3-D physical layout of the proposed 180 hybrid. (b) Cross section of the broadside-coupled ACPSs.

C. Crossover Junction In the proposed hybrid, a crossover is introduced in one of the line sections. This crossover makes it possible for the sum and difference ports to reside on one side, whereas the input (IN) and isolated (ISO) ports reside on the other, and thus leads to the advantage of easy cascading. For a thin and short crossover wire, it can be approximated simply by an inductor. In essence, a transmission-line section having a crossover junction can be modeled by an equivalent circuit shown in Fig. 4. Ideally, in order to have a minimum disruption, it is necessary for this transmission line of equivalent circuit to behave as close to a as possible. The corresponding values for impedance and under such circumstances can be calculated by equating matrix of each case and results in the following the equations: Fig. 7. Schematic-level simulations for the ideal and proposed reduced-size hybrid-ring. (a) Return loss and isolation. (b) Insertion losses.

(11a) (11b) (11c)

where and are normalized to . The three equations are not independent of each other, and any two equations among the three are valid for solution.

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Fig. 8. Schematic-level simulated results and experimental measurements of the prototype hybrid. (a) Return loss and isolation. (b) Insertion losses. (c) Phase difference between S 21 and S 31. (d) Phase difference between S 24 and S 34. (e) Amplitude balance between S 21 and S 31. (f) Experimental prototype.

Now, given a particular value of and (11b) gives

and

, solving (11a)

(13)

(12)

For example, with a 2-nH inductor, the line should have an impedance of and a length of 37.5 so that

YEUNG AND WANG: NOVEL 180 HYBRID USING BROADSIDE-COUPLED ACPSs

the equivalent circuit behaves like a transmission line of impedance . It is seen from (13) that only a limited range can be “compensated.” of inductance, namely, III. PHYSICAL LAYOUT Fig. 5(a) depicts the physical layout of the proposed 180 hybrid. Several points about this layout are worth mentioning. Firstly, the coupled ACPS section has a slightly different configuration from the one shown in Fig. 2. The two grounds actually reside at two opposite sides, not directly stacking together [see Fig. 5(b)]. Nonetheless, (9) and (10) can still be used to roughly estimate its even- and odd-mode impedances. Secondly, the transmission-line connecting port- and port-IN, and the one connecting port- and port-ISO lie on different layers of the PCB. Thirdly, a crossover is employed to flip around the locations of port- and port-ISO such that a straightforward cascading of two hybrids is possible. According to the analysis presented above, the crossover wire should be short and thin in order to minimize any undesired effect. Fig. 6 shows the metallization patterns on the top and bottom PCB layers of the proposed layout. Notice that subminiature A (SMA) connectors can directly connect to the transmission lines leading to the four input ports. With the necessary analytical models for broadsided-coupled ACPSs and crossover junction developed, it is now ready to design a schematic-level reduced-size hybrid ring based on these components. In this design, the branch line impedance rather than is used in order to have a good input matching. This leads to the even- and odd-mode impedances of 170.7 and 29.3 , respectively, for the coupled-line section. Therefore, given a 0.64-mm-thick substrate of and a 2.4-GHz operating frequency, the gapwidth, linewidth, , 0.77 mm , and and length should be 0.8 mm 13.1 mm respectively. In addition, the two transmission lines of an assumed 2-nH crossover junction should be and 37.2 in length. According to these parameter values, a schematic-level design using Agilent Technologies’ Advanced Design System (ADS) has been obtained and the corresponding simulation results are shown in Fig. 7. Notice that the built-in multilayer transmission line model in ADS has been used to model the broadsided-coupled ACPS section. Compared with and the ideal reduced-size hybrid-ring, the isolation of the proposed hybrid-ring amplitude balance are slightly worse. As seen from these simulation results, it is clear that the analytical model previously developed provides a good way to obtain an initial design. However, due to various approximations made in the model, this initial design should be further optimized to achieve better performances, and using a gradientbased optimization procedure in ADS, the final schematic-level design is obtained. Firstly, the characteristic impedance for two quarter-wave sections should be 70.7 . Secondly, the line section with a crossover junction should be 2 237.2 in length and 53.6 in impedance. Translating these values into physical dimensions for a prototype implementing on a 0.64-mmthick substrate of , the two transmission-line sec-

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tions connecting port- and port-IN, and connecting portand port-ISO should each be 12.5-mm long and 0.24-mm wide. The transmission line section with a crossover junction should be 2 5-mm long and 0.49-mm wide with an assumption of a 2-nH crossover wire. Finally, the broadside-coupled line section should be 12.3-mm long and has a linewidth and gapwidth of 1 and 0.3 mm, respectively. These three values are obtained after the optimization and are different from the values calculated analytically in Section II. IV. EXPERIMENTAL RESULTS An experimental prototype has been fabricated and the corresponding measured results are shown in Fig. 8. From the measurements, it is seen that the prototype achieves a 0.4-dB max3 phase difference (beimum amplitude balance and a 180 – ) within a 400-MHz bandwidth. In addition, a tween good isolation and an input port matching are obtained with both of them less than 20 dB. Notice that the bandwidth is limited by the crossover wire realization in this prototype. If a multilayered substrate technology such as low-temperature co-fired ceramic (LTCC) is used, the bandwidth can be wider. Notice that the difference between measurements and simulations around the center frequency is mainly due to the inaccurate assumption of the crossover wire inductance. It has been confirmed by a schematic-level-based simulation that when the inductance is lower than expected (in this case, 2-nH inductor), or will be split into two, just the single sharp notch of like those shown in the experimental measurements. To improve the prototype’s performances, including bandwidth, a precise control of the crossover wire is essential. V. CONCLUSION A novel 180 hybrid using broadside-coupled ACPSs has been proposed. The device is based on the reduced-size hytransmisbrid-ring configuration, which does not need a sion-line section and, therefore, has a smaller size than the traditional “rat-race” coupler. Due to the use of coplanar line technique, it also offers a convenient layout for cascading. Antenna beamforming networks, as well as other applications that require cascading of multiple hybrids, may find benefits from this device. Models for a broadside-coupled ACPS section and a crossover junction have been developed to assist the design. In addition, computer simulations and experimental measurements have been conducted to validate the proposed concept, showing its promising potential for many classical, as well as new applications. REFERENCES [1] S. March, “A wideband stripline hybrid ring,” IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 6, p. 361, Jun. 1968. [2] S. Rehnmark, “Wide-band balanced line microwave hybrids,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 10, pp. 825–830, Oct. 1977. [3] M. Murgulescu, E. Penard, and I. Zaquine, “Design formulas for generalized 180 hybrid ring couplers,” Electron. Lett., vol. 30, no. 7, pp. 573–574, Mar. 1994. [4] T. Wang and K. Wu, “Size-reduction and band-broadening design technique of uniplanar hybrid ring coupler using phase inverter for M(H)MIC’s,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 2, pp. 198–206, Feb. 1999.

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[5] C. Chang and C. Yang, “A novel broadband Chebyshev-response ratrace ring coupler,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 4, pp. 455–462, Apr. 1999. [6] C. Kao and C. Chen, “Novel uniplanar 180 hybrid-ring couplers with spiral-type phase inverters,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 412–414, Oct. 2000. [7] X. Wang and Y. E. Wang, “A “zoom-in” scanning array for wireless communications,” in IEEE Radio Wireless Symp. Dig., Jan. 2007, pp. 491–494. [8] R. N. Simons, Coplanar Waveguide Circuits, Components, and Systems. New York: Wiley, 2001. Lap K. Yeung (S’00–M’02) received the B.Eng. degree in electrical and information engineering from the University of Sydney, Sydney, Australia, in 1998, the M.Eng. degree in electronic engineering from the Chinese University of Hong Kong, Shatin, Hong Kong, in 2002, and is currently working toward the Ph.D. degree at the University of California at Los Angeles (UCLA). During 1999, he was with the Commonwealth Scientific and Industrial Research Organization (CSIRO), Sydney, Australia, where he was a Research Engineer involved in the numerical modeling of different antenna structures. From 2003 to 2006, he was with the Chinese University of Hong Kong, where he is involved in various LTCC multichip-module (MCM) designs and the development of numerical algorithms for analyzing multilayer embedded RF modules.

Yuanxun E. Wang (S’96–M’99) received the B.S. degree in electrical engineering from the University of Science and Technology of China (USTC), Hefei, China, in 1993, and the M.S. and Ph.D. degrees in electrical engineering from the University of Texas at Austin, in 1996 and 1999, respectively. From 1993 to 1995, he was a Graduate Researcher with USTC, where he was involved with numerical methods and millimeter-wave radar-based instruments. From 1995 to 1999, he was with the Department of Electrical and Computer Engineering, University of Texas at Austin, where he was a Graduate Research Assistant involved with radar scattering modeling and synthetic aperture radar (SAR) imaging. From 1999 to 2002, he was a Research Engineer and Lecturer with the Department of Electrical Engineering, University of California at Los Angeles (UCLA). In November 2002, he became an Assistant Professor with the Electrical Engineering Department, UCLA. He has authored or coauthored over 60 refereed journal and conference papers. He has been involved with novel experimental architectures and hardware implementations for high-performance antenna array and microwave amplifier systems with applications in wireless communication and radar sensors. A portion of his research also involves numerical modeling techniques for microwave circuits. His research interests feature the fusion of signal-processing and circuit techniques into microwave system design. Dr. Wang is a member of the International Society for Optical Engineers (SPIE).

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Frozen Modes in Parallel-Plate Waveguides Loaded With Magnetic Photonic Crystals Ryan A. Chilton, Student Member, IEEE, Kyung-Young Jung, Student Member, IEEE, Robert Lee, Senior Member, IEEE, and Fernando L. Teixeira, Senior Member, IEEE

Abstract—We examine parallel-plate magnetic photonic crystal (MPC) waveguides comprised of a periodic loading with two anisotropic layers and one ferrite layer on each period. It is shown that, at a specific design frequency 0 , parallel-plate MPC waveguides can support a Bloch mode with zero axial group velocity similar to the “frozen mode” regime of the prototypical 1-D MPC structure. When the proposed 2-D structure is illuminated with a properly polarized time–harmonic wave, near unity power transmission (coupling) into the frozen mode occurs, and field strength within the MPC becomes orders of magnitude larger than the incident field strength. The steady-state case is evaluated using both analytical tools and finite elements, while the finite-difference time-domain method is applied to evaluate the 2-D MPC transient response. Index Terms—Dispersion engineering, frozen mode, guidedwave propagation, magnetic photonic crystals (MPCs), numerical techniques in electromagnetics.

I. INTRODUCTION ISPERSION-ENGINEERED metamaterials have shown great promise as building blocks for novel devices with unique electromagnetic responses. Photonic crystals (periodic structures of contrasting dielectric media) are a particular wellknown class of dispersion-engineered materials. The electromagnetic-bandgap (EBG) properties of photonic crystals can be readily explored, for example, for waveguiding and/or containment (resonator) applications [1]. Recently, Figotin and Vitebsky [2] proposed a magnetic photonic crystal (MPC) structure that included a gyrotropic ferrite layer sandwiched between misaligned anisotropic dielectric layers. Though this periodic structure exhibited bandgap behavior, it also had a new and unique spectral feature: a stationary inflection point (SIP) within its dispersion curve. Such a SIP leads to a unidirectional Bloch mode, which propagates with vanishingly small group velocity and increasingly large amplitude [3]. This propagation regime was coined the frozen mode and has since been utilized by Mumcu et al. [4], [5], who

D

Manuscript received May 3, 2007; revised August 27, 2007. This work was supported in part by the Air Force Office of Scientific Research under Multiuniversity Initiative Grant FA 9550-04-1-0359, by the National Science Foundation under NSF Grant ECS-0347502, by the Ohio Superconductor Center under Grant PAS-0061 and Grant PAS-0110, and by Northrop Grumman under a fellowhip. The authors are with the ElectroScience Laboratory and Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]; [email protected]; lee@ece. osu.edu; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.910063

sought to exploit its large amplitude growth to create highly directive miniature antennas [6] and arrays [7]. Additional relevant work on MPCs has included detailed finite-element analysis [8] and studies aimed at improving the bandwidth of the frozen mode regime using chirping of the unit cells [9]. Reference [9] also discussed other examples of devices (an isolator and long true-time delay line), which exploit the unique properties of the MPC. More recently, a sensitivity analysis of MPCs under geometric (manufacturing) perturbations and versus frequency variations has been considered in [10] and [11]. References [2]–[11] contain promising numerical results, but are limited to a 1-D “optical film” interpretation of the MPC structure and require longitudinally biasing a thin ferrite sheet, which, in practice, is difficult. This paper considers 2-D parallel-plate MPC waveguides. The bias configuration is redesigned to avoid the need to longitudinally bias a thin ferrite sheet. A modal transfer matrix formulation demonstrates that this parallel-plate structure, with appropriately chosen constitutive materials, also supports a guidedwave analogous to the 1-D frozen mode. The analysis is done via a combination of both analytical and numerical methods. The dispersion curve and design parameters are obtained using a analytical approach based on Bloch mode solutions via transfer matrices. The finite-element method (FEM) is employed to obtain the time–harmonic solution for determining the transmission coefficient from air to MPC and the amplitude growth inside the semiinfinite MPC, and for fine tuning the frozen mode bandwidth via modifications on the layer thickness. The finitedifference time-domain (FDTD) method is used to provide insights into the transient response of the structure and as a tool for a sensitivity study of the MPC response with respect to perturbations on the central frequency of excitation and to ferromagnetic losses in the ferrite layers. The combination of different methods also allows for cross-validation of the results. II. MODAL ANALYSIS OF PARALLEL-PLATE WAVEGUIDES LOADED WITH ANISOTROPIC AND GYROTROPIC MEDIA A. Problem Description Consider the infinite periodic problem depicted in Fig. 1, where two slabs of misaligned anisotropic media ( and ) are combined with a layer of gyrotropic ferrite media ( ) to form a single unit cell. The choice of the coordinate system is such that the -axis is the axis of invariance (not the direction of propagation). The dielectric layers and are biaxial crystals [12], which are misaligned about the -axis. Their misalignment angles are equal in magnitude and opposite in direction,

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decoupled. The introduction of anisotropic media couples these two solution types together. However, decomposition into four or modes, where all fields have either modal distribution, is still possible. In this case, each mode possesses all six field components propagating with a common wavenumber . The and components of a single mode are given by (5) as follows:

(5) Fig. 1. Schematic of parallel-plate waveguide loaded with a periodic MPC.

with they are the same physical medium, but their crystal orientations yield distinct constitutive tensors in the – – frame (1). Their magnetic permeability is that of free space and

(1) The ferrite layer ( ) is biased along the -axis, yielding gyrotropic activity in the – -plane. Its dielectric contrast is assumed isotropic, but not necessarily equal to free space

Here, is a scalar “mode weight” and is the free-space wavenumber . The remaining (transverse to ) field components for this mode are found from Maxwell’s equations by differentiating and (for brevity, not performed here). The wavenumber is given by a quartic dispersion relation (6) as follows: (6) where

(2) In contrast to the 1-D MPC structure, here the axis of ferrite bias and biaxial misalignment is not the axis of propagation, but instead a transverse axis. This is a key step for developing closedform modal field solutions in both media types, and is also desirable from a manufacturing standpoint. The 1-D MPC required the magnetization of a “thin plate” of ferrite with large “demagnetization factor” [13]; the 2-D guided-wave MPC avoids this complication. Sections II-B and C will solve Maxwell’s equations (3) and (4) as follows within both media: (3) (4) The individual slab solutions will then be combined into a transfer matrix, which relates the fields at the “left” end of a unit cell to the fields at the “right” end of a unit cell. The Bloch mode wavenumbers supported by the infinite periodic structure can be extracted from this transfer matrix, and they will show the unique propagation characteristics of the parallel-plate loaded MPC.

Each of the four roots to (6) characterizes a unique mode. If the anisotropy is removed by setting , this dispersion relation recovers the isotropic case (the four modes reduce to forward/backward propagating and forward/backward propagating waves). In the most general case, the total field will be given by summing over both mode “order” and mode “index” . Each order is a complete solution of Maxwell’s equations, and can exist independently of the others; i.e., tangential field boundary conditions can be matched at each slab without resorting to coupling between modes with different , only modes with different are required. That is a small finite set and makes for a very tractable problem. It is hereon assumed that the guide is excited with only the fundamental modes so complete summation over is not necessary. Only summation over is needed to give the total fields written as

B. Anisotropic Dielectric Layers In a waveguide filled with isotropic dielectric media, the th-order modes ( - and -field components) and modes ( - and -field components) are

(7)

CHILTON et al.: FROZEN MODES IN PARALLEL-PLATE WAVEGUIDES LOADED WITH MPCs

Similar expressions (for brevity, not supplied here) can be and . Given an derived for the other tangential fields -coordinate and set of eigenweights , (7) can be rewritten in matrix notation to determine the total tangential material slab via invertible 4 4 fields anywhere in the weight matrix as

or denoting the field polarization state as (8) becomes (9) as follows:

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D. Bloch Mode Solution Via Transfer Matrices

(8)

and can be combined The weight matrices into a “transfer matrix,” which relates tangential fields at one end of the unit cell to tangential fields at the other end. Assume a pois impressed upon the structure at . larization state The weight matrix can be inverted to find the unique within the slab, which produce these imeigenweights pressed fields. These weights are then multiplied by to find the polarization state impressed at the other end of the slab at . This gives

,

(13)

(9)

This procedure can be repeated for the and slabs to find the , as shown in polarization state at (14). The “transfer matrix” for the entire unit cell, i.e., , emerges as the product of all these weight matrices

Another weight matrix exists for the slab, the only differuses a different set of constitutive parameters ence is that according to (1).

(14) where the transfer matrices for each slab are

C. Gyrotropic Ferrite Layer The solution within the ferrite slab is similar; it follows by modes is possible. The duality. An expansion into four modal wavenumbers are given by the quartic dispersion relation coefficients are given by (6) where the generating

(10) The total - and -fields are given by (11) as follows, the other fields can be found through Maxwell’s equations and differentiation:

Since the MPC structure is infinite periodic (with period ), Bloch (or Floquet) theorem requires that all field components are themselves periodic with some phase shift between each unit cell. The Bloch wavenumber (15) gives this phase shift as follows:

(15) Substitution of (14) into (15) and rearranging yields the familiar eigenpair statement (16) as follows:

(16) (11) with

i.e., the Bloch wavenumbers can be found from the eigen. Since is a 4 4 values of the transfer matrix matrix, it possesses four eigenvalues . Each one represents a permissible Bloch mode

(17) Using the weight matrix notation, the tangential field state can be found at any point within the ferrite slab from the eigenmode weights by the following modal summation:

(12)

E. Distinctions From the 1-D MPC Prototype Before proceeding, it is worthwhile to step back from the analysis and provide a brief synopsis of the basic similarities and differences between the proposed structure and the 1-D MPC prototype considered in [2] and [3].

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The 1-D MPC exhibits nonreciprocity through Faraday rotation, i.e., as plane waves impinge upon the layer, they are split into circular components and phase shifted by different lags that depend upon the direction of propagation. In addition to this well-known longitudinal bias case, when the static magnetization of an unbounded ferrite medium (1-D slab) is perpendicular to the axis of propagation, there is still a straightforward solution (the medium supports a linearly polarized plane wave). This solution does exist in the ferrite layer of the parallel-plate MPC: “perpendicular” mode described by Fuller it is the TEM solution of the [14]. It couples into the layers, but the physics of this configuration is uninteresting. This mode can only exhibit a symmetric band diagram, just like a regular photonic crystal. The 2-D MPC oblique/guided modes considered here fall into neither of these cases. In the coordinate frame of either ray of the waveguide mode, the static magnetization is applied at an arbitrary angle, and the material tensor has a complicated form [15]. Fuller provides the dispersion relation for this case, similar -orthoray – coordito (6) and (10), but solved in a ray nate system, instead of the orthobias -bias – frame used here. This configuration is rarely encountered in the literature. In fact, Fuller goes as far to remark that “there are very few, if any applications where the direction of propagation of an electromagnetic wave in a magnetised ferrite material is at an arbitrary angle to the direction of static magnetisation.” Similar observations can be made about the fields within the layers. One can tell these waves are not propagating like “ordinary” and “extraordinary” rays that are encountered while analyzing the 1-D MPC because their propagation constants depend upon all three optical indices. These differences in wave behavior make the phenomenology of the 2-D parallel-plate MPC quite distinct from the previously considered 1-D structures. However, the 2-D guided-wave MPC can still yield the most important features of the 1-D MPC, including spectral nonreciprocity [16]. In particular, with appropriately designed material parameters, construction of a Bloch dispersion, which exhibits a SIP, is still possible.

Fig. 2. Bloch wavenumber locus for an example 2-D MPC structure.

propagates with an extremely small group velocity. In 1-D, this mode is coined the frozen mode. 2) The frozen mode is unidirectional. Three degenerate eigenvalues are required to construct a SIP, and only four are available from the underlying fourth degree characteristic equation. Hence, a SIP cannot appear in both branches of the band diagram (which becomes spectrally asymmetric) and only has an effect upon waves propagating in one direction. 3) As a consequence of vanishingly small group velocity and nonzero power transmission, conservation of energy requires that the frozen mode exhibits dramatic amplitude increase as . Remarkably, we show here that 2-D MPC structures described by Fig. 1 can also exhibit band diagrams with a SIP akin to the 1-D MPC case. Fig. 2 depicts such an example, whose parameters are given in (19) as follows:

F. Frozen Mode Regime Example Many periodic structures loaded within parallel-plate waveguides can be analyzed using the transfer matrix formulation and will yield EBG/photonic bandgap (PBG) effects. What makes the MPC unique is the interesting locus that the Bloch wavenumber traces out away from the bandgap. A 1-D MPC can be constructed such that its dispersion relation exhibits a SIP at some design frequency . At such a point, the first and second derivatives of the dispersion relation are zero, while the third derivative is strictly nonzero, shown as follows in (18):

(18) A detailed treatment of the consequences of a SIP appear elsewhere [2], [3], but the key features are as follows. 1) Like the propagation behavior near a bandgap , the group velocity becomes vanishingly small as the SIP frequency is approached. However, power transmission is still nonzero at the SIP (no actual cutoff occurs). This indicates the existence of a guided Bloch wave mode, which

cm

cm cm

cm

(19)

Determining a set of constitutive parameters and dimensions that support a SIP is an iterative somewhat trial-and-error process. The first step is selecting an order and height such that the given mode propagates above cutoff. The optical indices, thicknesses, and misalignment angles between layers and are then picked such that a complete bandgap appears

CHILTON et al.: FROZEN MODES IN PARALLEL-PLATE WAVEGUIDES LOADED WITH MPCs

slightly above the design frequency. The thickness of the gyrotropic layer is then slowly increased from zero, which can introduce spectral nonreciprocity and a continuous deformation of the band diagram. If the curve deforms into something close to a SIP, continue tuning the dielectric misalignment angles and ferrite thickness, and a SIP can eventually be realized. If the addition of a thick ferrite does not yield anything like a SIP, start over with new optical indices and thicknesses. Since computing the band diagram by computer is fast, a designer can check many possibilities in a short turnaround time until a suitable diagram is found. The example above is of an academic nature, tuned for a SIP at 300 MHz so that all dimensions in meters are in wavelengths as well. Any structure can be scaled to arbitrary frequency, the major limitations being the availability of low-loss ferrites and anisotropic materials with large axial ratios at the frequency and ytof interest. In the microwave regime, rutile for trium–iron–garnet (YIG) for are promising choices. Section III will investigate the properties of this specific MPC structure (19), evaluating the claims of nonzero power transmission at the SIP and divergent energy density by examining the time–harmonic problem with a custom finite-element code. Some comparisons between the frozen mode and the more familiar bandgap Fabry–Perot modes will be made as well to illustrate why the MPC structure can be more attractive for implementing “slow-wave” devices.

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Fig. 3. Finite-element calculation of power transmission into semiinfinite MPC, illuminated by a mode with an optimized “elliptical” polarization.

and the basis function space are chosen identical. The weak form of the vector wave equation then writes

(21) where represents the testing function. Upon assembly, the fully discrete FEM system is of the form (22)

III. TIME–HARMONIC MPC RESPONSE USING FEM As presented in Section II, the 2-D parallel-plate MPC problem admits a modal solution and can be described without resorting to numerical techniques. Simulations presented here are provided to corroborate that analysis. Future investigations of derived structures that do not admit analytical solutions (nonuniformly biased ferrite layers, holes/defects in various layers, sensitivity studies) are planned. In those cases, the numerical codes developed here will be the principal investigation tools.

(22) where and are the stiffness and mass matrices, and is the forcing excitation from current sources (chosen to match the fundamental modal profile). The above linear system (22) is solved via LU factorization using the unsymmetrical multifrontal package UMFPACK [18].

A. Problem Discretization

B. Power Transmission Into Semiinfinite MPC

- and The MPC waveguide problem, which requires both -field components to be modeled simultaneously, is discretized over quadrilateral cells using a combination of edge elements (denoted ) for transverse fields and scalar nodal elements (denoted ) for the invariant field [17]. Either the electric field or magnetic field can be modeled (here, the magnetic field is chosen), shown as follows in (20), such that perfect electric conductor (PEC) walls are natural boundary conditions and require no effort to implement:

For the slow-wave frozen mode to be of any use, it must be possible to couple power from an incident guided mode into a semiinfinite MPC structure. To simulate a semiinfinite MPC using a finite grid size, a long (100-unit cell) lossless MPC is terminated with a short (20-unit cell) matching stub of a lossy MPC. The loss tangent of the matching section is gradually tapered with a quadratic polynomial profile. The matching section does a reasonable job of suppressing reflections off the MPC truncation, although some oscillations are present in Fig. 3. These oscillations are transmission windows of weak Fabry–Perot modes, standing wave resonances that are set up by small reflections off of the matching section. When the MPC structure is excited with a pure or incident wave, it exhibits a poor match. However, when an “elliptical” combination of these modes (a linear sum of and with specific phasor weights) is chosen, the power transmission coefficient can be nearly unity, as demonstrated in Fig. 3. This unity transmission distinguishes the frozen mode regime from Bloch mode propagation near an EBG: both cases

(20) denotes the number of mesh nodes, denotes the where number of mesh edges, and are the unknown degrees of freedom (amplitude coefficients). The testing procedure used here is the Galerkin alternative, where the testing function space

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H

Fig. 4. Magnitude of sampled at the lower guide wall as a function of frequency and position along the MPC loaded waveguide, in decibels relative to = 300 MHz is obincident wave. Large field intensity increase at served as a horizontal line.

f

!f

Fig. 5. Transmission coefficients for Fabry–Perot modes near the for a 250-unit-cell finite MPC.

f

band edge

exhibit vanishing group velocity, but the EBG tends to reject incident power (large mismatch with ever greater reflectivity as from below), while the frozen mode regime remains well matched for frequencies below or above . C. Amplitude Growth Within Semiinfinite MPC Assuming no material losses, there is a fundamental relation between transferred power , energy density , and group velocity , shown as follows in (23), from energy balance arguments: (23) This is upheld by both Bloch modes near the EBG and by the frozen mode. For a typical Bloch wave, as the operating frequency approaches the band edge , both group velocity and transferred power approach zero so that (23) remains balanced. In marked contrast, as the frozen mode frequency approaches the SIP frequency , the group velocity decreases, but the power transmission remains nonzero, as shown in Fig. 3. To balance (23), the energy density must grow large enough so that the product remains constant. Thus, close to , the energy density (and by extension, the field intensity) within the MPC can become orders of magnitude larger than that of the incident wave. Intuitively, when material losses are considered, dissipation inside the MPC will reduce below the maximum value permitted by (23). Fig. 4 depicts this “amplitude growth” within the same matched MPC structure previously considered. The field intensity is measured at and plotted as a function of frequency and position within the MPC waveguide. As , the frozen mode of the MPC structure exhibits field intensity roughly 15 dB above the incident field level. In theory, the field strength of the frozen mode can be arbitrarily large as the SIP is tuned flatter, but, in practice, material losses and the finite size of the MPC will curtail unbounded growth (finite ). References [4] and [5] contain studies of the effects of material losses upon frozen mode field strength.

H

Fig. 6. sampled at the lower guide wall as a function of frequency and position along the MPC loaded waveguide, in decibels relative to incident wave. Three (horizontal) frequency lines of the Fabry–Perot modes near the band edge also show large field intensity increase.

D. Comparison Between Fabry–Perot Modes and the Frozen Mode The frozen mode is not unique in yielding vanishingly small group velocity. As mentioned above, the nearby band edge at has the same effect, the difference being that poor power transmission coefficient near the band edge limits the ability to couple power into the waveguide. Still, if a finite structure is considered, reflections off the unmatched ends can set up standing wave Fabry–Perot modes, characterized by narrow windows of near unity transmission. Some of these windows can exist quite close to the band edge, as depicted in Fig. 5. At these near-band-edge Fabry–Perot windows, group velocity also becomes vanishingly small and power transmission is still nonzero, akin to the frozen mode at the SIP. From conservation of energy (23), fields within the MPC waveguide should exhibit very large intensity in this situation as well. Fig. 6 demonstrates this phenomenon, which is comparable to Fig. 4. To push individual modes closer to the band edge, and obtain an energy

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Fig. 7. Input impedance response for the frozen mode around f

=f .

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!

Fig. 8. Input impedance response for the first three near-band-edge f . Fabry–Perot modes as f

density that is a even comparison to the frozen mode, the length of the structure has to be increased. Note that such modes are possible even for a regular EBG structure, and neither anisotropy, nor gyrotropic properties are required to construct near-band-edge Fabry–Perot resonators. The question naturally arise as to why bother then with the additional complexity of the MPC structure. The answer to this question is that the frozen mode regime from an MPC exhibits a much superior (smoother and wider band) input impedance response. Although it is difficult to specify an input impedance for a structure that supports such complicated modes, consider , as follows in (24), and , as follows in (25), just as they would be defined for waveguide filled with only isotropic media:

(24) (25) where and are the typical field reflection coefficients measured in the incident field region. These can be obtained unambiguously using finite-element results. Figs. 7 and 8 depict the input impedance measured for both the frozen mode regime and the first few near-band-edge Fabry–Perot resonances, respectively. In Fig. 8, each Fabry–Perot mode exhibits a rapid swing in input impedance around the matched condition, while the frozen mode input impedance in Fig. 9 exhibits smoother behavior over a comparatively wider bandwidth. The frozen mode is undoubtedly a narrow band phenomenon, but this is due to the critical point nature of the SIP and not to impedance considerations. As will be shown in Section IV, this property allows for tuning the SIP (or perhaps more accurately, slightly detuning) to slow down incident waves at a wider frequency band and permit pulsed operation, as shown in Section IV.

=0

Fig. 9. Widening the SIP from a zero group velocity v , zero bandwidth f spectral feature (upper dispersion curve) to a window of small velocity propagation and a finite bandwidth (two lower dispersion curves).

1 =0

E. Tuning Frozen Mode Bandwidth Via Layer Thicknesses Prior observations [3] showed that for 1-D MPCs, perturbing layer thicknesses could detune the zero-bandwidth SIP and increase its bandwidth. This idea does extend to 2-D, and varying the -layer thickness has a strong effect upon the slope of the relation, as shown in Fig. 9. This improvement in bandwidth, however, does not come for free. Since the zero group velocity condition has been weakened, (23) indicates that smaller energy density (and field intensity) should be expected for the corresponding frozen mode. Fig. 10 depicts this tradeoff: as the SIP is detuned away from zero slope, the frozen mode response becomes wider in frequency, but weaker in intensity. Similar tradeoffs between group velocity and bandwidth were observed in previous investigations with 1-D MPCs [4], [9].

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where denotes or . In (29), is the PML conductivity profile along the -direction, . Typically, and . The parameters and are used to increase the absorption for evanescent waves if any. Maxwell’s equations (26) and (27) are discretized using central differences in space and time in a staggered grid (Yee’s scheme). They are completed by a discretization of the constitutive equations. For the dielectric layers, FDTD update equations follow directly from (1) and for brevity are not included here. In our simulations, the spatial interpolation on the staggered grids is not employed because it results in late-time instabilities at the interfaces of different layers. For layers, the time–harmonic expression for in (2) is not amenable to be directly used to derive time-domain update equations. To proceed, we instead write the permeability tensor using the magnetic dipole spectral response [13] Fig. 10. Peak H within a unit cell, as a function of frequency. The curves are parameterized by F -layer thickness, L , and L = 6 mm corresponds to the dispersion relation with a zero-slope SIP (upper curve in Fig. 9). As expected, there is a clear tradeoff between field intensity and bandwidth.

(30) where

IV. TRANSIENT MPC RESPONSE A. FDTD Formulation The transient analysis of the MPC can be done via a FDTD algorithm augmented to handle anisotropic dielectrics and gyrotropic ferromagnetic media. Here, we briefly discuss this extension. The current FDTD algorithm employs complex-frequency-shifted perfectly-matched-layer finite difference time domain (CFS–PML–FDTD). The CFS–PML–FDTD algorithm and PML formulations [19], which is based on leads to a natural decoupling of the equations involving the complex (anisotropic and dispersive) material parameters from the FDTD update equations involving finite-difference approximations of the spatial derivatives in a complex coordinate space [20]–[22]. The latter equations are independent of material parameters and are written as

(26) (27) where the modified

operator is given by [20], [22]

(28) The complex coordinates are defined through the following transformation [23]–[25]:

(29) with

In (30),

is the damping (loss) constant and , where is the gyromagnetic ratio, is the dc biasing magnetic field magnitude, and is the dc saturation magnetization. The coupled time-domain magnetic field equations are obtained by inverse Fourier transforming their frequency-domain counterparts obtained from (30). Standard central differencing are used for first- and second-order time derivatives, and central averaging for zerothorder derivatives. By solving the two resulting finite-difference equations simultaneously, explicit FDTD update equations for and can be obtained [10]. B. Simulation of MPC Responses Under Pulsed Excitations The transient response of the 2-D MPC structure depicted in Fig. 1 with material parameters specified by (19) is considered. In this case, MHz and MHz to match the steady-state permeability of layers at the design frequency MHz. The excitation pulse is a sine wave modulated by a Gaussian pulse with the fractional bandwidth of 0.05%. FDTD grid steps cm, cm, and a Courant factor of 0.9 are used. In the CFS-PML region, a quartic growth profile is chosen for both and , and a linear decay profile is chosen for . Moreover, , and , as suggested in [24], are used. To illustrate the unidirectional property, we consider a 2-D MPC with 400-unit cells illuminated by both a forward propagating pulse ( – – slab sequence) pulse and backward propagating pulse ( – – slab sequence). In both cases, the pulse propagates along the -direction, but the ordering of slabs within the MPC unit cell is permuted to test both left and

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Fig. 11. Snapshots of jE j due to a transient pulse incident on 2-D MPC along the forward direction (A1–A2–F layer arrangement). Ten snapshots are displayed at every 3.7 s along the 2-D MPC with 400-unit cells (the total physical size is 256 m). The values of jE j-field intensity are normalized to the incident pulse.

Fig. 13. Energy collected as a function of time, at two observation planes, used to calculate an effective rate of energy transport. The vertical axis is normalized by the total energy contained in the incident pulse.

Fig. 12. Snapshots of jE j due to a transient pulse incident on 2-D MPC along the backward direction (F –A2–A1 layer arrangement). Ten snapshots are displayed at every 3.7 s along the 2-D MPC with 400-unit cells (the total physical size is 256 m). The values of jE j-field intensity are normalized to the incident pulse.

right sides of the dispersion curve. The forward wave encounters a SIP, while the backwards wave does not (recall Fig. 2). within the 2-D MPC for Fig. 11 shows snapshots of the forward propagating pulse. It is observed for this pulse that the peak electric field magnitude increases by a factor of 4.5 ( 13 dB) relative to the incident field intensity. This is comparable to the amplitude growth observed in the time–harmonic case shown in Fig. 4. When the incident pulse enters the MPC in the forward direction, the wave is drastically slowed down and (spatially) compressed with a corresponding increase in field amplitude. However, because the group velocity changes sharply near the SIP in relative terms, the pulse propagation is also highly dispersive. The results in Fig. 11 also illustrate the spreading of the pulse due to such dispersion, as different spectral components propagate with different group velocities. when a wave impinges upon Fig. 12 shows snapshots of the – – ordered MPC. This probes the left arm of the dispersion relation, which has no SIP. As expected, in this case, the pulse propagates at much faster velocity and without significant amplitude growth. The backward propagating pulse arrives at the end of the MPC at a much earlier time, and then it is end of the MPC. Note that, upon reflected back towards the reflection, the pulse encounters an – – ordered MPC and then propagates at much slower speed and with the characteristic amplitude growth of the frozen mode.

To estimate the “average” pulse group velocity, compute the effective rate of energy transport by measuring the times at which half of the electromagnetic energy has flowed past two observation points, at unit cell 100 and unit cell 200 (the entire structure is 400-cells long) [10]. It must be pointed out that this is a position dependent estimation and not a characteristic parameter of the MPC. Fig. 13 shows the (accumulated) energy passing by the two observation planes. By dividing the difference in elapsed time to collect half of the total energy s) by the distance between the two observation ( planes ( m), the effective rate of energy transport for the forward propagating pulse is found to be approximately (considerably slower than vacuum propagation). A similar procedure is carried out for the backward direction. The time difference to collect half the energy at the two observation points is now s, which implies an effective rate of energy transport equal to (still slower than vacuum propagation, but in this case, because the MPC is partially made of electrically dense dielectrics). This significant difference in energy transport speed with respect to propagation direction is a clear manifestation of the unidirectional nature of the SIP. C. Sensitivity Analysis The MPC is a tuned resonant structure and is naturally sensitive to changes in input frequency, geometry, and ferromagnetic losses. Fig. 14 examines the effect of shifting the center frequency of the incident wave upon the peak field magnitude observed inside the structure at the frozen mode. As one would expect, the amplitude growth of decreases considerably as the frequency mismatch increases. For example, for a 0.1% relative frequency shift of, the peak is reduced to approximately 54% of the frequency-matched case. It is interesting to compare the input frequency sensitivity between the frozen mode and the Fabry–Perot mode. For a 250-unit cell finite MPC, the first Fabry–Perot resonance frequency is 307.061 MHz (see Fig. 5) and the 0.01% shifted frequency is located at the forbidden frequency band . Therefore, the electromagnetic energy of the Fabry–Perot mode is not transferred into the MPC for this

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TABLE I EFFECT OF FERROMAGNETIC LOSSES ()

Fig. 14. jE j (relative to the incident field) inside MPC when illuminated with frequency-shifted incident pulses.

Fig. 16. Peak value of jE j (relative to the incident field) observed inside an MPC loaded waveguide as a function of the total number of unit cells.

Fig. 15.

j

E

j

inside an MPC with layer thickness perturbations.

frequency mismatched case. This implies that the Fabry–Perot mode is much more sensitive to the excitation frequency than the frozen mode. Next, we perform the sensitivity analysis against the layer thickness. We intentionally perturb the layer thickness of 5% of unit cells. Since the bandwidths of the frozen mode and the Fabry–Perot mode are different from each other, we use a single frequency sine-wave excitation in order to fairly compare this geometry sensitivity between the frozen mode and Fabry–Perot mode. We evaluate the time-averaged value of spatial peak in a steady state. Fig. 15 illustrates the normalized (time averaged) against different amounts of thickness perturbation. As expected, the field amplitude decreases as the error in the layer thickness increases in both the frozen mode and Fabry–Perot mode. From Fig. 15, it is clearly illustrated that the Fabry–Perot mode is more sensitive to the geometrical perturbations than the frozen mode. Table I explores the sensitivity of the MPC frozen response to ferromagnetic losses by recording the peak value of inside the MPC with lossless layers and compare to

lossy layers with and , respectively. Although lossy MPCs yield lower peak field amplitudes as expected, this effect is not too detrimental and the ferromagnetic losses do not destroy the fundamental qualitative properties of the MPC response (wave slow-down and pulse compression). This is similar to what has been observed in the 1-D case [10]. As a final sensitivity study, we consider the effect of varying inthe total number of unit cells upon the peak value of side the MPC waveguide. The frozen mode is an “emergent” behavior, which depends upon the collective interaction of many unit cells, and using too few unit cells clearly invalidates the infinite periodicity assumption of Bloch/Floquet analysis. Fig. 16 quantifies how many layers are required to effectively realize the frozen-mode large-amplitude behavior. For the same MPC design in (19), as the MPC size is increased, the peak field amplitude grows, and using beyond 75 cells has little additional effect. The overshoots near 50 cells are observed due to Fabry–Perot resonances. V. CONCLUSION This paper has extended the MPC concept by showing that a guided-wave mode, analogous to the 1-D “frozen mode,” can exist within truncated slab structures composed of alternating layers of anisotropic dielectrics and gyrotropic ferrites. Unlike conventional EBG/PBG periodic structures, the Bloch modes of the MPC can exhibit vanishing group velocity and near-unity power transmission simultaneously. This results in superior input impedance performance over traditional near-bandgap Fabry–Perot photonic crystal resonators. In addition to analytical transfer matrix techniques, customized numerical methods

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(finite elements in the frequency domain and a anisotropic/gyrotropic FDTD implementation with complex frequency shifted-PML absorbers) were developed to characterize the transient and time–harmonic response of the structure. These numerical experiments confirm that the guided-wave MPC can exhibit the same key features as the 1-D optical slab MPC: a unidirectional frozen mode exists, which possesses vanishing group velocity, nonzero power transmission, and divergent energy density. Further numerical experiments were used to evaluate a candidate MPC specification, by measuring sensitivity to loss parameters, unit cell count required to realize maximum field amplitude, tunability of the SIP, and input impedance. REFERENCES [1] J. Joannopoulos, Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton Univ. Press, 2005. [2] A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 63, pp. 066609-1–066609-20, May 2001. [3] A. Figotin and I. Vitebsky, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B, Condens. Matter, vol. 67, pp. 165210-1–165210-20, Apr. 2003. [4] G. Mumcu, K. Sertel, J. Volakis, I. Vitebsky, and A. Figotin, “RF propagation in finite thickness nonreciprocal magnetic photonic crystals,” in Proc. IEEE AP-S. Int. Symp., Jun. 2004, vol. 2, pp. 20–25. [5] G. Mumcu, K. Sertel, J. Volakis, I. Vitebsky, and A. Figotin, “RF propagation in finite thickness unidirectional magnetic photonic crystals,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4026–4034, Dec. 2005. [6] G. Mumcu, K. Sertel, and J. Volakis, “Superdirective miniature antennas embedded within magnetic photonic crystals,” in Proc. IEEE AP-S. Int. Symp., Jul. 2005, vol. 2A, pp. 10–13. [7] G. Mumcu, K. Sertel, and J. Volakis, “Miniature antennas and arrays embedded within magnetic photonic crystals,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 168–171, 2006. [8] R. A. Chilton and R. Lee, “FEM Analysis of frozen mode regime inside chirped MPCs,” in Proc. IEEE AP-S Int. Symp., Jul. 2005, vol. 1A, pp. 717–720. [9] R. A. Chilton and R. Lee, “Chirping unit cell length to increase frozen mode bandwidth in nonreciprocal MPCs,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 473–480, Jan. 2006. [10] K.-Y. Jung, B. Donderici, and F. Teixeira, “Transient analysis of spectrally asymmetric magnetic photonic crystals with ferromagnetic losses,” Phys. Rev. B, Condens. Matter, vol. 74, pp. 165207-1–165207-11, Oct. 2006. [11] K.-Y. Jung, B. Donderici, and F. Teixeira, “PML-FDTD analysis of nonreciprocal magnetic photonic crystals with ferromagnetic losses,” in IEEE Int. Antenna Technol. Workshop Dig., Mar. 2006, pp. 333–336. [12] A. Yariv and P. Yeh, Optical Waves in Crystals, ser. Pure and Appl. Opt. New York: Wiley, 1984. [13] D. Pozar, Microwave Engineering. Hoboken, NJ: Wiley, 2005. [14] A. B. Fuller, Ferrites at Microwave Frequencies. London, U.K.: Peregrinus, 1987. [15] G. Tyras, “The permeability matrix for a ferrite medium magnetized at an arbitrary direction and its eigenvalues,” IRE Trans. Microw. Theory Tech., vol. 7, no. 1, pp. 176–177, Jan. 1959. [16] H. Suhl and L. Walker, “Topics in guided wave propagation through gyromagnetic media, Part II—Transverse magnetization and the nonreciprocal helix,” Bell Syst. Tech. J., vol. 33, pp. 939–986, Jul. 1954. [17] J. M. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993. [18] T. Davis and I. Duff, “An unsymmetric-pattern multifrontal method for sparse LU factorization,” SIAM J. Matrix Anal. Applicat., vol. 19, no. 1, pp. 140–158, 1997. [19] A. Zhao, “Generalized-material-independent PML absorbers used for the FDTD simulation of electromagnetic waves in 3-D arbitrary anisotropic dielectric and magnetic media,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 1511–1513, Oct. 1998.

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[20] W. Chew and W. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett., vol. 7, pp. 599–604, Sep. 1994. [21] F. Teixeira and W. Chew, “On causality and dynamic stability of perfectly matched layers for FDTD simulations,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 6, pp. 775–785, Jun. 1999. [22] F. Teixeira and W. Chew, “A general approach to extend Berenger’s absorbing boundary condition to anisotropic and dispersive media,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1386–1387, Sep. 1998. [23] M. Kuzuoglu and R. Mittra, “Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers,” IEEE Microw. Guided Wave Lett., vol. 6, no. 12, pp. 447–449, Dec. 1996. [24] J. Roden and S. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett., vol. 27, pp. 334–339, Dec. 2000. [25] E. Becache, P. Petropoulos, and S. Gedney, “On the long-time behavior of unsplit perfectly matched layers,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1335–1342, May 2004. Ryan A. Chilton (S’04) was born in Louisville, KY, in 1980. He received the B.S.C.S. and B.S.E.E. degrees from the University of Kentucky, Lexington, in 2002 and 2003, respectively, the M.S.E.E. degree from The Ohio State University, Columbus, in 2005, and is currently working toward the Ph.D. degree at The Ohio State University. He is currently a University Fellow with The Ohio State University. His research interests include “h–p” refinement techniques for time-domain FEMs, ultra-wideband antenna design, unstructured mesh generation techniques, and high-performance parallel computing.

Kyung-Young Jung (S’06) was born in Seoul, Korea, in 1974. He received the B.S. and M.S. degrees in electrical engineering from Hanyang University, Seoul, Korea, in 1996 and 1998, respectively, and is currently working toward the Ph.D. degree at The Ohio State University, Columbus. From 1998 to 2001, he was with the Research and Development Department, Mobile Telecommunication Terminal Division, Hyundai Electronics. From 2001 to 2004, he was with the Research and Development Laboratory, Curitel Communications. His research interests include numerical modeling for electromagnetic metamaterials, plasmonics, and nanophotonics.

Robert Lee (S’82–M’83–SM’01) received the B.S.E.E. degree from Lehigh University, Bethlehem PA, in 1983, and the M.S.E.E. and Ph.D. degrees from the University of Arizona, Tucson in 1988 and 1990, respectively. From 1983 to 1984, he was a Microwave Engineer with the Microwave Semiconductor Corporation, Somerset, NJ. From 1984 to 1986, he was a Member of the Technical Staff with the Hughes Aircraft Company, Tucson, AZ. From 1986 to 1990, he was a Research Assistant with the University of Arizona. During the summer from 1987 to 1989, he was with Sandia National Laboratories, Albuquerque, NM. Since 1990, he has been with The Ohio State University, Columbus, where he is currently a Professor and Chair of the Electrical and Computer Engineering Department. His major research interests are the analysis and application of finite methods to electromagnetics.

Fernando L. Teixeira (S’89–M’93–SM’04) received the B.S. and M.S. from the Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil, in 1991 and 1995, respectively, and the Ph.D. degree from the University of Illinois at Urbana-Champaign, in 1999, all in electrical engineering. From 1999 to 2000, he was a Post-Doctoral Research Associate with the Research Laboratory of Electronics, Massachusetts Institute of Technology (MIT). Since 2000, he has been with the Department of Electrical and Computer Engineering and the ElectroScience Laboratory, The Ohio State University (OSU), Columbus, where he is currently an Associate Professor. His current research interests include modeling of wave propagation, scattering, and transport phenomena for communications, sensing, and device applications. Dr. Teixeira is a member of Sigma Xi. He is an elected member of U.S. Commission B of URSI. He was the recipient of numerous awards for his research, including the CAREER Award presented by the National Science Foundation (NSF) in 2004 and the triennial Henry Booker Fellowship presented by the USNC/URSI in 2005.

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Surface Micromachined Microelectromechancial Ohmic Series Switch Using Thin-Film Piezoelectric Actuators Ronald G. Polcawich, Member, IEEE, Jeffrey S. Pulskamp, Daniel Judy, Prashant Ranade, Susan Trolier-McKinstry, Senior Member, IEEE, and Madan Dubey

Abstract—This paper presents results on a surface micromachined RF microelectromechanical switch that uses piezoelectric actuators. The switch uses solution chemistry-derived lead zirconate titanate thin films spun deposited onto a high-resistivity silicon substrate with coplanar waveguide transmission lines. Actuation voltages, applied via circuits independent of the RF circuitry, average less than 10 V, with switch operation demonstrated as low as 2 V. The series switch exhibits better than 20-dB isolation from dc up to 65 GHz and as large as 70 dB below 1 GHz. In the closed state, the switch has an insertion loss less than 1 dB up to 40 GHz, limited in this demonstration by substrate losses from the elastic layer used to stress control the piezoelectric actuators. Switching speeds for the different designs are in the range of 40–60 ms. Thermal sensitivity measurements show no change in isolation observed for temperatures up to 125 C. However, an increase in actuation voltage is required at elevated temperatures. Index Terms—Lead zirconate titanate (PZT), piezoelectric, reliability, RF microelectromechanical system (MEMS), switch, switching speed, temperature sensitivity.

I. INTRODUCTION VER SINCE the initial work by Hughes Research Laboratories produced a switch capable of operating up to 50 GHz [1], [2], RF microelectromechanical system (MEMS) switches have offered the potential for advances in low-loss phase shifters and phased arrays for miniaturized communication and radar systems. Following this research, a concentrated effort began in developing capacitive and ohmic switches in both series and shunt configurations. A vast majority of this research concentrated on electrostatic actuation as the means of closing the switch contacts. Radant, Raytheon, Rockwell Scientific, MEMtronics and MIT Lincoln Laboratories have been at the forefront of the RF MEMS switch development over the

E

Manuscript received October 31, 2006; revised June 18, 2007. This work was supported by the U.S. Army Research Laboratory. R. G. Polcawich is with the Advanced MicroDevices Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD 20783 USA, and also with the Materials Science and Engineering Department, Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). J. S. Pulskamp, D. Judy, P. Ranade, and M. Dubey are with the Advanced MicroDevices Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD 20783 USA. S. Trolier-McKinstry is with the Materials Science and Engineering Department and Materials Research Institute, Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.910072

Fig. 1. Depiction of the starting wafer composite thin-film stack (Si/elastic layer/Ti/TiO /Pt/TiO /PZT/Pt).

last few years [2]. Each of these switches has exhibited excellent performance characteristics and use voltages ranging from 30 to 100 V for actuation. Attempts to lower the actuation voltage with electrostatic switches generally rely on reducing either the mechanical stiffness of the released structures or the gap between the mechanical bridge/cantilever and the corresponding biasing pad. Reductions in the stiffness may limit the restoring force of the switch and can lead to stiction failures. Decreasing the electrode gap limits the RF performance (in particular, the isolation in the open state for series switches). Alternative actuation mechanisms (including thermal, magnetic, and piezoelectric) have been investigated in an attempt to lower the actuation voltage required while maintaining large restoring forces and excellent RF performance. Of these, piezoelectric actuation requires extremely low currents and voltages for operation along with the ability to close large vertical gaps and still allows for low microsecond operating speeds. Thus, it was selected for this study. There have been a limited number of publications describing use of piezoelectric thin films in dc microrelays or as RF MEMS switch actuators. Hoffmann et al. fabricated a lead zirconate titanate (PZT) thin-film actuator for a micromachined switch [3]. Working relays were demonstrated by Gross et al. using in-plane poled PZT film actuators [4]. These devices used a cantilever beam with a supporting elastic layer, a PZT thin film as the actuator, and patterned gold structures to act as the counter electrode. Lee et al. described a functional piezoelectrically actuated RF MEMS switch [5], [6]. This design utilized a cantilever that is perpendicular to the wave propagation direction

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Fig. 2. Fabrication flowchart. (a) Pattern top Pt. (b) Pattern PZT and bottom Pt. (c) Etch PZT to access bottom Pt. (d) Etch elastic layer to access silicon substrate. (e) Deposition of CPW. (f) Deposition of contact dimple metal. (g) Pattern sacrificial photoresist. (h) Deposition of bridge metal. (i) Pattern bridge metal. (j) Remove sacrificial layer. (k) Etch silicon substrate with XeF .

along the RF conductor of the coplanar waveguide (CPW) and relies on bulk micromachining of the silicon substrate. In contrast to that recently published study, this paper describes an integrated surface micromachining process that places thin-film PZT actuators within the lateral RF to ground-plane gap of a CPW transmission line. A robust low-voltage ( 10 V) MEMS ohmic contact series switch has been produced with high isolation and good insertion loss characteristics up through 65 GHz. II. FABRICATION Switch fabrication [7], [8] was done at the Specialty Electronic Materials and Sensors Cleanroom Facility, U.S. Army Research Laboratory, Adelphi, MD, and began with silicon substrate coating a high-resistivity 10 k with a plasma-enhanced chemical vapor deposited elastic layer (1000-Å SiO /500-Å Si N /3500-Å SiO ). The elastic layer was specifically designed to control the post fabrication deformation of the PZT actuators and will be described in Section III. After deposition, the elastic layer was annealed at 700 C in flowing N to eliminate trapped hydrogen [9]. Next, 160-Å titanium, 90-Å titanium dioxide, 1640-Å platinum, and 20-Å titanium dioxide was deposited at a substrate temperature of 500 C in a multichamber clustertool. The thin titanium dioxide layer on top of the platinum served to help template orientation for the PbZr Ti O (PZT) thin film [10], [11]. The PZT thin films were prepared via a metal organic solution chemistry process modified from the one outlined by Budd et al. [12]. Solutions were batched with 12% excess lead. A 5000-Å PZT thin film was prepared using a multideposition and anneal procedure. Films were crystallized at 700 C. Next, a 1050-Å platinum thin film was sputter deposited onto the PZT surface at 300 C. A schematic of the film stack is shown in Fig. 1.

The fabrication process, depicted in Fig. 2, began with the creation of the actuators by defining the top metal using argon ion-milling [see Fig. 2(a)]. Subsequently, the PZT and bottom electrode were patterned using argon ion-milling [see Fig. 2(b)]. To gain access to the bottom electrode, an H O HCl HF (2:1:0.04) wet etch was used to locally remove the PZT [see Fig. 2(c)]. Next, the structural layer was patterned with a reactive ion etch using CF , CHF , and He to open access to the silicon substrate for the eventual release etch [see Fig. 2(d)]. The CPW transmission lines were then patterned using liftoff to define the 200-Å titanium/7300-Å gold metallization [see Fig. 2(e)]. Contact dimples (4000-Å gold coated with 1000-Å platinum) were electron beam evaporated onto a portion of the CPW transmission line and patterned via liftoff [see Fig. 2(f)]. To create the switch contact mechanism, a photodefinable organic (Clariant AZ 5200 resist) was spun onto the substrate [see Fig. 2(g)] and cured at 200 C. Depressions directly above the contact dimples were subsequently etched into the top of the organic layer using a patterned photoresist layer and an oxygen plasma. Next, a 2- m-thick gold thin film was evaporated onto the surface of the wafer [see Fig. 2(h)]. To define the switch contact beams, argon ion-milling with a photoresist mask was utilized [see Fig. 2(i)]. To release the gold contact beams, an oxygen plasma was used to remove the organic sacrificial layer [see Fig. 2(j)]. The final step was to release the PZT actuators in a xenon difluoride etching system in which the silicon was isotropically removed from underneath the actuators [see Fig. 2(k)]. III. DESIGN Our initial attempts at developing a PZT RF MEMS switch utilized actuators that were located within the CPW ground planes with gold contact structures that spanned the full CPW width dimension. These designs proved unsuccessful largely

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Fig. 3. Predicted piezoelectric actuator deflection and contact force as a function of voltage.

due to the residual stress effects related to the long contact structure spans and the designs’ inherent requirement of multiple contact points. These issues were addressed by placing the actuators within the gap between the RF conductor and ground plane (RF gap) to minimize the compliances of the actuators and the contact structures, thus mitigating the stress deformation issues, and by implementing a single contact point design. Based on these changes, several successful device designs were developed through a close iterative process between the requirements of the mechanical, high-frequency electromagnetic, and fabrication design processes. Although the motivation for placement of the actuators within the RF gaps was initially inspired by mechanical design, high-frequency numerical modeling suggested an additional advantage of the placement. The undercut of the actuators within the RF gap removes a portion of the substrates and places a significant amount of air in the gap in the vicinity of the switch. This considerably changes the local characteristic impedance of the transmission line. Achieving 50 required approximately a 4- m gap in the air-dominated sections, limiting the potential for adding the actuators to the gap. However, the presence of the top and bottom Pt electrodes on the actuators allows the actuators to be absorbed into the CPW transmission line. Hence, the impedance is dictated by both the gap between the RF conductor and actuators and the gap from the actuators to the ground plane. This relieves the fabrication constraints on the PZT actuator and allows the actuators to remain in the RF conductor to the ground-plane gap. Electromechanical analytical and ANSYS finite-element analysis models were developed to analyze the performance of the actuators in terms of actuator displacement sensitivity, resonance characteristics, switching speed, temperature sensitivity, and contact behavior. The finite-element models suggested a switch capable of operating at low voltages with piezoelectric actuators could provide a reasonable contact force, leading to (see Fig. 3). The a contact resistance of approximately 2 contact resistance was computed assuming a diffusive transport and plastic deformation model [13], [14]. In order to design the switch for elevated frequencies, Ansoft’s High Frequency Structure Simulator (HFSS) was used

Fig. 4. HFSS simulation of a PZT series switch in the: (a) open state and (b) closed state.

Fig. 5. PZT switch equivalent-circuit model with parameters derived from Ansoft’s Q3D.

to simulate the switch design and to design and optimize the switch and transmission line geometries to ensure minimal disturbance to the RF transmission line impedance and loss characteristics. The results of the HFSS model are illustrated in Fig. 4. In addition, an equivalent-circuit model was developed for a PZT switch (see Fig. 5). Ansoft’s Q3D extractor was used to extract the capacitances and inductances for this model. The solid model that was used for the HFSS model was imported to Q3D and all of the CPW geometry was removed, leaving the geometry for the actuators, input and output cantilevers, and the switch contact pad. Table I shows the resulting component pais the input canrameters for the on and off states. Where is the output cantilever inductance and tilever inductance,

POLCAWICH et al.: SURFACE MICROMACHINED MICROELECTROMECHANCIAL OHMIC SERIES SWITCH USING THIN-FILM PIEZOELECTRIC ACTUATORS

TABLE I LIST OF THE NOMINAL COMPONENT VALUES FOR THE EQUIVALENT-CIRCUIT MODEL SHOWN IN FIG. 5

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TABLE II DIMENSIONAL INFORMATION FOR THE PZT SWITCH DESIGN

Fig. 6. Image outlining the critical features of the PZT series switch that requires two ohmic contacts for good RF performance.

is the contact pad inductance. Similarly, is the capaciis the catance from the input cantilever to the contact pad, pacitance from the output cantilever to the contact pad, and is the capacitance from the input cantilever to the output cantilever. Finally, and are the contact resistances from the input cantilever to the contact pad and from the output cantilever to the contact pad, respectively. These components encapsulate the essence of the switch model. The input and output CPWs were added to better match the measured results. In adand resistors were added to model the dition, the effects of the loss of the elastic layer (see Section V-A for discussion). The current design (see Fig. 6) decoupled the actuators from the center RF conductor. The actuators are coupled to one another with a common structural pad containing a conductive section. Two very short center RF cantilevered structures overhanging the structural pad allow the switch structure to close the series configured gap (see Table II for device dimensions). Upon actuation, the PZT actuators deflect the RF contact pad upward, enabling switch closure. The contact pad completes the RF circuit by making contact to the two cantilevers suspended above the RF contact pad. Two bias lines, one for each actuator, are electrically tied with air bridges over the RF conductor. The actuator bias lines allow the actuation voltage to be applied independent from the RF conductor. The bias air bridges spanning

Fig. 7. Illustration of the RF conductor width modifications to account for: (a) capacitive loading from the dc bias air bridges and (b) the changes to the capacitance within the RF gap from the air cavity underneath the actuators and CPW transmission line.

the RF conductor require the width of the conductor to be modified to maintain 50- impedance. Thus, the RF conductor is narrowed underneath the bias air bridges and widened in the regions where the PZT actuators were located (see Fig. 7). Thin-film PZT generates an in-plane compressive strain at large electric field values. The direction of actuation is dictated by two main parameters, i.e., the magnitude and sense of the , induced strain (force), transverse piezoelectric coefficient and the position of the geometric midplane of the piezoelectric thin film relative to the composite neutral axis (moment arm). If the geometric midplane of the piezoelectric layer lies above the neutral axis, the piezoelectric strain will deflect the actuator up, which is the desired actuation direction for the switch. Therefore, a critical design feature was the negative static deflection of the piezoelectric actuators and RF contact pad visible in Fig. 6. In order to stress engineer the structures, accurate values of the residual stress and elastic properties were required. The residual

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TABLE III AVERAGE RESIDUAL STRESS MEASURED ON EACH OF THE THIN FILMS USED IN THE COMPOSITE ACTUATOR FOR THE PZT SWITCH

stress values used are obtained from Stoney’s equation by measuring the change in wafer curvature following each film deposition with a Tencor FLX-2908 system [15]. The nominal values for each of the thin films used to create the PZT actuators are listed in Table III. The elastic properties for each of the films in the composite stack were obtained from microtensile specimens in collaboration with Dr. William Sharpe, The Johns Hopkins University, Baltimore, MD [16]. The location of the neutral axis was derived from the elastic properties of each layer using the transformed section method of composites [17]. The static deflections of the fabricated actuators were predicted using the method described by Pulskamp et al. [18]. The results indicated that placing the highly stressed silicon nitride layer towards the bottom of the composite elastic layer created the necessary negative deflection. The layer thicknesses are outlined in Table III.

Fig. 8. Switching speed measurement setup.

IV. EXPERIMENTAL PROCEDURE The PZT switch was characterized using a combination of dc and RF testing along with optical profilometry. Using an Electroglass autoprober configured with a voltage source meter, the actuation voltage of the switch was determined by calculating the median voltage between the high current and low current state of the switch as the voltage was swept from 0 to 15 V. A Cascade probe station combined with an HP 8510 network analyzer was used to characterize the RF performance from 45 MHz to 40 GHz. Measurements up to 50 GHz were made using an Agilent E8364A precision network analyzer combined with a Cascade Summit 10-K probe station configured with a thermal chuck. Measurements up to 65 GHz were made using an Agilent E8361A precision network analyzer at the U.S. Air Force Research Laboratory, Hanscom Air Force Base, MA. The static deformations of the actuators, as well as the RF contact pad as a function of voltage and temperature, were characterized with a Veeco NT1100 optical profilometer. In addition, a Polytec laser Doppler vibrometer (LDV) was used to characterize the resonance characteristics and transient actuation characteristics of the switch components. For the transient actuation measurements, single point data were taken on all critical features of the switch with separate actuation pulses. The locations collected were the midpoint and tip of each actuator, the left, center, and right sides of the RF contact pad, and the three corresponding dimple locations on both RF cantilevers, for a total of 13 individual scans. The switching time was measured as outlined in Fig. 8 with square waves for both the actuation and sensing waveforms. The actuation waveform was triggered by the sensing waveform with a delay of 1.9 s. The 250-Hz applied sensing waveform was

Fig. 9. Description of the actuation (solid line) and RF pulse (dashed line) used for reliability testing of switches under cold-switching conditions.

50 mV with a 25-mV dc offset. The 500-Hz actuation wavewith a 2.5-V dc offset. The actuation voltage form was 5 (CH 2) and the sensing voltage out from the switch, i.e., RF out, (CH 3) were placed into 50- loads at the oscilloscope, while the sensing waveform input went into a 1-M load (CH 1). The cycle reliability of the switches was characterized using a Gigatronics model 605 RF signal generator at 8 GHz with a 30-mW RF pulse applied for 62.5 s at an 8-kHz rate. The detector output was recorded under cold switching operation (i.e., the RF energy was off during the switch actuation) for both the open and closed states of the switch. The actuation waveform was supplied at 4 kHz with a 90 phase offset from the RF pulse (see Fig. 9). Devices were tested until there was no discernible change between the open and closed states. V. RESULTS AND DISCUSSION A. DC and RF Characterization After fabrication, the static deflection of the actuators, as well as the piezoelectric induced deformations of the actuators and RF contact pad was measured. The piezoelectric deformations of the actuators can be seen in the inset of Fig. 10. Beginning with an initial deflection of 5 m into the substrate, the actuators bend upward with increasing voltage until the RF contact pad makes contact with the RF cantilevers, thereby completing

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Fig. 10. S and vertical gap between the RF contact pad and RF out cantilever as a function of applied voltage. S -parameter data was recorded at 17 GHz to meet a military satellite communication application. The inset shows the deformations of a 150-m-long piezoelectric actuators in the PZT series switch. Note that this data was recorded on a low-resistivity Si substrate so the insertion loss was limited to approximately 4 dB by substrate induced transmission loss.

the electrical circuit and preventing further upward deflection of the actuators. The voltage required to close the switch contacts was generally 6–7 V. As the voltage was reduced, the piezoelectric deflection decreases and, at approximately 5 V, the switch returned to its open state (see Fig. 10). Another important characteristic to notice is that although the vertical contact gap between the RF contact pad and the RF-out cantilever is decreased by nearly 11 m at 4 V, the isolation at 17 GHz only changes 4 dB from 34 to 30 dB. This characteristic ensures a temperature stable off state [given that the initial curvature of the PZT actuators is a function of temperature (see Section V-E)]. The performance of the PZT switch at 5 V from 45 MHz to 65 GHz is plotted in Fig. 11. The switch exhibited an isolation as high as 60 dB below a few gigahertz and better than 20 dB above 40 GHz. The insertion loss was less than 1 dB up to 40 GHz and rises to 2 dB at 65 GHz. It should be noted that the insertion loss can be improved by increasing the contact force with increasing voltage with values less than 0.6 dB achievable up to 40 GHz with 8 V applied. The high insertion loss resulted from the elastic layer used to control the initial curvature of the PZT actuators. Examination of CPW transmission lines with varying dielectric thin films between the CPW and a high-resistivity silicon substrate revealed transmission losses as large as 1.25 dB/mm at 40 GHz (see Fig. 12). The transmission line loss using the elastic layer was nearly an order of magnitude higher than one using a silicon dioxide thin film (0.18 dB/mm). The elastic layer cannot be eliminated from the substrate because of its role in controlling the residual stress of the PZT actuators. However, the elastic layer can be thinned after the actuators have been patterned. In this process, the elastic layer may be thinned in the regions that define the CPW transmission line. If the elastic layer is thinned to 1000 Å, approximately half the transmission loss can be eliminated. Further thinning of the elastic layer to 300 Å resulted in a transmission loss of 0.2 dB/mm at 40 GHz, comparable to that

Fig. 11. Measured RF performance of the PZT switch. (a) Open. (b) Closed.

Fig. 12. Transmission loss of CPW transmission lines on different dielectrics on a high-resistivity silicon substrate.

obtained with a silicon dioxide thin film. The amount of thinning required appears to be related to the location of the silicon

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Fig. 13. Actuation voltage histogram for the as-fabricated (unpoled) PZT switches (device A01).

TABLE IV SUMMARY OF ACTUATION VOLTAGE AND RF PERFORMANCE FOR THE PZT SERIES SWITCH DESIGNS ON ONE WAFER

nitride. The exact amount of thinning will strongly depend on the width of the diffuse interface arising from the plasma enhanced chemical vapor deposition conditions and any diffusion from subsequent annealing. An examination of device wafers with an autoprober revealed better than 85% yield for any particular device geometry. A histogram of the dc actuation voltages required for switch closure is shown in Fig. 13. As-fabricated devices exhibited a skewed distribution with a median actuation voltage of 5 V and a standard deviation of 1.7 V. The median actuation voltage and RF performance for each of the designs are summarized in Table IV. As shown, there was little change in median actuation voltage or contact resistance between designs with different length actuators. Similarly, the A23 and A24 designs, which incorporate a pair of etch holes in the RF contact pad to facilitate a faster release etch, correspond well with the A02 and A03 designs. The A02 and A23 designs exhibited the best insertion-loss characteristics when actuated at 5 V. Note that the sample size for the RF measurements was very small (five devices) compared to the autoprobed dc results (90 devices). B. Piezoelectric Poling and Aging PZT thin films require a poling procedure to align the ferroelectric domains and increase the piezoelectric coefficient. Poling was generally done with an elevated voltage, 2–3 times

Fig. 14. Series of actuation voltage histograms of device A01 after poling. (a) First poling of 60 s. (b) Second poling of 60 s.

the coercive field applied for 10–15 min. However, the current configuration of the autoprober could only accommodate a short poling time of 60 s with 15 V (approximately seven times the coercive field). After an initial scan to pole the actuators, the wafer was rescanned and the median actuation voltage reduced to 3.9 V with a standard deviation of 1.1 V [see Fig. 14(a)]. In addition, the distribution of actuation voltages became tighter. The poling procedure was repeated again with very little change in voltage and further tightening of the distribution [see Fig. 14(b)]. After the second poling, several devices closed at less than 3 V, with the lowest device switching at 2 V. On poling, the actuators develop a small degree of residual deflection upward relative to the unpoled state, presumably due to changes in the PZT film domain state. Thus, it was essential to insure sufficient initial negative deflection so that the switches do not permanently close. The changes in actuator position with poling are shown in Fig. 15. Poling with 15 V at room temperature for 15 min results in a 1.2- m change in the initial position of the actuators, whereas poling at 125 C resulted in a 3.9- m

POLCAWICH et al.: SURFACE MICROMACHINED MICROELECTROMECHANCIAL OHMIC SERIES SWITCH USING THIN-FILM PIEZOELECTRIC ACTUATORS

Fig. 15. Actuator tip deflection as a function of poling conditions for three different device geometries with the actuators for device A03* having a length of 175 m.

change. In both cases, a negative actuator deflection profile remains after poling and little change to the actuation voltage was observed. Aging in ferroelectric thin films reduces the piezoelectric coefficient over time. Many PZT thin films have piezoelectric aging rates ranging from 3%–9% per time decade [19], [20]. As the piezoelectric coefficient decays over time, the actuation voltage should gradually increase with aging. Following the second poling procedure described above, the switches were aged overnight and rescanned after 15 h. The resulting histogram broadened slightly, with the median actuation voltage rising to 4.2 V with a standard deviation of 1.1 V (see Fig. 16). The aging characteristics of the switch actuators were also measured with the optical profiler over time. The deformations recorded in Fig. 17 illustrate that after 3 h there was approximately a 15% (1–1.5 m) change for hot poled (125 C) actuators and approximately 25% (1.6–2.5 m) change for room-temperature poled actuators. These relatively small changes in actuator tip deflection compared with the vertical contact gap (10–15 m) were consistent with the small increase in actuation voltage observed in Fig. 16. It should be mentioned that the increase in actuation voltage will only affect the first cycle of switch operation, as any further application of voltage will result in the poling of the PZT actuators, as mentioned above. C. Hold-Down Response The RF performance of the switch was also investigated with long hold times in the actuated state. As mentioned previously, continuous application of the actuation voltage increases the piezoelectric coefficient of the PZT through poling. The increased piezoelectric coefficient leads to an increase in the contact force, resulting in improvements to the insertion- and return-loss characteristics of the switch (see Fig. 18). The insertion loss improved from 1 to 0.5 dB over a period of nearly 46 h. Note that these values were the raw insertion-loss values, including losses contributed by the elastic layer. At the completion of the hold tests, the actuation voltage reduced

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Fig. 16. Actuation voltage histogram following aging of the PZT switches for 15 h.

Fig. 17. Normalized actuator tip deflection as a function of aging time for devices poled at room temperature and at 125 C.

from 5 to 1–2 V with several devices requiring a short negative 1.0 V) pulse to open. The use of the negative voltage ( voltage pulse can be eliminated by stress engineering the structural layer such that a more negative initial curvature exists in the actuators. D. Switching Time The resonance frequency of the switch actuators listed in Table V can be used to estimate the switching time. For all of the devices, the switching time should be approximately 10 s depending on the initial gap between the RF contact pad and RF cantilevers. However, measured times were on the order of 60 s for as-fabricated devices and approximately 40 s for poled samples (see Fig. 19). The initial contact generally occurred within 4–8 s for all designs, but final closure was delayed by bouncing between the contacts. The faster switching times for poled samples was a result of the reduced vertical gap after poling the actuators. A smaller gap decreased the velocity at impact, thereby reducing the settling time. Further

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Fig. 18. Hold-down characteristics of a PZT switch (A02 design) with 5 V applied. The data was taken at 17 GHz to meet a military satellite communication application.

Fig. 20. Illustration of the ripple in the contact resistance that resulted from the bouncing between the RF contact pad and RF cantilevers.

TABLE V RESONANCE FREQUENCIES FOR EACH DEVICE DESIGN

Fig. 21. Switch off speed for the PZT switch.

Fig. 19. Switching time to close the switch for device A02, post-poling 2 min at 5 V.

evidence of switch bouncing was the small oscillations in the sensing voltage observed for extended periods of time after contact (See Fig. 20). Furthermore, experiments under vacuum increased the time to switch closure past 100 s. In contrast to the switch on time, the switch off times were not limited by bouncing, although some degree of bouncing was observed in some devices (see Fig. 21). The switch off time average was approximately 0.5 s for all device designs.

To verify that the switch was indeed bouncing, the LDV was used to measure the velocity of the major components of the switch as a function of time for a single actuation pulse at atmospheric pressure. A series of measurements were taken with an applied voltage at the actuation voltage (5 V) and lower than the actuation voltage (0.5 V). The time response of the actuators with 0.5 V applied is shown in the inset of Fig. 22. The actuators rang for over 200 s for an applied voltage pulse of 1 ms. Similarly, the RF contact pad rang for over 200 s (see Fig. 22). As expected, the RF cantilevers show no sign of movement at 0.5 V. On applying 5 V, the actuator ringing again persisted for greater than 200 s, except that approximately the first 60 s has a different vibration response created each time the RF pad contacted the RF cantilevers (see inset of Fig. 23). In this case, the RF pad and both RF cantilevers exhibit a matching vibration response out past 200 s (see Fig. 23). The large velocity changes occurring out to approximately 60 s correspond well with the switching time measurements previously outlined. Methods to improve the switching time to less than 20 s are currently under investigation, including waveform actuation

POLCAWICH et al.: SURFACE MICROMACHINED MICROELECTROMECHANCIAL OHMIC SERIES SWITCH USING THIN-FILM PIEZOELECTRIC ACTUATORS

Fig. 22. LDV time scan for the switch actuators (inset) and the RF contact pad and RF cantilevers with a 0.5-V 1-ms pulse applied.

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Fig. 24. Thermal deformation (at 0-V dc bias) of PZT actuators from room temperature to 125 C.

Fig. 23. LDV time scan for the switch actuators (inset) and the RF contact pad and RF cantilevers with a 5.0-V 1-ms pulse applied.

control described by Uchino and Giniewicz [21]. Difficulties in this type of control arise from the interaction of two free vibrating entities. As the RF contact pad makes contact with each of the RF cantilevers, the interaction sends each cantilever into vibration. To minimize this vibration, we are concentrating on closing the initial gap between the RF contact pad and RF cantilevers, thereby limiting the velocity at initial impact. With proper control, modeling efforts suggest that switching speeds as low as 15–20 s can be obtained. E. Thermal Response This switch design relied upon stress engineering of a multilayer composite actuator to control the initial deflection. Differences in the coefficient of thermal expansion between the individual layers cause the actuator to move with temperature. The thermal deformation of the actuators was determined with the optical profilometer and compared to the ANSYS thermal modeling results displayed in Fig. 24. From 25 C to 125 C, the actuators deflect nearly 2 m away from the RF cantilevers, which is in good agreement with the ANSYS predictions. The results

Fig. 25. S values as a function of temperature and voltage for PZT switch A02 on a low-resistivity silicon substrate with the inset illustrating the off state S values as a function of temperature.

from Fig. 10 indicate that these relatively small thermal deformations should not affect the isolation. However, the negative displacements may increase the actuation voltage required for switch closure. Models of the switch at 55 C show the actuators deforming upward 1.5 m relative to their room-temperature position so that the actuator tip is approximately 0.5 m above zero displacement. Switch closure does not occur until the actuator tips are approximately 1.4 m above zero displacement. Thus the switch should operate normally, albeit with a smaller actuation voltage at the lower end of the military temperature specification. The RF characteristics, from 45 MHz to 50 GHz, versus temperature were measured from 25 C to 125 C. As anticipated from the optical profilometry and ANSYS predictions, the isolation remains unchanged over this temperature range (see inset of Fig. 25). The increased negative deflection of the actuators increased the actuation voltage to 15 V at 50 C (see Fig. 25).

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Fig. 26. Deformation of the width and length (inset) of the RF contact pad as a function of temperature with 0 V applied to the actuators.

Fig. 28. PZT MEMS switch cycle reliability for device A01 at 6 GHz. The frequency was chosen based on available testing equipment. The inset picture is of one of the contact dimples after cycling illustrating no apparent wear or contamination on the contact surface.

the temperature dependence. In addition, design modifications to the RF contact pad and RF cantilevers are being considered to reduce the temperature dependence. F. Reliability

Fig. 27. RF contact pad thermal deformation for the width and length (inset) of the pad with 15 V applied to the actuators.

At 100 C, the actuation voltage increased to 20 V. In addition to the actuator thermal deformation, the RF contact pad thermal deformations contribute to the overall switch performance with temperature. In the off state, the RF contact pad undergoes a smaller negative displacement along its length with increasing temperature than the actuators (see inset of Fig. 26). In addition, the width curvature of the RF contact pad changes from concave up to a more flattened structure at 125 C (see Fig. 26). These changes in the length and width curvature are suspected to be the main cause for the opening of the actuated switch at 100 C with 15 V applied. As shown in Fig. 27, the RF contact pad does not move from its average vertical position along its width, but it does flatten with increasing temperature. However, the tip of the RF contact pad (see inset of Figs. 26 and 27) undergoes a dramatic change along its length at 125 C for both the unactuated and actuated profiles. The behavior of the RF contact pad can be controlled to some degree by the residual stress characteristics of the elastic layer and the CPW metallization so as to minimize

Long cycle lifetimes are essential to transitioning MEMS switches into commercial and military systems. Previous studies on mechanical [22] and electrical (unipolar) [20] excitation of thin-film PZT yielded excellent lifetimes of up to billions of cycles with little degradation in mechanical strength and stiffness (at a strain level of 0.05%) and piezoelectric coefficient for unipolar fields up to five times the coercive field. Since the actuators do not appear to limit reliability, it was anticipated that the most prevalent failure mode for the PZT MEMS switches would be contact failures. Contact failures common to electrostatic switches include increased contact resistance from contamination build-up and shorting failures from microwelding of the contacts [14], [23]–[26]. The PZT switches exhibit a high contact resistance (open) degradation/failure after a cycle count in the low millions (see Fig. 28). In all cases, the actuators were examined with optical profilometry after failure, and in each case exhibited normal deformation with voltage. The failures appear to be related to the contacts. However, the contact dimples fail to illustrate any signs of wear or contamination (see inset of Fig. 28). Several possibilities are currently under investigation, including build-up of organic materials on the contacts, impact damage to the contact surfaces from the bouncing discussed in Section V-D, and material transfer between the contact surfaces. To address this problem, modified actuation waveforms, alternative contact materials, and design modifications are being considered for enhancing the performance of these switches. VI. CONCLUSION A surface micromachined RF MEMS switch using thin-film PZT actuators has been successfully demonstrated. The isola-

POLCAWICH et al.: SURFACE MICROMACHINED MICROELECTROMECHANCIAL OHMIC SERIES SWITCH USING THIN-FILM PIEZOELECTRIC ACTUATORS

tion characteristics are better than 20 dB up to 65 GHz and the insertion loss of the switch is less than 1 dB up to 40 GHz. The fabrication process is reliable and successfully produced devices with better than 85% yield with an as-fabricated median actuation voltage of approximately 5 V. Poling of the PZT actuators lowers the actuation voltage to a median of 3.8 V, with a low of 2 V being demonstrated. Thermal stability measurements show no change in the isolation observed from 25 C to 125 C. However, the actuation voltage increases to 20 V to maintain closure at 100 C. Changes in the stress balance in the actuators can be used to limit the increase in actuation voltage at elevated temperatures at the expense of the performance at lower temperatures. Lastly, these PZT switches have exhibited cycle lifetimes in the low millions of cycles with high resistance failures likely the result of contact contamination consistent with earlier research into electrostatically actuated ohmic contact switches.

ACKNOWLEDGMENT The authors would like to thank R. Piekarz, J. Conrad, and D. Washington, all of the U.S. Army Research Laboratory, Adelphia, MD, for their assistance in the fabrication of the devices. In addition, the authors thank J. R. Reid, U.S. Air Force Research Laboratory, Hanscom AFB, MA, for assistance in measuring the PZT switches above 40 GHz. The authors would also like to thank R. Kaul, U.S. Army Research Laboratory Emeritus Corps, for his many discussions regarding RF devices and circuits.

REFERENCES [1] L. E. Larson, R. H. Hackett, M. A. Melendes, and R. F. Lohr, “Micromachined microwave actuator (MIMAC) technology—A new tuning approach for microwave integrated circuits,” in Millimeter-Wave Monolithic Circuits Symp. Dig., Boston, MA, Jun. 1991, pp. 27–30. [2] G. Rebeiz, RF MEMS Theory, Design, and Technology. Hoboken, NJ: Wiley, 2003, pp. 1–121-156. [3] M. Hoffmann, H. Kuppers, T. Schneller, U. Bottger, U. Schnakenberg, W. Mokwa, and R. Waser, “Theoretical calculations and performance results of a PZT thin film actuator,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 50, no. 10, pp. 1240–1256, Oct. 2003. [4] S. J. Gross, S. Tadigadapa, T. N. Jackson, S. Trolier-McKinstry, and Q. Q. Zhang, “Lead-zirconate-titanate based piezoelectric micromachined switch,” Appl. Phys. Lett., vol. 83, pp. 174–6, 2003. [5] H. C. Lee, J. Y. Park, and J. U. Bu, “Piezoelectrically actuated RF MEMS DC contact switches with low voltage operation,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 202–204, Apr. 2005. [6] H. C. Lee, J. H. Park, J. Y. Park, H. J. Nam, and J. U. Bu, “Design, fabrication, and RF performance of two different types of piezoelectrically actuated ohmic MEMS switches,” J. Micromech. Microeng., vol. 15, pp. 2098–2104, 2005. [7] J. S. Pulskamp, R. G. Polcawich, and D. Judy, “Piezoelectric inline RF MEMS wwitches and method of manufacture,” U.S. Patent, Feb. 6, 2006, filed. [8] J. S. Pulskamp, R. G. Polcawich, and D. Judy, “RF MEMS series switch using piezoelectric actuation and method of fabrication,” U.S. Patent, Sep. 2006, filed. [9] M. S. Haque, H. A. Naseem, and W. D. Brown, “Residual stress behavior of thin plasma-enhanced chemical vapor deposited silicon dioxide films as a function of storage time,” J. Appl. Phys., vol. 81, pp. 3129–3133, 1997. [10] P. Muralt, T. Maeder, L. Sagalowicz, S. Hiboux, S. Scalese, D. Naumovic, R. G. Agostino, N. Xanthopoulos, H. J. Mathieu, L. Pattey, and E. L. Bullock, “Texture control of PbTiO and PZT thin films with TiO seeding,” J. Appl. Phys., vol. 83, no. 7, pp. 3835–3841, 1998.

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[11] P. Muralt, J. Baborowski, and N. Ledermann, , N. Setter, Ed., “PiezoO thin electric micro-electro-mechanical systems with PbZr Ti films: Integration and application issues,” in Piezoelectric Materials in Devices. Lausanne, Switzerland: EPFL, 2002, pp. 231–260. [12] K. Budd, S. Dey, and D. Payne, “Sol-gel processing of PbTiO , PbZrO , PZT, and PLZT thin films,” in Br. Ceram. Proc., 1985, vol. 36, pp. 107–121. [13] R. Holm, Electric Contacts: Theory and Applications. Berlin, Germany: Springer, 1969. [14] R. A. Coutu, P. E. Kladitis, K. D. Leedy, and R. L. Crane, “Selecting metal alloy electric contact materials for MEMS switches,” J. Micromech. Microeng., vol. 14, pp. 1157–1164, 2004. [15] G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond. A, Math. Phys. Sci., vol. 82, pp. 172–175, 1909. [16] W. Sharpe, Johns Hopkins University. Baltimore, MD, 2004. [17] J. Gere and S. Timoshenko, Mechanics of Materials. Boston, MA: PWS Publishing, 1997, pp. 400–403. [18] J. S. Pulskamp, A. Wickenden, R. Polcawich, B. Piekarski, M. Dubey, and G. Smith, “Mitigation of residual stress deformation in multilayer microelectromechanical systems cantilever devices,” J. Vac. Sci. Techol., vol. 21, pp. 2482–2486, 2003. [19] A. L. Kholkin, A. K. Tagantsev, E. L. Colla, D. V. Tayler, and N. Setter, “Piezoelectric and dielectric aging in Pb(Zr,Ti)O thin films and bulk ceramics,” Int. Ferroelect., vol. 15, pp. 317–324, 1997. [20] R. G. Polcawich and S. Trolier-McKinstry, “Piezoelectric and dielectric reliability of lead zirconate titanate thin films,” J. Mater. Res., vol. 15, pp. 2505–2513, 2000. [21] K. Uchino and J. Giniewicz, Micromechatronics. New York: Marcel Dekker, 2003, pp. 242–254. [22] I. Demir, A. L. Olson, J. L. Skinner, C. D. Richards, R. F. Richards, and D. F. Bahr, “High strain behavior of composite thin film piezoelectric membranes,” Microelect. Eng., vol. 75, pp. 12–23, 2004. [23] S. Majumber, N. E. McGruer, and G. G. Adams, “Study of contacts in an electrostatically actuated microswitch,” Sens. Actuators A, Phys., vol. 93, no. 1, pp. 19–26, 2001. [24] J. Schimkat, “Contact measurements providing basic design data for microrelay actuators,” Sens. Actuators A, Phys., vol. 73, pp. 138–143, 1999. [25] R. A. Coutu, P. E. Kladitis, R. E. Strawser, and R. L. Crane, “Microswitches and sputtered Au, AuPd, Au-on-AuPt, and AuPtCu alloy electric contacts,” IEEE Trans. Compon. Packag. Technol., vol. 29, no. 2, pp. 341–349, Jun. 2006. [26] N. E. McGruer, G. G. Adams, L. Chen, Z. J. Guo, and Y. Du, “Mechanical, thermal, and material influences on ohmic-contact-type MEMS switch operation,” in Proc. MEMS, 2006, pp. 230–233.

Ronald G. Polcawich (M’07) received the B.S. in materials science and engineering from Carnegie-Mellon University, Pittsburgh, PA, in 1997, and the M.S. and Ph.D. degrees in materials from the Pennsylvania State University, University Park, in 1999 and 2007, respectively. He is currently an Engineer with the Advanced MicroDevices Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD. His research focuses on RF MEMS devices, electronic scanning antenna, PZT thin films, MEMS fabrication, and microrobotics. Mr. Polcawich is a member of the Materials Research Society.

Jeffrey S. Pulskamp received the B.S. degree in mechanical engineering from the University of Maryland at College Park, in 2000. He is currently a Mechanical Engineer with the Advanced MicroDevices Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD. His current research focuses on RF MEMS devices, electronic scanning antenna, mechanical modeling of MEMS, and microrobotics.

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Daniel Judy received the B.S. degree in electrical engineering from the University of Maryland at College Park, in 1988, and the M.S. degree in electrical engineering from The Johns Hopkins University, Baltimore, MD, in 1990. He is currently an Electronics Engineer with the RF and Electronics Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD. His current research focuses on high-frequency simulation of RF MEMS devices, electronic scanning antennas, and MEMS switch reliability.

Prashant Ranade received the B.S. degree in mechanical engineering from the University of Maryland at College Park. He is currently an Engineer with General Technical Services, Wall, NJ, and working with the MicroDevices Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD. His current research focuses on MEMS device fabrication and testing.

Susan Trolier-McKinstry (M’92–SM’01) received the B.S., M.S., and Ph.D. degrees in ceramic science from Pennsylvania State University, University Park. She is currently a Professor of ceramic science and engineering and Director of the W. M. Keck Smart Materials Integration Laboratory, Pennsylvania State University. She has held visiting appointments with the Hitachi Central Research Laboratory, the U.S. Army Research Laboratory, and the École Polytechnique Federale de Lausanne. Her main research interests include dielectric and piezoelectric thin films, the development of texture in bulk ceramic piezoelectrics, and spectroscopic ellipsometry. Dr. Trolier-McKinstry is a Fellow of the American Ceramic Society. She is a member of the Materials Research Society. She is past-president of both the Keramos and the Ceramics Education Council. She is co-chair of the committee revising the IEEE Standard on Ferroelectricity. She is currently the presidentelect of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control (UFFC).

Madan Dubey received the Ph.D. degree in physics from Banaras Hindu University, Varanasi, India, in 1977. He is currently a Research Physical Scientist with the Advanced MicroDevices Branch, Adelphi Laboratory Center, U.S. Army Research Laboratory, Adelphi, MD, where he leads a team of researchers on the process and fabrication of piezoelectric MEMS and low- and no-power sensors. He was a Post-Doctoral Fellow with North Carolina State University, Raleigh, and was also a Research Engineer with the Research Triangle Institute, Research Triangle Park, NC.

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