IEEE MTT-V055-I03 (2007-03) [55, 03 ed.]

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MARCH 2007

VOLUME 55

NUMBER 3

IETMAB

(ISSN 0018-9480)

PAPERS

Linear and Nonlinear Device Modeling Volterra Behavioral Model for Wideband RF Amplifiers ......... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ C. Crespo-Cadenas, J. Reina-Tosina, and M. J. Madero-Ayora

449

Active Circuits, Semiconductor Devices, and ICs Design and Analysis of Transmit/Receive Switch in Triple-Well CMOS for MIMO Wireless Systems ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... . A. Poh and Y. P. Zhang A High-Performance CMOS Voltage-Controlled Oscillator for Ultra-Low-Voltage Operations . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... H.-H. Hsieh and L.-H. Lu A 15/30-GHz Dual-Band Multiphase Voltage-Controlled Oscillator in 0.18- m CMOS ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... H.-H. Hsieh, Y.-C. Hsu, and L.-H. Lu Design of Class E Amplifier With Nonlinear and Linear Shunt Capacitances for Any Duty Cycle ..... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . A. Mediano, P. Molina-Gaudó, and C. Bernal Analysis and Design of a Dynamic Predistorter for WCDMA Handset Power Amplifiers ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... . S. Yamanouchi, Y. Aoki, K. Kunihiro, T. Hirayama, T. Miyazaki, and H. Hida

493

Millimeter-Wave and Terahertz Technologies The Direct Detection Effect in the Hot-Electron Bolometer Mixer Sensitivity Calibration ....... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ..... S. Cherednichenko, V. Drakinskiy, T. Berg, E. L. Kollberg, and I. Angelov

504

Field Analysis and Guided Waves Modeling Effects of Random Rough Interface on Power Absorption Between Dielectric and Conductive Medium in 3-D Problem ........ ......... ........ ......... ......... ........ ......... ......... ........ ...... X. Gu, L. Tsang, and H. Braunisch Modeling of 3-D Surface Roughness Effects With Application to -Coaxial Lines ..... . M. V. Lukic´ and D. S. Filipovic

511 518

458 467 474 484

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) CAD Algorithms and Numerical Techniques Equivalent SPICE Circuits With Guaranteed Passivity From Nonpassive Models ....... .. A. Lamecki and M. Mrozowski Singular Tetrahedral Finite Elements for Vector Electromagnetics ....... ........ ......... ......... ........ ........ J. P. Webb Space-Mapping Optimization With Adaptive Surrogate Model .. ......... ........ ......... ..... S. Koziel and J. W. Bandler

526 533 541

Filters and Multiplexers Ceramic Layer-By-Layer Stereolithography for the Manufacturing of 3-D Millimeter-Wave Filters .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ... N. Delhote, D. Baillargeat, S. Verdeyme, C. Delage, and C. Chaput

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Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Heat Conduction in Microwave Devices With Orthotropic and Temperature-Dependent Thermal Conductivity ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. J. Ditri Low-Loss Patterned Ground Shield Interconnect Transmission Lines in Advanced IC Processes ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... L. F. Tiemeijer, R. M. T. Pijper, R. J. Havens, and O. Hubert Compact Left-Handed Transmission Line as a Linear Phase–Voltage Modulator and Efficient Harmonic Generator ... .. .. ........ ......... ......... ........ ......... ......... ........ .. H. Kim, A. B. Kozyrev, A. Karbassi, and D. W. van der Weide

555 561 571

Microwave Photonics An LTCC-Based Wireless Transceiver for Radio-Over-Fiber Applications ...... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ..... L. Pergola, R. Gindera, D. Jäger, and R. Vahldieck

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Biological, Imaging, and Medical Applications Development of a 2.45-GHz Local Exposure System for In Vivo Study on Ocular Effects ....... ........ ......... ......... .. .. .. K. Wake, H. Hongo, S. Watanabe, M. Taki, Y. Kamimura, Y. Yamanaka, T. Uno, M. Kojima, I. Hata, and K. Sasaki

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LETTERS

Comments on “TOA Estimation for IR-UWB Systems With Different Transceiver Types” ...... ........ .... S. Nadarajah Authors’ Reply ... ......... ........ ......... ......... ........ ......... ......... ........ I. Guvenc, Z. Sahinoglu, and P. V. Orlik

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Information for Authors

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IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $14.00, plus an annual subscription fee of $20.00 per year for electronic media only or $40.00 per year for electronic and print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE J. S. KENNEY, President L. BOGLIONI D. HARVEY S. M. EL-GHAZALY J. HAUSNER M. HARRIS K. ITOH

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Editors-In-Chief DYLAN WILLIAMS NIST Boulder, CO 80305 USA Phone: +1 303 497 3138 Fax: +1 303 497 3970 email: [email protected] AMIR MORTAZAWI Univ. of Michigan Ann Arbor, MI 48109-2122 USA Phone: +1 734 936 2597 Fax: +1 734 647 2106 email: [email protected]

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Associate Editors

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RICHARD SNYDER RS Microwave Company USA email: [email protected] RUEY-BEEI WU National Taiwan Univ. Taiwan, R.O.C. email: [email protected] ALEXANDER YAKOVLEV Univ. of Mississippi USA email: [email protected] T. LEE, Web Master

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Digital Object Identifier 10.1109/TMTT.2007.894108

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 3, MARCH 2007

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Volterra Behavioral Model for Wideband RF Amplifiers Carlos Crespo-Cadenas, Associate Member, IEEE, Javier Reina-Tosina, Associate Member, IEEE, and María J. Madero-Ayora

Abstract—This paper proposes a behavioral modeling approach for the description of nonlinearities in wideband wireless communication circuits with memory. The model is formally derived exploiting the dependence on frequency of the amplifier nonlinear transfer functions and reduce the number of parameters in a general Volterra-based behavioral model. To validate the proposed approach, a commercial amplifier at 915 MHz, exhibiting nonlinear memory effects, has been widely characterized using different stimuli, including two tones, quadrature phase-shift keying wideband code division multiple access, and 16-quadrature amplitude modulation signals with rectangular and root-raised cosine conforming pulses. The theoretical results have been compared with experimental data demonstrating that the model performance is comparable to the well-established memory polynomial model. Calculated and measured baseband waveforms, signal constellation, spectral regrowth and adjacent channel power ratio are tightly coincident in all cases, emphasizing the relevance of the proposed model. Index Terms—Behavioral models, microwave amplifiers, nonlinear memory effects, Volterra series.

I. INTRODUCTION

A

S A fundamental block in wireless communications systems, the power amplifier (PA) has undergone exhaustive study of its characteristics, in particular those related with nonlinear memory effects. Many efforts have been devoted to obtain behavioral models for microwave PAs, for which the output of this black-box method is predicted without knowledge of the nonlinear device internal structure. The goals of these approaches are, on the one hand, reduction of complexity maintaining an accuracy comparable to the results obtained with circuit-level simulations and, on the other hand, a simple method to extract the model parameters. An indicator of the importance of these approaches is the valuable work presented over the last years, e.g., [1]–[3], and the recent publication of an extensive revision related to this topic [4]. Exploiting the bandlimited character of wireless signals, PA description can be translated into an envelope representation and frequently has been deduced as Volterra series, a procedure that treats this problem in a strictly and an orderly way. However, one difficulty of amplifier modeling using this Volterra approach

Manuscript received August 8, 2006; revised October 26, 2006. This work was supported by the Spanish National Board of Scientific and Technological Research under Project TEC2004-06451-C05-03. The authors are with the Departamento de Teoría de la Señal y Comunicaciones, Escuela Superior de Ingenieros, Universidad de Sevilla, 41092 Seville, Spain (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.2006.890514

is its high computational complexity [5]. The great number of coefficients required for the description of systems with a strong nonlinearity and long memory has steered the work of many researchers in order to reduce the number of model parameters. Probably the most manageable solution to reduce the number of coefficients is the memory polynomial (MP) model proposed in [1], the structure of which presents a notable truncation in the number of parameters. Although the reduction obtained with this simplified Volterra model is important, the number of coefficients remains high, particularly in the case of amplifiers with long memory. To achieve a further reduction, an extension of the MP model with a sparse delay tap structure was proposed in [2]. It is not clear how the parameter reduction of these particular structures affects the attainable accuracy of the model owing to the possible importance of other underestimated terms. Specifically, the need for considering those neglected terms was the purpose of the novel structure reported in [3]. That new approach is based on the pruning of redundant kernels in the full Volterra-series model so that the coefficients with less effect on the output signal are discarded following an a posteriori procedure. Despite the significance of the cited models, it is desirable to develop an approach with an optimized number of coefficients, sustained on theoretical principles and with no need of a previous empirical selection. That was the aim of the authors’ initial study of an amplifier with one FET based on its simplified equivalent circuit. A third-order model was validated with experimental data and partial results were presented in [6]. In this paper, the authors introduce the demonstration of a fifth-order Volterra model for a general amplifier with bandwidth larger than the RF signal band. The approach allows the analysis of nonlinear memory effects from a model based on the transfer functions with no restrictions in the number or kind of nonlinear devices composing the amplifier. In Section II, the study of a wideband amplifier and the completion of its behavioral model based on a Volterra-series approach is described. The frequency independence of the amplifier response inside the RF bandwidth is exploited to reduce the order model and to reveal the dependence of coefficients extraction on the sampling rate. Section III describes the procedure of parameter extraction, which has been simplified because of order reduction in the model structure. Application of the method to a commercial amplifier and comparison with the MP model, and also with experimental data using different types of input signals, is presented to validate the demonstrated theoretical results. Finally, a generalization of the model to any order is proposed and some relevant statements are commented upon.

0018-9480/$25.00 © 2007 IEEE

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. In (2), is a vector with its first for components equal to and the remaining components are is a column vector having equal to baseband frequency components, is the transpose of , and is a column vector with its components equal to . The bandlimited condition of the input signal allows to neglect the integral outside two -dimensional boxes with length and centered at . Consequently, it is possible to substiwith an equivalent transfer function bandlimited into tute , from which the banthese -dimensional hypercubes dlimited equivalent Volterra kernels are obtained making use of a multidimensional inverse Fourier transform. This equivalent transfer function and the discrete-time corresponding kernel satisfy the relation

Fig. 1. General schematic of a nonlinear circuit.

II. VOLTERRA MODEL FOR A WIDEBAND AMPLIFIER

(3)

A. Volterra-Based Behavioral Model Background Let a general amplifier be represented by the circuit shown in Fig. 1. Nonlinearities are constituted by their linear components, included in the associated linear circuit, and nonlinear sources, which are assumed to be dependent on two control voltages and . Let the input be an RF current excitation and be the output voltage corresponding to the fundamental frequency zone, centered at . Making use of the fact that a wireless signal commonly has a bandwidth negligible with respect to the carrier frequency , the discrete time-domain complex envelope Volterra model for this general nonlinear system can be expressed as

(1) and are complex envelope samples of the input where represents the disand output RF signals, respectively, is an -dimensional crete Volterra kernels of order , and vector composed of the integer-valued delays [3]. Although a sampling rate equal to the input signal RF bandwidth is sufficient for memoryless nonlinear system identification [7], it should be increased according to the broadening of if aliasing has to be avoided. As a conthe output bandwidth , the sequence, for an adequate representation of the output sampling time has to be reduced correspondingly to . Equation (1) is a discrete-time Volterra series derived from the representation using the multidimensional nonlinear transfer [8]. In continuous-time form, the functions (NLTFs) th-order term of the output signal can be written as

(2)

Substitution in (2) allows to separate the integrals and to ob. Therefore, after sampling at tain the output component , (1) is immediately derived. instants The Volterra model (1) is a very general result, but it has a high degree of difficulty due to the large number of parameters and numerical operations involved [5]. The complexity of the form an problem is revealed in the fact that the kernels -dimensional grid defined by the discrete delays in each axis ; hence, it is desirable of the multidimensional space to reduce the number of these delays. One of the most extended methods proposed to achieve a more manageable number of parameters is the MP model described in [1]. For this model, the reduction in the number of coefficients is obtained by selecting only the delays positioned in the diagonal, i.e., the delays along the direction defined by . Moreover, if a sparse delay tap structure is adopted and only the most significant delays are retained following an a posteriori procedure, a further important cutback in the number of coefficients can be procured [2]. However, the model precision can be diminished due to the possible importance of nondiagonal terms. Following a more relaxed pruning approach, which also retains the terms near the diagonal, a more recent model was proposed with a consequent improvement in precision at the expense of a moderate increase in the number of coefficients [3]. Although the MP model has proven to be effective and the reduction of coefficients is considerable, the lack of a theoretical justification originates the need of these empirical-based methods. Additionally, important issues as the adequate sampling rate or the dependence of the discrete kernels on this sampling rate in (1) should be addressed by a behavioral model. B. NLTFs for a Wideband Amplifier A formal reduction in the number of coefficients in (1) can be obtained under the only assumption of a wideband amplifier, i.e., an amplifier with a passband larger than the RF signal bandwidth. This supposition does not introduce any important loss of generality since many wireless amplifiers presenting nonlinear memory effects have frequency responses essentially constant

CRESPO-CADENAS et al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS

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Fig. 2. Illustrative example to show frequency dependence of the NLTFs. (a) Elementary nonlinear network. (b) Associated linear network excited by appropriate nonlinear currents and spectrum of the generated second-order transfer functions. (c) The same associated linear network producing the third-order transfer functions and their related spectrum.

in their respective RF signal bands, see, e.g., [2]. Under this assumption, completion of the particular frequency dependence for a typical circuit can be acof the transfer functions complished by a combination of the probing method and the nonlinear currents method, a widespread procedure [10], [11]. For the general circuit of Fig. 1, the vector formed with the NLTFs of order relating the voltages of the independent ports of the associated linear network can be obtained by using the following equation: (4) is the admittance matrix of the associated linear where is a vector with the spectral components network and of the nonlinear currents exciting the independent ports. The bandlimited condition of the wireless signal allows to extend the wideband amplifier assumption to all the harmonic zones so that the admittance matrix can be approximated by for at all relevant values of . For an th-order approximation, the band of interest in the first harmonic zone should be supposed here to be . , the transfer functions For clarity, let consider only relating the input with the voltage at the port of one nonlinearity, typically a voltage-controlled current source, conductance, or capacitance. To illustrate the procedure, an elementary circuit with a nonlinear current source is shown in Fig. 2 and a sketched summary of the method is attached. In the example, a nonlinear current source dependent on voltage is considered as the main nonlinearity. For each order, the associated linear network is excited by appropriate nonlinear currents in order to obtain the transfer functions relating voltage with the input. Spectra show the prevalent components for orders 2 and 3.

The wideband condition of the amplifier allows to approxias a coefficient independent of mate the linear function baseband frequencies. The same is true for the transfer function . relating the linear part of the output with the input can depend on the The spectral components function , as is the case of a nonsum of baseband frequencies linear capacitance, or is independent of baseband frequencies for other nonlinearities. In any case, since the components of the admittance matrix have this same dependence, the second-order depends on (with NLTF and ) in the dc zone, and can be considered as a constant coefficient in the second-harmonic zone (see Table I). Notice that if other nonlinearities are present, their contribution is superposed at the same frequencies and, therefore, the type of dependence remains unchanged. Both in the fundamental frequency and in the third-harmonic zones, the admittance matrix is a function only of the carrier frequency because the baseband frequency dependence of the is due to the spectral compothird-order transfer function nents of . Taking into account that this dependence comes , in the fundamental frequency zone, it is possible from to determine two types of terms: a baseband frequency-independent term and terms of the form . In the third-harmonic zone, this transfer function does not present dependence on baseband frequencies. The type of dependence described above is in agreement with previously published results [12]–[16]. Although the extension of the analysis focused to the deduction of closed-form expressions for higher order transfer functions is almost beyond the bounds of possibility, keeping track of their frequency dependence is a more feasible exercise. Next, this study is widened to higher order NLTFs.

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TABLE I ~ ( ) TYPES OF BASEBAND FREQUENCY DEPENDENCE FOR K

TABLE II ~ ( ) TYPES OF BASEBAND FREQUENCY DEPENDENCE FOR H

These results can be exploited to reduce the number of parameters in the behavioral model (1) without the addition of any other restriction. C. Kernels of the Reduced-Order Behavioral Model

The cause of frequency dependence in is twofold. On the one hand, the admittance matrix can be approximated by in the dc zone, and by a frequencyindependent function in the second and fourth harmonic zones. presents several terms On the other hand, the component with products of the transfer functions , and . Summarizing, in the dc zone, the fourth-order transfer function is composed by one term that can be expressed , a second type of terms by , and a third type of terms with the form . It has been considered that and , for and with and (see Table I). Note that is represented as the , and in the same product of two separated functions and form, is denoted by the product of the functions and . In these expressions, is a generic function with the specific dependence on the sum of four baseband frequencies and are generic functions dependent on the sum of two baseband frequencies. The significance of this particular frequency dependence is discussed below. In the second-harmonic zone, there are two type of terms, and , and in the fourth-harmonic zone, i.e., there is only one (frequency independent) term, i.e., . The explicit dependence has been omitted in the components that are only functions of the carrier frequency. These arguments are sufficient to deduce the frequency behavior of the relevant and . output transfer functions has a frequency dependence simThe transfer function , discussed above. In the case of , considering ilar to that the admittance matrix does not introduce any frequency deor, equivalently, pendence, its behavior is determined by to . Recalling that the zone of inby the functions terest is the fundamental frequency zone, the relevant transfer functions can be represented by the six types of terms shown in Table II. Observe that now there are four different types of terms to with a complete dependence on all the frequencies from .

1) Third-Order Kernel: Based on the previous deductions, the third-order term of the output voltage is obtained by and substituting in (2) the corresponding components . The first type generates the memoryless term, and the second type gives rise to the generic expression

(5) Although this is a double integral, the transfer function is a onedimensional function, a fact that can be explicitly displayed with and a change to the new variables for which so that

(6) , the integral in Relying on the bandlimited assumption of is negligible outside any bandwidth , allowing the definition of an equivalent transfer function confined to this band. Making use of the Fourier transform and for frequencan be expressed in terms of its discrete cies inside impulse response (7) should be, at least, half the so that the sampling time symbol period. Substituting in (6) and changing now to the original variables, the two integrals become separable as follows:

(8) Finally, after adding the memoryless part and sampling at in, the third-order term of the output complex envestants

CRESPO-CADENAS et al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS

lope can be expressed in a discrete-time form (9)

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Substituting in (12) and changing to the original variables, the , four integrals are now separable and, sampling at instants the output in discrete-time form can be written as

When the memoryless term is included in (9), we obtain the following expression for the third-order coefficients: (14) with

for (10)

(15)

2) Fifth-Order Kernel: According to the previous discussion, the fifth-order transfer function is composed of six different types of terms, which produce a particular set of kernels after substitution in (2). As in the third-order transfer function, one , remaining of the five integrals involved gives rise again to in this case a quadruple integral, for which the first type of terms can be computed directly to contribute only to memoryless nonso linear effects. The second type of terms depends on that only two integrals can be computed directly and the other two can be handled as in the third-order case producing one-dimensional kernels of the type

, related with the third type , It is immediate to note that is a particular case of this result in which the transfer function is directly separable. , which involves the More revealing is the fourth type sum of all frequencies so that it presents the highest degree of symmetry and produces terms given by

for

(16) with (17)

(11) For the other four types, it is possible to write a generic fifthorder transfer function with a frequency dependence given by , where and are defined around and , respectively. Let consider the different properties of symmetry that this function can present, beginning with the more general condition corresponding to the fifth and sixth types of terms. At a first glance, the multiple integral in (2) is negli. However, gible outside a four-dimensional cube of length the particular symmetry of involves a bidimensional frequency dependence that is exhibited clearly after the change of and variables as follows:

(12) is negligible outside any square of length and can be substituted by its equivalent function. It is possible to express this bandlimited function as a relation between the corresponding discrete-time kernels Therefore,

(13)

Notice that, for this particular coefficients, the sampling rate should be at least four times the symbol rate. In conclusion, let us observe that the demonstrated Volterra behavioral model incorporates a substantial reduction in the number of parameters, when compared to (1), with the only assumption of a wideband amplifier. Surprisingly, the described representation exhibits an exclusively “out of diagonal” structure, different to other well-known published behavioral models [1], [2]. To corroborate these new results, an amplifier has been tested and the model parameters have been extracted from experimental data, as will be discussed in Section III. III. MODEL PARAMETERS EXTRACTION AND VALIDATION The commercial amplifier MAX2430 manufactured by MAXIM Integrated Products Inc., Sunnyvale, CA, has been modeled with the current structure. It is a wideband amplifier at 915 MHz; however, in the experimental characterization with two tones separated by 2 MHz, the amplifier exhibited an asymmetry in the intermodulation distortion (IMD) products, a clear indication of the existence of nonlinear memory effects. The measurement setup used in this study is basically the same as that presented in [15]. However, the excitations taken into account are diverse in order to test the proposed model with a wide variety of signals. In particular, standard two-tone, as well as digitally modulated signals like quadrature phase-shift keying (QPSK) wideband code division multiple access (WCDMA), 16-quadrature amplitude modulation (QAM) with rectangular pulses and root-raised cosine pulses have been used as input stimuli. These signals have been loaded in the internal memory of an SMIQ02B signal generator with built-in

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(a)

Fig. 3. Normalized error as a function of the input level. Proposed VBW model (squares and solid line) and MP model (circles and dotted–dashed line). (b)

arbitrary waveform facility and the E4407B spectrum analyzer with a modulation analysis option has been used to acquire the baseband signal at the amplifier’s output.

Fig. 4. Normalized complex envelope in-phase component for a 16-QAM signal. (a) Rectangular pulses. (b) Root-raised cosine pulses. Acquired data: dots. VBW model: solid line. MP model: dotted–dashed line.

A. 16-QAM Signal With Rectangular Pulses As was demonstrated in Section II, it is necessary to use a sampling rate of approximately four times the symbol rate if parameters up to the fifth order have to be extracted. In the case of root-raised cosine modulating pulses, each sample is dependent on previous and future samples, included those many symbols away. Even in the case of a memoryless amplifier, the output will display memory and it is, therefore, reasonable to use for nonlinear characterization modulating pulses without inter symbol interference, i.e., with length no longer than a symbol period. Consequently, the use of an RF signal modulated with rectangular pulses as an input stimulus guarantees that memory effects, if present in the output, have been caused by the nonlinear memory of the device. The modulation format is also relevant because it is well known that for phase-shift keying (PSK) signals, a third-order model can capture some of the higher order nonlinear characteristics and produce degeneration in the parameter-extraction process [9]. According to the above considerations, a 915-MHz carrier modulated with a random train of rectangular pulses at 2 Msymb/s and a 16-QAM format was selected as the first sounding signal. The arbitrary waveform generator can handle up to 40 Msamp/s so that the shape of the pulses was rather rectangular. Since in the recovery part the setup has a sampling rate of 15 Msamp/s, the acquired output signals were sampled at 7.5 samples per symbol, amply suitable for signal representation and fifth-order parameters extraction. A fifth-order Volterra behavioral model for wideband amplifiers (VBW) was extracted from the acquired output complex envelope samples through the minimization of the average normalized mean square error (NMSE) between measured and modeled outputs. A first result is shown in Fig. 3, where the normalized error is plotted as a function of the input level. The error is represented as a dotted line for a memoryless model and the dashed line

corresponds to the results for a third-order VBW model with a samples. In both cases, the error grows up memory of as the amplifier enters in a more nonlinear condition, indicating in the first case the presence of nonlinear memory and, in the second case, that not all the nonlinear memory coefficients have been extracted. On the contrary, the fifth-order VBW model depicted in solid line shows a very low error, constant in all the range of input levels, which is an evident confirmation of a correct parameters identification of the memory nonlinear model. To corroborate the validity of the current results, the well-established MP model described in [2] was extracted with a generous number of delays, and taken as a reliable reference. The error for this model with seven delays is also represented in Fig. 3 (dashed–dotted line), demonstrating a similar performance with respect to the VBW model. Furthermore, in the range of higher levels, the most relevant to this context, the VBW model outperforms the MP model. The corresponding time-domain in-phase component for an input level of 11 dBm is shown normalized in Fig. 4(a). The acquisitions (dots), the prediction of the MP model (dashed–dotted line), and the prediction of the present VBW model (solid line) have been represented. The average NMSE of the VBW model is 30.5 dB, representing 1 dB of improvement compared to the MP model. The vector representation of the modeled complex envelope is plotted with dots in Fig. 5 and compared with the input envelope (squares) and the acquired output (crosses). B. Two-Tone Signal Another set of experiments was performed using an input signal formed by two tones of equal magnitude and phase, with a frequency separation of 2 MHz, and measuring the upper and lower third-order IM products. When the amplifier is working in

CRESPO-CADENAS et al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS

Fig. 5. Vector representation of the 16-QAM signal with rectangular pulses. Input signal: squares. Acquired data: crosses. VBW model: dots.

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Fig. 7. Output spectrum of a 16-QAM signal with 2 Msymb/s. Root-raised conforming pulses. P = 11 dBm. Spectrum analyzer trace (dots) and VBW model prediction (solid line).

0

measured and calculated lower IM3 at 14 dBm is caused by setup limitation at low input levels. On the contrary, the significant coincidence inside the faithful range is also revealed when measured and predicted asymmetries are compared, as is shown in Fig. 6(b). C. 16-QAM Signal With Root-Raised Cosine Pulses (a)

(b) Fig. 6. Measured and simulated results for IM3 when tone spacing is 2 MHz. Acquired data: triangles. VBW model: solid line.

the linear mode, the IM products are negligible compared to the nonsystematic errors of the measurement setup so that model parameters extracted from acquisitions at low signal levels are irregular. This is not specially inconvenient because the interest is in the range of high signal levels, where the nonlinear effects are more relevant and model parameters can be reliably extracted. For that reason, after rejection of meaningless data, the experimental points represented in Fig. 6 belong to levels near the 1-dB compression point. In Fig. 6(a), the measured output of the fundamental tones and the third-order intermodulation (IM3) products are represented with marks. Also in Fig. 6, the results of the extracted model are depicted via a solid line, reflecting a remarkable correspondence with the acquired data. According to the previous discussion, the difference between

Another type of sounding signal employed in the extraction process has been a carrier at 915 MHz modulated in a 16-QAM format with a 2-Msymb/s train of symbols using root-raised cosine pulses. The acquired normalized in-phase component and the corresponding waveform obtained with the extracted model are represented in Fig. 4(b) (dots and solid line, respectively). For comparative purposes, the waveform obtained with the MP model is also depicted in Fig. 4(b) (dashed–dotted line). As a further test, the extracted model was used to predict the spectrum of the signal and the adjacent-channel power (ACP) in order to be compared with other alternative measurements using the conventional spectrum analyzer without the acquisition facility. The results are plotted in Figs. 7 and 8 using marks for the experimental data and solid lines for the modeled output. It is worth noting that the model was first extracted from an experimental acquisition of baseband samples and served to predict the output signal spectrum. Although the marks represent a measurement process independent of the acquisition, the prediction is favorably compared in Fig. 7. The second figure corroborates this outcome showing a good match between the ACP measured and calculated. The prediction is also able to estimate adequately the ACP asymmetry between upper and lower channels. As a reference, the dashed–dotted line represents the results of the MP model. D. QPSK-WCDMA Signal Finally, a carrier at 915 MHz modulated with a WCDMA signal compliant with the Universal Mobile Telecommunications System (UMTS) standard was employed as input signal. In this case, the condition of four times the symbol rate to correctly

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Fig. 8. In-channel and ACP. Acquired data: triangles. VBW model: solid lines. MP model: dotted–dashed lines.

Fig. 10. ACP measured and calculated with the proposed method. Acquired data: triangles. VBW model: solid lines. TABLE III ACPR FOR A QPSK-WCDMA SIGNAL AT 3.84 Ms/s

of the fact that the acquisitions were accomplished beyond the limits of theoretical accuracy, a satisfactory agreement is revealed. For the sake of comparison, the reference MP model is also included, attaining equivalent results. IV. FINAL DISCUSSION AND CONCLUSIONS

Fig. 9. Spectral regrowth of a W-CDMA signal with 3.84 Msymb/s. P 10 dBm. Spectra of acquired data (dots) and model output (solid line).

0

=

identify fifth-order parameters, 4 3.84 Msymb/s, just exceeds the sampling rate, 15 Msamp/s.1 However, the model has been able to adequately extract the parameters, as can be verified by comparing the acquired and modeled spectra, which are shown in Fig. 9 with dots for the experimental data and solid line for the calculations with the extracted model. Again, the current approach allows a reliable prediction of ACP and its asymmetry from the acquired spectra, as can be confirmed in Fig. 10, in which the measured data (triangles) and the model results (solid lines) are represented. An alternative method was used to measure the ACP with an input level of 10 dBm and the measured data are presented in Table III for the two adjacent channels. In Table III, the calculations with the proposed VBW model are also shown. In spite 1Considering that with respect to the modulation format, the term chip ultimately corresponds to a symbol, in this context, we use the term symbol instead of chip.

This study has demonstrated a new Volterra approach to model wideband amplifiers with nonlinear memory. The main characteristic of the current behavioral model is that it has been formally derived starting from a conventional nonlinear circuit analysis and makes it possible to propose the extension of its structure to give the following equation:

(18) This expression has a remarkable difference with respect to the MP model consisting in the absence of the so-called “diagonal terms.” Although only the assumption of a frequency-independent response has been necessary to obtain the new model, the huge number of coefficients associated to the general discretetime Volterra series has been drastically reduced. The parameter order reduction can be quantified for an example in which delays is considered. Taking a fifth-order model with into account symmetry considerations, 244 coefficients are necessary with the general Volterra model. As a reference, recall that, in the MP model, a total of 12 diagonal coefficients are needed, and a total of 54 or 133 coefficients form the model or , with the “near-diagonality” structural restriction respectively [3]. Instead, the current model would need 21 coefficients. It is worth observing that a fair comparison between

CRESPO-CADENAS et al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS

the previous models and the current VBW would only be possible if some procedure of pruning to optimize the number of coefficients were also included. In relation to memoryless nonlinear systems, the introduced analysis states that identification may be achieved by sampling at the symbol rate, which is in accordance with previously published results [17]. However, it is also demonstrated that in the in case of systems with nonlinear memory, an increase of the sampling rate is necessary to adequately identify th-order parameters. Another notable conceptual issue is the exhibited with respect to the relation of the model coefficients adopted sampling rate, revealing a somewhat involved structure with terms of the same order showing different dependence on this parameter, e.g., (15) and (17). To contrast with the theoretical results, a commercial amplifier was characterized using four different types of waveforms and the coefficients of the model were extracted using an experimental setup with acquisition facilities. The model predictions were compared with the well-established MP behavioral model and performance is very similar, indicating the validity of the current procedure. Requiring a search algorithm for an abundant number or delays, the reference behavioral model exhibits somewhat better prediction of the output characteristics in the low power range. On the contrary, the introduced Volterra-based wideband behavioral method uses more delays without requiring a search process and outperforms the polynomial model in the range of powers where nonlinear effects are more significant. An important requirement for a Volterra-based model is its ability to manage different types of signals. It has been revealed by observing the performance of the demonstrated model with input stimuli as diverse as a two-tone signal, a 3GPP WCDMA signal, and 16-QAM signal with rectangular and root-raised cosine pulses. The results were very satisfactory in all the cases. Due to space limitations, only the most relevant preliminary results have been presented. At the moment, the model is being tested with the experimental data in order to evaluate consistence at diverse measurement conditions and its ability to manage input signals with different bandwidths and power levels. REFERENCES [1] J. Kim and K. Konstantinou, “Digital predistortion of wideband signals based on power amplifier model with memory,” Electron. Lett., vol. 37, no. 23, pp. 1417–1418, Nov. 2001. [2] H. Ku and J. S. Kenney, “Behavioral modeling of nonlinear RF power amplifiers considering memory effects,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2495–2504, Dec. 2003. [3] A. Zhu and T. J. Brazil, “Behavioral modeling of RF power amplifiers based on pruned Volterra series,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 12, pp. 563–565, Dec. 2004. [4] J. C. Pedro and S. A. Maas, “A comparative overview of microwave and wireless power amplifier behavioral approaches,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1150–1163, Apr. 2005. [5] J. Tsimbinos and K. V. Lever, “Computational complexity of Volterra based nonlinear compensators,” Electron. Lett., vol. 32, no. 9, pp. 852–854, Apr. 1996. [6] C. Crespo-Cadenas, J. Reina-Tosina, and M. J. Madero-Ayora, “Volterra series approach to behavioral modeling: Application to an FET amplifier,” in IEEE Asia–Pacific Microw. Conf., Yokohama, Japan, Dec. 2006, vol. 1, pp. 445–448.

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[7] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing. New York: Wiley, 2000. [8] D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1984. [9] C. H. Cheng and E. J. Powers, “Optimal Volterra kernel estimation algorithms for a nonlinear communication system for PSK and QAM inputs,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 147–163, Jan. 2001. [10] J. J. Bussgang, L. Ehrman, and J. W. Graham, “Analysis of nonlinear systems with multiple inputs,” Proc. IEEE, vol. 62, no. 8, pp. 1088–1119, Aug. 1974. [11] J. C. Pedro and N. Borges, Intermodulation Distortion in Microwave and Wireless Circuits. New York: Artech House, 2003. [12] J. F. Sevic, K. L. Burger, and M. B. Steer, “A novel envelope-termination load-pull method for ACPR optimization of RF/microwave power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, Jun. 1998, pp. 723–726. [13] N. Borges and J. C. Pedro, “A comprehensive explanation of distortion sideband asymmetries,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2090–2101, Sep. 2002. [14] J. Brinkhoff and A. E. Parker, “Effect of baseband impedance on FET intermodulation,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 1045–1051, Mar. 2003. [15] C. Crespo-Cadenas, J. Reina-Tosina, and M. J. Madero-Ayora, “Phase characterization of two-tone intermodulation distortion,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 1505–1508. [16] C. Crespo-Cadenas, J. Reina-Tosina, and M. J. Madero-Ayora, “IM3 and IM5 phase characterization and analysis based on a simplified newton approach,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 321–328, Jan. 2006. [17] J. Tsimbinos and K. V. Lever, “Input Nyquist sampling suffices to identify and compensate nonlinear systems,” IEEE Trans. Signal Process., vol. 46, no. 10, pp. 2833–2837, Oct. 1998. Carlos Crespo-Cadenas (A’99) was born in Madrid, Spain. He received the Physics degree from the University of Havana, Havana, Cuba, in 1973, and the Doctor degree from the Polytechnique University of Madrid, Madrid, Spain, in 1995. Since 1998, he has been an Associate Professor and currently teaches lectures on radio communications in the area of signal theory and communications with the Universidad de Sevilla, Seville, Spain. His current interests are nonlinear analysis applied to wireless digital communications and to microwave monolithic integrated circuits (MMICs).

Javier Reina-Tosina (S’98–A’03) was born in Seville, Spain, in May 1973. He received the Telecommunication Engineering and Doctor degrees from the Universidad de Sevilla, Seville, Spain, in 1996 and 2003, respectively. Since 1997, he has been with the Departamento de Ingeniería Electrónica, Universidad de Sevilla. His current research interests include MMIC technology, nonlinear analysis of active microwave devices, and integration of information technologies in biomedicine.

María J. Madero-Ayora received the Telecommunication Engineering degree from the Universidad de Sevilla, Seville, Spain, in 2002, and is currently working toward the Doctor degree in telecommunication engineering at the Universidad de Sevilla. Since 2003, she has been with the Departamento de Teorí a de la Señal y Comunicaciones, Universidad de Sevilla. Her research interests are in the area of nonlinear analysis of active microwave devices and measurement techniques for nonlinear communication systems.

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Design and Analysis of Transmit/Receive Switch in Triple-Well CMOS for MIMO Wireless Systems Andrew Poh and Yue Ping Zhang

Abstract—This paper presents the design and analysis of an RF transmit/receive switch in CMOS for multiple-input multiple-output (MIMO) wireless systems. First, both full-range and abridged quad-pole quad-throw (4P4T) switch architectures for MIMO systems with four antenna elements are comparatively studied. The full-range 4P4T switch is selected for circuit implementation because of its more switching states and lower insertion loss. The series connection and body-floating techniques are then employed in the circuit design to achieve an acceptable performance. Fabricated in 0.12- m triple-well CMOS with an effective die area 0.35 0.19 mm2 , the 4P4T switch exhibits less than 2.7-dB insertion loss and higher than 20-dB isolation over the frequency range from 2 to 10 GHz. It also attains measured power-handling capability more than 25 dBm at 2.4 and 5.8 GHz. Index Terms—CMOS, multiple-input multiple-output (MIMO), transmit/receive (T/R) switch, wireless communications.

I. INTRODUCTION

M

ULTIPLE-INPUT multiple-output (MIMO) technology is currently being considered for use in cellular communication broadband wireless access, as well as for wireless local area networks [1]. MIMO technology is enabled by the presence of multiple transmit antennas and multiple receive antennas in the communication link. Spatial diversity and spatial multiplexing are two mechanisms by which MIMO enhances capacity and robustness of the link. Spatial diversity represents the existence of multiple signal paths between the multiple transmit and receive antennas that fade independently. As such, the probability of having all the signal strength falling below detection threshold would be very low. In other words, the probability of having all the antennas in adverse locations is significantly reduced with the presence of multiple receive antennas. Therefore, spatial diversity improves the robustness of the link. Spatial multiplexing involves the transmission and reception of multiple data streams from multiple antennas at either the same time or in the same frequency spectrum. The multiple antennas are primarily used to split the different data streams at the receiver. Spatial multiplexing facilitates an increase of data rate without the requirement for a larger Manuscript received June 10, 2006; revised September 21, 2006. A. Poh was with the Integrated Systems Research Laboratory, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798. He is now with the Institute of Microelectronics, Singapore 117685 (e-mail: [email protected]). Y. P. Zhang is with the Integrated Systems Research Laboratory, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890510

frequency spectrum as the gains arise from a resolution of parallel spatial paths in the channel. Regardless of whether spatial diversity or spatial multiplexing is used, the key problem of any MIMO system is an increased complexity, which often translates to higher cost. Hybrid selection is a popular method in overcoming the complexity issues. It involves a reduction in the number of active antennas used in the course of transmission or reception. The hybrid selection scheme necessitates the use of a transmit/receive (T/R) switch. The T/R switch has been designed in GaAs for MIMO systems [3]. To the best of our knowledge, the design of the T/R switch in CMOS for MIMO systems has not yet been reported. It is known that the design of the T/R switch in CMOS is rather desirable for high-level integration of MIMO systems. There have been several T/R switches designed in CMOS for non-MIMO systems [4]–[11]. Key figures-of-merit of a T/R switch includes insertion loss, isolation, and power-handing ca. Repability measured by the 1-dB compression point garding the performance of insertion loss and isolation, the effect of substrate resistance is studied in [4] and [5], where low insertion loss is obtained by minimizing the substrate resistance and dc biasing the T/R nodes. A high isolation is achieved using CMOS silicon-on-insulator technology [6]. In both cases, however, the power-handling capability is limited due to the parasitic capacitance and source/drain junction diodes. Thus, techniques of body floating are developed for higher power-handling capability. An LC-tuned substrate bias technique is firstly reported in [7], where the bulk is not separated from the substrate. Using an on-chip inductor can tune the bulk of the switching transistor to be floating at certain frequencies. 28-dBm in the transmit mode is obtained. The disadvantages of this approach are the design complexity and large silicon area consumed. Taking advantage of the modern triple-well CMOS process, the idea of body floating can be simply realized by using a large resistor to bias the bulk [8]. As resistors are intrinsically wideband, the power-handing improvement of this approach is also wideband. 20-dBm was achieved at 5.8 GHz. Another approach to the power-handing improvement is using stacked transistors; however, insertion loss will be degraded and has to be compensated, e.g., by the special depletion layer extended transistor process [9]. A 15-GHz T/R switch is reported in [10], the impedance matching network was employed to improve the power handling, while the isolation performance is degraded. The power-handling capability can also be improved by using differential architectures [11]; 3-dB power-handing capability improvement can be obtained.

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Fig. 2. Abridged 4P4T architecture [3].

Fig. 1. Full-range 4P4T architecture [3].

This paper presents the first T/R switch designed in CMOS for MIMO systems. Section II evaluates two T/R switch architectures for MIMO systems. Section III describes the circuit design in a 0.12- m triple-well CMOS. Section IV discusses the experimental results. Finally, Section V summarizes the conclusions. II. SWITCH ARCHITECTURES FOR MIMO SYSTEMS Traditionally, a T/R switch serves to assign a single antenna between the transmitter and receiver according to the time division duplexing command. A MIMO system, however, shifts the paradigm to which the T/R switch functions. It utilizes multiple antennas to attain true full duplexing and, therefore, does not require the switch to specify the transmitting or receiving operations. The number of antenna elements used in the MIMO system dictates the configuration of the T/R switch. It is suggested that the antenna elements for a base station be limited to a moderate number such as four, as a large number of antennas pose environmental problems [1], [2]. In addition, it is also predicted that a maximum of four dual polarized patch antennas with half-wavelength spacing can be exploited for terminal operations. However, there are concerns that space constraints in handsets may pose a challenge to the use of more than one antenna. Nevertheless, we believe that four antenna elements are proper for most MIMO systems. Two 4P4T T/R switch architectures have been proposed for MIMO systems with four antenna elements [3]. Unfortunately, neither has been thoroughly evaluated. Here, the two architectures are comparatively studied according to their complexity and performance. Fig. 1 shows the full-range 4P4T architecture. The switch exhibits a total of 16 states determined by four to . The architecture allows a full-range bias controls, switching states, which means that there is no redundant port. Fig. 2 shows the abridged quad-pole quad-throw (4P4T) architecture. The abridged 4P4T architecture offers a reduction in complexity through a decrease in the number of bias controls. However, the reduction is made at the expense of switching

states. There are eight possible states for the architecture; however, the number of usable states is limited to only four. The redundant four states involve the direct connection of a transmitter to the receiver of a different set, thereby eliminating any possibilities of use for on-chip calibration operations. As such, the limitations of the switch architecture present several implications. Foremost, the configuration only allows for a single transmitting and receiving operation at any one time, which implies that the switch is only capable of single-in–single-out operations. Another concern is that individual transmitting and receiving operations are each limited to two antennas, which imposes constraints on the degrees of freedom, thereby severely crippling the exploitation of antenna diversity advantages. Both architectures perform differently for insertion loss and isolation. The insertion loss of the full-range 4P4T architecture is smaller than that of the abridged 4P4T architecture because the signal passes through an additional transistor in the abridged 4P4T architecture. The isolation of the full-range 4P4T architecture is lower than that of the abridged 4P4T architecture maximally by 3 dB because there are two off-state transistors in parallel in the full-range 4P4T architecture, while there is only one off-state transistor in the abridged 4P4T architecture connecting the transmitter and the receiver of the same set. Our simulations show that the insertion loss of the full-range 4P4T architecture is 0.7 dB better than that of the abridged 4P4T architecture, but the isolation of the full-range 4P4T architecture is 1.5 dB worse than that of the abridged 4P4T architecture. Both architectures perform similarly for power-handling capability because they share the same mechanisms limiting power handling. Based upon the above evaluation, the full-range 4P4T architecture is selected for circuit implementation in this study. The full-range 4P4T architecture can support both a one-uplink–three-downlink antenna operation as well as a two-uplink– two-downlink antenna operation. III. CIRCUIT DESIGN IN TRIPLE-WELL CMOS The circuit was designed using Cadence SpectreRF in the 0.12- m triple-well CMOS from ST Microelectronics, Grenoble, France. Two circuit techniques known as body floating and series connection of transistors were employed to enhance the switch performance. It is shown that an increase in body resistance can improve the insertion loss and that the floating of the body can improve the power handling [8]. The body-floating technique is not viable with bulk CMOS, as it involves manipulation to the substrate conditions. With triple-well CMOS, the body is isolated from the substrate. Therefore, a resistor can be introduced to the body contact of the transistor to float the body. Besides improving the power handling, the introduction of resistors to the body contact

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Fig. 4. Annotated cross-sectional view of triple-well nMOS transistor. Fig. 3. Series-connected transistors for the switch.

transistor, the insertion loss is given by serves to reduce the insertion loss by diminishing capacitive coupling losses through the body contact. Furthermore, the body resistors function to eliminate the coupling between transistors that increases the insertion loss in the configuration of series-connected transistors. The series connection of transistors is a potential method to increase the power handling. A major setback for the technique lies with the fact that it yields a significant increase in the insertion loss. The series connection of transistors is viable with triple-well CMOS as the addition of resistors to the body contact can prevent losses through the body, thereby facilitating an equal voltage division, as well as manageable insertion losses. It was highlighted that the series connection of transistors results in a different impedance to ground at each successive drain/source node [9]. The asymmetry produces an unequal splitting of the RF voltage causing the transistor nearest to the signal source to experience the largest voltage and breakdown first. Again, the concern is unwarranted with triple-well CMOS as the body is floated at RF and isolated from the substrate. Hence, the series connection of transistors does not exhibit any symmetry issues reported in conventional bulk CMOS. Fig. 3 shows the series-connected transistor structure proposed for the switch. As shown, both the gate and body nodes and . The are biased using individual large resistors use of individual resistors is necessary to prevent signal coupling between the series-connected transistors. To date, only the advantages of implementing resistors at the gate node have been widely covered in the literature [4]. Increasing the number of series-connected transistors effectively improves the power handling and isolation, while compromising the insertion loss. Therefore, it is expected that there exists an optimum number of series-connected transistors for which the switch performance is optimized. The size of a transistor is a critical aspect of design as it determines the on-state resistance, as well as the parasitic capacitance. Foremost, the design of the switch should use the minimum length transistors so as to minimize the on-state resistance for each transistor in the proposed series-connected configuration. It is known that there exists an optimum transistor width for insertion loss. With reduction of the capacitive losses due to the introduction of the body resistance, the insertion loss of the switch can be attributed predominantly to the on-state resistance. Based upon the assumption that the capacitive coupling through the n-well isolation is negligible, for a series-connected

(1) and the isolation is given by (2) and such that , defines the number of series-connected transistors used in the switch, is the source/load impedance, and others have their usual meanings. Fig. 4 depicts an annotated cross-sectional view of a triplewell nMOS transistor with the relevant parasitic components and introduced resistors. In a triple-well implementation, there exist two pn-junction diodes, one between the p-well and deep n-well and the other between the deep n-well and p-substrate. These two junction diodes have to be maintained in reverse bias in order to prevent a breakdown in isolation between the p-well, deep n-well, and p-substrate. is also introduced As seen in Fig. 4, a large resistor to the deep n-well node in addition to the resistors connected that are to the gate and body nodes. The large resistor serves to float the deep n-well node at RF. This allows the RF voltage level in the deep n-well to adjust according to the RF voltage swing in the p-well, thereby ensuring that the p-well to deep n-well junction diode is always in reverse bias. The capacitance between the p-substrate and deep n-well junction can be kept minimum by isolating each transistor with an individual deep n-well. For our implementation, each transistor is placed into a p-well of area 211.5 m within a deep n-well area of 345.83 m , yielding a capacitance of 171 fF between the p-well and deep n-well interface and 71-fF capacitance between the deep n-well and p-substrate interface. As such, the effective series parasitic capacitance between the p-well and p-substrate can be neglected and, therefore, the body node of the transistor can be considered as a floating node at RF. To reiterate, the power-handling limitations of conventional CMOS switches have mainly been attributed to either the unintentional forward biasing of the source/drain-to-body junction diodes of the off-state transistors or the turning on of the off-state switch transistors by a gate–drain/source voltage that where

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Fig. 5. Equivalent circuit of an N -series-connected transistor switch in the off state. (For the 20-finger 285-m off-state transistor used: C gs = 145 fF, C gd = 144 fF, Cdb = 2:6 aF, Csb = 2:8 aF, C ds = 15 aF).

is greater than the magnitude of the threshold voltage [9]. How- Fig. 6. Variation of the insertion loss with respect to the transistor width and ever, it can be shown that the power-handling limitation of the gate resistance at N = 1. switch implemented with a floating body and series-connection technique would predominantly depend upon the voltage at which the off-state switch transistors are turned on. sistor remains off as long as As illustrated in Fig. 5, the drain and source nodes of the off-state -series-connected transistor switch are coupled by a (4) series of capacitance. In most cases, the drain/source-to-body capacitances and the drain–source capacitance are relatively Thus, -series-connected transistors remain off as long as smaller compared to the drain/source–gate capacitances and, therefore, can be neglected. Hence, the overall isolation can (5) be attributed mainly to the drain/source–gate capacitance. The voltage across the overall drain and source node will be divided equally by symmetry and, therefore, the effective This leads to the maximum input voltage swing that can be toldrain/source–gate voltages across each transistor will be re- erated, which is duced significantly. Hence, the gate–source voltage at which (6) the off-state transistors are turned on limits the power handling. By this analysis alone, the power handling would be indepenSince the maximum handling power is determined by dent of transistor width as long as the resistance of the on-state switch is kept small. It is well known that the application of dc dBm (7) bias to both the source and drain nodes can also boost power handling [4]. However, the main drawback of applying a dc bias to both the source and drain node would be the increase Equation (3) can be obtained by substituting (5) into (7). in the on-state resistance in the on-state switch. When smaller An insight to the design of the switch is provided below. The transistor width is used, the on-state resistance may increase initial analysis is based upon the optimization of a SPDT switch. to significant values, which may result in a significant increase It is known that the insertion loss increases as the frequency of to the equivalent input impedance seen at the input of the operation increases as the capacitive loss mechanisms predomswitch. The increase in equivalent input resistance would result inate at higher frequencies. The capacitive loss partially occurs in a larger input voltage swing and, hence, the switch would through the gate of the transistor. The gate capacitance increases compress at a lower input power than expected. Hence, it is as the transistor width increases; therefore, the gate capacitance desirable to keep the on-state resistance as low as possible appears as a path for the high-frequency signals, thereby reand, therefore, larger transistor width is desirable. The series sulting in increased loss. Introducing a resistor to the gate of connection of transistors allows for the achieving of lower the transistor can reduce the loss. Fig. 6 illustrates the variation on-state resistance through a selection of transistors with larger of the insertion loss with respect to the transistor width and gate width while maintaining an acceptable isolation. The 1-dB resistance at 10.6 GHz. The insertion loss is suppressed with inseries-connected transistor creasing gate resistance. However, the insertion loss increased compression for an optimized switch can be estimated by significantly when the gate resistance was approximately 100 . This is consistent with the maximum point indicated in [4], whereby the maximum insertion loss is predicted to occur when dBm (3) the gate resistance is equivalent to the impedance of the total loss within an series network. As capacitance such, it is ideal to use a large gate resistance to achieve lower where is the effective dc voltage between the gate and insertion loss. A typical value of 10 k is recommended. Howsource of the th transistor in the series connection. The deriva- ever, it is also observed from Fig. 7 that the capacitive loss still tion of the above equation is as follows. A single off-state tran- dominates at larger width even after 10-k gate resistance has

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Fig. 7. Variation of the insertion loss with respect to the transistor width and = 10 k and = 1. frequency at

R

N

Fig. 8. Variation of the isolation with respect to the transistor width and gate resistance at = 1.

N

been introduced, which indicates that there is another capacitive path for loss. As seen in Fig. 8, the isolation is generally degraded by 40 dB with the introduction of the gate resistor. This is due to the fact that the resistor floats the gate node at RF. Hence, the loss mechanism in the off state through the gate node is eliminated. As such, the isolation is determined mainly by the coupling of the parasitic gate capacitance. It can be observed from Fig. 9 that the isolation mechanism is purely capacitive in nature, as there is no convergence for isolation at different frequencies. It should be mentioned that the maximum power handling is improved by 3 dB to 11.5 dBm with the inclusion of the gate resistor for the . case of As mentioned earlier, there is another capacitive coupling loss mechanism that affects the switch insertion loss. Fig. 8 shows that the introduction of the body contact resistance is able to suppress this capacitive loss mechanism. Similar to that observed in Fig. 10, the insertion loss increases considerably with the body resistance of 100 . As explained earlier, it loss effect, whereby may be attributed to the series the impedance of the loss capacitance is equivalent to the body resistance. Therefore, the body resistance of 10 k is selected. As observed in Fig. 11, the insertion loss is constant throughout all frequencies. Hence, the capacitive coupling

Fig. 9. Variation of the isolation with respect to the transistor width and = 10 k and = 1. frequency at

R

N

Fig. 10. Variation of the insertion loss with respect to the transistor width and body resistance at = 10 k and = 1.

R

N

Fig. 11. Variation of the insertion loss with respect to the transistor width = = 10 k and = 1. and frequency at

R

R

N

loss mechanisms have been suppressed entirely by the gate resistance of 10 k and the body resistance of 10 k . As such, the on-state resistance solely characterizes the insertion loss. In comparison with the earlier results, the addition of the body resistance reduces isolation by 10 dB. This is indicative that the loss mechanism through the gate node in the off state is more significant than that through the body. Nevertheless, the isolation mechanism is shown to depend upon the capacitive coupling via both the gate and body capacitance.

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Fig. 12. Variation of the insertion loss with respect to the transistor width and the number of series-connected transistors at R = R = 10 k .

Fig. 14. Variation of the insertion loss with respect to transistor width and frequency.

Fig. 13. Variation of the isolation with respect to the transistor width and the number of series-connected transistors at R = R = 10 k .

Fig. 15. Variation of the isolation with respect to the transistor width and frequency.

Fig. 12 shows the variation of the insertion loss with respect to the transistor width and the number of series-connected tranV, and V. It is sistors at 10.6 GHz, observed that although the insertion loss increases as the number of series-connected transistors is increased, the difference does not appear significant when the large transistor width is used. Fig. 13 shows the variation of the isolation with respect to the transistor width and the number of series-connected transistors V, and V. It is evident at 10.6 GHz, that the isolation is improved by approximately 5 dB for every additional series-connected transistor used. The power handling is also improved as the number of series-connected transistors is increased while maintaining a constant isolation close to 20 dB , the power-handling capability reaches at 10.6 GHz. For 28 dBm. The performance of the full-range 4P4T switch using singleswitches is nonviable. The replacement of transistor switches with the switches of four the single-transistor resolves the performance series-connected transistors problems. Fig. 14 shows the variation of the insertion loss with respect to the transistor width and the frequency of operation. It is seen that the insertion loss depends slightly on the frequency. Fig. 15 shows the variation of the isolation with respect to the transistor width and the frequency of operation. The isolation for the 4P4T switch was simulated between the transmitter

and receiver ports of the same set as they yield the worst case isolation. As shown, the minimum isolation of 20 dB can be achieved by using transistors of larger widths. The transistor width to yield an isolation of 20 dB at 10.6 GHz is approximately 270 m with a control on voltage of 1.2 V and a source/drain bias of 0 V. With the control on voltage increased to 1.8 V and the source/drain bias to 0.6 V, the transistor width can be increased up to 310 m in order to maintain an isolation of 20 dB at 10.6 GHz. Transistors of width 285 m are finally chosen for the switch, as it yields an isolation of more than 20 dB at 10.6 GHz. The power-handling capability is significantly improved, as compared with the 4P4T switch using the single-transistor switches. The implementation of the proposed techniques of floating body and series connection showed that power handling could be improved by at least 13 dB. The application of 1.8-V control voltage with 0.6-V drain/source biasing produced an additional 9-dB improvement. Periodic steady-state simulation for the 4P4T switch indicated a 1-dB compression point of approximately 28 dBm, which concurs with the value that was estimated using (3). In order to fully appreciate the power-handling improvements brought about the techniques utilized, consider the case with reference to Fig. 1, when a 50- source is connected to the transmitter A (TXA) port and the ports are all matched to 50 . The

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Fig. 16. Simulated waveforms of the RF voltage swing at the input with an input power of 30 dBm for transistors with a width of 60 m and 285 m, respectively.

Fig. 17. Simulated waveforms of the RF voltage swing at the junction between the p-well and deep n-well of each 285-m transistor with an input power of 30 dBm for SW2.

drain/source node voltage ( ) was biased at 0.6 V and the control voltage was set to 1.8 V. Switch SW1 is on and switch SW2 is off. The input signals at the nodes of each transistor in the switches where simulated. As shown in Fig. 16, for an input signal of 30 dBm, the voltage swing at the input of the switch comprising of switches of transistors of 60- m width is larger than that for the switch with transistors of 285- m width. This is because the increased on-state resistance of the switch with smaller transistor width has significantly increased the equivalent input resistance looking into the switch. Therefore, a larger input RF swing develops at the input and, hence, the switch with smaller transistor width would compress with a lower input power. Fig. 17 illustrates the voltage between the p-well (body) and deep n-well node for the case with the transistor width of 285 m for SW2. As observed, the junction is never forward biased as the voltage swings are well below the turn-on voltage for the junction diode of 0.57 V. From Figs. 18 and 19, the gate–drain and gate–source voltages are shown to be relatively equal for all the series-connected transistors concurring with our analysis that the input voltage swing is divided equally between the gate–source/drain capacitances. Compression has occurred as the off-state transistors are turned on, as seen in Figs. 18 and 19.

Fig. 18. Simulated waveforms of the RF voltage swing of the gate–drain voltage of each 285-m transistor with an input power of 30 dBm for SW2.

Fig. 19. Simulated waveforms of the RF voltage swing of the gate–source voltage of each 285-m transistor with an input power of 30 dBm for SW2.

Fig. 20. Die microphotograph of the full-range 4P4T switch in 0.12-m triplewell CMOS.

IV. MEASUREMENT RESULTS The switch was fabricated in a 1.2-V two-poly six-metal 0.12- m triple-well CMOS process. The switch die microphotograph is shown in Fig. 20. The overall die size is 0.98 0.5 mm owing to the large number of pads. The effective area without a pad is only 0.35 0.19 mm . The switch measurements were performed on-wafer. The control on voltage was 1.8 V and the source/drain bias was 0.6 V. Fig. 21 shows the simulated and measured insertion loss

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TABLE I PERFORMANCES OF MIMO T/R SWITCHES

Fig. 21. Simulated and measured insertion loss.

insertion loss, comparable isolation, but worse power-handling capability. V. CONCLUSION

Fig. 22. Simulated and measured isolation.

This paper has described the design and analysis of an RF T/R switch in CMOS for MIMO wireless systems. Both fullrange and abridged 4P4T switch architectures for MIMO systems with four antenna elements were comparatively studied. The full-range 4P4T switch has 16 switching states and lower insertion loss was, therefore, selected for circuit implementation. Both series connection and body-floating techniques were employed in the circuit design of the full-range 4P4T switch to achieve an acceptable performance. The 4P4T switch was fabricated in 0.12- m triple-well CMOS with an effective die area 0.35 0.19 mm . The 4P4T switch achieved less than 2.7-dB insertion loss and higher than 20-dB isolation over the frequency range from 2 to 10 GHz. It was also found that the measured power-handling capability was more than 25 dBm at 2.4 and 5.8 GHz. ACKNOWLEDGMENT The authors wish to thank L. W. Ming, Nanyang Technological University, Singapore, for his support in measurements. REFERENCES

Fig. 23. Measured insertion loss versus input power level.

from antenna 1 (ANT1) to receiver A (RXA). The measured insertion loss is less than 2.7 dB over the frequency range from 2 to 10 GHz. Fig. 22 compares the simulated and measured isolation between transmitter A (TXA) and RXA. The isolation is 20 dB better over the frequency range from 2 to 10 GHz. The simulations agree reasonably with the measurements. The power-handling capability was measured at 2.4 and 5.8 GHz, respectively. The maximum 25-dBm input power was limited by our testing setup. Estimated from Fig. 23, the switch should have power-handling capability over 25 dBm. Table I shows the comparison of this full-range 4P4T switch in CMOS with the abridged 4P4T switch in GaAs at 5.8 GHz. It is seen that the full-range 4P4T switch in CMOS achieves lower

[1] A. F. Molisch and M. Z. Win, “MIMO systems with antenna selection,” IEEE Micro, vol. 5, no. 1, pp. 46–56, Mar. 2004. [2] D. Gesbert, M. Shafi, D. S. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: An overview of MIMO space-time coded wireless systems,” IEEE J. Select. Areas Commun., vol. 21, no. 4, pp. 281–302, Apr. 2003. [3] C.-H. Lee, B. Banerjee, and J. Laskar, “Novel T/R switch architectures for MIMO applications,” IEEE MTT-S Int. Microw. Symp. Dig., pp. 1137–1140, Jun. 2004. [4] F.-J. Huang and K. O, “A 0.5-m CMOS T/R switch for 900-MHz wireless applications,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 486–492, Mar. 2001. [5] Z. Li, H. Yoon, F.-J. Huang, and K. O, “5.8-GHz CMOS T/R switches with high and low substrate resistance in a 0.18-m CMOS process,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 1, pp. 1–3, Jan. 2003. [6] C. Tinella, J. M. Fournier, D. Belot, and V. Knopik, “A high-performance CMOS–SOI antenna switch for the 2.5–5-GHz band,” IEEE J. Solid-State Circuits, vol. 38, no. 7, pp. 1279–1283, Jul. 2003. [7] N. A. Talwalkar, C. P. Yue, H. Gan, and S. S. Wong, “Integrated CMOS transmit-receive switch using LC-tuned substrate bias for 2.4-GHz and 5.2-GHz applications,” IEEE J. Solid-State Circuits, vol. 39, no. 6, pp. 863–870, Jun. 2004. [8] M.-C. Yeh, Z.-M. Tsai, R.-C. Liu, K.-Y. Lin, Y.-T. Chang, and H. Wang, “Design and analysis for a miniature CMOS SPDT switch using body-floating technique to improve power performance,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 31–39, Jan. 2006.

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[9] T. Ohnakado, S. Yamakawa, T. Murakami, A. Furukawa, E. Taniguchi, H. Ueda, N. Suematsu, and T. Oomori, “21.5-dBm power-handling 5GHz transmit/receive CMOS switch realized by voltage division effect of stacked transistor configuration with depletion-layer-extended transistors (DETs),” IEEE J. Solid-State Circuits, vol. 39, no. 4, pp. 577–584, Apr. 2004. [10] Z. Li and K. O, “15-GHz fully integrated nMOS switches in a 0.13-m CMOS process,” IEEE J. Solid-State Circuits, vol. 40, no. 11, pp. 2323–2328, Nov. 2005. [11] Y. P. Zhang, Q. Li, W. Fan, C.-H. Ang, and H. Li, “A differential CMOS T/R switch for multi-standard applications,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 8, pp. 782–786, Aug. 2006.

Andrew Poh received the B.E. and M.Sc. degrees from Nanyang Technological University, Singapore, in 2001 and 2006, respectively. He is currently a Research Engineer with the Institute of Microelectronics, Singapore. His research interests are mainly focused on the design of RF integrated circuits.

Yue Ping Zhang received the B.E. and M.E. degrees from the Taiyuan Polytechnic Institute and Shanxi Mining Institute, Taiyuan University of Technology, Shanxi, China, in 1982 and 1987, respectively, and the Ph.D. degree from the Chinese University of Hong Kong, Hong Kong, in 1995, all in electronic engineering. From 1982 to 1984, he was with the Shanxi Electronic Industry Bureau. From 1990 to 1992, he was with the University of Liverpool, Liverpool, U.K. From 1996 to 1997, he was with City University of Hong Kong. He has taught at the Shanxi Mining Institute (1987–1990) and the University of Hong Kong (1997–1998). In 1996, he became a Full Professor with the Taiyuan University of Technology. He is currently an Associate Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has authored or coauthored numerous publications in the field of radio science and technology across seven IEEE societies. He is listed in Marquis’s Who’s Who, Who’s Who in Science and Engineering, and Cambridge IBC 2000 Outstanding Scientists of the 21st Century. He serves on the Editorial Board of the International Journal of RF and Microwave Computer-Aided Engineering and was a Guest Editor of its “Special Issue on RF and Microwave Subsystem Modules for Wireless Communications.” His research interests include propagation of radio waves, characterization of radio channels, miniaturization of antennas, design of RF integrated circuits, and implementation of wireless communications systems. Prof. Zhang has delivered numerous invited papers/keynote addresses at international scientific conferences. He has organized/chaired dozens of technical sessions of international symposia. He serves on the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He was the recipient of the 1990 Sino-British Technical Collaboration Award for his contribution to the advancement of subsurface radio science and technology. He was the recipient of the Best Paper Award presented at the Second International Symposium on Communication Systems, Networks and Digital Signal Processing, 1 Bournemouth, U.K. He was also the recipient of a 2005 William Mong Visiting Fellowship from the University of Hong Kong.

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A High-Performance CMOS Voltage-Controlled Oscillator for Ultra-Low-Voltage Operations Hsieh-Hung Hsieh, Student Member, IEEE, and Liang-Hung Lu, Member, IEEE

Abstract—In this paper, a novel circuit topology of voltage-controlled oscillators (VCOs) suitable for ultra-low-voltage operations is presented. By utilizing the capacitive feedback and the forward-body-bias (FBB) technique, the proposed VCO can operate at reduced supply voltage and power consumption while maintaining remarkable circuit performance in terms of phase noise, tuning range, and output swing. Using a standard 0.18- m CMOS process, a 5.6-GHz VCO is designed and fabricated for demonstration. Consuming a dc power of 3 mW from a 0.6-V supply voltage, the VCO exhibits a frequency tuning range of 8.1% and a phase noise of 118 dBc/Hz at 1-MHz offset frequency. With an FBB for the cross-coupled transistors, the fabricated circuit can operate at a supply voltage as low as 0.4 V. The measured tuning range and phase noise are 6.4% and 114 dBc/Hz, respectively. Index Terms—Capacitive feedback, forward body bias (FBB), ultra-low power, ultra-low voltage, voltage-controlled oscillators (VCOs).

Fig. 1. Forecast of the CMOS supply voltage by International Technology Roadmap for Semiconductors (ITRS) 2004.

I. INTRODUCTION

Being a crucial part in the RF front-ends, the voltage-controlled oscillator (VCO) is considered as one of the most power-consuming components. For low-power applications, CMOS oscillators operating at a supply voltage lower than 1 V were reported [7]–[11]. However, most of the circuits suffer from reduced output swing and degraded phase noise due to the limitations on the supply voltage. In this study, a VCO topology suitable for ultra-low-power and ultra-low-voltage operations is presented. By incorporating the capacitive feedback and the forward-body-bias (FBB) technique in the cross-coupled CMOS VCO, remarkable circuit performance in terms of the phase noise, frequency tuning range, and output swing can be achieved. Using a 0.18- m CMOS technology, a 5.6-GHz VCO is implemented for demonstration. The remainder of this paper is organized as follows. Section II describes the proposed circuit topology and the design considerations. The circuit design and the experimental results of the 5.6-GHz CMOS VCO are presented in Sections III and IV, respectively. Finally, a conclusion is given in Section V.

W

ITH THE emerging applications such as wireless personal area networks (WPANs), wireless sensor networks and RF identifications (RFIDs), the development of low-cost and low-power RF integrated circuits (RFICs) for short-range communications have attracted great attention over the past few years. For the circuit implementations, the CMOS technology appears to be particularly well suited due to its unparalleled advantages in the fabrication cost and system-level integration. Unfortunately, the inherently low transconductance of the MOSFETs at higher frequencies have impeded the evolution of low-power designs to RF front-ends. In order to overcome the limitations, various design methodologies and circuit techniques have been proposed [1]–[4]. Among these approaches, low-voltage circuit operation is one of the most promising solutions. In addition to the power considerations, a reduced supply voltage is also an inevitable trend for CMOS designs as well. With the continuous shrinking in the transistor feature size, a proportional down-scaling of the supply is required to ensure the gate–oxide reliability [5]. The forecast of the supply voltage for CMOS circuits within the next decade [6] is shown in Fig. 1, where a low-voltage operation of 0.4 V is anticipated by the end of 2020. Therefore, there exists an urgent need to develop low-voltage circuit techniques for high-performance RFICs at multigigahertz frequencies.

Manuscript received July 10, 2006; revised November 20, 2006. This work was supported in part by the National Science Council under Grant 94-2220-E002-026 and Grant 94-2220-E-002-009. The authors are with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.891471

II. PROPOSED VCO TOPOLOGY The schematic of the proposed VCO is shown in Fig. 2. In order to reduce the required supply voltage and to eliminate additional noise contribution, the tail current transistor in a conventional cross-coupled VCO is replaced by on-chip inductors [12]. For an enhanced voltage swing under an ultra-low supply voltage, the capacitive-feedback technique is employed [13]. Due to the use of the on-chip inductor and the feedback loop and , the drain and source voltages can established by swing above the supply voltage and below the ground potential. Consequently, the output swing of the VCO is enhanced, leading

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where

The circuit oscillates if the loop gain is unity, which corresponds at the oscillation frequency to a voltage gain

Fig. 2. Schematic of the proposed VCO topology.

(4) With proper arrangement, (4) yields

(5) and

(6) Provided be simplified as

Fig. 3. Simplified half-circuit model of the proposed VCO.

in typical design cases, (5) can

(7) to a superior close-in phase noise. Since the varactors are employed in the source terminals of the cross-coupled transistors, a more effective controlled mechanism of the tank resonance is presented. Therefore, a reasonable frequency tuning range can be achieved even with a reduced voltage range for the controlled . To further investigate the proposed VCO, detailed signal circuit analysis is provided as follows.

From (7), the oscillation frequency can be approximated by (8) Based on (6) and (8), the required transconductance tain the oscillation is given by

to sus-

A. Startup Conditions In order to derive the startup conditions and the oscillation frequency, the equivalent half-circuit of the proposed VCO is shown in Fig. 3, where and represent the losses of the and , respectively. Note that the losses on-chip inductors of the inductors are typically modeled by a series resistance. In the equivalent circuit, the narrowband approximation is emand ployed to simplify the analysis, and the shunt resistance can be estimated by (1) (2) and are the equivalent series resistances of where and , respectively. Besides, the transistor parasitic capacitances, which are much smaller than the values of and , are neglected. From the small-signal analysis, the transfer funcand is given by tion between (3)

Assuming that can be simplified as

and

(9) , (8) and (9)

(10) (11) has more inFrom (10), it is clear that the variation of than does. As a result, the capacitance fluence on is realized by varactors in this design, leading to a reasonable tuning range for the VCO under ultra-low-voltage operations. is desirable to enhance Note that a large capacitance value for the frequency tuning range. However, the required transconducand tance to sustain the oscillation increases with the ratio of . Therefore, a design tradeoff is established between the frequency tuning range and the power consumption of the VCO . Once the capacitance ratio is by the circuit parameter determined, the required transconductance of the cross-coupled

HSIEH AND LU: HIGH-PERFORMANCE CMOS VCO FOR ULTRA-LOW-VOLTAGE OPERATIONS

Fig. 6. Actual and modeled drain current waveform of the transistor

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M.

Q

Fig. 4. Required transconductance to satisfy the startup conditions for various inductor factors.

Fig. 7. Predicted and simulated peak-to-peak amplitude at the VCO output.

Fig. 5. Conceptual illustration of the drain and source voltage due to the capacitive feedback.

source voltages and can be obtained by the voltage divider of and . Thus, is approximated by the maximum drain current with the transistor operating in the nonsaturated region

transistors can be estimated by (11). Assuming an inductance value of 2 nH and an oscillation frequency of 5 GHz, the refor various inquired transconductance as a function of ductor quality factors are shown in Fig. 4, which provides useful design guidelines for the proposed circuit topology. B. Output Voltage Swing In the proposed VCO circuit, a capacitive feedback is formed by capacitors and . Due to the in-phase relationship provided by the capacitive feedback and the use of on-chip inductors, the drain and source voltage can swing above the supply voltage and below the ground potential, as illustrated in Fig. 5. Consequently, the close-in phase noise benefits from the enhanced voltage swing at the VCO output. To evaluate the performance enhancement of this technique, the output voltage swing of the VCO is derived from the time-domain waveform of the , as shown in Fig. 6. For simplicity, the pedrain current riodic drain current is modeled by a square wave with a period and an amplitude of . Note that, at the quiesof and cent point, the transistors are biased at . The maximum drain current occurs when reaches its peak value. Assuming that the the gate voltage amplitude of output oscillating signal is , the gate voltages and are and , respectively, while the

(12) and is the threshold voltage of where , the fundamental the MOSFET. From the Fourier series of current component is given by (13) and the fundamental voltage amplitude is (14) is the load resistance. From (12) and (14), a simplified where expression of the VCO output swing is given by (15) According to (15), the predicted VCO output amplitude as a is depicted in Fig. 7 along with the values function of obtained from the circuit-level simulation. It is obvious that an output swing significantly larger than the supply voltage can be achieved by the proposed capacitive feedback.

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Fig. 8. Simulated phase noise at various temperatures with an offset frequency of 1 MHz.

Fig. 11. Proposed methodology for the design of the low-voltage VCO. Fig. 9. I–V characteristics of the MOSFET with and without FBB. TABLE I CIRCUIT PARAMETERS OF THE VCO

as [15] Fig. 10. Simulated threshold voltage and drain current of the MOSFET with FBB.

C. Phase Noise Since the proposed VCO is operated under an ultra-low dc voltage, the cross-coupled transistors are potentially biased in the weak-inversion region. Therefore, the drain noise of the transistors is no longer dominated by the thermal noise as [14] (16) where is the channel conductance with , and is the thermal noise coefficient. Instead, the drain noise is expressed

(17) is the drain current, is the thermal voltage, and where is the weak inversion slope factor. From (17), includes not only the contribution from the thermal noise, but also that of the shot noise since the drain current consists of both drift and diffusion components as the gate overdrive approaches . Due to the fact that is generally greater for low-power operation, the cross-coupled tranthan sistors may contribute more noise to the tank as the supply voltage decreases. Therefore, it imposes a fundamental limitation on the phase noise of the VCO for ultra-low-power and ultra-low-voltage applications.

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Fig. 12. Microphotograph of the fabricated VCO. Fig. 15. Measured and simulated VCO characteristics with a supply voltage sweeping from 0.5 to 0.7 V.

Fig. 13. Measured and simulated tuning characteristics of the VCO with a supply voltage of 0.6 V. Fig. 16. Measured and simulated tuning characteristics of the VCO with a supply voltage and body bias of 0.4 V.

Fig. 14. Measured close-in output spectrum of the VCO with a supply voltage of 0.6 V.

For the conventional LC-tank VCOs operating at a reduced supply voltage, a significant degradation in the phase noise is inevitable due to the limited signal power at the output. In the proposed VCO topology, the output swing can be effectively enhanced by the capacitive feedback, leading to an improved close-in phase noise. Furthermore, due to the similarity to the Colpitts oscillator, the VCO circuit also benefits from the cyclostationary noise effect [16]. To further investigate the influence of the capacitive feedback, the simulated phase noise as a funcis shown in Fig. 8. It is noted tion of the capacitance ratio that lower phase noise can be achieved by increasing the capacitance ratio. However, as shown in (11), the performance improvement in the phase noise is accompanied by an increase

Fig. 17. Measured close-in output spectrum of the VCO with a supply voltage and body bias of 0.4 V.

in the required transconductance, which implies higher power consumption of the VCO circuit. In addition to the capacitance ratio, the temperature also influences the VCO phase noise. The simulated results at various temperatures are illustrated in Fig. 8 as well, indicating an average 4-dB degradation in phase noise as the temperature increases from 25 C to 50 C. D. FBB For deep-submicrometer MOSFETs, the threshold voltage is no longer constant, but influenced by circuit parameters such as gate length, channel width, and drain-to-source voltage due

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TABLE II PERFORMANCE SUMMARY OF THE LOW-VOLTAGE AND LOW-POWER VCOS

to the short-channel and narrow-channel effects [17]. Typically, transistors with a large channel width and a minimum gate length exhibit a reduced , which is preferable for low-voltage operations. In this VCO topology, the fundamental limitation on the supply voltage is imposed by the threshold voltage of the cross-coupled transistors. To further reduce the supply voltage, the FBB technique is adopted as shown in Fig. 9. For a MOSFET device, the threshold voltage is governed by the body effect as [18]

(18) where is the threshold voltage for V, is a physical parameter with a typical value of 0.3 V, is the substrate doping, and is the permittivity of silicon. By applying a forward bias voltage to the body through a current-limiting re, the effective threshold voltage is thus reduced while sistor maintaining a minimum forward junction current between the body and the source terminals. The simulated effective threshold m voltage and the drain current of a MOSFET with m are demonstrated in Fig. 10, indicating a and threshold voltage reduction more than 100 mV due to the FBB technique. III. CIRCUIT DESIGN Fig. 11 shows the design procedure of the high-performance VCO. The circuit design starts with the inductors and . For simplification, both inductors are chosen to be identical as , which are optimized for a high- factor at the frequencies of interest. Once the inductance is determined, the required value of can be obtained from the designated oscillation frequency by (10)

(19)

By defining the capacitance ratio and are given by

, the values of (20) (21)

The circuit performance of the VCO is strongly influenced by the design parameter . As indicated in (11) and (15), the required transconductance of the cross-coupled transistors and the associated output swing increase with . Considering the design specifications such as the phase noise, output swing, tuning range, and power consumption, the optimum is thus determined and the circuit parameters including the capacitance values and the transistor aspect ratio can be calculated accordingly. Finally, a global optimization is performed and design iterations may be needed to satisfy the required circuit performance. Following the design procedures, the parameters of the VCO design are tabulated in Table I. Based on the narrowband approximation in (1) and (2), the calculated value of is 633 in this particular case. The resulting values for and are 1.2 10 and 1.06 10 , respecin tively, which validate the assumption the theoretical derivation of the proposed VCO. IV. EXPERIMENTAL RESULTS The proposed VCO is implemented in a standard 1P6M 0.18- m CMOS process provided by a commercial foundry. With an optimum layout for the RF performance, the -channel MOSFETs in the deep n-well exhibit a maximum oscillation up to 60 GHz. As for the passive components, frequency top AlCu metallization layer of 2- m thickness is available for on-chip inductance while metal–insulator–metal (MIM) capacitors with oxide intermetal dielectric are also provided. Fig. 12 shows a microphotograph of the fabricated circuit with a chip area of 0.55 0.9 mm including the pads. In order to enhance the quality factor and to ensure the layout symmetry for fully differential operations, center-tapped spiral inductors

HSIEH AND LU: HIGH-PERFORMANCE CMOS VCO FOR ULTRA-LOW-VOLTAGE OPERATIONS

with a metal width of 15 m are employed for the required inand . In addition, the accumulation-type devices ductance are used as the varactors. For test purposes, the VCO outputs are buffered by open-drain nMOS transistors to drive the external 50- load. To evaluate the circuit performance, the chip was mounted on an FR4 test board and measured with an Agilent E4407B spectrum analyzer. When operating at a supply voltage of 0.6 V, the VCO core consumes a dc power of 3 mW. Fig. 13 shows the measured tuning characteristics. As the controlled voltage sweeps from 0 to 0.6 V, the 5.6-GHz VCO exhibits a frequency tuning range of 8.1%. The tuning range can be increased to 12.2% with a maximum controlled voltage of 1.8 V. Fig. 14 shows the close-in output spectrum of the VCO with a controlled voltage of 0.3 V. The measured output power and phase noise at 1-MHz offset are 1 dBm and 118.3 dBc/Hz, respectively. For ultra-lowvoltage operations, the sensitivity to the supply voltage is also investigated. As sweeps from 0.5 to 0.7 V, the output characteristics of the VCO are illustrated in Fig. 15, exhibiting a reasonable performance deviation with respect to the variation in the supply voltage. By employing the FBB technique, the VCO can operate at a supply voltage as low as 0.4 V. With a supply voltage and a body bias of 0.4 V, the VCO core consumes a dc power of 1 mW. Fig. 16 shows the measured tuning characteristics, indicating a 6.4% tuning range as the controlled voltage sweeps from 0 to 0.4 V. The close-in output spectrum of the VCO with a 0.2-V controlled voltage is shown in Fig. 17. The measured phase noise is 114.4 dBc/Hz at 1-MHz offset, while the output power is 9 dBm. The performance of the VCO is summarized in Table II along with the results from previously published low-voltage and low-power CMOS VCO circuits. V. CONCLUSION Using a 0.18- m CMOS technology, a VCO operating at a reduced supply voltage is presented. The performance of the fabricated circuit is characterized with a supply voltage from 0.6 to 0.4 V. Due to the use of the capacitive feedback and the FBB technique, significant performance improvement in terms of output swing, frequency tuning range, and phase noise is demonstrated, making it extremely attractive for ultra-low-power and ultra-low-voltage RF applications. ACKNOWLEDGMENT The authors would like to thank the National Chip Implementation Center, Hsinchu, Taiwan, R.O.C., for chip fabrication, and K.-S. Chung, Realtek Semiconductor Corporation, Hsinchu, Taiwan, R.O.C., and Y.-C. Huang, National Taiwan University, Taipei, Taiwan, R.O.C., for technical support. REFERENCES [1] P. Choi et al., “An experimental coin-sized radio for extremely lowpower WPAN (IEEE 802.15.4) application at 2.4 GHz,” IEEE J. SolidState Circuits, vol. 38, no. 12, pp. 2258–2268, Dec. 2003. [2] S. Verma et al., “A 17-mW 0.66- mm direct-conversion receiver for 1-Mb/s cable replacement,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2547–2554, Dec. 2005. [3] N.-J. Oh and S.-G. Lee, “A CMOS 868/915 MHz direct conversion ZigBee single-chip radio,” IEEE Commun. Mag., vol. 43, no. 12, pp. 100–109, Dec. 2005.

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[4] T.-K. Nguyen et al., “A low-power CMOS direct conversion receiver with 3-dB NF and 30-kHz flicker-noise corner for 915-MHz band IEEE 802.15.4 ZigBee standard,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 735–741, Feb. 2006. [5] J. B. Kuo and J.-H. Lou, Low-Voltage CMOS VLSI Circuits. New York: Wiley, 1999. [6] “International Technology Roadmap for Semiconductors,” Semiconduct. Ind. Assoc., (2004 ed.). [Online]. Available: http://www.public. itrs.net/ [7] H.-H. Hsieh, K.-S. Chung, and L.-H. Lu, “Ultra-low-voltage mixer and VCO in 0.18- m CMOS,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2005, pp. 167–170. [8] K. Kwok and H. C. Luong, “Ultra-low-voltage high-performance CMOS VCOs using transformer feedback,” IEEE J. Solid-State Circuits, vol. 40, no. 3, pp. 652–660, Mar. 2005. [9] A. Fakhr, M. J. Deen, and H. deBruin, “Low-voltage, low-power and low phase noise 2.4 GHz VCO for medical wireless telemetry,” in Can. Elect. Comput. Eng. Conf., May 2004, vol. 3, pp. 1321–1324. [10] A. H. Mostafa, M. N. El-Gamal, and R. A. Rafla, “A sub-1-V 4-GHz CMOS VCO and a 12.5-GHz oscillator for low-voltage and high-frequency applications,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 48, no. 10, pp. 919–926, Oct. 2001. [11] M. Harada et al., “2-GHz RF front-end circuits in CMOS/SIMOX operating at an extremely low voltage of 0.5 V,” IEEE J. Solid-State Circuits, vol. 35, no. 12, pp. 2000–2004, Dec. 2000. [12] N. Troedsson and H. Sjolamd, “An ultra low voltage 2.4 GHz CMOS VCO,” in IEEE Radio Wireless Conf., Aug. 2002, pp. 205–208. [13] H. Wang, “Oscillator circuit having maximized signal power and reduced phase noise,” U.S. Patent 6 229 406, May 8, 2001. [14] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. New York: Cambridge Univ. Press, 1998. [15] F. Gatta et al., “A 2-dB noise figure 900-MHz differential CMOS LNA,” IEEE J. Solid-State Circuits, vol. 36, no. 10, pp. 1444–1452, Oct. 2001. [16] R. Aparicio and A. Hajimiri, “A noise-shifting differential Colpitts VCO,” IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1728–1736, Dec. 2002. [17] Y. Tsividis, Operation and Modeling of the MOS Transistor, 2nd ed. New York: Oxford Univ. Press, 1999. [18] B. Razavi, Design of Analog CMOS Integrated Circuits. New York: McGraw-Hill, 2001. [19] M. N. El-Gamal et al., “Very low-voltage (0.8 V) CMOS receiver frontend for 5 GHz RF applications,” Proc. Inst. Elect. Eng.—Circuits, Devices, Syst,, vol. 149, no. 5/6, pp. 355–362, Oct.–Dec. 2002. [20] M.-D. Tsai, Y.-H. Cho, and H. Wang, “A 5-GHz low phase noise differential Colpitts CMOS VCO,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 5, pp. 327–329, May 2005. Hsieh-Hung Hsieh (S’05) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree in electronic engineering at National Taiwan University. His research interests include the development of low-voltage and low-power RFICs, multiband wireless systems, RF testing, and monolithic microwave integrated circuit (MMIC) designs.

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

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A 15/30-GHz Dual-Band Multiphase Voltage-Controlled Oscillator in 0.18-m CMOS Hsieh-Hung Hsieh, Student Member, IEEE, Ying-Chih Hsu, and Liang-Hung Lu, Member, IEEE

Abstract—A multiphase oscillator suitable for 15/30-GHz dual-band applications is presented. In the circuit implementation, the 15-GHz half-quadrature voltage-controlled oscillator (VCO) is realized by a rotary traveling-wave oscillator, while frequency doublers are adopted to generate the quadrature output signals at the 30-GHz frequency band. The proposed circuit is fabricated in a standard 0.18- m CMOS process with a chip area of 1.1 1.0 mm2 . Operated at a 2-V supply voltage, the VCO core consumes a dc power of 52 mW. With a frequency tuning range of 250 MHz, the 15-GHz half-quadrature VCO exhibits an output power of 8 dBm and a phase noise of 112 dBc/Hz at 1-MHz offset frequency. The measured power level and phase noise of the 30-GHz quadrature outputs are 16 dBm and 104 dBc/Hz, respectively. Index Terms—Coplanar striplines (CPSs), frequency doublers, half-quadrature output phases, rotary traveling-wave oscillators.

I. INTRODUCTION Fig. 1. Proposed dual-band multiphase VCO architecture.

M

ULTIPHASE voltage-controlled oscillators (VCOs) are widely used in both wired and wireless communication systems. In the pursuit of increasing carrier frequencies and higher data rates to satisfy the emerging application standards, the implementation of high-frequency signal sources with multiple phases is of crucial importance. Due to the superior device characteristics, III–V compound semiconductors were preferred in the realization of high-frequency oscillators [1], [2]. With recent advances in the deep-submicrometer fabrication and beyond 100 GHz [3], technology, transistors with [4] are available in a standard CMOS process. High-frequency CMOS VCOs have been proposed to operate at frequencies in tens of gigahertz. However, most of the reported circuits provide single-ended or differential oscillation signals [5]–[10]. It is still a challenging task to implement quadrature-phase CMOS VCOs operating at millimeter-wave frequencies [11]–[13]. In this paper, a fully integrated dual-band multiphase VCO is presented. Using a 0.18- m CMOS process, a prototype circuit is implemented to operate at the 15/30-GHz frequency bands. The proposed circuit architecture is described in Section II. Theoretical analysis and circuit design of the 15-GHz half-quadrature VCO and the 30-GHz quadrature VCO are presented in Manuscript received July 19, 2006; revised November 8, 2006. This work was supported in part by the National Science Council under Grant 94-2220E-002-026 and Grant 94-2220-E-002-009. The authors are with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890518

Sections III and IV, respectively. The experimental results of the fabricated circuit are shown in Section V. II. VCO ARCHITECTURE In a conventional VCO topology, quadrature output phases are realized by two parallel-coupled LC-tank oscillators [14], [15]. Due to the simplicity in the circuit design, it is widely used for applications at multigigahertz frequencies. As the operating frequency increases beyond 10 GHz, the parasitics from the coupling transistors contribute additional capacitive loading to the LC tank. Consequently, a small inductance is required to maintain a high resonant frequency of the tank, which might cause significant offset in the oscillation frequency due to small deviations in the inductance value. Moreover, a large varactor is not preferred in high-frequency designs to compensate for the process variation. Therefore, it is not practical to implement a high-frequency multiphase VCO with the active coupling technique. In order to overcome the limitations on the multiphase VCO designs, a dual-band circuit architecture is presented. Fig. 1 shows the conceptual illustration of the proposed architecture, which is composed of a half-quadrature VCO at the fundamental frequency and four doublers to generate the quadrature phases at the second harmonic. Since the fundamental VCO is oscillating at half of the maximum operating frequency, the stringent restrictions on the circuit implementation are effectively alleviated, resulting in better performance in terms of the phase

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Fig. 2. Schematic of the 15-GHz half-quadrature CMOS VCO based on a rotary traveling-wave oscillator.

noise, tuning range, and output power. As for the quadrature outputs at the second harmonic, the additional noise contribution from the doublers can be minimized by careful circuit design such that a phase noise lower than what an oscillator at twice of the fundamental frequency would give can be achieved, especially for VCOs operating at frequencies close to the transistor cutoff. Therefore, the proposed architecture is employed to realize the 15/30-GHz dual-band multiphase VCO in a standard CMOS process.

Compared with all-NMOS circuit implementations, the complementary architecture provides a higher transconductance at the same bias current. In addition, a more symmetric oscillation waveform can be obtained, resulting in a reduction in the noise for better phase-noise performance up-conversion of [17]. For an enhanced tuning range of the VCO circuit, pMOS devices are used as the varactors in the resonator. Finally, the half-quadrature outputs of the fundamental oscillator are buffered by open-drain stages to drive the 50- load of the test instruments.

III. 15-GHz HALF-QUADRATURE VCO B. Theoretical Analysis A. Circuit Topology To achieve the required half-quadrature output phases with low close-in phase noise, a rotary traveling-wave topology [16] is employed for the 15-GHz fundamental oscillator. Fig. 2 shows the complete schematic of the fully integrated VCO with all on-chip components. In this design, the oscillation frequency is determined by the resonator, which is composed of a coplanar stripline (CPS) structure and the capacitances from the transistors and the varactors. A cross-connection is included in the CPS to form a closed loop for the required reverse feedback. Since no termination is needed, limitations on bandwidth and impedance mismatch can be alleviated. In order to ensure the odd-mode operation of the CPS and to compensate for the loss from the resonator, four identical complementary cross-coupled inverters are utilized in a symmetric manner.

1) Modeling: The CPS is a uniplanar transmission line, which has been widely used in microwave and millimeter-wave applications [18], [19]. With two adjacent metal lines running in parallel on the same substrate surface, the CPS possesses the advantages of low insertion loss, small dispersion, less discontinuity parasitics, and low sensitivity to the substrate thickness [18]. Since the resonator of the fundamental oscillator is realized by a CPS structure in a closed-loop form, a circuit model is introduced for the detailed analysis of the VCO. Fig. 3(a) illustrates the equivalent circuit of the unloaded and CPS [20], where the parameters are per-unit-length quantities. Note that represents is the the self-inductance of the individual metal lines and capacitance between these two lines. The CPS attenuation is and , accounting for the conductor and modeled by

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approximation is used in (5) and (6) for simplification, resulting in (7) (8) Based on the unloaded CPS model, a distributed equivalent circuit of the rotary traveling-wave VCO including the lumped loading elements is constructed as shown in Fig. 3(b). Note that and represent the overall shunt capacitance and conductance per-unit-length, respectively, defining between the two metal lines. From the geometry of the proposed oscillator, and can be expressed as [22] (9) (10)

Fig. 3. Distributed circuit model of: (a) the unloaded CPS and (b) the proposed rotary traveling-wave VCO.

dielectric losses, respectively. Since the two metal lines are in close proximity, the mutual inductance is represented by the . coupling coefficient Based on the equivalent circuit, a small-signal analysis is performed to characterize the propagation parameters of the CPS for both even- and odd-mode operations. The even-mode operation corresponds to a common-mode signal. By equalizing the potential in both signal paths, the characteristic impedance and the propagation constant for even-mode operation can be derived as (1) (2) and . Aswhere suming that and , the low-loss approximation in [21] applies and (1) and (2) can be expressed as (3)

where is the length of the CPS segment between two consecand are the utive cross-coupled inverters. Note that and parasitics from the cross-coupled inverters, while account for the shunt capacitance and conductance from the varactors, respectively. Due to the use of the cross-coupled inverters, the signal propagating in the CPS of the oscillator is in the differential mode. Therefore, propagation parameters in the similar forms of (7) and (8) are derived for the rotary traveling-wave oscillator, and the resulting characteristic impedance and propagation are given by constant (11)

(12) where and . The derivations in (11) and (12) are thus used for analysis and design of the 15-GHz halfquadrature VCO. 2) Start-Up Condition and Oscillation Frequency: The equivalent circuit of the rotary traveling-wave oscillator is shown in Fig. 4(a). By treating the circuit as a closed-loop feedback system, the startup conditions to initiate the VCO oscillation are given by

(4) As for the odd-mode operation, it is analyzed by providing a differential signal in the CPS. The characteristic impedance and propagation constant are given by (5) (6) where

and . Similar to the even-mode operation, the low-loss

(13) (14) where is the loop gain and is the oscillation frequency. In order to evaluate the startup conditions, is derived by breaking the loop between the points and . Fig. 4(b) illustrates the equivalent open-loop circuit model of the rotary traveling-wave oscillator. Note that the transconductance contributed from the cross-coupled inverters are modeled by ideal current sources to simplify the analysis while terminations with a characteristic impedance of

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Fig. 4. Equivalent: (a) closed- and (b) open-loop circuit model of the proposed rotary traveling-wave VCO.

are included in the CPSs to sustain the propagation mode in the closed-loop case. As the wave propagates through the CPSs, amplified components are superimposed at the points where the cross-coupled inverters locate. For an odd-mode operation, the waves of the two lines have identical amplitude with a phase difference of 180 . As a result, the voltage at the first inverter node can be expressed as

Finally, the loop gain of the rotary traveling-wave oscillator is and as defined as the ratio of (18) From (18), the startup conditions in (13) and (14) can be expressed as (19)

(15) where is the transconductance provided by the inverter stage and represents the effective load impedance at the inverter output. Similarly, the voltages at the loading nodes of the following inverter stages can be expressed as

(20) where is an arbitrary odd integer. Furthermore, the required transconductance and the fundamental oscillation frequency are determined by (21)

(16) (22) where given by

and , and the voltage amplitude at point

is

(17)

From (21), it is noted that the startup condition can be alleviated by increasing the characteristic impedance of the CPS. may increase with , However, the line attenuation at as indicated in (12), due to the increasing value of

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Fig. 6. Schematic of the complementary cross-coupled inverters with a pMOS tail current.

TABLE I CIRCUIT PARAMETERS OF THE CROSS-COUPLED INVERTERS

Fig. 5. Simulated results of: (a) the characteristics impedance and (b) the quality factor for the unloaded CPS at 15 GHz with various w and s.

higher frequencies. A tradeoff has to be made in determining the CPS parameters and the transistor sizes for an optimum VCO design, especially for high-frequency applications. In the design of the rotary traveling-wave oscillator, varactors are included in the cross-coupled inverters to provide the required frequency tuning. By varying the equivalent capacitance to with the controlled of the varactors from voltage, the tuning range of the VCO can be estimated by

(23)

C. Circuit Design The design of the rotary traveling-wave oscillator starts with the geometry of the CPS structure. To minimize the conductor and the substrate losses, top metal layer with a thickness of 2 m provided by the CMOS technology is utilized for the implementation of the CPS. The most important design parameters for the CPS are the linewidth and the line spacing , which predetermine the values of and . Based on the full-wave electromagnetic (EM) simulation, the extracted values of the CPS parameters for various and at 15 GHz are shown in Fig. 5.

In consideration of the -factor and the layout dimensions, m, m is employed a CPS structure with for the design. Once the width and the spacing of the CPS are determined, the required length can be estimated by (22). In this particular design, is chosen to be 250 m to achieve an oscillation frequency of 15 GHz. As for the inverter stages, the complementary cross-coupled pairs with a pMOS tail current, as shown in Fig. 6, are adopted to compensate the losses of the resonator. The transistor sizes and bias currents are selected according to (21) to provide sufficient transconductance while maintaining minimum parasitic capacitance and reasonable power consumption. Table I summarizes the device parameters in this design. Due to the use of the complementary inverter stages, the overall capacitance in (23) , leading to a reduced VCO tuning range is dominated by at the fundamental frequency. With the design values utilized for the active and the passive components, the circuit parameters of the loaded CPS are extracted and summarized in Table II. From (21)–(23), to satisfy the startup condition is the calculated value for 150 mA/V, while the estimated oscillation frequency ranges from 15.6 to 15.9 GHz. Compared with the circuit-level simulation, it is noted that the theoretical analysis provides a first-order prediction for the design of the rotary traveling-wave oscillators with sufficient accuracy.

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TABLE II EXTRACTED CIRCUIT PARAMETERS OF THE LOADED CPS

Fig. 8. Small-signal equivalent circuit of the push–push doubler.

and differential fundamental signals are applied to the input terminals of the push–push stage, the drain currents of the transistors are given by Fig. 7. Schematic of the quadrature VCO using push–push doublers.

(25) IV. 30-GHz QUADRATURE VCO (26)

A. Circuit Topology With the half-quadrature output phases provided at the fundamental frequency, the 30-GHz quadrature VCO is realized by four push–push stages as the frequency doublers. Complete circuit schematic of the frequency doublers are shown in Fig. 7. The generation of the harmonics is based on the nonlinear variation in the transconductance of the push–push stages. To maximize the conversion gain, the transistors are in the commonare employed source configuration, while spiral inductors for the purposes of output matching at the second harmonic frequency. The 15-GHz fundamental signals are directly coupled to the inputs of the push–push stages without dc blocks or level shifters. In addition, open-drain buffers are used at the 30-GHz quadrature outputs to drive the 50- input impedance of the testing instruments.

From (25) and (26), the ac current at the doubler output is (27) Note that the fundamental components are eliminated at the output node due to the push–push operation, while the currents are constructively added at the second harmonic frequency. If is chosen to resonate with the parasitic the inductance value capacitance at 2 , the conversion gain of the frequency doubler can be approximated by

(28)

B. Theoretical Analysis 1) Conversion Gain and Phase Noise: In order to evaluate the output power at the second harmonic, the conversion gain of the frequency doublers is derived. Fig. 8 shows the schematic and the equivalent circuit of the frequency doubler. Considering that the drain current of a MOSFET is expressed by its Taylor series as (24)

By taking the velocity saturation into account, the drain current of a MOSFET in the saturation region is given by [23] (29) is the threshold voltage, is the gate–oxide capaciwhere is the saturation velocity, is the mobility, tance per area, is the mobility degradation factor, and and are the channel

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Fig. 9. Simulated conversion gain of the doublers as a function of the gate bias voltage with various transistor aspect ratios width/length (W/L) (m/m).

width and length, respectively. The coefficient series is derived as

of the Taylor

(30)

Fig. 10. Microphotograph of the fabricated 15/30-GHz dual-band multiphase VCO.

Finally, the conversion gain of the frequency doubler can be obtained by plugging (30) into (27) as follows: (31) Expression (31) provides a guideline for the design of the frequency doublers in terms of the conversion gain. In order to achieve a high conversion gain for enhanced output power at the second harmonic, MOSFETs with a large channel width and a minimum gate length are desirable, while a low overdrive voltage is required to maximize the nonlinearity of the devices. In addition to the conversion gain, the excess phase noise at the second harmonic output is also investigated. Provided a conversion gain of 0 dB for the frequency doubler, the output phase noise is theoretically increased by 6 dB in an ideal case [24]. However, due to the losses of the matching networks and the noise sources from the transistors, the noise power density at a specific offset frequency is generally higher than the theoretical prediction in a practical design. Furthermore, the output power level of the frequency doubler should be taken into account, as well in the evaluation of the phase noise at the second harmonic frequency. 2) Phase and Amplitude Errors in the Fundamental Inputs: In the derivations of the conversion gain, it is assumed that the fundamental signals are equal in amplitude with a phase difference of 180 . However, the phase and amplitude errors due to device mismatch and layout asymmetry are inevitable in practical circuit implementations. The influence on the doubler performance is evaluated by introducing a phase error and an amplitude error in the differential signals at the fundamental frequency as

(32) (33)

Fig. 11. Measured and simulated tuning characteristics of: (a) the fundamental and (b) second harmonic outputs.

From (32) and (33), the drain currents of the transistors are

(34)

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Fig. 12. Measured: (a) output spectrum and (b) close-in phase noise of the 15-GHz half-quadrature VCO.

(35) The ac current at the doubler output is obtained by summing (34) and (35) (36) and . It is noted where that the output current consists of components at the fundamental and the second harmonic frequencies due to the phase and amplitude error. Therefore, the fundamental rejection ratio can be evaluated by (36). Provided that the phase error is relatively small, it is clear that the fundamental rejection ratio is inversely proportional to the phase error (37) As indicated in (37), the fundamental rejection ratio decreases by 6 dB as the phase error doubles. Thus, careful design and symmetric layout are required to ensure sufficient suppression of the fundamental component at the second harmonic output. C. Circuit Design Four push–push stages are utilized as the frequency doublers to generate the quadrature output at 30 GHz from the 15-GHz half-quadrature signals. The simulated conversion gain of the

Fig. 13. Measured: (a) output spectrum and (b) close-in phase noise of the 30-GHz quadrature VCO.

doublers as a function of the gate bias voltage is illustrated in Fig. 9. In this particular design, transistors with m and m are employed. To maximize the conversion gain, 0.32-nH spiral inductors are used to resonate with the parasitic capacitance of the output node at 30 GHz. Based on the circuit simulation, the frequency doublers with the open-drain buffers exhibit a reasonable conversion gain for the second harmonic component providing an output power level of 15 dBm. V. EXPERIMENTAL RESULTS The proposed dual-band multiphase VCO is implemented in a standard 0.18- m CMOS process. Fig. 10 shows a die photograph of the fabricated circuit with a chip area of 1.1 1.0 mm . In order to minimize the phase error among the multiphase outputs, a symmetrical layout is used for the design. On-wafer probing was performed to characterize the performance of the VCO at the 15- and 30-GHz frequency bands, while the losses from the measurement setup were calibrated and deembedded in the experimental results. The output spectrum and phase noise were measured by a 50-GHz spectrum analyzers. Operated at a supply voltage of 2.0 V, the VCO core consumes a dc power of 52 mW. As the controlled-voltage sweeps from 0 to 2 V, the tuning characteristics, including the oscillation frequency and the output power, of the fundamental and second harmonic outputs are shown in Fig. 11(a) and (b), respectively. With a frequency tuning range of 250 MHz, the

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TABLE III PERFORMANCE SUMMARY OF THE HIGH-FREQUENCY MULTIPHASE VCOS

15-GHz half-quadrature VCO exhibits an output power ranging from 8 to 9 dBm. The measured output power of the doublers at 30 GHz is 16 dBm. Fig. 12 shows the measured output spectrum and close-in phase noise of the 15-GHz fundamental oscillator, indicating a phase noise of 112 dBc/Hz at 1-MHz offset frequency. The measured wideband spectrum and the close-in phase noise at the second harmonic output are shown in Fig. 13(a) and (b), respectively. With careful circuit design of the frequency doublers, the 30-GHz output signal exhibits a deembedded fundamental rejection of 19.79 dB and a phase noise of 104 dBc/Hz with an offset frequency of 1 MHz. The performance of the proposed circuit along with results from the state-of-the-art multiphase VCOs are summarized in Table III for comparison. VI. CONCLUSION This paper has presented a dual-band multiphase VCO using 0.18- m CMOS technology. By employing a rotary travelingwave oscillator and four push–push doublers, the fabricated circuit provides half-quadrature output phases at 15 GHz and quadrature oscillating signals at 30 GHz. Since the resonator of the VCO core is designed at half of the maximum operating frequency, stringent design constrains are alleviated. As a result, a manageable design and superior performance can be achieved by the proposed VCO topology. ACKNOWLEDGMENT The authors would like to thank National Chip Implementation Center (CIC), Hsinchu, Taiwan, R.O.C., for chip fabrication, and National Nano Device Laboratories (NDL), Hsinchu, Taiwan, R.O.C., and Y.-C. Huang, National Taiwan University, Taipei, Taiwan, R.O.C., for measurement support. REFERENCES [1] Z. Lao, J. Jensen, K. Guinn, and M. Sokolich, “80-GHz differential VCO in InP SHBTs,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 407–409, Sep. 2004. [2] K. W. Kobayashi, “A 108-GHz InP–HBT monolithic push–push VCO with low phase noise and wide tuning bandwidth,” IEEE J. Solid-State Circuits, vol. 34, no. 9, pp. 1225–1232, Sep. 1999. [3] L. F. Tiemeijer et al., “A record high 150 GHz f max realized at 0.18 m gate length in an industrial RF-CMOS technology,” in Int. Electron Devices Meeting Tech. Dig., Dec. 2001, pp. 10.4.1–10.4.4.

[4] H. M. J. Boots, G. Doornbos, and A. Heringa, “Scaling of characteristic frequencies in RF CMOS,” IEEE Trans. Electron Devices, vol. 51, no. 12, pp. 2102–2108, Dec. 2004. [5] C. Cao and K. K. O, “A 140-GHz fundamental mode voltage-controlled oscillator in 90-nm CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 10, pp. 555–557, Oct. 2006. [6] ——, “Millimeter-wave voltage-controlled oscillators in 0.13-m CMOS technology,” IEEE J. Solid-State Circuits, vol. 41, no. 6, pp. 1297–1304, Jun. 2006. [7] J. Lee, J.-Y. Ding, and T.-Y. Cheng, “A 20-Gb/s 2-to-1 MUX and a 40-GHz VCO in 0.18-m CMOS technology,” in IEEE VLSI Circuits Tech. Symp. Dig., Jun. 2005, pp. 136–139. [8] H. Shigematsu et al., “Millimeter-wave CMOS circuit design,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 472–477, Feb. 2005. [9] A. P. van der Wel et al., “A robust 43-GHz VCO in CMOS for OC-768 SONET applications,” IEEE J. Solid-State Circuits, vol. 39, no. 7, pp. 1159–1163, Jul. 2004. [10] L. M. Franca-Neto, R. E. Bishop, and B. A. Bloechel, “64 GHz and 100 GHz VCOs in 90 nm CMOS using optimum pumping method,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2004, pp. 444–445. [11] A. W. L. Ng and H. C. Luong, “A 1 V 17 GHz 5 mW quadrature CMOS VCO based on transformer coupling,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2006, pp. 711–712. [12] F. Ellinger and H. Jackel, “38–43 GHz quadrature VCO on 90 nm VLSI CMOS with feedback frequency tuning,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1701–1703. [13] A. Natarajan, A. Komijani, and A. Hajimiri, “A fully integrated 24-GHz phased-array transmitter in CMOS,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2502–2514, Dec. 2005. [14] A. Rofougaran, J. Rael, M. Rofougaran, and A. Abidi, “A 900-MHz CMOS LC-oscillator with quadrature outputs,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 1996, pp. 392–393. [15] H. C. Choi, S. B. Shin, and S.-G. Lee, “A low-phase noise LC-QVCO in CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 540–542, Nov. 2004. [16] J. Wood, T. C. Edwards, and S. Lipa, “Rotary traveling-wave oscillator arrays: A new clock technology,” IEEE J. Solid-State Circuits, vol. 36, no. 11, pp. 1654–1665, Nov. 2001. [17] A. Hajimiri and T. H. Lee, “Design issues in CMOS differential LC oscillators,” IEEE J. Solid-State Circuits, vol. 34, no. 5, pp. 717–724, May 1999. [18] Y.-H. Suh and K. Chang, “Coplanar stripline resonators modeling and applications to filter,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 5, pp. 1289–1296, May 2002. [19] H. Krishnaswamy and H. Hashemi, “A 26 GHz coplanar striplinebased current sharing CMOS oscillator,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2005, pp. 127–130. [20] G. Le Grand de Mercey, “A 18 GHz rotary traveling wave VCO in CMOS with I/Q outputs,” in IEEE Eur. Solid-State Circuits Conf., Sep. 2003, pp. 489–492. [21] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005.

HSIEH et al.: 15/30-GHz DUAL-BAND MULTIPHASE VCO IN 0.18- m CMOS

[22] H. Wu and A. Hajimiri, “Silicon-based distributed voltage-controlled oscillators,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 493–502, Mar. 2001. [23] C. Yu, J. S. Yuan, and H. Yang, “MOSFET linearity performance degradation subject to drain and gate voltage stress,” IEEE Trans. Device Mater. Rel., vol. 4, no. 4, pp. 681–689, Dec. 2004. [24] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. [25] D. Baek, J. Kim, D. Kang, and S. Hong, “Low phase noise band frequency multiplied and divided MMIC VCOs using InGaP/GaAs HBT technology,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 2193–2196. [26] S. Ko et al., “ - and -bands CMOS frequency sources with -band quadrature VCO,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2789–2800, Sep. 2005. [27] S. Hackl et al., “A 28-GHz monolithic integrated quadrature oscillator in SiGe bipolar technology,” IEEE J. Solid-State Circuits, vol. 38, no. 1, pp. 135–137, Jan. 2003.

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Ying-Chih Hsu was born in Taichung, Taiwan, R.O.C., in 1981. He received the B.S. degree in electronics engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 2003, and the M.S. degree in electronics engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2005. His research interests include RF integrated circuits and monolithic microwave integrated circuit (MMIC) designs.

X

Hsieh-Hung Hsieh (S’05) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree in electronic engineering at National Taiwan University. His research interests include the development of low-voltage and low-power RF integrated circuits, multiband wireless systems, RF testing, and monolithic microwave integrated circuit (MMIC) designs.

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

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Design of Class E Amplifier With Nonlinear and Linear Shunt Capacitances for Any Duty Cycle Arturo Mediano, Senior Member, IEEE, Pilar Molina-Gaudó, Senior Member, IEEE, and Carlos Bernal, Student Member, IEEE

Abstract—One of the main advantages of class E amplifiers for RF and microwave applications relies on the inclusion of a shunt capacitance in the tuned output network. At high frequencies, this capacitance is mainly provided by the output parasitic capacitance of the device with perhaps a linear external one for fine adjustments. The device’s output capacitance is nonlinear and this influences the design parameters, frequency limit of operation, and performance of the class E amplifier. This paper presents a design method for the class E amplifier with shunt capacitance combining a nonlinear and linear one for any duty cycle, any capacitance’s nonlinear dependence parameters, and any loaded quality factor of the tuned network. Nonlinear design with possibly different duty cycles is of relevance to maximize power or, alternatively, frequency utilization of a given device. Experimental, simulated, and compared results are presented to prove this design procedure. Index Terms—Class E amplifier, duty cycle, high efficiency, nonlinear shunt capacitance, RF power.

I. INTRODUCTION LASS E amplifiers [1] are advantageous networks for high-efficiency RF amplifiers because of the inclusion in the output tuned network of a capacitor shunting the device . As frequency increases, the parasitic capacitance of the device dominates the shunt capacitance. This capacitance is nonlinear and can be expressed by1

C

(1)

with being the capacitance at zero voltage, being the built-in potential (generally ranging from 0.5 to 0.9), and being the grading coefficient of the pn-junction. Several authors acknowledge the importance of designing class E amplifiers taking into account this nonlinear capacitance, and a few approaches have been published. The analytical solution for a restricted set of conditions was started by Chuand duty cycle was also 0.5. A more dobiak in [2] for Manuscript received August 8, 2006; revised October 3, 2006. The authors are with the Department of Electronics Engineering and Communications and the Power Electronics and Microelectronics Group, Aragón Institute for Engineering Research, University of Zaragoza, 50018 Zaragoza, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2006.890512 1Although this work is primarily aimed at power MOSFETs, analysis may be applied to other microwave devices such as MESFETs, HBTs, pseudomorphic HEMTs (pHEMTs), etc.

Fig. 1. Class E resonant tuned circuit with losses.

practical example with similar restrictions is presented in [3] and further discussed in [4], allowing combinations of linear and nonlinear shunt capacitances, something already seen in [5]. Numerical approaches expand the method for a variety of grading coefficients and several other more realistic situations [6], [7], but the degree of freedom presented in this paper is novel. This method is based on the computation of an equivalent of the device’s nonlinear one, as defined linear capacitance in [8] and [9]. The linear equivalent includes both the linear external capacitor and the nonlinear parasitic contribution. As a consequence of this definition, the frequency limit of the amplifier can be improved [10], [11]. The advantage of such a capacitance is the ability to account for the nonlinearities in classical with the equivalent value, exdesigns by mere substitution of cept for some effects (such as the maximum drain peak voltage) that are recalculated. The form factor is defined to describe . the role played by the nonlinear counterpart in In this paper, we present a class E design method for the circuit depicted in Fig. 1, including the nonlinear output capacitance of the device, valid under the following conditions. Condition 1) Any duty cycle. This is important because for a given device and frequency of operation, maximum output power may be obtained at a different duty cycle than 50%. Additionally, to maximize the frequency of a device in class E for a given output power, optimum is 33% [10]. Condition 2) Efficiency is 100%, thus, the zero-voltagecondition is satisfied, but switching (ZVS) the zero-voltage-derivative-switching (ZVDS) condiis not mandatory (if satisfied, tion

0018-9480/$25.00 © 2007 IEEE

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optimum operating conditions will occur, if not, nominal switching conditions). Losses in all the network components and in the device may also be considered, in which case, will not be 100%. Losses are accounted for in the performance, but not modifying the design values of the class E components. and the grading coeffiCondition 3) The built-in potential cient in (1) are not fixed and can be chosen by the designer. Guidelines on how to obtain them are provided. Condition 4) Loaded quality factor of the tuned output network is not necessarily high and, again, can be chosen by the designer. The resulting circuit parameters and performance of the class E amplifier are numerically obtained. A good number of representative results are presented graphically and in tables. To prove this method, three different verification approaches are included. II. EQUIVALENT CAPACITANCE AND FORM FACTOR Utilizing the nonlinear description of the capacitance in a completely analytical description of the class E amplifier has proven unfeasible unless a good number of constraints are imposed in the assumptions. obtained and used in this The equivalent capacitance [8] paper is specifically defined as the constant (thus, linear) capacitance that substituted for the nonlinear intrinsic capacitance produces the same nominal operating conditions (ZVS) at the instant of turn on that would occur with the real device’s capacitance, maintaining the values of the rest of the amplifier’s elements. Such a linear equivalent is also used in [3], but in this case, the ZVDS condition is always assumed, which is a more particular case of the one used here.2 With this definition, classical design methods may be used, substituting the nonlinear capacitance for its equivalent one. Nevertheless, and even though the equivalent capacitance yields the same switching conditions as the real capacitance, the voltage waveform across the device with a nonlinear capacitance is different during the OFF interval. The nonlinear nature of the capacitance increases the voltage peak across the device, even though the switching occurs under the same conditions and at the same instant. The increase factor depends on the fraction of the total capacitance that is provided by the device. This detail has to be taken into account by designers to select transistors with a higher breakdown voltage in order to maintain safe operating conditions for the amplifier and to protect the circuit from over-voltages due to nonlinearities. The design method presented in this paper provides the new peak voltage value normalized by the supply voltage. To quantify the percentage of nonlinearity in , we define the form factor as the quotient of the equivalent capacitance and the theoretical value of (2) 2This analysis could also be restricted to include the condition of zero voltage derivative at turn on and, therefore, calculate the optimum linear equivalent capacitance. On the other hand, the nominal operating mode has been adopted in order to achieve greater generality.

Fig. 2. Equivalent capacitance with two different supply voltages.

with (3) where is completely linear (classical analysis is com, it means that is completely propletely valid). If ; i.e., equals zero and provides the duced by whole necessary . To estimate the equivalent capacitance, it is necessary to know the device’s output capacitance response [see (1)], in this sense, every single parameter influencing would also influence ; therefore, the supply voltage and the form factor play a relevant role in . Mathematically, (4) A. Supply Voltage Influence on According to the mathematical model chosen to represent the voltage dependence of , the higher the supply voltage , the lower the equivalent capacitance because will reach lower values due to broader voltage excursions. In Fig. 2, two operating situations have been plotted with two different and , each one leading to a different supplies, i.e., and voltage waveform and to a different peak voltage ( ). The excursion to a higher voltage value in the second case yields a reduction in the equivalent capacitance value. B. Form Factor

Influence on

The equivalent capacitance also depends on the contribution of the nonlinear capacitance to the total required capacitance . This contribution is exactly what is characterized by the form factor . Thus, the higher the value of (the closer it is to unity), the greater the influence of the nonlinearities and, therefore, the higher the voltage peak across the device. The equivalent capacitance increases at the same rate as the contribution

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Fig. 3. Drain voltage waveform for three different form factors. solid line.  = 0:55: gray line.  = 1: dotted line.



:

= 0 05:

TABLE I CLASS E CIRCUIT VALUES FOR THREE DIFFERENT FORM FACTORS [10]

as a function of V for Q = Fig. 4. Equivalent normalized capacitance C D = 0:5 for  = 0:5 (solid line) and  = 1 (dashed line), and for ) and n = 0:5( ). In all cases, two different values of n: n = 0:3( V = 0:6.

5 and

of the device capacitance increases in relation to the total . Fig. 3 illustrates three different responses of a class E amplifier with the same output nonlinear capacitance (e.g., the same device) and the same supply voltage, but at three different frequencies. At very low frequencies, 150 kHz in the example, the nonlinear device capacitance is a negligible part of the total capacitance and the linear capacitance dominates. On the other hand, at 2 MHz, the nonlinear device output capacitance dominates the total capacitance required for optimal class E operation. The actual values are given in Table I. The switching conditions remain the same, but the equivalent capacitances, the form factors, peak voltages, and waveforms are different. C. Computing the Linear Equivalent Capacitance To compute the equivalent capacitance, a numerical statespace description of the class E amplifier has been programmed, allowing for nonlinear capacitance and including losses in all the circuit elements [12]. To compute the equivalent capacitance, the algorithm comprises the following steps. Step 1) First design a completely linear and ideal class E, obtaining a linear by means of classical results (e.g., [13]). for a partially Step 2) Substitute the linear capacitance nonlinear one (thus, needs to be known) and an external one. Change the nonlinearity constant and iterate until equivalent nominal switching conditions are obtained (ZVS). Alternatively, also include ZVDS for the optimum equivalent. To do this, and have to been known. To estimate those values, use a reverse curve-fitting method, if any information of variation of output capacitance with voltage is provided in the datasheet, or obtain those values from measurements [14] if otherwise. Step 3) This computed nonlinear capacitance will be the equivalent of the previous linear one. This equiv-

0 0

00

alent capacitance is a function of the values of , , , and that have been considered. For simplicity in this paper, a good number of equivalent capacitances have been computed and are presented in the tables included in the Appendix for a wide range of supply voltages and form factors and for several values of and . A clarifying integrates in the explanation on how the computation of a design process is provided in Section IV. III. CIRCUIT ANALYSIS The circuit of the class E amplifier analyzed is presented in Fig. 1. The derivation of the equations considers the following assumptions. is large enough so that • The inductance of the choke coil current may be considered constant. • The shunt capacitance is considered linear, but it consists of the linear equivalent of the nonlinear output capacitance and the external capacitor. Therewith, nonlinearities are taken into account. Let the starting point of this analysis be the amplifiers response constants [13], which are defined by

(5) with from (3). The nominal operating waveforms of the amplifier depend on these constants. Thus, different circuit paramand yield the eters of the output tuning network very same waveforms under nominal operating conditions if the and remain constant. These preceding parameters amplifier response constants are and dependent. for (2), Substituting (6)

MEDIANO et al.: DESIGN OF CLASS E AMPLIFIER WITH NONLINEAR AND LINEAR SHUNT CAPACITANCES FOR ANY DUTY CYCLE

Fig. 5. Equivalent normalized capacitance C as a function of the form factor  for Q = 5; V = 0:6; and n = 0:5. (a) For several values of V : V = 1 :5 V ( ); V = 3V( ); V = 6V( ); V = 12 V ( ); V = 24 V ( ). In all these cases, D = 0:5. (b) Fixed supply voltage of 12 V and several duty cycles: D = 0:25 (dashed line), D = 0:5 (solid line), and D = 0:75 (dotted line).

0r0

00

0 0

0}0

030

with being the loaded quality factor of the output network in conduction.3 A certain calculated for a particular amplifier with a specific load and operation frequency will still be valid for other and recases provided that main constant in all of them. A. Parameter Normalization A normalization of the equivalent capacitance by the factor proves very interesting because is a mere linear scaling factor in the characterization of the linear equivalent capacitance (1). Thus, the following normalization can be applied: (7) 3

Q

to Q

= =

! L =R !L =R

, where ! = 1=(L [13].

C

). The expression may be related

487

Fig. 6. (a) Normalized drain peak voltage v as a function of  for D = 0:25 (dashed line), D = 0:5 (solid line), and D = 0:75 (dotted line) and for ) for V = 12 V and Q = 2( ); Q = 5( ) and Q = 10( as a function of V for n = 0:5. (b) Normalized drain peak voltage v for the particular case of D = 0:5 and Q = 5 and for  = 1 (solid line),  = 0:5 (dashed line), and  = 0:1 (dotted line) and for n = 0:3( ) and n = 0:5( ). In all cases, V = 0:6.

0r0

00

0 0

00

0}0

Including this in (6),

(8) In general, the normalizing equations can be defined as follows. 1) Normalized frequency (9) 2) Normalized equivalent capacitance (10)

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TABLE II VERIFICATION BY COMPARISON WITH [4]

Fig. 8. Class E amplifier with losses built for experimental verification.

Fig. 7. Class E design process using numerical results in the tables. V is the maximum voltage withstood by the transistor (given by manufacturers in the datasheet).

By combining them all and substituting in (6), the following expression is obtained: (11)

and constant and solving the problem Maintaining leads to the results determining the design numerically for parameters. We have obtained a set of equations with a signifiand recant generality, provided that the values of main constant. This is important because it makes the results and of (which deindependent of actual load resistance pends on the exact member of a device family). The normalization might also be extended to voltage and current waveforms, defining the normalized voltage or current as the value of the voltage across or current through a node divided by the value of or , respectively. The normalized results of the numerical design method will be presented according to this nomenclature. Some results are shown here in the form of graphs. In Fig. 4, with is shown for two different the dependence of form factors and two different values. The more linear the capacitance (lower ), the lower the equivalent capacitance. As increases, the device’s output capacitance also decreases and . Fig. 4 graphically shows the effect described in so does Section II-A, demonstrating that the higher the supply voltage, the lower the equivalent capacitance. Fig. 5(a) shows the depenwith for several values of supply voltage. In dence of Fig. 5(b), the same dependence is shown, but for three different duty cycles. Fig. 6(a) and (b) shows the results for the normalas a function of and , respectively. ized peak voltage The first one shows that the peak varies strongly with the duty

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TABLE III SIMULATED AND EXPERIMENTAL VERIFICATION CIRCUIT

cycle and does not vary significantly with . In general, the peak voltage value is higher for increasingly nonlinear situations, as expected. Fig. 6(b) highlights that, except for very low values of supply voltage, the normalized peak value does not depend and the more nonlinear (higher form factor, on the actual higher ), the higher the peak. IV. DESIGN PROCEDURE Generally, the specifications of a class E amplifier are: frequency, intended output power, and available supply voltage, . Generally, and sometimes also output harmonic content duty ratio is not predetermined by external conditions, although some constraints may apply depending on the available driver. If no external constraints apply, the idea is to select a duty cycle equal to 0.3 to maximize the frequency utilization of a device or, alternatively, use [15] to maximize output power depending on losses. At this point, there are two possible lines of action. The first one is to use the tables that include numerical results in the or are not Appendix. If the exact parameter values for listed in them, interpolating values are still applicable with good results. How to design a class E amplifier using this method is summarized in the flowchart of Fig. 7. Secondly, if the parameter values are very different to those in the tables, or high accuracy needs to be achieved, additional numerical results need to be obtained and the method needs to be numerically programmed. To do so, the following steps apply. with the method proposed in Step 1) Calculate Section II-C. Iterate for a good number of values depending on and some possible variation of with (10). This yields the constraints. Obtain first column of the table. Step 2) Compute response constants with (5) and derive with (11). This gives the second part of the table. Step 3) Compute peak voltage values and divide by to obtain the third part of the numerical tables.

Fig. 9. Waveforms of drain voltage v and gate voltage in SPICE simulated (dotted line) and experimental (solid line) test circuit for the design example in Table III.

V. VERIFICATION Three different procedures are investigated to verify this design method. The first one uses the particular results obtained in [4] to compare with a similarly specified design. In the second one, a circuit is designed and simulated in SPICE, and in the third case, the simulated circuit is built and tested to add experimental results to this paper, further proving the method. A. Comparison With Other Analysis In [4], an analytical method to design class E amplifiers with a combination of nonlinear and linear shunt capacitances is presented. The duty cycle is fixed to 0.5, as well as the grading coefficient , and the quality factor of the tuned output network is high, so, output is a sine wave. Making these particularizations in our design procedure and using similar specifications, the design values obtained (Table II) are exactly the same, except for the equivalent capacitance of the IRF510 transistor (something not calculated in [4]), which

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TABLE IV NUMERICAL RESULTS FOR DESIGN PROCEDURE FOR

was important to a priori determine the external capacitance value with precision. Peak voltage value, as well as output

Q

=5

power and other performance parameters, are equal in both cases.

MEDIANO et al.: DESIGN OF CLASS E AMPLIFIER WITH NONLINEAR AND LINEAR SHUNT CAPACITANCES FOR ANY DUTY CYCLE

B. Simulated and Experimental Results Fig. 8 shows the class E amplifier that has been designed with this method, simulated and built for MHz, , , and output power of 1 W. Precise component values and are shown in Table III. The device used is PolyFET’s P123 has been exLDMOS and the nonlinear dependence of tracted from datasheet information. In the design, the amplifier optimum load resistance was 24 based on the method (theory) presented in this paper. A value of 16.2 was obtained when performing SPICE simulations and experiments. The difference between theory and experiment is the inclusion of on resistance and component losses in the SPICE simulation [16]. The results have been simulated using SPICE. The circuit has been built and tested. The ZVS was achieved straight away at the desired 100-MHz frequency without any need of optimization loops in the design process. The expected efficiency (72.7%) is almost exactly achieved (71.4%) and could be improved with a lower on resistance device. Fig. 9 shows the results obtained in the simulation for the drain voltage waveform compared to the oscilloscope captured plot for the same waveform in the experiment. A specific D-variable driver has been designed and built for this purpose. VI. CONCLUSION In this paper, a novel and straightforward design method has been presented for class E amplifiers for any combination of nonlinear and linear capacitances shunting the device, duty cycle, output harmonic content, and possible nonlinear dependence. Losses in all the elements may also be included to predict the performance. The advantage of this method is that, to account for the nonlinearities, an equivalent linear capacitance is computed. This equivalent capacitance can be directly substituted in any other class E design method, except for a few parameters that need to be recalculated. Guidelines to calculate this capacitance are provided. Some representative results are summarized in graphs and additional results are presented in tables. To verify the method, three different alternatives have been tested, which are: 1) comparison with results of an existing less general analysis; 2) simulation, and 3) experimental verification; all of them yielding positive results.

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REFERENCES [1] N. O. Sokal and A. D. Sokal, “Class E—A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE J. Solid-State Circuits, vol. SC-10, no. 3, pp. 168–176, Jun. 1975. [2] M. J. Chudobiak, “The use of parasitic nonlinear capacitors in class-E amplifiers,” IEEE Trans. Circuits Syst. I, Fundam. Theory and Appl., vol. 41, no. 10, pp. 941–944, Dec. 1994. [3] T. Suetsugu and M. K. Kazimierckzuk, “Comparison of class-E amplifier with nonlinear and linear shunt capacitance,” IEEE Trans. Circuits Syst. I, Fundam. Theory and Appl., vol. 50, no. 8, pp. 1089–1097, Aug. 2003. [4] ——, “Analysis and design of class-E amplifier with shunt capacitance composed of nonlinear and linear capacitances,” IEEE Trans. Circuits Syst. I, Fundam. Theory and Appl., vol. 51, no. 7, pp. 1261–1268, Jul. 2004. [5] A. Mediano, “Contribution al estudio de los amplificadores de potencia de RF clase E. influencia de la capacidad de salida del dispositive active,” (in Spanish) Ph.D. dissertation, Dept. Electron. Commun. Eng., Univ. Zaragoza, Zaragoza, Spain, 1997. [6] C. Chan and C. Toumazou, “Design of class-E power amplifier with nonlinear transistor output capacitance and finite DC feed inductance,” in Int Circuits Syst. Symp., Sydney, Australia, Jun. 2001, pp. 1129–1132. [7] P. Alinikula, D. K. Choi, and S. Long, “Design of class-E power amplifier with nonlinear parasitic capacitance,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 2, pp. 114–119, Feb. 1999. [8] N. O. Sokal and R. Redl, “Power transistor output port model,” RF Des., vol. 10, pp. 45–48, Jun. 1987. [9] A. Mediano, P. Molina, and J. Navarro, “Class E RF/microwave power amplifier: Linear ‘equivalent’ of transistor’s nonlinear output capacitance, normalized design and maximum operating frequency vs. output capacitance,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, 2000, pp. 783–786. [10] P. Molina-Gaudo, C. Bernal, and A. Mediano, “Design technique for class E RF/MW amplifiers with linear equivalent of transistor’s output capacitance,” in Proc. IEEE Asia–Pacific Microw. Conf., Dec. 2005, vol. 2, 4 pp. [11] A. Mediano and P. Molina, “Frequency limitation of a high-efficiency class E tuned power amplifier due to a shunt capacitance,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1999, pp. 363–366. [12] J. M. Burdío and A. Martinez, “A unified discrete-time state-space model for switching converters,” IEEE Trans. Power Electron., vol. 10, no. 6, pp. 694–707, Nov. 1995. [13] M. Kazimierczuk and K. Puczko, “Exact analysis of a class E tuned power amplifier at any and switch duty cycle,” IEEE Trans. Circuits Syst., vol. CAS-34, no. 2, pp. 149–159, Feb. 1987. [14] P. Molina-Gaudo, “A contribution to nonlinear class-E amplifier device modelling and parameter extraction,” Ph.D. dissertation, Dept. Electron. Commun. Eng., Univ. Zaragoza, Zaragoza, Spain, 2004. [15] D. Kessler and M. K. Kazimierckzuk, “Power losses and efficiency of class E power amplifier at any duty ratio,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 51, no. 9, pp. 1675–1689, Dec. 1996. [16] N. Sokal, “Class E RF power amplifiers,” QEX/Commun. Quarterly Mag., vol. 46, no. 12, pp. 2220–2225, Jan./Feb. 2001.

Q

APPENDIX NUMERICAL RESULTS IN TABLES In Table IV, an extensive set of numerical results for the design process are given. The data provided covers a good number of representative examples for three different values , five possible values of , eight possible values of supply voltage, and ) and for and two possible values of ( and . These are only examples provided here for simplicity, but any particular combination of all the parameters mentioned previously can be numerically computed with and are this design procedure. The tables for directly obtainable from the authors upon request. ACKNOWLEDGMENT The authors would like to thank PolyFET, Camarillo, CA, for their generous donation of transistor samples.

Arturo Mediano (M’98–SM’06) received the M.Sc. and Ph.D. degrees in electrical engineering from the University of Zaragoza, Zaragoza, Spain, in 1990 and 1997, respectively. Since 1992, he has been a Professor with special interests in RF (HF/VHF/UHF) and electromagnetic interference (EMI)/electromagnetic compatibility (EMC) design for telecommunications and electrical engineers. From 1990, he has been involved in design and management responsibilities for research and development projects in the RF field for communications, industry, and scientific applications. His research interest is focused on high-efficiency switching-mode RF power amplifiers, where he possesses experience in applications like mobile communication radios, through-earth communication systems, induction heating, plasmas for industrial applications, and RF identification (RFID). Dr. Mediano is an active member of the MTT-17 (HF/VHF/UHF technology) Technical Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) since 1999.

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Pilar Molina Gaudó (S’98–M’99–SM’05) received the M.Sc. (equivalent) degree in telecommunications engineering and Ph.D. degree in electronic engineer from the University of Zaragoza, Zaragoza, Spain, in 1997 and 2004, respectively. From 1995 to 1996, she was a Visiting Student with the Technical University of Munich. Since 2000, she has been an Assistant Professor with the University of Zaragoza, where her research concerns the area of power amplifiers for HF/UHF/VHF bands. Dr. Molina-Gaudó was a member of the IEEE Women in Engineering Committee (2001–2005). She was an elected Region 8 student activities vice-chair (2003–2004) and a member of the Region 8 Committee, the R8-OpCom, and the IEEE RAB Student Activities Committee (2003–2004). She is counselor of the Student Branch at her the University of Zaragoza. She was member of the 2004 and 2005 IEEE History Committee and is current member of the IEEE New Initiatives Committee and other subcommittees.

Carlos Bernal (S’03) received the B.Sc. degree in electronics engineering and M.Sc. degree in industrial engineering from the University of Zaragoza, Zaragoza, Spain, in 1997 and 2000, respectively, and is currently working toward the Ph.D. degree at the University of Zaragoza. He is currently an Assistant Professor with the Department of Electronics and Communications, University of Zaragoza, where he is currently involved in the field of high-frequency resonant power inverters and direct digital synthesizers.

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Analysis and Design of a Dynamic Predistorter for WCDMA Handset Power Amplifiers Shingo Yamanouchi, Yuuichi Aoki, Member, IEEE, Kazuaki Kunihiro, Tomohisa Hirayama, Takashi Miyazaki, and Hikaru Hida, Member, IEEE

Abstract—This paper presents a dynamic predistorter (PD), which linearizes the dynamic AM–AM and AM–PM of a wideband code division multiple access handset power amplifier (PA). The dynamic PD allows an adjacent channel leakage power ratio (ACPR) improvement of 15.7 dB, which is superior to conventional PDs that linearize static AM–AM and AM–PM. The dynamic PD was designed using an HBT generating nonlinearity, a short circuit at the baseband ( 4 MHz), and a load circuit for the HBT at the RF fundamental band ( 1.95 GHz). Volterra-series analysis was performed to understand the mechanism of the dynamic PD. The analysis revealed that the short circuit at the baseband enabled the dynamic PD generating third-order intermodulation distortion (IMD3) with opposite phase to the fundamental tone (i.e., antiphase IMD3). The antiphase IMD3 allows dynamic gain compression, which linearizes the dynamic gain expansion of a PA with low quiescent current. The analysis also revealed that the IMD3 amplitude of the dynamic PD can be adjusted by load impedance at the RF fundamental band, which enables the gradient of dynamic AM–AM and AM–PM to be optimized to linearize the PA. The fabricated two-stage InGaP/GaAs HBT PA module with the dynamic PD exhibited an ACPR of 40 dBc and a power-added efficiency of 50% at an average output power of 26.8 dBm with a quiescent current of 20 mA. Index Terms—Adjacent channel leakage power ratio (ACPR), intermodulation distortion (IMD), linearization, nonlinear distortion, power amplifiers (PAs), predistorter (PD).

Fig. 1. Principles of linearization. (a) Predistortion based on AM–AM and AM–PM that are nonlinearities represented in time domain. (b) Predistortion based on IMD that is nonlinearity represented in frequency domain.

I. INTRODUCTION

HE DESIGN of highly linear and highly efficient power amplifiers (PAs) has been a critical issue in recent high-speed wireless communication systems with bandwidth-efficient modulation formats, such as wideband code division multiple access (WCDMA). The WCDMA system imposes stringent requirements on the linearity of handset PAs to minimize the adjacent channel leakage power ratio (ACPR) and maintain modulation accuracy. The WCDMA handset PAs are also required to operate at low quiescent current to improve power efficiency to maximize the standby and talk time of handsets [1], [2]. However, low quiescent-current operation inevitably results in nonlinear distortions in both AM–AM (gain distortion) and AM–PM (phase distortion), which degrades the ACPR. Consequently, the development of linearization

T

Manuscript received September 12, 2006; revised December 3, 2006. S. Yamanouchi, Y. Aoki, K. Kunihiro, T. Hirayama, and T. Miyazaki are with the NEC Corporation, Kawasaki 211-8666, Japan (e-mail: [email protected]). H. Hida is with the NEC Electronics Corporation, Kawasaki 211-8666, Japan. Digital Object Identifier 10.1109/TMTT.2006.890515

techniques that will improve the ACPR of WCDMA-handset PAs with low quiescent current is urgently required. Various linearization techniques such as feedback [3], [4], feed-forward [5]–[7], and predistorters (PDs) [8]–[10] have been proposed. The feedback technique is not suitable for wideband applications such as WCDMA due to problems with instability [3], [4]. The feed-forward technique allows the use of WCDMA base-station PAs reducing the ACPR on the order of 20 dB [5]–[7]. However, this technique suffers from the bulky size and highly dissipated power of the control circuits, which prevent its use in handset applications. The PDs can be implemented with analog circuits having low dissipated power in compact monolithic microwave integrated circuits (MMICs). As we can see from Fig. 1(a), the PD has AM–AM and AM–PM inverse to that of the nonlinear PA, which compensates for nonlinearity. Conventional PDs [8]–[10] have been designed based on the static AM–AM and AM–PM obtained by single-tone measurements. The static AM–AM and AM–PM cannot accurately model the nonlinearity in wideband applications because they do not contain complete information about the dependence of the nonlinearity on the modulation bandwidth [11]–[13]. As a result, conventional PDs

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for WCDMA handset PAs have limited improvements in ACPR to only several decibels (e.g., 5 dB [9] and 2.5 dB [10]). The dependence of nonlinearity on the modulation bandwidth has usually been characterized by intermodulation distortion (IMD) with the tone spacing sweep of a two-tone input signal. Consequently, some PDs have been designed to generate IMD that cancels out the IMD from a PA, as shown in Fig. 1(b), which allowed the ACPR to be improved by a factor of 10 dB (e.g., 9 dB for code division multiple access (CDMA) [14] and 13.4 dB for WCDMA [15]). However, these PDs [14], [15] have not still been suitable for handset PAs because this technique also suffers from the bulky size of the circuits for generating IMD. The dynamic AM–AM and AM–PM obtained by modulatedinput-signal measurements have recently also been used to characterize nonlinearity that depends on the modulation bandwidth [11]. We, therefore, have developed a dynamic PD that compensated for dynamic (not static) AM–AM and AM–PM to overcome the limitations with conventional PDs [16]. We have demonstrated that the dynamic PD for WCDMA-handset PAs has improved the measured ACPR by 11 dB, which allowed better linearity than that of conventional PDs [16]. We have also demonstrated that the developed PD was implemented in a compact MMIC. In this paper, we describe the detailed analysis and design of a dynamic PD for a WCDMA-handset PA. Section II discusses the dynamic PD design to generate third-order intermodulation distortion (IMD3) with the required amplitude and phase for nonlinear compensation. Volterra-series analysis was performed to understand how the dynamic PD generates IMD. We derive the relationship between IMD and dynamic AM–AM and AM–PM, which allows discussing the dynamic PD design to exhibit dynamic gain compression and phase expansion that is inverse nonlinearity to a PA with low quiescent current. The operation of the dynamic PD was validated by experiment in Section III.

Fig. 2. Circuit diagram of developed dynamic PD.

Fig. 3. Circuit model of dynamic PD for Volterra-series analysis.

The nonlinear element in Fig. 3 represents the HBT Q4 in Fig. 2. The nonlinear element exhibits the following characteristics: (1) (2)

II. DESIGN OF DYNAMIC PD A. Volterra-Series Analysis of Dynamic PD Fig. 2 is a circuit diagram of the dynamic PD we developed. It employs an InGaP/GaAs HBT, i.e., Q4, as the nonlinear element that generates IMD3 with the required amplitude and phase for nonlinear compensation. The load circuit for Q4 at the RF funand damental band ( 1.95 GHz) that was comprised of was used to optimize the IMD3 amplitude of the dynamic PD. A , of 2 k is part of the bias circuit for Q4. The resistor, i.e., load circuit impedance at the RF fundamental band is dominated of the order of several picofarads. The short by capacitor and circuit at the baseband ( 4 MHz) that comprises series enables HBT Q4 to generate IMD3 with opposite phase to the fundamental tone (i.e., antiphase IMD3). A capacitor, i.e., , of 0.1 F is used to reduce the baseband impedance. An , is used for the RF choke. inductor, i.e., We analyzed the circuit model of the dynamic PD in Fig. 3 using Volterra-series expansion [17] to find out how it operated.

(3) where is the input voltage and is the output current of the nonlinear element, and are quiescent conis the ac component of the voltage, and is ditions, represents the th degree coefficient of that of the current. nonlinear conductance, which can be extracted from small- and large-signal characteristics of the nonlinear element as V , and V . and in Fig. 3 represent the impedances of the short circuit at the baseband and the load circuit for Q4 at the can be apRF fundamental band in Fig. 2, respectively. at the baseband and to proximated to at the RF fundamental band. can be approximated to at the baseband and to at the RF fundamental band. in Fig. 3 represents the input impedance of a PA that is connected to the RF output port of the PD (RFout) in Fig. 2.

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The small-signal load impedance viewed from the HBT-Q4, i.e., , is given as (4) in Fig. 3 represents the source impedance viewed from the RF input port (RFin), which is connected to an input matching network. The small-signal source impedance viewed , is given as from the load circuit for Q4, i.e., (5) Fig. 4. IMD3 generated by third-order nonlinear processes.

Nonlinear response to multiple input signals can be described by nonlinear transfer functions so-called Volterra kernels [17]. of the dynamic The th-order Volterra kernels PD in Fig. 3 are derived as

(6)

(7)

Fig. 5. IMD3 generated by second-order nonlinear processes.

(8) where where

in (8) represents the sum of all the permutations of and are defined as

.

(9) (10) We calculate the nonlinear response to a two-tone input signal of (11) with the frequency and

carrier center frequency is a mixing term of the frequency sets and derived by is a mixing the third-order nonlinear processes in Fig. 4, term of the fundamental tone and the second-order harmonics and derived by the second-order nonlinear processes in Fig. 5, and are mixing terms of the and fundamental tone and IMD2 derived by the second-order nonlinear processes in Fig. 5, and are given as respectively.

(11)

is

the

(15)

where is the maximum amplitude. The complex spectrum of at the fundamental tone ( at and output voltage at ) and IMD3 ( at and at ) with the two input signal can be calculated using the Volterra kernels of (6)–(8)

(16) (17) (18)

(12) where (13)

(14)

is the tone spacing of the two-tone input signal is the total impedance of parallel and , * denotes the complex conjugate, and is defined as

(19)

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The in (19) can be approximated under the conjugate matching condition at the RF fundamental band (i.e., ) as

(20) The derived IMD3 in (13)–(18) can be well approximated in the low-signal power region to neglect more than fifth-order nonlinearity. from the fundamental tone is The relative phase of calculated from (12) and (15) under the conjugate matching conas dition (21) Thus, the relative phase of is antiphase (i.e., . and from the fundaThe relative phases of mental tones are calculated from (12), (17), and (18) as

(22)

(23) The relative phases of

and

are in-phase (i.e., under

the condition that the baseband impedance does not exhibit negative resistance. The amplitude ratio of to is given by (15) and (16) as (24) is calculated as 0.028 from (24) using of of V , and a simulated of 21 . can then be neglected com. pared with The amplitude ratio of and to is given by (15), (17), and (18) as V

(25) With the short circuit at the baseband that reduces and can be reduced so . In such cases, the they are neglected compared with total IMD3 generated from the dynamic PD is dominated by . Without the short circuit at the baseband, on the other hand, and are calculated as 1.3 of that from (25) using a . In such cases, the total IMD3 is dominated equals and . by The dynamic PD can consequently only generate antiphase , thanks to the short circuit at the baseband IMD3, i.e.,

Fig. 6. Simulated dependence of Z

jj

Z

on frequency at baseband.

and . As disthat eliminates in-phase IMD3, i.e., cussed in Section II-C, antiphase IMD3 allows the dynamic PD to exhibit dynamic gain compression. The IMD3 amplitude of the dynamic PD has to be optimized to compensate for the nonlinearity of PA. IMD3 amplitude is a monotone increasing function of , as shown by (15) and (20). RF fundamental-band impedance can be designed using at the load circuit for Q4. can then be optimized by . The IMD3 amplitude Thus, Volterra-series analysis of the dynamic PD indicates that the IMD3 amplitude can be designed with the load circuit for Q4 at the RF fundamental band, and that the IMD3 phase can be designed with the short circuit at the baseband. B. Simulated IMD of Dynamic PD We simulated the IMD of the dynamic PD in Fig. 2 to validate the characteristics obtained by Volterra-series analysis. We will now discuss the relationship between the IMD3 phase . Fig. 6 shows the and the baseband impedance simulated at the baseband of the dynamic PD. on frequency at the baseband The dependence of and can be well fitted to the total impedance of parallel as (26) k and where F. We categorized the frequency into three regions acon frequency. We cording to the dependence of Hz in Fig. 6, where the frequency defined region I as was much less than the transition frequency of kHz. In region I, can be . We defined reapproximated to kHz in Fig. 6, where the frequency is much gion III as higher than the of 3.4 kHz. In region III, baseband impedance was reduced to less than 16 . Region II was Hz kHz between regions I and III. defined as Figs. 7 and 8 show the simulated dependence of IMD ampliof the tude and phase of the dynamic PD on tone spacing two-tone input signal, respectively. The simulated IMD agreed

YAMANOUCHI et al.: ANALYSIS AND DESIGN OF DYNAMIC PD FOR WCDMA HANDSET PAs

Fig. 7. Simulated two-tone output spectrum amplitude of dynamic PD. Average power of two-tone input signal was 8 dBm. The calculated IMD using (12)–(18) was also shown for comparison.

0

497

j

j

Fig. 9. Simulated dependence of Z (f ) and Z (f ) =Re(Z (f )) on C .

Fig. 10. Simulated IMD3 relative amplitude and phase of dynamic PD. Average power of two-tone input signal was 8 dBm. The calculated IMD amplitude using (12)–(18) was also shown for comparison.

0

Fig. 8. Simulated two-tone output spectrum phase of dynamic PD. Average power of two-tone input signal was 8 dBm. The calculated IMD using (12)–(18) was also shown for comparison.

0

well with the calculated IMD using (12)–(18). As can be seen from Fig. 8, the IMD3 relative phase from the fundamental tone kHz where it corresponded to region was antiphase at and were supIII because the in-phase IMD3 . The pressed by reducing baseband impedance relative phase of IMD3 was in-phase at Hz where it and corresponded to region I because the in-phase IMD3 were dominant due to the higher baseband-impedance of 4.7 10 . These results indicated that the reduced baseband impedance achieved by the short circuit at the baseband enabled antiphase IMD3 to be generated. We will next discuss the relationship between the IMD3 in the load circuit at the RF fundamental amplitude and band. Fig. 9 shows the simulated results for RF funda. As we can see from this mental-band impedance was decreasing monotonically figure, . Fig. 10 shows the simulated dependence with increasing of IMD3 relative amplitude and phase from the fundamental

. As we can see from this figure, tone of the dynamic PD on the simulated IMD3 relative amplitude was decreasing mono, as well as , tonically with increasing which was similar to the calculated IMD using (12)–(18). , which The IMD3 relative phase was slightly changed by allowed us to design the IMD3 amplitude separately from the IMD3 phase. C. Relationship Between IMD and Dynamic AM–AM and AM–PM The relationship between IMD and dynamic AM–AM and AM–PM is discussed here, which allowed us to design dynamic AM–AM and AM–PM from the IMD discussed in Section II-B. and the complex enThe RF-modulated output signal can be described as velope of the RF output signal

(27) (28)

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where is the voltage amplitude and is the phase . The RF-modulated input signal of the complex envelope and the complex envelope of the RF input signal can also be described in the same manner as (27) and (28). and dynamic phase shift of a deDynamic gain vice-under-test (DUT) are defined as

(29) (30) Fig. 11. Relationship between IMD3 relative phase and behavior of dynamic AM–AM and AM–PM.

where and are the real parts of the source and load impedance of the DUT, respectively. The dynamic AM–AM and and AM–PM are evaluated as variations in dynamic gain that are functions of dynamic input power phase shift (i.e., ). We will now discuss dynamic AM–AM and AM–PM with a and the two-tone input signal. The two-tone input signal are given as complex envelope of the input signal

signal can be calculated by substituting the complex envelope in in (29) as (32) and (35) for

(36) (37) (38)

(31) (39) (32) The dynamic power of the two-tone input signal in (32) as tained from

where signal as

is the average input power of the two-tone input

is ob(40) (33)

The dynamic phase shift can also be calculated by substituting the complex envelope for in (30) as

The RF output signal with the two-tone input signal is given as the Fourier transformation of the complex spectrum as follows: (41) (34) and represent the th-order IMD where at the upper and lower bands, respectively. From (34), the output is obtained as complex envelope (35) As we can see from Figs. 7 and 8, the IMD of the dynamic PD was symmetric in the lower and upper bands (i.e., ) in region III ( kHz) that accounted for 97% of the WCDMA bandwidth of 3.84 MHz. Therefore, in the following discussion, we have assumed that IMD is symmetric. We have also assumed that because was 40 dB less than in region III, as with the two-tone input shown in Fig. 7. The dynamic gain

(42) (43) (44) Equations (36)–(44) represent the relationship between IMD and dynamic gain (AM–AM) and phase shift (AM–PM) with a , two-tone input signal. Under conditions of the coefficients and are much less than and , respectively. The gradients of dynamic AM–AM and AM–PM are and , respectively. The signs for then dominated by and depend on the IMD3 relative phase , as indicated by (38) and (43). The relationship between the IMD3 relative phase and the behavior of dynamic AM–AM and AM–PM can then be summarized by Fig. 11.

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Fig. 12. Simulated static (single-tone signal) and dynamic (two-tone and WCDMA signal) AM–AM and AM–PM of dynamic PD. Average input power of the modulated signal (two-tone and WCDMA) is 8 dBm.

Fig. 13. Simulated static (single-tone signal) and dynamic (WCDMA signal) AM–AM and AM–PM of dynamic PD. Average input power of WCDMA signal is 8 dBm.

The coefficients and are proportional to IMD3 am, as indicated by (38) and (43). The gradients of dyplitude namic AM–AM and AM–PM are then monotonically increasing with IMD3 amplitude. The IMD depends on the tone spacing of the two-tone input signal, as seen in Figs. 7 and 8. With the dependence of IMD on tone spacing, dynamic AM–AM and AM–PM also depend on tone spacing due to the relationship between IMD and dynamic AM–AM and AM–PM.

As we can see from Fig. 8, region III accounted for 97% of the WCDMA bandwidth of 3.84 MHz. Dynamic AM–AM and AM–PM with the WCDMA signal were, therefore, dominated by those with the two-tone signal in region III. In fact, dynamic AM–AM and AM–PM with the WCDMA signal agreed well with those with the two-tone signal at 4-MHz tone spacing in region III, as shown in Fig. 12. However, dynamic AM–AM and AM–PM with the WCDMA signal did not agree with those of the two-tone signal with 40-Hz tone spacing in region I ( Hz) because it only accounted for 0.003% of the WCDMA bandwidth. In Fig. 12, dynamic AM–AM and AM–PM with a two-tone signal at 40-Hz tone spacing in region I agreed well with static AM–AM and AM–PM with a single-tone signal because a twotone input signal converges into a single-tone input signal as tone spacing approaches 0 Hz. Thus, the discrepancy between static AM–AM and AM–PM with a single-tone signal and dynamic AM–AM and AM–PM with a WCDMA signal was caused by the existence of regions I and III, i.e., the dependence of IMD on tone spacing. The dynamic gain compression characteristics with the WCDMA signal can be exhibited by the short circuit at the baseband that reduced baseband impedance to generate antiphase IMD3 with wide tone spacing (i.e., in region III). Fig. 13 shows the dependence of the dynamic AM–AM and in the load circuit at the AM–PM of the dynamic PD on RF fundamental band. The gradients of dynamic AM–AM and AM–PM were decreasing monotonically with increasing because the IMD3 amplitude was also monotonically decreasing, as shown in Fig. 10. We can then optimize the grato linearize dient of dynamic AM–AM and AM–PM with a nonlinear PA.

0

D. Simulated AM–AM and AM–PM of Dynamic PD We simulated the dynamic AM–AM and AM–PM of the dynamic PD to validate the operation. Dynamic AM–AM and AM–PM can be calculated from the complex envelopes, as was done in (29) and (30). The complex envelopes with a two-tone input signal were calculated from (32) and (35) using the complex spectrum that was obtained by the harmonic-balance (HB) simulation of Agilent ADS. The complex envelopes with the WCDMA input signal were calculated with the co-simulation of the system-level simulation and the envelope simulation of ADS. High-degree polynominal functions were fitted to the simulated dynamic AM–AM and AM–PM with the WCDMA signal. Fig. 12 shows the simulated dynamic AM–AM and AM–PM of the dynamic PD in Fig. 2 with the two-tone input signal Hz and MHz) and WCDMA input signal. It also ( shows the simulated static AM–AM and AM–PM of the dynamic PD with a single-tone input signal. As shown in Fig. 12, the dynamic AM–AM and AM–PM of the dynamic PD with a two-tone signal at 4-MHz tone spacing exhibited gain compression and phase expansion. This agreed with the behavior that was deduced from Fig. 11 and the IMD3 kHz) in Fig. 8. relative phase of 140 in region III ( Dynamic AM–AM and AM–PM with a two-tone signal at 40-Hz tone spacing, on the other hand, exhibited gain expansion and phase compression, which agreed with the behavior deduced from Fig. 11 and the IMD3 relative phase of 30 in region I ( Hz).

0

III. EXPERIMENTAL VALIDATION Here, we validate the effect of the dynamic PD by measurement. We fabricated an integrated circuit (IC) integrated with the dynamic PD and driver stage Q1. This IC was fabricated using the InGaP/GaAs HBT process. We also fabricated a two-stage

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Fig. 14. Circuit diagram of two-stage PA with dynamic PD. Two-stage PA was fabricated in a 4

2 4 mm module with a resin substrate.

Fig. 15. Fabricated two-stage PA module with dynamic PD.

PA module with the dynamic PD shown in Fig. 14. There is a photograph of the two-stage PA module with the dynamic PD in Fig. 15. The PA module was fabricated from a 4 4 mm resin substrate, matching networks implemented by using surface-mounted passive components, and an InGaP/GaAs HBT IC integrated with the dynamic PD, the driver stage Q1, and the power stage Q2. We measured dynamic AM–AM and AM–PM with the WCDMA signal of the IC and the PA module using a measurement setup with a vector signal analyzer (VSA). The output signal of the DUTs was down-converted to the complex envelope signal in the VSA measurement setup, and its time-domain data were captured by the VSA. Dynamic AM–AM and AM–PM were obtained by plotting the captured complex envelope signals. High-degree polynominal functions were fitted to the measured dynamic AM–AM and AM–PM, revealing nonlinear distortion. Static AM–AM and AM–PM with a single-tone signal were measured using a setup with a network analyzer. The characteristics of the IC were measured using an impedance tuner to set the source and load impedance. The characteristics of the PA module were measured in a 50measurement setup. Fig. 16 shows the measured and simulated static and dynamic AM–AM and AM–PM of the IC integrated with the dynamic PD and driver stage Q1. The IC integrated with the dynamic PD exhibited a measured dynamic gain compression of 0.4 dB and

Fig. 16. Measured static (single-tone signal) and dynamic (WCDMA signal) AM–AM and AM–PM of dynamic PD with driver stage. Average input power of WCDMA signal is 11.5 dBm. Simulated characteristics are shown for comparison [16].

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a dynamic phase expansion of 5.8 with WCDMA signal at an input power from 17.0 to 9.0 dBm. On the other hand, the IC exhibited a measured static gain expansion of 1.3 dB and a static phase compression of 11.6 with a single-tone signal at an input power of 17.0 from 9.0 dBm. Thus, the fabricated dynamic PD exhibited dynamic gain compression and phase expansion with the WCDMA signal and static gain expansion and phase compression with the single-tone signal, as discussed in Section II. Fig. 17 shows the measured and simulated AM–AM and AM–PM of the fabricated two-stage PA module with the dis. The PA abled PD by cutting the emitter node of the HBT module exhibited a measured dynamic gain expansion of 1.0 dB and phase compression of 6.9 with the WCDMA signal at an input power from 15.0 to 7.0 dBm because the PA module mA and was operated with low quiescent current (

YAMANOUCHI et al.: ANALYSIS AND DESIGN OF DYNAMIC PD FOR WCDMA HANDSET PAs

Fig. 17. Measured static (single-tone signal) and dynamic (WCDMA signal) AM–AM and AM–PM of two-stage PA module with disabled PD. Average input power of WCDMA signal is 9.6 dBm. Simulated characteristics are shown for comparison [16].

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Fig. 19. Measured ACPR of two-stage PA module with and without dynamic PD. Simulated characteristics are shown for comparison [16].

Fig. 20. Measured average gain and PAE of two-stage PA module with dynamic PD. Simulated characteristics are shown for comparison [16]. Fig. 18. Measured static (single-tone signal) and dynamic (WCDMA signal) AM–AM and AM–PM of two-stage PA module with dynamic PD. Average input power of WCDMA signal is 8.4 dBm. Simulated characteristics are shown for comparison [16].

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mA). The PA module exhibited the static AM–AM and AM–PM similar to the dynamic AM–AM and AM–PM. Fig. 18 shows the measured and simulated AM–AM and AM–PM of the fabricated two-stage PA module with the dynamic PD. As shown in Fig. 18, the two-stage PA module with the dynamic PD exhibited a measured dynamic gain deviation of 0.04 dB and a dynamic phase deviation of 1.2 at an input power from 15.0 to 7.0 dBm. Thus, the dynamic gain deviation and phase deviation of the PA module with the dynamic PD were reduced compared with that of the PA module with the disabled PD. Fig. 19 shows the measured and simulated ACPR at 5-MHz offset of the fabricated two-stage PA module with the dynamic PD and those of the PA module with the disabled PD. As we can see from Fig. 19, the measured ACPR of the PA with the

dynamic PD was improved by 15.7 dB at an average output power of 24.4 dBm compared to that of the PA with the disabled PD. As a result, the PA module with the dynamic PD exhibited high linearity with an ACPR of less than 40 dBc at an average output power of less than 26.8 dBm thanks to the dynamic PD, even though the PA was operated with the low quiescent current mA and mA). ( Fig. 20 shows the measured and simulated average gain and power-added efficiency (PAE) with the WCDMA signal of the fabricated two-stage PA module with the dynamic PD. The PA module also exhibited high performance with an average gain of 28.4 dB and a PAE of 49.9% at an average output power of 26.8 dBm in measurement. As shown in Figs. 16–20, the measured results agreed well with the simulated results. These results validated the dynamic PD design based on theory and simulation in Section II, and also demonstrated that the PA design using the simulation was useful to fabricated PAs that achieved the required performances.

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IV. CONCLUSION This paper has presented a dynamic PD that linearized the dynamic AM–AM and AM–PM of a WCDMA handset PA, which allowed greater ACPR improvements than conventional PDs that linearize static AM–AM and AM–PM. The dynamic PD was comprised of an HBT generating nonlinearity, a short circuit at the baseband, and a load circuit for the HBT at the RF fundamental band. Volterra-series analysis revealed that the short circuit at the baseband enabled the dynamic PD generating antiphase IMD3. The analysis also revealed that the IMD3 amplitude of the dynamic PD could be optimized by the load impedance at the RF fundamental band. We analytically derived the relationship between IMD3 and dynamic AM–AM and AM–PM with a two-tone input signal. The analysis demonstrated that the behavior of dynamic AM–AM and AM–PM with the two-tone input signal was dominated by the IMD3 relative phase. Dynamic gain compression was specially enabled by antiphase IMD3. The simulation demonstrated that the dynamic PD could exhibit the dynamic gain compression with a WCDMA signal by antiphase IMD3 in wide tone spacing, which was enabled by the short circuit at the baseband. The analysis and simulation also revealed that the gradients of the dynamic AM–AM and AM–PM were monotonically increasing with the IMD3 amplitude that could be optimized by a capacitor in the load circuit at the RF fundamental band. We could then design the dynamic PD to exhibit dynamic gain compression characteristics with an optimized gradient to linearize the nonlinear PA that exhibited gain expansion due to low quiescent-current operation. The measurements demonstrated that the dynamic PD exhibited dynamic gain compression and phase expansion, which were the inverse distortion characteristics of the PA. We also experimentally demonstrated that the two-stage PA module with the dynamic PD exhibited fewer dynamic gain deviations and phase deviations than those by the PA with the disabled PD. The measured ACPR of the PA with the dynamic PD was improved by 15.7 dB compared to that of the PA with the disabled PD. The two-stage PA module with the dynamic PD exhibited an measured ACPR of 40 dBc, an average gain of 28.4 dB, and a PAE of 49.9% at an average output power of 26.8 dBm with a total quiescent current of 19.7 mA. Thus, the dynamic PD allowed the PA with low quiescent current to exhibit high linearity, gain, and PAE, which are required for WCDMA handset PAs.

ACKNOWLEDGMENT The authors wish to thank M. Akita, K. Nakai, and Dr. Y. Hasegawa, all with the NEC Electronics Corporation, Kawasaki, Japan, for their assistance with chip fabrication and testing. The authors would also like to thank Dr. T. Maeda, Dr. S. Tanaka, and Dr. N. Sumihiro, all with the NEC Corporation, Kawasaki, Japan, and Dr. N. Iwata, Dr. H. Hirayama, and Dr. T. Noguchi, all with the NEC Electronics Corporation, Kawasaki, Japan, for their encouragement throughout this paper’s research.

REFERENCES [1] S. Shinjo, K. Mori, H. Ueda, A. Ohta, H. Seki, N. Suematsu, and T. Takagi, “A 20 mA quiescent current CV/CC parallel operation HBT power amplifier for W-CDMA terminals,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Jun. 2002, pp. 249–252. [2] G. Hau, S. Caron, J. Turpel, and B. MacDonald, “A 20 mA quiescent current 40% PAE WCDMA HBT power amplifier module with reduced current consumption under backoff power operation,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Jun. 2005, pp. 243–246. [3] M. Faulkner, “Amplifier linearization using RF feedback and feedforward techniques,” IEEE Trans. Veh. Technol., vol. 47, no. 1, pp. 209–215, Feb. 1998. [4] S. Pipilos, Y. Papananos, N. Naskas, M. Zervakis, J. Jongsma, T. Gschier, N. Wilson, J. Gibbins, B. Carter, and G. Dann, “A transmitter IC for TETRA systems based on a Cartesian feedback loop linearization technique,” IEEE J. Solid-State Circuits, vol. 40, no. 3, pp. 707–718, Mar. 2005. [5] Y. Yang, Y. Y. Woo, and B. Kim, “Optimization for error-canceling loop of the feedforward amplifier using a new system-level mathematical model,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 475–482, Feb. 2003. [6] K.-J. Cho, J.-H. Kim, and S. P. Stapleton, “A highly efficient Doherty feedforward linear power amplifier for W-CDMA base-station applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 292–300, Jan. 2005. [7] J. Legarda, J. Presa, E. Hernandez, H. Solar, J. Mendizabal, and J. A. Penaranda, “An adaptive feedforward amplifier under “maximum output” control method for UMTS downlink transmitters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2481–2486, Aug. 2005. [8] K. Yamauchi, K. Mori, M. Nakayama, Y. Mitsui, and T. Takagi, “A microwave miniaturized linearizer using a parallel diode with a bias feed resistance,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2431–2435, Dec. 1997. [9] G. Hau, T. B. Nishimura, and N. Iwata, “A highly efficient linearized wideband CDMA handset power amplifier based on predistortion under various bias conditions,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1194–1201, Jun. 2001. [10] J. Kim, M.-S. Jeon, J. Lee, and Y. Kwon, “A new ‘active’ predistorter with high gain and programmable gain and phase characteristics using cascode-FET structures,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2459–2466, Nov. 2002. [11] W. Boesch and G. Gatti, “Measurement and simulation of memory effects in predistortion linearizers,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1885–1890, Dec. 1989. [12] H. Ku, M. D. McKinley, and J. S. Kenney, “Quantifying memory effects in RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2843–2849, Dec. 2002. [13] S. Boumaiza and F. M. Ghannouchi, “Thermal memory effects modeling and compensation in RF power amplifiers and predistortion linearizers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2427–2433, Dec. 2003. [14] J. Yi, Y. Yang, M. Park, W. Kang, and B. Kim, “Analog predistortion linearizer for high power RF amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 1511–1514. [15] J. Cha, J. Yi, J. Kim, and B. Kim, “Optimum design of a predistortion RF power amplifier for multicarrier WCDMA applications,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 655–663, Feb. 2004. [16] Y. Aoki, S. Yamanouchi, K. Kunihiro, T. Miyazaki, T. Hirayama, and H. Hida, “50% PAE 20-mA quiescent current W-CDMA power amplifier with on-chip dynamic-gain linearizer,” in IEEE Radio Wireless Symp. Dig., Jan. 2006, pp. 251–254. [17] J. J. Bussgang, L. Ehrman, and J. W. Graham, “Analysis of nonlinear systems with multiple inputs,” Proc. IEEE, vol. 62, no. 8, pp. 1088–1119, Aug. 1974. Shingo Yamanouchi was born in Ehime, Japan, on May 10, 1974. He received the B.S. and M.S. degrees in applied physics from the University of Tokyo, Tokyo, Japan, in 1997 and 1999, respectively. In 1999, he joined the NEC Corporation, Kawasaki, Japan, where he has been engaged in research and development of InGaP/GaAs HBT PAs for mobile and wideband local area network (WLAN) applications. Mr. Yamanouchi is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan.

YAMANOUCHI et al.: ANALYSIS AND DESIGN OF DYNAMIC PD FOR WCDMA HANDSET PAs

Yuuichi Aoki (M’03) was born in Yamanashi, Japan, on March 22, 1974. He received the B.S. degree in electrical and computer engineering from Kanazawa University, Ishikawa, Japan, in 1996, and the M.S. degree in electrical and electronic engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1998. In 1998, he joined the NEC Corporation, Kawasaki, Japan, where he is currently an Assistant Manager with the System Devices Research Laboratories. In 2005, he was a Visiting Researcher at with the Interuniversity Microelectronics Centre (IMEC), Leuven, Belgium. He has been engaged in the research and development of high-performance RF integrated circuits (RFICs) for wireless communications.

Kazuaki Kunihiro received the B.S. and M.S. degrees in applied physics from the Tokyo Institute of Technology, Tokyo, Japan, in 1988 and 1990, respectively, and the D.E. degree in quantum engineering from Nagoya University, Nagoya, Japan, in 2004. In 1990, he joined the NEC Corporation, Kawasaki, Japan, where he has been engaged in research and development of GaAs FET, GaN FET, and InGaP/GaAs HBT devices and their circuit applications for optical communications and RF PAs. His current interests include RF front-end architectures, and their IC/module implementation for wireless communications. Dr. Kunihiro is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan.

Tomohisa Hirayama received the B.S. and M.S. degrees in applied physics from the University of Tokyo, Tokyo, Japan, in 1994 and 1996, respectively. In 1996, he joined the NEC Corporation, Kawasaki, Japan, where he is currently an Assistant Manager with the System Devices Research Laboratories. He has been engaged in research and development of SiMOSFET, HBT, and heterojunction FET (HJFET) PAs for mobile applications. Mr. Hirayama is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan.

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Takashi Miyazaki was born in Ibaraki, Japan, on April 30, 1972. In 1991, he joined the NEC Corporation, Kawasaki, Japan, where he has been engaged in research and development of InGaAs/AlGaAs lasers for optical communications and InGaP/GaAs HBT PAs for mobile and WLAN applications. His current interests include tunable lasers for optical communications.

Hikaru Hida (M’92) received the B.E. and M.E. degrees in electronics from Osaka University, Osaka, Japan, in 1980 and 1982, respectively, and the D.E. degree from the University of Tokyo, Tokyo, Japan, in 1995. In 1982, he joined the NEC Electronics Corporation, Kawasaki, Japan, where he has been engaged in the development of new high-speed GaAs heterojunction FETs (HJFETs), analog and digital large-signal integrations (LSIs), and cutting-edge technology of Si CMOS e-DRAM application-specific integrated circuits (ASICs). His current interest is in the research and development of RF-CMOS LSI core design and GaAs MMIC/RF modules for mobile, WLAN, and wireless personal area network (WPAN) applications, as well as that of high-speed CMOS SERDES and 10.40-Gb/s GaAs and/or InP ICs for optical communications. Dr. Hida is a member of the IEEE Solid-State Circuits Society.

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The Direct Detection Effect in the Hot-Electron Bolometer Mixer Sensitivity Calibration Sergey Cherednichenko, Vladimir Drakinskiy, Therese Berg, Erik L. Kollberg, Fellow, IEEE, and Iltcho Angelov, Member, IEEE

Abstract—We investigate an error in the noise temperature measurements of the hot-electron bolometer mixers caused by the so-called “direct detection effect.” The effect originates in the changing of the mixer parameters when the mixer is loaded on calibration black body sources at different temperatures (300 and 77 K). A correction factor was obtained from the mixer output power versus the bias current dependence, measured by; 1) the local oscillator (LO) power tuning; 2) mixer heating; and 3) application of an external RF source. Furthermore, the direct detection effect was assessed by elimination of the heterodyne response using a LO frequency, which is far off the mixer RF band. We show that the direct detection effect can be mitigated by using an isolator between the mixer and the IF amplifier. Index Terms—Hot-electron bolometer (HEB), IF matching, mixer, noise temperature, terahertz.

I. INTRODUCTION

S

UPERCONDUCTING niobium–nitride (NbN) hot-electron bolometer (HEB) mixers [1] are used in many radio astronomical instruments, e.g., APEX, TELIS, and SOFIA [2]–[4]. The Heterodyne Instrument for Far Infrared (HIFI) of the Herschel Space Observatory [5] employs 1.4–1.7- and 1.6–1.9-THz NbN HEB mixers [6] with a double-sideband (DSB) noise temperature of 900–1100 K. In order to obtain high-power resolution, precise mixer noise temperature measurements are required (with an error 1%). On the Herschel Observatory, the mixer calibration (i.e., the noise temperature measurements) will be done using 10- and 100-K “cold”–“hot” sources, placed in the receiver beam path after the dual-way diplexer (the telescope signal is split for two mixers of orthogonal polarizations). However, on the ground-based instruments, 300-K/77-K sources are frequently used. The calibration sources are made of highly absorptive materials (e.g., SiC), which are set at a precisely controlled temperature. In a bandwidth of 1 THz, the power of 3 nW can be estimated from

Manuscript received June 20, 2006; revised October 23, 2006. This work was supported by the Swedish National Space Board under the frame of the HEB Mixers for Herschel Space Observatory Project. S. Cherednichenko, V. Drakinskiy, and E. L. Kollberg are with the Microwave and Terahertz Technology Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Göteborg, Sweden (e-mail: [email protected]; [email protected]; [email protected]). T. Berg was with the Microwave and Terahertz Technology Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Göteborg, Sweden. She is now with Sensys Traffic AB, 550 02 Jönköping, Sweden (e-mail: [email protected]). I. Angelov is with the Microwave Electronic Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Göteborg, Sweden (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.891470

the Planck law (in a single spatial mode). This is not negligible compared to the local oscillator (LO) power incident on the mixers (100–200 nW). The electron temperature in the NbN LO power. An IF film is proportional to the absorbed RF signal is generated by electron temperature (hence, resistance) modulation caused by the mixing of the LO and the RF waves. Therefore, for different RF input powers (for the cold and hot sources), the HEB mixer is at different electron temperatures and, hence, has different gain and noise. When the HEB mixer is voltage biased to the resistive state, the change of the electron temperature is seen as a change of the mixer bias current, the scale of which depends on the HEB volume, critical current, and RF bandwidth of the antenna. The shift of the HEB’s bias current is traditionally called “a direct detection effect” since it is just this current modulation that is used for the response readout of all bolometric incoherent detectors. Although the linearity of the HEB mixer’s response has been verified up to a 1000-K input load [7], the direct detection effect modifies the -factor and introduces a systematic error in the receiver noise temperature calibration. In this paper, we discuss several techniques to calibrate out the direct detection effect. In Section II, we describe the direct detection effect and discuss the method of the noise temperature correction found in literature. In Section III, we propose a new method to account for the direct detection affect, and compare it to the earlier published method. Section IV discusses an influence of the mixer IF impedance on the low noise amplifier (LNA) gain, and its role in the direct detection effect. II. DIRECT DETECTION EFFECT A. Noise Temperature Error HEB mixers are made of ultrathin superconducting NbN (sometimes NbTiN) films (3–4 nm). The bolometer itself is very small (in our case, 2 0.1 m ). In a quasi-optical HEB mixer, it is integrated with a planar antenna on a low-loss substrate (in our case, high resistive silicon), which is placed on a silicon elliptical (or spherical) lens [8]. This paper is based on experimental data obtained with one of the HIFI band 6 at 1.63 THz is low mixers [6]. The DSB noise temperature shown in Fig. 1 as a function of the IF. The mixer noise temperature is measured with the -factor technique (source temperatures of 300 K/77 K). The -factor, corresponding to the noise temperature of 1000 K, is approximately 0.8 dB (see Fig. 2). We calculated the -factor error, which leads to 1%, 5%, and 10% of the noise temperature error, as a function of (Fig. 2). From this figure, we can see that the -factor shall be measured with an accuracy better than 0.01 dB

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CHEREDNICHENKO et al.: DIRECT DETECTION EFFECT IN HEB MIXER SENSITIVITY CALIBRATION

Fig. 1. HIFI band 6 low HEB mixer noise temperature at 1.63-THz LO frequency across the IF band. The dashed line notes the Herschel performance baseline. The noise temperature is corrected for the input optics loss (air loss: 0.6 dB, vacuum window: 0.7 dB, IR filter (at 77 K): 0.3 dB, IR filter (at 10 K): 0.3 dB).

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Fig. 3. 1.6-THz DSA HEB mixer response versus RF frequency (squares) and an approximation curve (solid).

Fig. 4. Current–voltage curves and mixer output power versus bias voltage at different LO power levels. P = 290 K. The LO power changes by 50% from the lowest IV to the highest IV (LO power grows in the direction of the arrow).

The integration limits are 0.5 and 3 THz, and

Fig. 2. Y -factor (300 K/77 K) error corresponding to the 1%, 5%, and 10% of the noise temperature error; and Y -factor (dashed) as functions of the receiver noise temperature.

(2) in order to achieve a error as low as 1% at K. It relaxes to 0.035 dB for a error of 5%. In this paper, we will not discuss a -factor error introduced by the measurement equipment, but rather caused by the measurement technique. B. RF Power Coupling to the Mixer Single-mode black-body power coupled to the HEB mixer is defined by the antenna RF bandwidth and the optical path loss. A 1.6-THz double-slot antenna (DSA) is used for the HIFI Band 6 Low mixers. The antenna is used in its low-impedance resonance in order to facilitate an efficient matching to the HEB mixer, whose normal state impedance (resistance) is of the order of 50–100 . The low-impedance resonance is much broader compared to the high-impedance resonance, and it reaches a relative bandwidth of 30% [9]. A Fourier transform spectrometer (FTS) was used to measure the DSA RF bandwidth [10] (see Fig. 3, symbols). The obtained curve was approximated with a , and the black-body power coupled to polynomial function the HEB mixer was calculated as

is the single-mode black-body power spectral density. With the estimated optical losses (air loss: 0.6 dB, vacuum window: 0.7 dB, IR filter (at 77 K): 0.3 dB, IR filter (at 10 K): 0.3 dB), the power coupled to the mixer from the 300 K (77 K) source is approximately 2.8 nW (1.6 nW). For the used 2 m 0.1 m size HEB mixer at 4.2-K bath temperature, the LO power incident on the Si lens is of the order of 100–200 nW. The LO power increases the electron temperature in the HEB from the bath temperature (4.2 K) up to the K. A set of superconductor transition temperature curves and a set of corresponding mixer output power versus bias voltage at several LO power levels are shown in Fig. 4. The to the lowest LO power increases by 50% from the highest IV, shown in this figure. Equivalently, the change from the cold (77 K) source to the hot (290 K) source increases the mixer curve, as shown electron temperature , and changes the in Fig. 5. The higher input load corresponds to the higher dc resistance, i.e., to the lower current (the mixer is voltage biased). C.

(1)

-Factor Measurements

In the IF band of 1 Hz, the mixer output power consists of the down-converted input signal (in the upper and lower sidebands),

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Fig. 5. Current–voltage curves under the hot (300 K, dashed line) and cold (77 K, solid line) sources. The shown bias area corresponds to the minimum mixer noise temperature.

and the mixer output noise (3) Here, is the Boltzmann constant, and is the single-sideband (SSB) mixer gain. Taking derivative of (3), the IF output change, switches from 77 to 300 K, can be found as follows: when (4) where is a contribution to the from the direct detection effect (bias current shift). Since the mixer is voltage biased, both and are caused by the (see Fig. 5). For an ideal HEB mixer change of bias current ), both and are (no direct detection effect, is caused by the change of the heterodyne zero. In this case, response. The mixer DSB noise temperature is obtained from the -factor measurement (5) Please note that the mixer gain and the mixer output noise are the same for both 300- and 77-K sources. The righthand term in (5) is obtained from the middle term by division . In this case, the of both the numerator and denominator by and . following relations hold: , then and . Therefore, (5) shall If be written as (6a)

). Using the -factor, obtained from (6), to calculate the mixer noise temperature from (5) will result in a (small) error, which is caused by the bias current shift (i.e., due to the direct detection effect). The magnitude of the direct detection effect in the -factor measurements depends on the bias current shift , and how strong the and dependences are. Ultimately, the current shift is proportional to the power, coupled to the mixer from the 300-K (77-K) sources, and inversely proportional to the optimal LO power. For the discussed mixer, is 0.2 A at the bias point corresponding to the lowest mixer noise temperature (shown in Fig. 5). The direct detection effect is pronounced in small-volume HEB mixers with broadband antennas (e.g., a spiral antenna where the bandwidth is more than 1 THz). However, for radio astronomical observations, the 1% accuracy of the noise temperature measurements is often required. In this case, even with a narrower band antenna (e.g., a DSA), the direct detection effect might still degrade the -factor measurements. Therefore, a thorough study of the direct detection effect in the small volume HEB mixers is needed in order to achieve the required calibration accuracy. One way of taking the direct detection effect into account was suggested in [11]. In [11], the increase of the electron temperature (i.e., reduction of the bias current of the voltage-biased HEB mixer), when the 77-K source switches to the 300-K source, was compensated with an equivalent reduction of the LO power. Therefore, the bias current was kept constant. Obtained in this and were used for calculation of the -factor and way, the noise temperature using (5). This method was also applied in [12]. However, in both studies, it has not been taken into account that the change of the LO power leads to a change of the mixer gain. Section II-D discusses this problem. D. Correction for the Direct Detection Effect Within both the lumped HEB model [1] and the distributed is a function of HEB model [13], the mixer output noise . The the electron temperature (for the given mixer), i.e., mixer gain depends on both the electron temperature and the LO power. The electron temperature changes via application of the LO power. Therefore, the LO power influences the gain directly . Consider and via changing the electron temperature , and a situation when the 77-K source, with an LO power , correspond to the the 300-K source, with an LO power same bias current. In this case, (6a) modifies to

(7) In (7), the parentheses denote the function dependence. With great confidence (based on the existing HEB models), we can and write that (8)

A combination of (4) and (6a) results in (6b) Indices 300 and 77 refer to the calibration source temperatures. In (6a), since they correspond to the different bias currents (related to the different electron temperatures,

Indeed, a small change of the LO power has the same effect on the electron temperature as a small change of the input noise signal (from the calibration sources) since both signals (one is monochromatic and the other is broadband) belong to the terahertz range. All other HEB parameters (dc resistance, IF impedance, etc.) will also be the same. If the mixer gain is a

CHEREDNICHENKO et al.: DIRECT DETECTION EFFECT IN HEB MIXER SENSITIVITY CALIBRATION

Fig. 6. Receiver output power (P ) versus bias current (I ) at a fixed bias voltage. The bias current was changed by changing the mixer bath temperature (filled symbols), by applying an RF heater (open symbols), and by changing the LO power (solid line).

function of the electron temperature only (i.e., ), (7) transforms to (5), and the mixer noise temperature can be obtained in this way. However, as we have mentioned above, the mixer gain is a function of the LO power even for a constant electron temperature. Therefore, . Using (8) and (7), it results in

(9) When is known, then one can calculate the mixer noise corresponds to temperature from (9). Please note that the same bias current (the same electron temperature), but to and . Both [11] and [12] the different LO powers, as soon as . In Section III, we propose a assume method that does not require this simplification. We also compare the new method against the results in [11] and [12]. III. MEASUREMENTS OF THE NOISE TEMPERATURE ERROR Initially, we investigated the mixer output power versus the at a constant LO power (and a constant bias bias current voltage). The bias current was changed in two ways, which are: 1) by increasing the mixer bath temperature (a resistive heater) and 2) by applying an RF power at a frequency far off the mixer signal band (an RF heater). In both cases, the LO frequency was 1.63 THz (an optically pumped far infrared (FIR) laser). The measurement procedure was as follows. For the mixer bias voltage of 0.6 mV, the LO power was tuned to set the bias current to 50 A. Using a resistor on top of the mixer unit, the mixer bath temperature was then slightly increased so that the bias current reduces by 5 A. The corresponding mixer output power 1 2 was recorded. The same measurements were repeated at 45 and 40 A (Fig. 6, filled symbols). Similar measurements were performed at 45 and 40 A with an RF heater, a 600-GHz backward-wave oscillator (BWO) (Fig. 6, open symbols). The BWO radiation was inserted into the LO beam path with a second beam splitter. The RF power is absorbed by the electrons in the NbN film, i.e., it heats the electrons in the same way as the calibration sources do. We exclude a direct heating of the mixer unit by the BWO

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radiation due to high reflectivity of terahertz radiation from the gold-plated housing. The silicon lens absorption is very low as well (the lens is made of high resistive silicon). Therefore, the radiation is coupled to the mixer via the antenna. With regard to the heat balance equations [14], the RF heating changes the input power to the electron subsystem, whereas the resistive heater changes the bath temperature. For comparison, we also when the LO power is tuned (similar to [11] measured and [12]). This is shown in Fig. 6 with a long solid line. From . When the Fig. 6, it follows that for all cases, calibration sources switch from 77 to 300 K, the bias current . Therefore, , reduces, i.e., and the direct detection effect reduces the -factor [see (6b)]. Using the method of [11], the -factor is measured assuming for no direct detection effect (5). A correction (caused by the current shift) is then applied. In our case, is approximately 0.15 dB A [from Fig. 6]. Since A (Fig. 5), then dB (i.e., the direct detection effect reduces the actual -factor). The receiver noise temperature (including the input optical loss) is 1700 K. Therefore, from Fig. 2, it follows that the -factor error of 0.03 dB corresponds to the noise temperature error of 7%. The A is caused by the current change of nW nW nW signal from 300-K/77-K calibration sources. The HIFI calibration sources are at the temperatures of 100 and 10 K. Since in HIFI the optical loss from the calibration is 0.74 nW, refersources to the mixer is very low, then enced to the silicon lens input. This is a factor of 2 lower than for the laboratory receiver. Assuming that the HEB is a linear , which detector for the discussed input loads, then results in dB. Since, in this case, the -factor is referred to the silicon lens (not to the calibration sources as for the laboratory receiver, i.e., through the air, the vacuum window, and the infrared (IR) filters), then the receiver noise temperature is approximately 1000 K (calibrated for the laboratory receiver input losses). From Fig. 2, we obtain that the -factor error of 0.015 dB corresponds to the noise temperature error of 2%. The resistive heating of the HEB mixer was used around three bias points (three LO power levels). As it follows from Fig. 6 is similar to the one obtained when (filled symbols), the LO power was tuned. It just confirms the fact that perforK) is not very mance of the NbN HEB mixer (with sensitive to a small bath temperature change [7]. It is, however, different when the RF heating is applied (Fig. 6, open symbols). The RF heating increases the electron temperature in the same way as the LO, but does not contribute to the conversion gain as (at a fixed bias current) the LO. The mixer output power is applied, is higher (see Fig. 6) when only the LO power compared to the case when a combination of both the LO power is applied . and the RF heating In other words, is bigger in case of the RF heating (LO power is constant) compared to the LO power tuning case. Since the current shift, caused by the direct detection effect, is very small, an LO source with a very good power stability is needed in order to perform the precise -factor calibration. Note that our method does not even require that the simplificain (9) is retion (8) holds. For a more general case,

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Fig. 7. HEB IV curves at the optimal LO power (lower) and slightly underpumped, at the 1.6-THz LO (crosses) and at the 2.6-THz LO (solid).

placed with , where is obtained using is obtained the RF heating (with a constant LO power), and by monitoring the bias current with 300- and 77-K sources. The actual -factor can then be calculated using (6b). In Section IV, we will discuss the role of the mixer to the amplifier matching in the direct detection effect. IV. MIXER TO LNA MATCHING When no direct detection is present, the IF signal change is caused by the heterodyne response. Therefore, the corresponding -factor is [see (5)]. As is seen from Fig. 6, . Therefore, the direct detection effect reduces the -factor. Of course, during the astronomical observations, the input signal power will be much less than the power from the calibration sources. However, it is important to understand the scale of the direct detection effect on the HEB calibration in order to achieve precise power measurements of the astronomical sources. , which is A possible solution would be to measure caused by the bias point shift only, i.e., when the heterodyne . From Fig. 3, it is seen response is eliminated that, at 2.6 THz, the DSA-HEB response drops nearly to zero. The FIR laser, which we used as the LO source, has a 2.6-THz line with an output power of a few milliwatts. We could pump our mixer at this frequency with the same 3- m Mylar beam splitter as for the 1.6-THz experiment (i.e., the input optical loss will remain the same, as will the current shift ). The HEB curves for these two LO frequencies are very close to each other, as is seen from Fig. 7. This is because both frequencies are much higher than the energy gap frequency of the used thin GHz, where NbN superconducting film ( is the Boltzmann constant, K is the superconducting transition temperature, and is the Planck constant). However, the direct detection effect is determined by the power of the calibration sources coupled to the mixer. For a particular sample, it is defined by the antenna bandwidth and by the optical path loss in the antenna RF band. Indeed, we observed is the same for both LO frequencies of 2.6 and 1.6 THz. that From (4), we see that when the mixer gain is zero, is determined by the direct detection effect only. Therefore, the

(or , measured -factor [see (6b)] shall be if no direct detection effect is present). Surprisingly, at 2.6 THz, we observed a positive -factor of the order of 0.05–0.1 dB (i.e., ). Since in Section III we have concluded that the direct detection effect ) reduces the -factor (when estimated from by at least 0.03 dB, it results in the -factor (measured -factor plus direct detection correction) of the order of 0.08–0.13 dB. It corresponds to a DSB mixer noise temperature – K. Since the antenna-HEB response at of 2.6 THz is at least 13 dB lower than at 1.6 THz (Fig. 3), a noise K K is expected temperature higher than at 2.6 THz. We shall note that Band 6 mixers of the HIFI instrument will operate without IF isolators between the mixers and the first stage LNAs. The reason for that is nonavailability of cryogenic isolators for the 2.4–4.8-GHz band. In our measurements, we used an Alcatel HIFI prototype LNA without an isolator as well. Therefore, the LNA input is directly loaded with the HEB mixer, whose IF impedance changes in a wide range through the discussed IF band [15]. An important parameter, which defines the at the opHEB IF impedance, is differential resistance ’s from Fig. 4, the changes eration point. As for the with the bias current. It causes the HEB impedance to change. Using a model of an InP HEMT amplifier, we estimate how sensitive the LNA gain is to the input load impedance. The calculations show that the LNA gain increases by 0.6 dB when the input load impedance increases from 50 to 60 . Of course, the impedance change associated with the direct detection effect is probably not more than 1 . However our modeling was quite simplified as well. It shall also mean that the direct detection error introduced into the -factor shall be IF dependent. In order to verify it experimentally, we measured the -factor at 2.6 THz with an isolator between the mixer and LNA and without it. 4–8-GHz cryogenic isolators are commercially available. The IF frequency was 4 GHz, i.e., in the overlap of the LNA band (2.4–4.8 GHz) and the isolator band. Since the input loss of the isolator is less than 0.3 dB, it does not add to the receiver noise. Without the isolator, the -factor was 0.05 dB, whereas with the isolator the measured -factor was below the accuracy of the measurements (i.e., 0.01 dB). Finally, we measured the LNA gain versus the mixer impedance. For this purpose, a 20-dB directional coupler was connected between the mixer and LNA. A monochromatic microwave probe signal was inserted to the LNA via the coupler. The mixer port to the LNA port transmission is dB. The probe signal port to the LNA port transmission is dB. The probe signal port to the mixer port dB, i.e., the probe signal leakage to the mixer can is be neglected. On the output of the LNA, the probe signal was approximately 20 dB above the mixer output power. The mixer bias voltage was fixed at 0.6 mV, while the current was tuned by the LO power. We observed that, as the mixer current increased, the LNA gain decreased as 0.1 dB A at 4 GHz. However, with the isolator between the coupler and LNA (see the above paragraph), the corresponding value was 0.004 dB A at the same frequency. Therefore, with the direct detection effect of A, a positive -factor of 0.025 dB is expected due

CHEREDNICHENKO et al.: DIRECT DETECTION EFFECT IN HEB MIXER SENSITIVITY CALIBRATION

to the LNA gain change (without the isolator). It is still smaller than the observed 0.05–0.1 dB during the 2.6-THz test. Since no -factor is observed when the isolator is used, we can argue that the remaining effect comes from a change of the HEB to the IF line mismatch factor, which is caused by the same bias current shift. V. CONCLUSION We have analyzed different techniques for correction of the measured HEB mixer noise temperature for the direct detection effect. The effect is observed as a bias current shift (when the mixer is voltage biased) when the 77-K calibration source is switched to the 300-K source. This effect is proportional to the RF bandwidth of the antenna, and inversely proportional to , obtained when the LO the mixer size (LO power). Using power was tuned (method of [11]), it can be estimated that for 300-K/77-K sources, the mixer noise temperature reduces by approximately 7% after the direct detection correction is taken into account. Using this method, the noise temperature error for the Band 6 Low mixers of the HIFI instrument (Herschel Space Observatory) of 2% is expected when the 100- and 10-K calibration sources are used (assuming no coupling loss from the calibration sources to the mixer). We show that it is more relwhen the LO power is kept constant, evant to measure while the bias current is changed by the application of an RF compared heater. This technique results in a higher to the approach from [11]. Eliminating the heterodyne response, by pumping the mixer with an LO source far off the mixer RF band, we observe that, A, no mixer output despite the bias current shift of power shift is seen when an isolator is used between the mixer and LNA. Without the IF isolator, the mixer output power shift is of the order of 0.05–0.1 dB. We obtained a qualitative explanation of this fact by an influence of the mixer impedance on the LNA gain. In order to obtain a quantitative agreement with the experimental data, a thorough investigation of the HEB mixer IF impedance versus mixer bias current is needed. ACKNOWLEDGMENT The authors would like to thank Dr. D. Meledin, Chalmers University of Technology, Göteborg, Sweden, Dr. P. Dieleman, SRON-Groningen, Groningen, The Netherlands, and Prof. G. Gol’tsman, Moscow State Pedagogical University (MSPU), Moscow, Russia, for helpful discussions. REFERENCES [1] E. M. Gershenzon, G. N. Gol’tsman, I. G. Gogidze, A. I. Elant’ev, B. S. Karasik, and A. D. Semenov, “Millimeter and submillimeter range mixer based on electronic heating of superconducting films in the resistive state,” Sov. Phys. Supercond., vol. 3, no. 10, pp. 1582–1597, Oct. 1990. [2] V. Belitsky et al., “Heterodyne single-pixel facility instrumentation for the APEX telescope,” Proc. SPIE, vol. 6275, pp. 62750G1–62750G9, Jun. 2006. [3] R. W. Hoogeveen et al., “Superconducting integrated receiver development for TELS,” Proc. SPIE, vol. 5978, pp. 59781F–59781J, Oct. 2005.

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[4] J. Zmuidzinas and P. L. Richards, “Superconducting detectors and mixers for millimeter and submillimeter astrophysics,” Proc. IEEE, vol. 92, no. 10, pp. 1597–1616, Oct. 2004. [5] J. C. Pearson et al., “Terahertz frequency receiver instrumentation for Herschel’s Heterodyne Instrument For Far Infrared (HIFI),” Proc. SPIE, vol. 4850, pp. 650–661, Mar. 2003. [6] S. Cherednichenko, M. Kroug, P. Khosropanah, A. Adam, H. Merkel, E. Kollberg, D. Loudkov, B. Voronov, G. Gol’tsman, H.-W. Huebers, and H. Richter, “1.6 THz HEB mixer for far infrared space telescope (Hershel),” Physica C, vol. 372, pp. 427–431, Aug. 2002. [7] S. Cherednichenko, M. Kroug, H. Merkel, E. Kollberg, D. Loudkov, K. Smirnov, B. Voronov, G. Gol’tsman, and E. Gershenzon, “Local oscillator power requirement and saturation effects in NbN HEB mixers,” in Proc. 12th. Int Space Terahertz Technol. Symp., San Diego, CA, Feb. 2001, pp. 273–286. [8] G. M. Rebeiz, “Millimeter-wave and terahertz integrated circuit antennas,” Proc. IEEE, vol. 80, no. 11, pp. 1748–1770, Nov. 1992. [9] M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antennas Propag., vol. AP-33, no. 6, pp. 600–607, Jun. 1985. [10] D. Loudkov, P. Khosropanah, S. Cherednichenko, A. Adam, H. Merkel, E. Kollberg, and G. Gol’tsman, “Broadband Fourier transform spectrometer (FTS) measurements of spiral and double-slot planar antennas at THz frequencies,” in Proc. 13th. Int. Space Terahertz Technol. Symp., Cambridge, MA, Mar. 2002, pp. 373–383. [11] S. Svechnikov, A. Verevkin, B. Voronov, E. Menschikov, E. Gershenzon, and G. Gol’tsman, “Quasioptical phonon-cooled NbN hot-electron bolometer mixer at 0.5–1.1 THz,” in Proc. 9th. Int. Space Terahertz Technol. Symp., Pasadena, CA, Mar. 1998, pp. 45–53. [12] J. J. A. Baselmans, A. Baryshev, S. F. Reker, M. Hajenius, J. R. Gao, T. M. Klapwijk, Y. Vachtomin, S. Maslennikov, S. Antipov, B. Voronov, and G. Gol’tsman, “Direct detection effect in small volume hot electron bolometer mixers,” Appl. Phys. Lett., vol. 86, pp. 163503-1–163503-3, 2005. [13] H. F. Merkel, P. Khosropanah, D. W. Floet, P. A. Yagoubov, and E. L. Kollberg, “Conversion gain and fluctuation noise of phonon-cooled hot-electron bolometers in hot-spot regime,” IEEE Trans. Microw. Theory Tech, vol. 48, no. 4, pp. 690–699, Apr. 2000. [14] N. Perrin and C. Vanneste, Phys. Rev. B, Condens. Matter, vol. 28, no. 9, pp. 5150–5159, Nov. 1983. [15] D. Meledin, P.-Y. E. Tong, R. Blundell, N. Kaurova, K. Smirnov, B. Voronov, and G. Gol’tsman, IEEE Trans. Appl. Supercond., vol. 13, no. 2, pp. 164–167, Jun. 2003.

Sergey Cherednichenko was born in Mariupol, Ukraine, in 1970. He received the Diploma degree in physics (with honors) from Taganrog State Pedagogical Institute, Taganrog, Russia, in 1993, and the Ph.D. degree in radio physics from Moscow State Pedagogical University, Moscow, Russia, in 1999. Since then he has been with the Department of Microtechnology and Nanoscience, Chalmers University of Technology, Göteborg, Sweden. His research interests include terahertz heterodyne receivers, detectors, antennas, thin superconducting films, and material properties at terahertz frequencies.

Vladimir Drakinskiy was born in Kurganinsk, Russia, in 1977. He received the Diploma degree in physics (with honors) from the Armavir State Pedagogical Institute, Armavir, Russia, in 2000. Since 2002, he has been with the Department of Microtechnology and Nanoscience, Chalmers University of Technology, Göteborg, Sweden. His research interests include microfabrication, detectors for submillimeter and terahertz ranges and superconducting thin films.

Therese Berg was born in Västervik, Sweden, in 1978. She received the M.S. degree in engineering physics and Licentiate of Engineering degree from the Chalmers University of Technology, Göteborg, Sweden, in 2002 and 2005, respectively. She was involved with the design and characterization of terahertz detectors and flight hardware qualification. She is currently with Sensys Traffic AB, Jönköping, Sweden.

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Erik L. Kollberg (M’82–SM’83–F’91) was born in Stockholm, Sweden, in 1937. He received the Ph.D. degree from the Chalmers University of Technology, Göteborg, Sweden, in 1970. In 1979, he became a Full Professor with the Chalmers University of Technology. From 1967 to 1987, he was the Head of the group that developed low-noise receivers for the Onsala Space Observatory telescopes. He was acting Dean of Electrical and Computer Engineering from 1987 to 1990. In 1995, he founded the Chalmers Center for High Speed Electronics (CHACH). He has been a Guest Professor with Ecole Normal Superieure, Paris, France, and with California Institute of Technology, Pasadena. He has been performing research on millimeter-wave and submillimeter-wave devices and low-noise receivers, including maser amplifiers, Schottky diodes, and superconductor–insulator–superconductor (SIS) mixer receivers. He has also been involved with harmonic multipliers and is the inventor of the heterostructure barrier varactor diode. His current main research interests are in the areas of millimeter-wave and terahertz devices and applications, in particular, HEB mixers. Dr. Kollberg is a member of the Royal Swedish Academy of Science and the Royal Swedish Academy of Engineering Sciences. He was the recipient of

the Microwave Prize presented at the 12th European Microwave Conference, Helsinki, Finland. He was also the recipient of an Honorary Ph.D. degree presented by the Technical University of Helsinki, Helsinki, Finland, in 2000. In 1983–1984, he chaired the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Chapter in Sweden.

Iltcho Angelov (M’89) was born in Burgas, Bulgaria. He received the M.Sc. degree (Hon.) in electronics and Ph.D. degree in physics and mathematics from Moscow State University, Moscow, Russia, in 1969 and 1973, respectively. From 1969 to 1991, he was with the Institute of Electronics, Bulgarian Academy of Sciences, Sofia, Bulgaria, as a Researcher, Research Professor, and Head of the Department of Microwave Solid State Devices. Since 1992, he has been with the Chalmers University of Technology, Göteborg, Sweden. His main interests are in microwave and millimeter-wave device modeling, and low-noise and nonlinear circuit design.

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Modeling Effects of Random Rough Interface on Power Absorption Between Dielectric and Conductive Medium in 3-D Problem Xiaoxiong Gu, Leung Tsang, Fellow, IEEE, and Henning Braunisch, Senior Member, IEEE

Abstract—We study the effects of a random rough surface on the power absorption between a dielectric and conductive medium in a 3-D configuration where the surface height varies in both horizontal directions. The analytic small perturbation method of second order and numerical T-matrix method are used. The absorption depends on the root mean square height, correlation length, and correlation function of the random rough surface. A closed-form expression of power absorption enhancement factor is obtained from small perturbation method of second order. Results show that the T-matrix method agrees with the small perturbation method for rough surfaces with a small slope. We further compare the 3-D results to the previous 2-D results and show significant difference. The power absorption enhancement factor exhibits saturation for the Gaussian correlation function, but not for the exponential correlation function. Index Terms—Perturbation methods, power absorption, rough surfaces, t-matrix method.

I. INTRODUCTION

T

HE SURFACE of metal conductors, especially in highspeed interconnects and microelectronic packaging based on organic materials, is artificially (e.g., chemically) roughened to enhance the interfacial adhesion between the dielectric and the conductive medium. It has been shown by measurement that the practical topological features may have peak to valley distances beyond 5 m [1], [2]. Such roughness of the surface can cause significant effects on conductor loss at microwave frequencies due to the skin effect in classical electrodynamics. Experiments by Tanaka and Okada [3] demonstrated the decrease of effective conductivity of different copper foils by as much as 50%–70% in the multigigahertz region due to surface roughness. One key application of the predictive capability developed here is in computer-aided design (CAD) of insertion-loss-limited interconnects in high-performance computing platforms. It can also be instrumental for guiding package and board substrate technology development, i.e., for making the tradeoff between thermomechanical reliability (adhesion), electrical performance (loss), and cost.

Manuscript received September 6, 2006; revised November 14, 2006. This work was supported by the Intel Corporation. X. Gu and L. Tsang are with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]. edu; [email protected]). H. Braunisch is with Components Research, Intel Corporation, Chandler, AZ 85226 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.891689

The underlying physics focus on the analysis of the interactions of electromagnetic waves with the rough surface. In his classic paper [4], Morgan solved a quasi-static eddy-current problem for a periodic ridge structure in a 2-D problem where the surface height varies only in one horizontal direction. Morgan computed the power absorption enhancement factor, which determines the additional losses due to surface geometry. Hammerstad and Bekkadal [5] fitted Morgan’s results by an empirical formula, which is currently the most common model of quantifying the impact of conductor surface roughness on ohmic loss. Other studies of rough surface effects on conductor loss include a series of studies published by Holloway and Kuester [6], [7], where the emphasis was based on the proper enforcement of boundary conditions at the rough interface to avoid expensive numerical computation of electromagnetic fields. Proekt and Cangellaris [8] and Wu and Davis [9] also analyzed the increase of resistance of conductors due to surface roughness. However, in the above papers, only a periodic ridge structure was considered as the rough interface, which may misrepresent the roughness-induced loss. Most research to date on the roughness effect on conductor loss is limited to 2-D problems. Biot [10] and Wait [11] proposed a 3-D rough surface model in which the roughness is represented by hemispherical bosses on a conducting plane. However, in their theory, only the bosses may have finite conductivity, while losses in the plane are absent. Furthermore, it is difficult to choose parameters such as the number and size of the bosses, as well as their relative positions. The distribution of such hemispherical bosses may not resemble accurately the real physical rough surface occurring on interconnect structures [12]. The Rayleigh–Rice technique [13], [14] is a well-known approach to study wave scattering by rough surfaces. This method is based on perturbation theories and it assumes that the height of the surface roughness is small compared to a wavelength and also that the slope of the roughness is small. Research by Sanderson [15] shows the Rayleigh–Rice method in agreement with Morgan’s numerical solution in case of a 2-D periodic triangular-groove structure. More recently, we applied the Rayleigh–Rice small perturbation method of second order to a 2-D problem and derived a closed-form formula of power absorption enhancement factor [16], [17] in which the 2-D formula utilizes a model of random rough surfaces. All our previous results [16], [17] were based on a 2-D problem with the assumption that the surface height varies only in one horizontal direction. In reality, typical surface roughness

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is most likely to have 3-D random profiles. In this paper, we develop a model of power absorption for a 3-D problem where the surface height varies in both horizontal directions. A 3-D random rough surface model is used to characterize the physical interface between a dielectric and a highly conductive medium. The characteristics include root mean square (rms) height, correlation length, and correlation function. The effects of surface roughness on power absorption are then analyzed by considering an incident plane wave impinging on the interface. The absorption is calculated by two methods: the analytic small perturbation method of second order and the numerical T-matrix method. The results of absorption based on the 3-D small perturbation method are in terms of the spectral density of the random rough surface. Characterization of random rough surfaces in a 3-D problem is discussed in Section II. Sections III and IV include formulation and solution of analytic and numerical methods. Section V illustrates numerical results followed by discussion. II. RANDOM ROUGH SURFACE For a 3-D problem of random rough surface, the height funcis treated as a stationary random process. The twotion point ensemble average of the random process is (1) where is the correlation function and is the rms surface height. Two common correlation functions are the Gaussian correlation function with and exponential correlation function with , where is the correlation length. The exponential correlation profile appears significantly rougher than that for the Gaussian correlation function. In generating the roughness profiles, we , which is the use the spectral density function Fourier transform of the correlation function. The spectral density of the Gaussian correlation function is given by and that of the exponential correlation function by [18]. III. DERIVATION OF 3-D FORMULA BASED ON SMALL PERTURBATION METHOD OF SECOND ORDER Here, we apply the small perturbation method of second order and derive a closed-form 3-D formula of power absorption enhancement factor due to the rough interface between a dielectric and a highly conductive medium. This study is an extension of previous work, in which we studied wave scattering at the interface between two dielectric regions for remote-sensing applications [19]. Here, the simplification takes advantage of an assumption that the conductivity of the lower conducting medium is much greater than that of the dielectric medium. In typical dielectric-metal layers occurring on the interconnect and package, the magnitude of the wavenumber in the conductor is a few thousand times larger than that in the dielectric medium at microwave frequencies.

Fig. 1. Random rough interface between dielectric and conductor in a 3-D problem.

A. Emissivity of a Dielectric-Conductor Interface In a two-media problem, as shown in Fig. 1, let and denote the permittivity of the upper and lower half-space, respectively. Consider a plane electromagnetic incident upon the wave and azimuthal interface with an incident altitudinal angle as the phasor notation. Note angle . Here, we use , , , that , and , where and are the wavenumbers in the upper and lower media. refers to the unit polarization vector of the incident wave. The rough interface is , where characterized by a random height function is a random function with zero mean . , , and Define wave vectors and let , , and . The emissivity (equivalent to absorptivity based on the reciprocity principle) of a vertically polarized (TM) wave and a horizontally polarized (TE) wave, respecis given by applying the tively, in the direction of small perturbation method to second order [19]

(2)

(3) where the second and third terms on the right-hand side represent the reflectivity of power due to coherent scattering, and the fourth term represents the reflectivity of power due to incoherent is the spectral density function of the scattering. Here, and are the Fresnel reflection coeffirough surface, cients of the TM and TE wave, and , , , , , and are the first- and second-order scattering coefficients. Next we consider the case of our interest where the lower medium is a highly conductive metal such as copper. On such a metal surface, the rms height and correlation length are in the order of micrometers. The wavenumber in the dielectric is typically in the order of cm . Also note that is

GU et al.: MODELING EFFECTS OF RANDOM ROUGH INTERFACE ON POWER ABSORPTION BETWEEN DIELECTRIC AND CONDUCTIVE MEDIUM

of order 10 , and is of order 10 . The power absorption due to incoherent and coherent scattering is analyzed in Sections III-B–D.

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to (4). Repeating the steps above also leads to an incoherent . reflectivity of order C. Absorption Due to Coherent Scattering

B. Reflection Due to Incoherent Scattering Let and denote the fourth terms on the righthand side of (2) and (3), representing the respective incoherent reflectivity of the TM and TE wave. Due to the integration limits , and are of the same order as , of and . Consider therefore, a random rough surface with a Gaussian correlation function in and . The spectral for the term density can then be approximated as (4)

Since the incoherent reflectivity is negligible in case that , the power absorption of the rough interface is solely determined by coherent scattering waves. Unlike incoherent scattering, coherent scattering covers evanescent waves. and have limits Therefore, the scattering coefficients to for and . To evaluate the inof integration from and , we assume that , tegration of , , and . Taking and as first- and second-order smallness, we obtain, to first order, (11)

The following approximations are also valid for incoherent scattering: , , and .

(12)

and as first- and second-order smallHere we take ness, respectively. Approximating to first order, we obtain (5)

(13)

(6) (7)

(14) Further, to first order of

with (11) and (12), we obtain (15)

(8) Putting (4)–(8) into yields

and

and integrating over

(9)

(10) , In (9) and (10), note that , and integral are all in the order of 1. Therefore, and are of order and negligible since and are in the order of micrometers, whereas is in the order of cm . For a random rough surface with exponential correlation function, similar the spectral density can be approximated as

(16) Note that , where denotes the skin depth in the conductor with . Here, is the frequency and and are the permeability and conductivity of the conductive medium, respectively. Combining (11) and (13), (12) and (14), approximating them to first order and removing imaginary terms, we obtain

(17)

(18) Substituting (15) and (17) into (2), (16) and (18) into (3), and making use of the property , we then obtain the absorptivity of the TM and TE wave due to coherent scattering

(19)

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magnetic field in the conductor. Next we define surface field unknowns as follows: (20)

(22) (23)

D. 3-D Power Absorption Enhancement Factor The power absorption enhancement factor is a ratio of the power loss dissipated in a conductor with a rough surface compared to that dissipated in the same conductor with a smooth boundary. For a smooth surface, the power absorptivity of the TM and TE wave are given by (15) and (16). Thus, the enhanceand ment factor can be obtained by taking the ratio of , or and as follows:

where , , and is the unit normal vector pointing into the conductor. To apply the numerical T-matrix method, we use the periodic and in the - and boundary condition with the period -direction. This is a valid approximation in random rough surface scattering provided that the period contains a number of as the th patch of the incorrelation lengths. Define finite periodic surface, where and are integers from to . Let denote a point on . The periodic boundary condition indicates that the field unknowns and are also periodic and, thus, can be represented by the following Fourier series:

(21) The 3-D formula of the absorption enhancement factor in (21) has the same form for the TM and TE wave and is independent of angle of incidence; an arbitrary can be chosen. Please note that, intuitively, the application to guided waves on quasi-TEM interconnect structures and loss predic. tion in planar circuits corresponds to the TM case as Similar to the 2-D closed-form formula of the power absorption enhancement factor that we derived in [16] and [17] based on the small perturbation method, (21) also depends on the rms height and spectral density of the rough surface, as well as the skin depth in the conductor. The integrand of (21) asymptotically as and become approaches large. Thus, the integral is convergent for both the Gaussian and exponential correlation function. If we make the rough surface profile uniform in the direction to reduce the 3-D problem becomes to a 2-D problem, then the spectral density , where is a Dirac delta function. Substituting into (21) leads back to the 2-D formula [see (38)]. Also notice that unlike for 2-D problems, Morgan’s assumption of a constant surface magnetic field is no longer valid for 3-D rough interfaces. It is necessary to take into account both dielectric and conductor media to analyze the field scattering on the interface. In [12], we show that applying the 3-D small perturbation method of second order by forcing constant magon the interface leads to an erroneous form netic field in (21) is replaced by of enhancement factor where .

(24)

(25) where

,

, , and . Here, the various and are the Fourier series coefficients of the components of the unknown surface fields and . To solve for these unknown coefficients, we need six matrix equations in total. From Huygen’s principle and the extinction theorem for electrical fields in the two media [19], [20], we first obtain four matrix equations as follows:

(26)

(27)

IV. NUMERICAL APPROACH USING T-MATRIX METHOD Here, we apply a numerical T-matrix method to solve for the power absorption of the rough interface. Let denote the rough interface. and are the wave impedance of the dielectric and conductor, respectively. Consider a normal incident plane wave , and and are the electric and

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and

(29) ,

where and

, and

are given by Fig. 2. Power absorption ratio: surface with Gaussian correlation function (h = 1 m) with varying correlation length.

(30) and

(31) From (22) and (23), we have is equivalent to

, which

Fig. 3. Power absorption ratio: surface with Gaussian correlation function (h = 0:75 m) with varying correlation length.

(32)

(25). We can then compute the power absorption for a rough and in the - and -direction surface with length of

(33) Taking Fourier series expansion on both sides of (32) and (33) gives the last two equations

(34)

(35) where are the Fourier series coefficients of the surface height function defined by

(37) and In the numerical implementation, we take in the six matrix equations. The sursuch that face discretization is chosen as in (36). To calculate the average power absorption, we use a Monte Carlo simulation approach. We generate a large number of realizations of 3-D profiles. Solving the T-matrix equations, we then calculate the absorption ratio for every realization and the average absorption is computed. For the simulation results shown in Section V, 500 realizations are used. V. RESULTS AND DISCUSSION

(36)

A. Results of 3-D Small Perturbation Method of Second Order

Thus, we have six matrix equations, i.e., (26)–(29), (34) and (35), to solve for six unknowns , , , , , and , which, in turn, gives the surface fields and from (24) and

In the following examples, we assume a conductor with the S/m and a diconductivity of pure copper electric with a relative permittivity of 4.0. Figs. 2–4 illustrate the results of power absorption ratio between rough surface and smooth surface based on the 3-D formula of small perturbation

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Fig. 4. Power absorption ratio: surface with exponential correlation function (h = 1 m) with varying correlation length.

Fig. 6. 3-D versus 2-D: surface with exponential correlation function (h = 1 m, l = 2 m); SPM2: small perturbation method of second order.

Fig. 7. 3-D SPM2 versus T-matrix: surface with Gaussian correlation function (l = 3:0 m); SPM2: small perturbation method of second order. Fig. 5. 3-D versus 2-D: surface with Gaussian correlation function (h = 1 m, l = 2 m); SPM2: small perturbation method of second order.

method. In Fig. 2, the results are for a Gaussian correlation funcm. The correlation length varies tion with rms height from 2.0 to 3.5 m. We note that the absorption ratio increases with frequency. It also increases when the correlation length gets smaller. In Fig. 3, the results are repeated for the case of m. The absorption ratios are smaller than those of Fig. 2 because a smaller rms height gives a smoother surface. In Fig. 4, the results are illustrated for surfaces with exponential correlation functions exhibiting larger absorption than surfaces with Gaussian correlation function. The results of Figs. 2–4 show that the absorption ratios saturate for Gaussian correlation function because of the finite rough surface area. However, the absorption ratios do not saturate for exponential correlation function because the surface contains multiscale roughness since the spectral density of the exponential correlation function decays slowly with and . The results also demonstrate that the absorption depends on all three of the roughness characteristics: rms height, correlation length and correlation function.

B. 3-D and 2-D Comparison for Small Perturbation Method of Second Order In [16], we followed Morgan’s assumption by enforcing constant magnetic fields on a 2-D rough interface and applied small perturbation method only in the conductor region. The

Fig. 8. 3-D SPM2 versus T-matrix: surface with Gaussian correlation function (l = 3:5 m); SPM2: small perturbation method of second order.

power absorption ratio is obtained by the following closed-form formula:

(38) Figs. 5 and 6 compare the power absorption ratio using the 3-D and 2-D perturbation method of second order based on Morgan’s boundary condition. The absorption ratios are illustrated for surfaces with Gaussian correlation function and exm and m). The ponential correlation function ( results show more significant power absorption by rough surface with 3-D configuration than those with 2-D configuration.

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C. 3-D Small Perturbation Method and T-Matrix Method Comparison Figs. 7 and 8 compare 3-D analytic results based on the small perturbation method of second order with numerical T-matrix results. Note that the validity of the T-matrix method requires the slopes of the surface profile to be much smaller than unity. Therefore, the correlation length has to be much larger than the rms height . The modeled surface profiles are for the Gaussian m, m, correlation function with rms height m, m. The numerical and correlation length T-matrix results are in good agreement with the analytic 3-D results of small perturbation method for rough surfaces with small slope.

[15] A. E. Sanderson, “Effect of surface roughness on propagation of the TEM mode,” in Advances in Microwaves. New York: Academic, 1971, vol. 7, pp. 1–57. [16] L. Tsang, X. Gu, and H. Braunisch, “Effects of random rough surface on absorption by conductors at microwave frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 221–223, Apr. 2006. [17] X. Gu, L. Tsang, H. Braunisch, and P. Xu, “Modeling absorption of rough interface between dielectric and conductive medium,” Microw. Opt. Technol. Lett., vol. 49, pp. 7–13, Jan. 2007. [18] L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theory and Applications. New York: Wiley, 2000, vol. 1, ch. 9, pp. 389–407. [19] L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves: Advanced Topics. New York: Wiley, 2001, vol. 3, ch. 1, pp. 18–60. [20] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, 1995, ch. 8, pp. 430–433.

VI. CONCLUSION In this paper, we use a 3-D random surface model with correlation functions and spectral densities to characterize different roughness profiles. The key result is the closed-form 3-D formula of the power absorption enhancement factor based on the small perturbation method of second order, which can be used in conjunction with an interconnect model with a perfectly smooth conductor surface to quantify the impact of surface roughness on conductor loss. The spectral densities in the 3-D model can be further obtained from measured height profiles. We are currently investigating the extraction of surface spectral densities and correlation between loss measurements and the theoretical model.

Xiaoxiong Gu received the B.S. degree from Tsinghua University, Beijing, China, in 2000, the M.S. degree from the University of Missouri, Rolla, in 2002, and the Ph.D. degree from the University of Washington, Seattle, in 2006, all in electrical engineering. He is currently a Research Staff Member with the IBM T. J. Watson Research Center, Yorktown Heights, NY. His research interests include characterization of high-speed interconnect and microelectronic packaging, signal and power integrity, and computational electromagnetics.

REFERENCES [1] C. S. Chang and A. P. Agrawal, “Fine line thin dielectric circuit board characterization,” IEEE Trans. Comp., Packag., Manuf. Technol., vol. 18, no. 4, pp. 842–850, Dec. 1995. [2] G. Brist, S. Hall, S. Clouser, and T. Liang, “Non-classical conductor losses due to copper foil roughness and treatment,” in Proc. Electron. Circuits World Conv. 10, Anaheim, CA, Feb. 22–24, 2005, pp. S19-2–1. [3] H. Tanaka and F. Okada, “Precise measurements of dissipation factor in microwave printed circuit boards,” IEEE Trans. Instrum. Meas., vol. 38, no. 2, pp. 509–514, Apr. 1989. [4] S. P. Morgan, Jr., “Effects of surface roughness on eddy current losses at microwave frequencies,” J. Appl. Phys., vol. 20, pp. 352–362, Apr. 1949. [5] E. O. Hammerstad and F. Bekkadal, Microstrip Handbook. Trondheim, Norway: Univ. Trondheim, 1975, pp. 4–8. [6] C. L. Holloway and E. F. Kuester, “Power loss associated with conducting and superconducting rough interfaces,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1601–1610, Oct. 2000. [7] ——, “Impedance-type boundary conditions for a periodic interface between a dielectric and a highly conducting medium,” IEEE Trans. Antennas Propag., vol. 48, no. 10, pp. 1660–1672, Oct. 2000. [8] L. Proekt and A. C. Cangellaris, “Investigation of the impact of conductor surface roughness on interconnect frequency-dependent ohmic loss,” in Proc. Electron. Compon. Technol. Conf., New Orleans, LA, May 27–30, 2003, pp. 1004–1010. [9] Z. Wu and L. E. Davis, “Surface roughness effect on surface impedance of superconductors,” J. Appl. Phys., vol. 76, no. 6, pp. 3669–3672, Sep. 1994. [10] M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys., vol. 28, no. 12, pp. 1455–1463, Dec. 1957. [11] J. R. Wait, “Guiding of electromagnetic waves by uniformly rough surfaces—Part I,” IRE Trans. Antennas Propag., vol. 7, no. 12, pp. S154–S162, Dec. 1959. [12] X. Gu, “Modeling effects of random rough surface on conductor loss at microwave frequencies,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Washington, Seattle, WA, 2006. [13] L. Rayleigh, The Theory of Sound. New York: Macmillan, 1929. [14] S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math., vol. 4, pp. 351–378, 1951.

Leung Tsang (S’73–M’75–SM’85–F’90) received the S.B., S.M., and the Ph.D. degrees in electrical engineering and computer science from the Massachusetts Institute of Technology (MIT), Cambridge, in 1971, 1973, and 1976, respectively. He is currently a Professor and Associate Chairman of Education with the Electrical Engineering Department, University of Washington, Seattle, where he has taught since 1983. From 2001 to 2004, he was on leave from the University of Washington, during which time he was a Professor Chair and Assistant Head with the Department of Electronic Engineering, City University of Hong Kong. He coauthored Theory of Microwave Remote Sensing and Scattering of Electromagnetic Waves (Volumes 1–3) (Wiley, 2000, 2001, 2001). His current research interests include remote sensing and applications, signal integrity of interconnects, computational electromagnetics, and wireless communications. Dr. Tsang is a Fellow of the Optical Society of America. He is currently the president of the IEEE Geoscience and Remote Sensing Society. From 1996 and 2001, he was the editor-in-chief of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. He was the recipient of the 2000 IEEE Geoscience and Remote Sensing Society Outstanding Service Award. He was also a recipient of the 2000 IEEE Third Millennium Medal.

Henning Braunisch (S’99–M’01–SM’06) received the M.S. degree in electrical engineering from Michigan State University, East Lansing, in 1995, the M.S. degree in electrical engineering from the University of Hanover, Hanover, Germany, in 1996, and the Ph.D. degree in electrical engineering and computer science from the Massachusetts Institute of Technology (MIT), Cambridge, in 2001. He is currently with Components Research, Intel Corporation, Chandler, AZ. His research interests and expertise are in advanced microelectronic packaging, applied electromagnetics and acoustics, physics-based signal processing, and forward and inverse modeling. Dr. Braunisch is an alumnus of the German National Merit Foundation (1992–1996), the Department of State Foreign J. W. Fulbright Graduate Student Program (1994–1995), and the German Academic Exchange Service (1997–1998). He is a member of the Institute of Physics (MInstP), London, U.K.

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Modeling of 3-D Surface Roughness Effects With Application to -Coaxial Lines Milan V. Lukic´, Student Member, IEEE, and Dejan S. Filipovic, Member, IEEE

Abstract—Effects of 3-D surface roughness on the propagation constant of transverse electromagnetic transmission lines are calculated using finite-element method software by solving for the fields inside conductors. The modeling is validated by comparison with available literature results for the special case of 2-D surface roughness and by simulations using the finite-integration technique. Results for cubical, semiellipsoidal, and pyramidal indentations, as well as rectangular, semicircular, and triangular grooves in conductor surfaces are presented. A developed surface roughness model is applied to rectangular -coaxial lines. It is shown that roughness contributes up to 9.2% to their overall loss for frequencies below 40 GHz. Index Terms—Attenuation constant, conductor losses, finite-element method (FEM), surface roughness, transmission lines.

I. INTRODUCTION

T

Fig. 1. Scanning electron microscope (SEM) photograph of several -coaxial lines built using the PolyStrata process [10].

Manuscript received September 19, 2006; revised November 10, 2006. 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]). Digital Object Identifier 10.1109/TMTT.2007.891688

The assumptions that need to be satisfied for the validity of this method are that the height of the surface roughness is small compared with a wavelength and that the height scale of the roughness is small compared with its width scale. Several authors have recently studied periodic [5], [6], as well as random [7] 2-D roughness profiles using different methods. Holloway and Kuester [5] calculated the power loss associated with 2-D periodic conducting and superconducting rough interfaces using a generalized impedance boundary condition, which they derived by applying the method of homogenization. Matsushima and Nakata [6] utilized the equivalent source method to numerically study periodic rectangular, triangular, and semielliptical grooves both transverse and parallel to current flow. In [7], the effects of random 2-D transverse surface roughness are analyzed by using two methods: the analytic small perturbation method and the numerical method of moments. In this paper, effects of periodic 3-D surface roughness on a of TEM transmission lines complex propagation constant are studied using the commercial finite-element method (FEM) software High Frequency Structure Simulator (HFSS) [8] by solving for fields inside conductors [9]. The modeling approach is described, thoroughly validated, and used for the analysis of several 2-D and 3-D roughness profiles. It is also applied to calculate the excess loss due to the surface micromachined air-filled rectangular -coaxial lines [10]–[12], shown in Fig. 1. This paper is organized as follows. • Section II describes the proposed modeling approach and demonstrates its validity by comparison with analytical and

HE scattering of electromagnetic waves from rough surfaces is a research subject studied by numerous authors for over a century. A comprehensive review of the relevant research literature is available in [1]. The analysis of the interaction between electromagnetic waves and rough surfaces is typically performed either by analytical methods or by numerical simulations. In 1949, Morgan [2] used the finite-difference method to solve a quasi-static eddy-current problem for a 2-D periodic rough surface. He computed the ratio of the power loss dissipated in a conductor with a rough surface ( ) to that dissipated . This power in the same conductor with a smooth surface is equal to the ratio of the attenuation conloss ratio stant due to conductor loses of a TEM transmission line with to that for the same line with rough conductor surfaces . Both rectangular and equilatsmooth conductor surfaces eral triangular roughness profiles were analyzed. The case of the current flowing transverse to the grooves is fully treated, is given for the while only a high-frequency asymptote for case of the current flowing parallel to the grooves. Two decades later, Sobol [3] applied Morgan’s theory to microstrip lines and for equilateral triangular grooves transverse to calculated current flow as a function of the root-mean-square (rms) value for roughness. Sanderson [4] used the Rayleigh–Rice method to study the effect of periodic grooves transverse to the current flow on propagation of the TEM mode in a parallel-plate line.

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•

•

• •

•

available literature results for the special case of 2-D surface roughness. Section III presents studies for the effects of several 2-D roughness profiles, specifically periodic rectangular, semicircular, and triangular grooves both transverse and parallel to current flow. A formula for equilateral triangular grooves transverse to current flow is derived and compared to the commonly used formulas. Section IV presents results for the effects of various 3-D roughness shapes, specifically periodic cubical, semiellipsoidal, and pyramidal indentations in conductor surfaces. The FEM results are compared with those obtained using finite-integration technique within the CST Microwave Studio [13]. Section V gives a brief discussion of the effects of 3-D surface roughness on the phase constant. In Section VI, the developed model is applied to compute the excess loss due to the surface roughness of a micromachined recta-coax line. Measured data are also presented. Section VII summarizes the presented study.

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Fig. 2. Convergence of  versus normalized thickness of the meshed surface conductor region for square and circular coaxial lines and a parallel-plate line.

II. MODELING A. Modeling Approach To evaluate the effects of an arbitrary 3-D surface roughness shape on amplitude and phase of the traveling TEM wave, we utilized inherent capabilities of the FEM implemented within the HFSS. Specifically, metal is modeled as a dielectric so that the mesh is created inside conductors with rough surfaces. Consequently, a natural boundary condition, which is the continuity of tangential components of the electric and magnetic field, is enforced at the rough interface between the conductor and dielectric. A short section of a transmission line with length is simulated using the FEM, and its complex propagation constant is extracted from the -parameter using the equation . Herein, is computed with respect to the characteristic impedance of the line obtained from the 2-D eigensolution of the line’s cross section. It is well known that high-frequency fields penetrate only into the shallow surface region of a thick good conductor. To reduce the size of the computational domain, the regions of the conductor where the fields are negligible are treated as perfect electric conductor. Thus, the mesh is generated and the fields are calculated only in the thin surface regions of the conductors. To determine their thickness, several TEM transmission lines with smooth conductor surfaces are modeled. Fig. 2 shows simulated results for the attenuation constant at 26 GHz as a function of normalized thickness of the meshed surface conductor region for a square coaxial line, circular coaxial line, and parallel-plate line. The separation between the conductors is 75 m for all three lines. Also shown are anafor the circular coaxial and parallel plate lytical values for lines of 0.167 and 0.149 dB , respectively. The attenuation constant of 0.183 dB for the square coax is obtained by the Schwartz–Christoffel conformal mapping in conjunction with Wheeler’s incremental inductance rule [11]. The relative deviations of the simulated results with respect to the referenced results are also given. As seen, the conductor thickness of

Fig. 3. Comparison between the literature [5] and our FEM-based modeling approach for rectangular grooves transverse (H -wave) and parallel to current flow (E -wave). Additional validation curves can be found in [9].

amounts to approximately 10% relative deviation. On the other hand, the results converge to 99.5%, and 99.9% of reference and values when the conductor thickness is approximately , respectively. In the subsequent simulations, we have used -thick meshed conductor regions underneath the grooves and indentations. Notice that the convergence history for all three analyzed TEM lines is almost the same. In this study, the size of the computational domain was greatly reduced by using symmetry. Specifically, 1/8th of a square coax, a small wedge-shaped part of a circular coax, and a very narrow section of a parallel-plate line are simulated. The size of the computational domain for the parallel-plate line is significantly smaller than for the other two lines, particularly much smaller than for the square coax. For this reason and since the convergence histories of the lines are virtually indistinguishable, the parallel-plate lines with rough conductor surfaces are studied in the following sections. B. Validation The modeling approach is validated using Holloway and Kuester’s results [5] for rectangular grooves in conductor surfaces. The results for versus for the grooves transverse ( -wave) and parallel to current flow ( -wave) are shown in Fig. 3, and are in good agreement with the reference results. Note that Matsushima and Nakata [6] have also found their results to be in good agreement with those in [5]. For the case of grooves transverse to current flow, it is observed

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Fig. 4. One period of 2-D surface roughness profiles. (a) Rectangular. (b) Semicircular. (c) Triangular. The profiles are symmetric with respect to the dashed line.

that the ratio approaches an asymptotic value as gets very large, i.e., at very high frequencies. Low-frequency currents flow below the grooves, thus the roughness effects are insignificant. However, at high frequencies, the current and fields in the conductor are forced near the surface, and a larger fraction of the total current is affected by the roughness [5]. The interested reader is referred to [6, Fig. 3], which shows contour lines of field intensity inside a conductor for grooves transverse and parallel to current flow. For transverse grooves, the high-frequency currents are forced to follow the path of the asymptote roughness profile, thus an estimate for the can be calculated as the ratio of the path length on the rough surface to that of the smooth surface. For example, for the two -wave curves in Fig. 3, the estimated asymptotes, computed , are 2 and 3. as

Fig. 5. Relative change of the attenuation constant  versus normalized rms value for roughness for equilateral triangular grooves transverse to current flow. The solid line, denoted by f , represents the FEM results obtained with our modeling approach.

III. 2-D ROUGHNESS A. 2-D Profiles The modeling approach explained in Section II is applied here to study several 2-D surface roughness profiles. Fig. 4 shows one period of the profiles with geometrical parameters ( and ) for rectangular, semicircular, and triangular grooves. For can be any roughness profile, the rms value for roughness expressed in terms of its dimensional parameters. For example, for an equilateral triangular profile with , and for a rectangular profile. A detailed description of calculations of can be found in [2]. The results for the grooves transverse and parallel to current flow are presented in Sections III-B and C. B. Grooves Transverse to Current Flow Increased conductor losses due to surface roughness are usually computed by [14] (1) is the normalized rms roughness. This formula was where obtained by curve-fitting measured data sets obtained with microstrip lines. It also agrees very well with the theoretical predictions in [3], derived for the equilateral triangular grooves transverse to the current flow. Expression (1) is often implemented for other transmission lines and roughness shapes. For example, in [15], it is used for a micromachined recta-coax, while its validity for any transmission line is argued in [16]. Roughness effects on the line losses are also computed by [17]

(2)

Fig. 6. Relative change of the attenuation constant  versus normalized roughness period for rectangular, semicircular, and triangular grooves in conductor surfaces transverse to current flow: h = g=2.

obtained from the measurements of the quality factor of the cavity resonators. By applying the modeling approach from Section II on the same roughness profiles for which (1) is derived, somewhat different results are obtained. Specifically, (3) fits the computed results, shown in Fig. 5, to within 1.5% as follows:

(3) Expressions for given by (1) and (2) are also plotted in Fig. 5. As seen, our results agree with (1) to within 1.6% for smaller than one skin depth, and within 4.5% for larger ’s. Although somewhat more complicated than (1) and (2), the derived formula (3) can be easily implemented in any software formulated to account for a finite conductivity of the metals. Note that all three formulas have the small skin depth limit of 2 equal to the ratio of the path length on the rough surface to that of the smooth surface. as a function of Fig. 6 shows simulation results for normalized roughness period for transverse rectangular, semi. For circular, and triangular grooves of Fig. 4 with , the high-frequency asymptotes for calculated as the ratio of the path lengths for rectangular, semicircular, and triangular profiles are approximately 1.8, 1.46, and 1.33, re, these asymptotes are approximately spectively. For

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Fig. 8. Unit cells of periodic: (a) cubical, (b) semiellipsoidal, and (c) pyramidal indentations in a conductor surface. The depth for all three shapes is denoted by h and, for clarity, it is shown only for cubical indentations. Fig. 7. Relative change of the attenuation constant  versus normalized roughness period for rectangular, semicircular, and triangular grooves in conductor surfaces parallel to current flow: h = g=2.

1.75, 1.43, and 1.31. Notice that in these studies, the normaland are varied simultaneously since ized parameters , and that the parameter is equal to 0.25 equal to 0.75 and 0.8, respectively. The effect of and 0.2 for on for transverse rectangular grooves the parameter is given in [5, Fig. 7]. We have for a constant value of obtained the same results; however, here we only provide some relevant observations. First, the high-frequency asymptote for for this case is , as explained above, and thus, . However, at low frequencies, the it does not depend on is much larger for larger values of . This could ratio be intuitively expected because wider metallic walls, of width , forming the transverse grooves, represent a larger obstacle for the current flowing underneath them. Namely, the current gets more diverted into wider walls traveling longer paths, and thus, creating more additional loss. It can be seen that, in Fig. 6, equal to 0.75 and 0.8 intersect for rectanthe curves for has smaller gular and semicircular grooves. Specifically, corresponding to much smaller values for larger values of and slightly larger . For rectangular grooves, values of this can be expected based on the above observations. Namely, causes an increase in , as seen in the slightly larger Fig. 3, but the decrease of due to much smaller is more dominant. It is interesting that a similar behavior is observed for semicircular grooves, but not for triangular grooves. at high frequencies is largest for It is also observed that rectangular and smallest for triangular grooves, while the situation is exactly the opposite at low frequencies. C. Grooves Parallel to Current Flow as a function of the normalized Simulation results for roughness period for parallel rectangular, semicircular, and triequal to angular grooves of Fig. 4 are shown in Fig. 7 for equal to 0.25 and 0.2. As for the trans0.75 and 0.8, i.e., verse grooves, low-frequency current flows beneath the grooves and only a small percent of the total current “sees” the roughness, resulting in a small additional power loss above that of a smooth conductor [5]. However, as the frequency increases, the current gets more diverted into the metallic walls, of width , forming the parallel grooves. A drawing of contour lines of field intensity inside a conductor that illustrates this can be found in [6, Fig. 3(b) and (d)]. The effect of the parameter on for parallel rectangular grooves for a constant value of

Fig. 9. Comparison between the FEM and finite-integration technique results for relative change of the attenuation constant  for cubical, semispherical, and pyramidal indentations in conductor surfaces: g=p = 0:75, h = g=2.

is given in [5, Fig. 10], which we have also demonstrated in [9, Fig. (2b)]. An important observation from that at all frestudy was that narrower walls cause larger quencies. This could be intuitively expected, especially at very high frequencies at which the current emerges to the uppermost part of roughness walls while the conductor surface right underneath the grooves gets depleted of current. It is then also exis largest for rectangular and pected that the effect on smallest for triangular grooves, as seen in Fig. 7. This is so because the width of the walls for triangular grooves increases linearly from to as the depth increases from 0 to , while for rectangular grooves it remains constant. IV. 3-D ROUGHNESS The effect of periodic cubical, semiellipsoidal, and pyramidal indentations in the conductor surface on the line attenuation constant is studied here. A. 3-D Profiles The geometrical parameters of studied 3-D roughness shapes, as well as the reference coordinate system, are shown in Fig. 8. The indentations are periodic in the - and -direction with periodicity . Their depth is denoted by for all three shapes, though this dimension is only depicted in Fig. 8(a) for clarity. An excellent agreement between the FEM and finite-integration technique results is demonstrated in Fig. 9. It is noticed that for the same values of the roughness period and the parameters and , the effect on is largest for cubical and smallest for pyramidal indentations.

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Fig. 10. Relative change of the attenuation constant  for cubical indentations in conductor surfaces: g g g.

=

=

Fig. 11. Relative change of the attenuation constant  for cubical indentations h. with varying width g and g

=

Fig. 12. Relative change of the attenuation constant  for cubical indentations g . with varying length g for two values of g and h

=

B. Cubical Indentations The results for cubical indentations are plotted in Figs. 10–12. In the study presented in Fig. 10, the base of the cube is square and two values of filling factor , i.e., 0.75 and 0.9, are considered. Four heights, i.e., and , are taken for each value. If the current flows in the -direction in Fig. 8, then the metallic walls parallel to the – -plane represent the obstacle like the walls of transverse grooves studied above, and the walls parallel to the – -plane are like those of parallel grooves. It is then expected that the relative change of the attenuation constant is larger for larger , as seen in Fig. 10. It values of the normalized dimension

is also observed that the small skin-depth asymptote for is predominantly determined by the value of and less by . On the contrary, for large skin depths, i.e., at lower frequencies, is primarily dependent on and . Note that the widths of walls is virtually independent on parallel to the – - and – -plane are equal and varied simultaneously. According to the above discussed studies of the effects at low freof the transverse and parallel wall widths on results in the increase of the conquencies, an increase of in the transverse walls and its decrease in tribution to the parallel walls. The two effects compensate and, as a result, is almost independent of at low frequencies. Howis not so important ever, at high frequencies, the parameter is for transverse grooves as it is for parallel grooves and , as seen in Fig. 10. larger for larger The effects of varying either width or length of the cubical indentations when the other dimensions are fixed are demonstrated in the next two studies. The - and -field are polarized in the - and -direction, respectively, thus the TEM wave propagates in the negative -direction. The results for for varying (fixed and ) and varying (fixed and ) cases are shown in Figs. 11 and 12, respectively. Notice that, for the former case, the indentations merge into rectangular grooves transverse to the current flow . Similarly, for the later case, the indentations when merge into rectangular grooves parallel to the current flow when . Dashed lines in Figs. 11 and 12 are the results for corresponding 2-D roughness profiles. , is virtually As seen in Fig. 11, for for small . Note that the high-frequency unaffected by , is estiasymptote of 2.2, which is calculated as mated for for transverse grooves of normalized height . Eventually, for very close to unity (see curve ), the results for start to approach for these for the corresponding transverse grooves. Notice that, at , for transverse grooves low frequencies, when is significantly lower than for cubical indentations for less than 0.95. Only for very close to unity do low-frefor cubical indentations become as quency values of small as for corresponding transverse grooves. For very close to unity, the cubical indentations are almost merged into rectangular grooves, except for thin metallic membranes of periodically located across the grooves. The thickness above results then imply that the effect of such thin membranes is very significant at all frequencies. on The results of a varying case are given in Fig. 12 for of 0.6 and 0.8. It is observed that the convergence toward the results for the corresponding 2-D roughness profile is much faster than for the above case of varying . This implies that, for this orientation of the grooves, the effect of the thin metallic membranes periodically located across the grooves is small, especially at low frequencies. C. Semiellipsoidal and Pyramidal Indentations Effects of the height of semiellipsoidal and pyramidal indenfor 100% filling factor are shown tations on in Figs. 13 and 14, respectively. It is observed that for the same is much higher for the former case. Roughly, height,

´ AND FILIPOVIC: MODELING OF 3-D SURFACE ROUGHNESS EFFECTS WITH APPLICATION TO -COAXIAL LINES LUKIC

Fig. 13. Relative change of the attenuation constant  for semiellipsoidal indentations in conductor surfaces for different heights of indentations.

Fig. 14. Relative change of the attenuation constant  for pyramidal indentations in conductor surfaces for different heights of indentations.

about a twice-larger height is needed for pyramidal indentations . It is observed that the simfor about the same value of ulated curves for the semiellipsoidal indentations were not as smooth as for other results here. This is attributed to the small geometrical errors occurring because tetrahedras are used curves can in meshing of elliptical curvatures. Smoother be obtained if special care is taken in creating the mesh. Actual are shown via dots simulation results for the case in Fig. 13 and are replaced with the solid line denoted by 0.8, cases. likewise for the other D. Discussion Results presented here clearly demonstrate that the 3-D roughness can be accurately modeled with either the FEM or finite-integration technique. It is also shown that the level of increase in conductor losses is dependent on the shape and distribution of indentations. The simulations here have typically used approximately 3 GB of random access memory. It took on average 1 h of running time on a 2.8-GHz Intel Pentium D processor for each of the curves in Figs. 9–14. The memory requirements for longer lines with realistic 3-D rough profiles impose severe modeling restrictions. Nevertheless, the studies and plots presented here can be used very effectively in estimating the additional losses due to the similar imperfections in the conductor surfaces.

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Fig. 15. Comparison between the FEM and finite-integration technique results for the relative change of the phase constant for cubical, semispherical, and : , h g= . pyramidal indentations in conductor surfaces: g=p

= 0 75 = 2

Fig. 16. Comparison between the measured and simulated jS j for a 1-cm-long recta-coax line. denotes the rms value for roughness of the recta-coax conductor surfaces, as shown in Fig. 17. The circle represents the line loss at 25 GHz extracted from a half-wavelength transmission line resonator measurement.

1

V. PHASE CONSTANT Simulation results for the ratio of the phase constant of the to that for the same line line with rough conductor surfaces as a function of the norwith smooth conductor surfaces malized period for cubical, semispherical, and pyramidal indentations are shown in Fig. 15. Note that, for this study, the separation between the conductors of a parallel-plate line was 50 m. As seen, FEM and finite-integration technique results agree very well. It is also noticed that the effect on is largest for cubical and smallest for pyramidal indentations. Presented results show the small effect roughness will have on the line electrical length, nevertheless the utility of the modeling approach for determining this parameter (if needed) is clearly demonstrated. VI. MEASUREMENTS Here, the developed roughness model is used to compute the excess loss of -rectangular coaxial lines shown in Fig. 1. The measured result for the averaged insertion loss of several recta-coax lines is given in Fig. 16 along with relevant simulations. The measurement setup includes an HP-8510C network analyzer with Cascade Microtech 150- m-pitch coplanar waveguide (CPW) microwave probes and a Cascade Summit 9000 probe station. An external short-open-load-thru calibration implementing CPW on an alumina substrate is used for the two-port calibration. Effects of the launch sections, whose design can be found in [18], are deembedded using the results

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TABLE I VALUES OF  = FOR INDIVIDUAL CONDUCTOR SURFACES OF THE RECTA-COAX LINE SHOWN IN FIG. 17, AND CONTRIBUTION OF THE SURFACE ROUGHNESS LOSSES AND DIELECTRIC LOSSES TO THE TOTAL LINE LOSS

=

Fig. 17. (a) Cross section of realized recta-coax line. (b) Cross section of the line as simulated by FEM with values for rms roughness for individual conductor surfaces. (c) 3-D view of the simulated section of the recta-coax.

from the back-to-back launches. Also given in Fig. 16 is the line loss at 25 GHz, as extracted from the -coaxial transmission line resonator measurements. Shown in Fig. 17(a)–(c) is a photograph of a cross section of the fabricated line, the cross section of the line with periodic dielectric supports for the inner conductor as modeled with FEM, and a 3-D view of the simulated section of the recta-coax. Inner and outer conductor sizes are 100 100 m and 250 250 m , respectively. Dielectric supports embedded in the sidewalls are 18- m thick and 100- m wide. They span the distance of 250 m between the sidewalls of the line and are periodically placed along the line with periodicity of 700 m. Their dielectric constant and are 2.85 and 0.045, respectively. The release holes are needed to enable the removal of sacrificial photoresist, and their designed dimensions in the sidewalls and top walls are 75 200 and 100 200 m , respectively. The line from Fig. 17 is first simulated assuming that all conductor surfaces are perfectly smooth. This is denoted by in Fig. 16. Note that Schwartz–Christoffel conformal mapping in conjunction with Wheeler’s incremental inductance rule [11] can alternatively be used for calculation of the losses for the line with smooth conductors. However, the incremental inductance rule cannot be used to estimate the additional losses due to surface roughness. These losses are taken into account, via FEM, in the second simulation of Fig. 16, and the obtained curve for the magnitude of the -parameter is denoted by . The loss contribution due to roughness is also plotted and, as expected, it monotonically increases with frequency, rising to 9.2% of the total loss at 40 GHz. Since the bandwidth for a single TEM mode operation of this recta-coax is approximately 467 GHz [11], the

roughness effects would become more pronounced at frequencies beyond the measured frequency range. In the above simulation, the bottom side of the outer conductor and the top side of the inner conductor are perfectly smooth. These conditions arise from the ability of the fabrication process to polish these horizontal copper surfaces to the optically fine smoothness. RMS roughness values of the remaining conductor surfaces shown in Fig. 17(b), as provided by the Mayo Foundation, are used in the following way. Owing to the inherent features of the FEM, the effects of surface roughness are modeled through a modified conductivity of the individual conductor surfaces. Specifically, the conduc, where tivity of a particular rough copper wall is S m is the bulk conductivity of the smooth conis determined from ductor. The parameter the measured rms roughness values and (3) obtained by the presented method. Its values for several frequencies within the measured frequency range are given in Table I. The skin depth at 40 GHz is approximately 0.33 m; hence, below this frequency. Since (3) and (1) agree within 1.6% for , (1) would give very similar values for these values of the parameter . Also shown in Table I are the fractions of the total loss due to the roughness and dielectric loss. The effect of the release holes is also studied and it is found that their loss contribution is marginal. These results show that, up to 40 GHz, the impact of the roughness on the overall losses of -coaxial lines is below 10%. However, these effects should be taken into account at higher frequencies when an accurate determination of the line losses is needed. Note that the direct solving for the fields inside the rough conductors for the realistic recta-coax line, shown in Fig. 17, is impractical. This is due to the lack of symmetries and associated excessive memory requirements needed to accurately mesh the interior of the metal walls. However, as demonstrated here, the roughness effects determined based on the simulations of a small section of a parallel-plate line, by the method described in Section II, can be readily applied. VII. CONCLUSION In this paper, the FEM has been used to calculate the effect of 3-D surface roughness on a propagation constant of TEM transmission lines. The surface region of conductors has been modeled as a dielectric so that the mesh is created inside the conductor volume of a rough profile. Consequently, a natural boundary condition enforced at the rough interface between the conductor and dielectric has insured its proper modeling. Modeling has been validated by comparison with available literature results for the special case of 2-D surface roughness profiles

´ AND FILIPOVIC: MODELING OF 3-D SURFACE ROUGHNESS EFFECTS WITH APPLICATION TO -COAXIAL LINES LUKIC

and by simulations of 3-D roughness shapes using the finite-integration technique. The results for periodic cubical, semiellipsoidal, and pyramidal indentations, as well as rectangular, semicircular, and triangular grooves in conductor surfaces have been presented. The developed roughness model has been applied to compute excess losses of the surface micromachined rectangular -coaxial line. ACKNOWLEDGMENT The authors would like to thank Prof. Z. Popovic´ , Prof. E. Kuester, Dr. S. Rondineau, and K. Vanhille, all with the University of Colorado at Boulder, G. Potvin and D. Fontaine, both with BAE Systems, Nashua, NH, Dr. C. Nichols and Dr. D. Sherrer, both with the Rohm and Haas Company, Blacksburg, VA, Dr. W. Wilkins and Dr. V. Sokolov, both with the Mayo Foundation, Rochester, MN, Dr. J. Evans, Defense Advanced Research Projects Agency–Microsystems Technology Office (DARPA–MTO), Arlington, VA, and E. Adler, Army Research Laboratory (ARL), Adelphi, MD, for useful discussions and support. The authors are also thankful to B. Brim, Ansoft Corporation, Boulder, CO, for many useful discussions. REFERENCES [1] K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media, vol. 11, pp. R1–R30, 2001. [2] S. P. Morgan, “Effect of surface roughness on eddy current losses at microwave frequencies,” J. Appl. Phys., vol. 20, pp. 352–362, 1949. [3] H. Sobol, “Application of integrated circuit technology to microwave frequencies,” Proc. IEEE, vol. 59, no. 8, pp. 1200–1211, Aug. 1971. [4] A. E. Sanderson, “Effect of surface roughness on propagation of the TEM mode,” in Advances in Microwaves. New York: Academic, 1971, vol. 7, pp. 1–57. [5] C. L. Holloway and E. F. Kuester, “Power loss associated with conducting and superconducting rough interfaces,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1601–1610, Oct. 2000. [6] A. Matsushima and K. Nakata, “Power loss and local surface impedance associated with conducting rough interfaces,” Elect. Commun. Jpn., vol. 89, no. 1, pt. 2, pp. 1–10, Jan. 2006. [7] L. Tsang, X. Gu, and H. Braunisch, “Effects of random rough surface on absorption by conductors at microwave frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 221–223, Apr. 2006. [8] “HFSS v10.0 User Manual,” Ansoft Corporation, Pittsburgh, PA, 2005. [9] M. Lukic and D. S. Filipovic, “Modeling of surface roughness effects on the performance of rectangular -coaxial lines,” in 22nd Annu. Progr. Electromagn., Miami, FL, Mar. 2006, pp. 620–625. [10] D. W. Sherrer and J. J. Fisher, “Coaxial waveguide microstructures and methods of formation thereof,” U.S. Patent Applicat. US 2004/ 026390A1, Dec. 30, 2004.

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[11] 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 2006. [12] 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. [13] “CST Microwave Studio User Manual, Version 2006.0.0,” CST GmbH, Darmstadt, Germany, 2006. [14] E. Hammerstad and O. Jensen, “Accurate models for microstrip computer-aided design,” in IEEE MTT-S Int. Microw. Symp. Dig., Washington, DC, May 1980, pp. 407–409. [15] J. R. Reid, E. D. Marsh, and R. T. Webster, “Micromachined rectangular-coaxial transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3433–3442, Aug. 2006. [16] D. M. Pozar, “Transmission line theory,” in Microwave Engineering. Hoboken, NJ: Wiley, 2005, ch. 2, p. 86. [17] S. Groiss, I. Bardi, O. Biro, K. Preis, and K. Richter, “Parameters of lossy cavity resonators calculated by the finite element method,” IEEE Trans. Magn., vol. 32, no. 3, pp. 894–897, May 1996. [18] 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. Milan V. Lukic´ (S’02) received the Dipl. Eng. degree in electrical engineering from the University of Banjaluka, Banjaluka, Bosnia and Herzegovina, in 1998, the M.S.E.E. degree from the University of Mississippi, University, in 2002, and is currently working toward the Ph.D. degree at the University of Colorado at Boulder. His research interests include multilayered rectangular waveguide dyadic Green’s functions, mode matching, conformal mapping (CM), transmission lines, and antennas. Mr. Lukic´ was the recipient of the 2002 Graduate Achievement Award presented by the University of Mississippi, and the 1998 Gold Medal presented by the University of Banjaluka.

Dejan S. Filipovic (S’99–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 an Assistant Professor with the University of Colorado at Boulder. His research interests are in the development of millimeter-wave components and systems, multiphysics modeling, antenna theory and design, as well as in computational and applied electromagnetics.

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Equivalent SPICE Circuits With Guaranteed Passivity From Nonpassive Models Adam Lamecki and Michal Mrozowski, Senior Member, IEEE

Abstract—In this paper, a new fast technique of passivity enforcement of a nonpassive rational model is introduced. The technique disturbs poles of the model to restore passivity in such way that the frequency response of a device being modeled is preserved. The passivity enforcement procedure is defined as an optimization routine with the gradients of the cost function evaluated using the theory of matrix perturbation. The rational model can be based either on passive (electromagnetic simulations, measurements) or nonpassive (surrogate models) data. In the second case, the proposed technique can lead to a parameterized SPICE networks. Some advanced examples are given to show the application of proposed approach in interconnect, packaging, and signal integrity analysis. Index Terms—Equivalent circuits, passivity enforcement, rational models, SPICE networks, strict passivity, surrogate models.

I. INTRODUCTION SPICE circuit simulator has became an industry standard for design of analog, digital, microwave, and mixed analog–digital circuits. The presence of active elements and mixed digital-analog parts in modern systems implies that circuit analysis is carried out in the time domain. The SPICE simulator allows one to simulate circuits described by a lumped elements (like resistors, capacitors, inductors, diodes, and transistors), which is enough for analysis of low-frequency circuits, but it is not enough when the operation frequency increases, which is the case of microwave and millimeter-wave devices. As the frequency increases, the simple lumped models are not sufficient and more complex models are required that take into account parasitic effects like signal delay, distortion, reflections, ringing, and crosstalk or frequency-dependent losses of passive elements and parasitic radiation. Such parasitic effects can be predicted using electromagnetic (EM) simulators, which operate mainly in the frequency domain. As a result, a tool is required that can connect two separate environments: full-wave simulators that generate frequency tabulated data and lumped-circuit simulators that operate in the

A

Manuscript received June 15, 2006; revised September 24, 2006. This work was supported by the Polish State Committee for Scientific Research under Contract 3 T11D 024 30, by the Foundation for Polish Science under the Domestic Grants for Young Scientists Scheme, and under the Polish-Flanders Scientific Bilateral Programme. The authors are with the Department of Electronics, Telecommunication and Informatics, Gdan´sk University of Technology, Gdan´sk, Poland (e-mail: adam. [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.2006.890520

time domain. Several solutions to this problem were reported. To generate a passive macromodel from frequency tabulated data, one can use a model order-reduction techniques that use Krylov-subspace methods, as in [1]–[3]; however, these techniques can preserve passivity only if input data is passive. One can introduce constraints that help to ensure the passivity directly when rational representation of admittance parameters is created [4]; however, these constraints are not sufficient to ensure passivity for all frequencies. Another approach proposed in [5] is derived from the pole-residue form of rational functions and can generate a passive macromodel over infinite frequency, but the enforcement problems might appear when real poles occur. A different solution, presented in [6], enforces the passivity condition at discrete frequencies; therefore, the resulting macromodel can be nonpassive at frequencies that were not taken into account during passivity enforcement. Guaranteed passivity can be achieved by applying either the method based on convex optimization [7] or an iterative procedure that enforces passivity while minimizing the distortion of the time-domain response [8], [9]. However, in several practical applications (like filter or other resonant circuits), one may be interested in controlling the accuracy of the frequency response in a specific frequency band. The convex optimization approach can only handle problems with a low number of states [7]. In both techniques, residues are perturbed, which implies that for multiport circuits, the number of variables to be adjusted is large. Additionally, for strong violation of passivity, the method of [8] may lead to significant distortion of the frequency response of the circuit, as shown in [10]. In this paper, we propose a new technique, which has the following three important features: • controls the accuracy of frequency response while enforcing the passivity (as in [7]); • can enforce the passivity even if passivity of input data is strongly disturbed; • is easy to implement. The proposed approach gives a passive model as a result of the optimization procedure, as in [7], but uses different passivity conditions and an optimization approach. What is even more important is that the technique can handle problems with a much higher number of states. The proposed technique can be utilized to construct a passive SPICE equivalent circuit from a nonpassive rational model based on either passive or nonpassive frequency tabulated data. The second case seems especially interesting due to the possibility of construction of passive SPICE networks that are derived from the response of nonpassive surrogate models [12]–[14]. Combining the proposed technique with a surrogate model leads to a parameterized SPICE network. The

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LAMECKI AND MROZOWSKI: EQUIVALENT SPICE CIRCUITS WITH GUARANTEED PASSIVITY FROM NONPASSIVE MODELS

time of computing of a surrogate model response is insignificant compared to the time required by the EM solver. The proposed technique of passivity enforcement also converges fast. As a result, the total time of creation of the SPICE circuit that has the accuracy of EM solver is a few orders of magnitude shorter than EM simulation of the circuit response at a single frequency. The resulting SPICE networks with guaranteed passivity can be very useful in many applications such as analysis and tuning of RFIC devices at the post-layout stage of design.

II. TECHNIQUE DETAILS In general, a response of a linear time-invariant circuit can be represented in the form of a rational function. To find a rational model, one can apply either a vector fitting technique [15]–[17] or direct interpolation scheme described in [18]. As a result, one obtains the rational model in the form (1) where is the residue of the scattering matrix element connected with the th pole and is the function order. All the elements of the scattering matrix can be fitted with the same set of common poles (real and complex). A rational representation obtained as a result of approximation schemes can ensure stability and high accuracy of the model in a wide frequency band. However, they cannot ensure passivity, which is essential in time-domain analysis. A. State–Space Model Construction At first, the rational representation is converted to a time domain state–space model (macromodel) [19] (2) (3) where is the state matrix, relates the input variables to state variables, relates the state variables to output variables, relates the inputs directly to the outputs, is the state vector, is the input vector, and is the output vector. Both time- and frequency-domain models are connected by a Laplace transform. Applying the transform to rational representation (1), one can realize the time-domain model [19]. B. Passivity Enforcement The passivity enforcement procedure can be divided into the following three major elements: • identification of the frequency bands where the model is not passive; • correction of the model parameters (poles and/or residues) in the optimization loop; • implementation of the constraints that minimize disturbance of the frequency response.

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1) Passivity Test: Matrices , , , and of a state–space model can be used to perform the passivity test of the model [8], [20]. Let us consider a Hamiltonian matrix

(4) where and . As presented in [8], the state–space model is guaranteed to be passive only if matrix has no imaginary eigenvalues. It is also shown that is the eigenvalue of and a maximum singular value if of scattering matrix crosses value 1 at , then is a frequency that denotes the crossover from a nonpassive frequency band to a passive one. Using this criterion, one can detect if the model is passive and detect the frequency bands in which the passivity condition is violated. Such a passivity test is much more useful than a standard procedure of testing if the scattering matrix is bounded real in the frequency domain [8]. The eigenvalue problem has to be solved only once and the solution provides complete information about passivity of the model in the entire frequency band. Using the information, one can define the optimization procedure that restores the passivity of the model. 2) Model Correction: A possible way to enforce model passivity at all frequencies is to perturb the state–space model to make the imaginary eigenvalues of complex, simultaneously minimizing the deformation of the model response. The solution of such a correction, based on perturbation of model residues and minimizing the distortion of the time-domain response, was proposed in [8]. An alternative is to optimize the location of common poles and/or residues and preserve the frequency response of the model. It should be noted that the optimization relying only on poles significantly reduces the number of varivariables comparing to in case ables—it needs only of the residues. In fact, if the complex pole representation is , the disturbance of both the real and imaginary parts causes both a shift of the pole on the frequency plane and introduces some additional loss to the circuit that helps to preserve the passivity. For this reason, we have decided to base our passivity enforcement procedure on the perturbation of poles. Let us introduce vector

(5)

where element is the width of the th frequency band of passivity violation. The procedure of passivity restoration can be organized as a min–max optimization problem with the location of common poles as variables. The goal of the optimization is to . In particular, if condition minimize the maximum value of is fulfilled for , the model is guaranteed to be passive for every . Such a definition of an optimization goal gives a passive state–space model, but the response of the obtained model can differ from the original one. To assure preservation of the response, some additional conditions are applied.

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Fig. 1. Electric network that realize k th input/output port and j th state of state–space model.

3) Preserving the Frequency Response: Let be and the -port scattering matrix computed at frequency be the scattering matrix obtained from state–space model. To minimize the distortion of the frequency-domain response due to passivity enforcement associated with the perturbation of the poles, an additional condition is imposed as follows: (6) at a set of discrete frequency points . Parameter is defined as an acceptable tolerance between the reference data and state–space model response and allows one to control the accuracy of the model after the passivity enforcement. The condition presented above enforces the absolute error for the created passive model to be less than . On the other hand, sometimes one can be interested to enforce the relative error of created model be less than . In this case, the condition should be modified to (7) Both conditions are implemented as nonlinear inequality constraints of min–max optimization. Additionally, the accuracy does not have to be the same for every . One can set a different value of at each frequency , setting the different response accuracy over different frequency bands. 4) Additional Considerations: In this paper, the optimization is performed as a min–max routine. However, it has to be noted that another optimization algorithm can be applied to minimize provided it can handle nonlinear constraints. the values of As stated above, the procedure successfully concludes if all of are set to zero. The proposed procedure is the values of an optimization one and it is not guaranteed to converge with arbitrary accuracy . Therefore, in case the optimization does not converge to a passive model, it is recommended to relax the accuracy parameter in (6) or (7).

C. Fast Gradients With Matrix Perturbation Theory The proposed approach of passivity enforcement allows one to compute an analytic gradient of the goal function. Let us analyze a perturbed unsymmetrical eigenvalue problem [21] (8) is a perturbation matrix, is a right eigenvector of where is an eigenvalue of the perturbed the perturbed matrix, and matrix. Assume that is a simple eigenvalue of and ( ) is the right (left) eigenvector corresponding to . Assuming the is small, one can obtain the first-order perturbation matrix of the perturbed matrix as approximation of eigenvalue (9) The above formula can be utilized for fast computation of the gradient of the goal function. To obtain full gradient information, one needs to compute a set of pure imaginary eigenvalues and corresponding right eigenvector only once. In the case of optimization relying only on poles (not residues) of the model, the perturbation of pole location influences only matrix , therefore, (10) Finally, the sensitivity of the eigenvalue of pole can be computed as

on the perturbation

(11) where is a matrix of the size of perturbation matrix . elements

with

III. SPICE NETWORK CONSTRUCTION Once the passive state–space model of a device has been constructed, it can be realized as an electric circuit using resistors,

LAMECKI AND MROZOWSKI: EQUIVALENT SPICE CIRCUITS WITH GUARANTEED PASSIVITY FROM NONPASSIVE MODELS

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Fig. 3. Cross section of the BGA package as simulated in Agilent’s ADS Momentum.

Fig. 2. Top view of simulated 96-pin BGA package.

capacitors, and voltage/current-controlled sources [19]. In general, for an -port device, the network have a form shown in state subcircuits. Fig. 1. It consists of input/output and Additional input/output circuits are added using the relations between incident and reflected waves and voltage and curat the th port rent amplitudes (12) (13) where

is the characteristic impedance of the th port. IV. APPLICATIONS

Fig. 4. Maximum singular value of scattering matrix of BGA package before (1 1 1) and after (—) passivity enforcement.

A. Packaging Application The passivity enforcement technique described above can be applied to create an equivalent circuit directly from a device response, which is obtained from an EM simulator or measurements. Fig. 2 shows a top view of a ball grid array (BGA) package with 96 pins. The package was simulated with Agilent’s Momentum RF EM simulator over a frequency range from dc up to 10 GHz. The package structure and localization of the package ports is presented in Fig. 3. A rational model of the response of 1/4 the device (one side of the package, 48 ports) was subsequently created. The order and the state–space of the rational model was equal model had 288 states. This is by far a larger number of states than the method using the convex optimization approach can handle [7]. Despite that the input data were passive, the resulting rational model was not, as shown in Fig. 4. To create the passive equivalent circuit of the package, the proposed technique was utilized. The total time of optimization was 5 min using a 1.5-GHz PC. The poles were perturbed and, consequently, the number of optimized variables was six, compared to 13 824 in the case when one perturbs residues. Fig. 5 shows very good agreement between selected scattering parameters for the created passive model and input data.

Fig. 5. Comparison of selected scattering parameters of BGA package before (1 1 1) and after (—) passivity enforcement.

B. Resonant Circuit—Microwave Filter As mentioned in Section I, the proposed technique allows one to create a passive SPICE network even if the input data are not

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Fig. 6. Input nonpassive data (1 1 1) and results of passivity enforcement with proposed technique (—) and presented in [8] (1 — 1).

Fig. 8. Input nonpassive data (1 1 1) and results of passivity enforcement with proposed technique (—) of RAMBUS FlexIO channel.

C. Measurements Application

Fig. 7. Maximum singular value of scattering matrix of microwave filter before (1 1 1) and after (—) passivity enforcement.

passive. To illustrate this, the proposed technique is applied to a set of nonpassive scattering parameters of a microwave hairpin filter. The data are strongly nonpassive, the maximum singular value of the scattering matrix in-band of the filter passband reaches 1.06. The same data were used to create a passive network using the technique presented in [8]. The results are shown in Fig. 6. In Fig. 7, a maximum singular value of the scattering matrix before and after passivity enforcement is presented. With the proposed approach, one gets a passive network with the response very close to the original nonpassive data. The alternative technique gives a model with higher error, which proves that the condition for minimization of time-domain response distortion may not be optimal for nonpassive input data. Additionally, the filter is a resonant circuit, which is very sensitive to the location of poles. The time of passivity enforcement was 2.5 s on a 1.5-GHz laptop PC. This example shows that even in the case of resonant circuits, the perturbation of the poles proposed procedure yields passive circuits with satisfactory accuracy.

The proposed technique can also be used to create a passive SPICE network based on measured data. To prove its usefulness, we have created a SPICE circuit based on measurement data of a RAMBUS FlexIO processor bus [11]. The FlexIO processor bus is a chip-to-chip interface technology that offers data rates of 400 MHz up to 8.0 GHz. The channel consists of two 12-in-long meandered lines made on an FR4 printed circuit board (PCB) substrate and two PCB/package vias [11]. It is a complex structure and hard to simulate with an EM solver, therefore, the performance of the design had to be verified by measurements. With the proposed technique, a passive equivalent circuit of the bus can be created directly from measured frequency-domain data. The circuit can be then used for signal integrity analysis of the design. The rational model of the transfer function was created in frequency band from dc up to 20 GHz using the vector fitting technique described in [16]. The transfer function of the circuit is very complex, therefore, the rational function that approximates poles. The rational model is not it has a high order passive in the frequency range of 58–157 and 188–237 MHz. The proposed technique was utilized to restore the passivity of the model, which took 5 min on a 1.5-GHz PC. The comparison of input data and the final passive model is shown in Fig. 8. D. Parameterized SPICE Networks SPICE circuits can also be generated from a surrogate model. A surrogate model is derived from full-wave simulations and allows one to quickly evaluate an approximate response of the element for a combination of geometric or physical parameters. Due to the mathematical nature of the approximation procedures, surrogate models are nonpassive. In this context, the method for SPICE circuit construction from nonpassive data proposed in this paper proves to be particularly powerful, as the passivity enforcement can be done almost instantaneously, while minimizing the disturbance of the circuit response in the frequency domain. Thus, by combining the surrogate modeling

LAMECKI AND MROZOWSKI: EQUIVALENT SPICE CIRCUITS WITH GUARANTEED PASSIVITY FROM NONPASSIVE MODELS

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Fig. 9. Proposed generation scheme of parameterized passive SPICE circuits for time-domain analysis.

Fig. 11. Frequency response of nonpassive (1 1 1) and passive (—) model of a three-conductor line bend. Fig. 10. Three-conductor line bend. (a) Top view of the modeled structure with model parameters. (b) 3-D view of double-bend structure.

TABLE I RANGE OF MODEL PARAMETERS

with our passivity enforcement solution, one gets parameterized SPICE circuits. The proposed scheme is presented in Fig. 9. The physical parameters are the input data. The nonpassive response is evaluated using a surrogate model and then transformed to a state–space model. The passivity enforcement is done on the fly. The passive state–space model is then transformed to the SPICE model. The whole procedure takes only seconds and, therefore, can be used in an optimization loop. Parameterized SPICE circuits are applicable, e.g., in the design of mixed analog–digital circuits and interconnects. To present the proposed method for this application and provide some insight regarding the speed of computation, let us consider a three-conductor microstrip-line bend on a thin-film substrate for multichip module-deposited (MCM-D) solutions, as shown in Fig. 10. A four-variate surrogate model of scattering parameters of the structure was created using the technique of [12] and Agilent’s ADS Momentum EM simulator. As a result, we obtained a mathematical model of scattering parameters of a structure in a form of a rational function (14) where and are a multivariate polynomials. The range of input model parameters is presented in Table I. The resulting model was used to compute the response of the double bend of the multiconductor line, as shown on Fig. 10, with dimensions mm, mm, and mm. The passive

Fig. 12. Maximum singular value of scattering matrix of nonpassive (1 1 1) and corrected (—) model of a three-conductor line bend.

TABLE II SAMPLE TIMING OF SPICE MODEL CONSTRUCTION (MATLAB)

SPICE circuit was then created by applying the above-described technique. The order of the rational model was set as and the state–space model had 36 states. Fig. 11 shows a frequency response of the model before and after passivity enforcement—very good agreement can be seen in the entire frequency band. The passivity check in the frequency domain is shown in Fig. 12. The total time of a passive SPICE model construction for a specified structure dimensions is only 4 s on a 1.5-GHz PC. Timing details are presented in Table II.

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V. CONCLUSIONS A new technique for creation of passive SPICE circuits from nonpassive models has been described in this paper. The technique is very fast and preserves the frequency response of the circuit. The procedure is defined as a min–max optimization problem with analytic gradients and nonlinear constraints. The authors’ implementation of the technique will soon be available online.1

ACKNOWLEDGMENT The authors would like to thank to Prof. T. Dhaene and D. Deschrijver, both with the University of Antwerp, Antwerp, Belgium, for their implementation of the vector fitting technique. The authors also thank W. Beyene, RAMBUS Inc., Los Altos, CA, for providing FlexIO bus measurements data used in Section IV.

REFERENCES [1] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: Passive reduced-order interconnect macromodeling algorithm,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 17, no. 8, pp. 645–654, Aug. 1998. [2] R. Achar, P. K. Gunupudi, M. Nakhla, and E. Chiprout, “Passive interconnect reduction algorithm for distributed/measured networks,” IEEE Trans. Circuits Syst., vol. 2, no. 4, pp. 287–301, Apr. 2000. [3] D. Saraswat, R. Achar, and M. S. Nakhla, “Passive reduction algorithm for RLC interconnect circuits with embedded state–space systems (PRESS),” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2215–2226, Sep. 2004. [4] D. Saraswat, R. Achar, and M. S. Nakhla, “A fast algorithm and practical considerations for passive macromodeling of measured/simulated data,” IEEE Trans. Adv. Packag., vol. 27, no. 1, pp. 57–70, Feb. 2004. [5] S.-H. Min and M. Swaminathan, “Construction of broadband passive macromodels from frequency data for simulation of distributed interconnect networks,” IEEE Trans. Electromagn. Compat., vol. 46, no. 4, pp. 544–558, Nov. 2004. [6] B. Gustavsen and A. Semlyen, “Enforcing passivity for admittance matrices approximated by rational functions,” IEEE Trans. Power Syst., vol. 16, no. 1, pp. 97–104, Feb. 2001. [7] C. P. Coelho, J. Phillips, and L. M. Silveira, “A convex programming approach for generating guaranteed passive approximations to tabulated frequency-data,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 23, no. 2, pp. 293–301, Feb. 2004. [8] S. Grivet-Talocia, “Passivity enforcement via perturbation of Hamiltonian matrices,” IEEE Trans. Circuits Syst., vol. 51, no. 9, pp. 1755–1769, Sep. 2004. [9] D. Saraswat, R. Achar, and M. S. Nakhla, “Global passivity enforcement algorithm for macromodels of interconnect subnetworks characterized by tabulated data,” IEEE Trans. Very Large Scale Integr. (VLSI) Syst., vol. 13, no. 7, pp. 819–832, Jul. 2005. [10] A. Lamecki and M. Mrozowski, “Passive SPICE networks from non-passive data,” presented at the 16th Int. Microw., Radar, Wireless Commun. Mikon Conf., 2006. 1[Online].

Available: http://mwave.eti.pg.gda.pl/passenf.html

[11] W. T. Beyene, J. Feng, N. Cheng, and X. Yuan, “Performance analysis and model-to-hardware correlation of multigigahertz parallel bus with transmit pre-emphasis equalization,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3568–3577, Nov. 2005. [12] A. Lamecki, P. Kozakowski, and M. Mrozowski, “Efficient implementation of the Cauchy method for automated CAD-model construction,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 7, pp. 268–270, Jul. 2003. [13] J. De Geest, T. Dhaene, N. Fache, and D. De Zutter, “Adaptive CADmodel building algorithm for general planar microwave structures,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1801–1809, Sep. 1999. [14] A. H. Zaabab, Q. J. Zhang, and M. Nakhla, “A neural network modeling approach to circuit optimization and statistical design,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 6, pp. 1349–1558, Jun. 1995. [15] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1052–1061, Jul. 1999. [16] B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del., vol. 21, no. 3, pp. 1587–1592, Jul. 2006. [17] T. Dhaene and D. Deschrijver, “Generalised vector fitting algorithm for macromodelling of passive electronic components,” Electron. Lett., vol. 41, pp. 299–300, Mar. 2005. [18] A. G. Lampérez, T. K. Sarkar, and M. S. Palma, “Generation of accurate rational models of lossy systems using the Cauchy method,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 10, pp. 490–492, Oct. 2004. [19] R. Achar and M. S. Nakhla, “Simulation of high-speed interconnects,” Proc. IEEE, vol. 89, no. 5, pp. 693–728, May 2001. [20] S. Boyd, V. Balakrishnan, and P. Kabamba, “A bisection method for norm of a transfer matrix and related problems,” computing the Math. Contr. Signals Syst., vol. 2, pp. 207–219, 1989. [21] G. H. Golub and C. F. Van Loan, Matrix Computation. Baltimore, MD: The Johns Hopkins Univ. Press, 1996.

H

Adam Lamecki was born in Inowroclaw, Poland, in 1977. He received the M.S.E.E. degree from the Gdan´ sk University of Technology (GUT), Gdan´ sk, Poland, in 2002, and is currently working toward the Ph.D. degree at the GUT. His research interests include surrogate models and their use in the computer-aided design (CAD) of microwave devices, SPICE equivalent circuits, RFIC/monolithic microwave integrated circuits (MMICs), computational electromagnetics, and filter design. Mr. Lamecki was a recipient of a Domestic Grant for Young Scientists Award founded by the Foundation of Polish Science in 2006.

Michal Mrozowski (S’88–M’90–SM’02) received the M.S.E.E., Ph.D., and D.Sc. degrees (with honors) in microwave engineering from the Gdan´ sk University of Technology (GUT), Gdan´ sk, Poland, in 1983, 1990 and 1994, respectively. Since 2001, he has been a Full Professor with the GUT. His interests are computational electromagnetics and field theory. He has authored or coauthored over 50 reviewed journal papers and two monographs on computational electromagnetics and guided EM waves. Prof. Mrozowski is a member of the Electromagnetics Academy. He is chairman of the IEEE Polish joint Aerospace and Electronic Systems (AES)/Antennas and Propagation (AP)/Microwave Theory and Techniques (MTT) Chapter.

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Singular Tetrahedral Finite Elements for Vector Electromagnetics Jon P. Webb, Member, IEEE

Abstract—The electromagnetic fields at reentrant edges made of conductor are generally singular and the polynomial basis functions used in traditional finite elements do not model the fields well in these regions. Two new tetrahedral elements are introduced with basis functions that incorporate the known asymptotic variation of the fields with distance from the edge. The basis is split into gradient and rotational parts. Additional vertex functions can be added to improve the conditioning of the global matrix system that arises in the time–harmonic case. Results are presented for the scattering parameters of a number of waveguide discontinuities. The new elements reduce the error considerably compared to the Whitney element, and generally outperform the second-order edge element, even though that has more degrees of freedom.

Fig. 1. (r; '; z ) coordinate system with its origin O on the singular edge. The z -axis is parallel to the edge and perpendicular to the page.

Index Terms—Finite-element methods, scattering parameters, waveguide discontinuities.

reentrant features resulting from dielectrics are not considered, though it is to be hoped that the ideas introduced here can be extended to cover these cases.

I. INTRODUCTION

II. REQUIREMENTS FOR THE BASIS

EENTRANT edges and corners commonly occur inside microwave devices. They are nearly always represented as perfectly sharp in numerical models because to do otherwise would involve considerable additional modeling complexity. However, in general, the electromagnetic field tends to infinity near perfectly sharp features of this kind, and any numerical method designed to predict the fields within the device has to be able to cope with this. The finite-element method, which is widely applied in the frequency domain to the analysis of microwave devices, generally uses a polynomial representation of the field inside each element. Necessarily, this representation remains finite everywhere and does not model the field near sharp features well. Since poor modeling of the singularities often leads to inaccurate global quantities, such as resonant frequencies or scattering parameters, the mesh needs to be highly refined in the vicinity of sharp features. In many cases, the bulk of the computational effort may be due to the fine meshes in these regions. Consequently there have been several proposals for special finite elements that incorporate different nonpolynomial basis functions and are better able to represent the singular field behavior. The earliest work of this kind in electromagnetics was scalar in nature, but more recently, it has been extended to the vector case, though only in 2-D [1]–[6]. This paper is concerned with the 3-D vector case. The focus is on edges formed from the perfect electric conductor (PEC). Reentrant corners (points) and

Fig. 1 shows the cross section of a reentrant PEC edge of , and a cylindrical coordinate system based on it. angle The field behavior near such an edge can be very complicated, particularly when other singular features are nearby. In [6] and the references cited therein, series expansions of the field are given. The leading term of these expansions corresponds to the asymptotic form of the electric field [7]

R

Manuscript received August 15, 2006; revised November 9, 2006. This work was supported by the Natural Sciences and Research Council of Canada. The author is with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC, Canada H3A 2A7 (e-mail: jon.webb@mcgill. ca). Digital Object Identifier 10.1109/TMTT.2006.890523

as

(1)

. For all singular where is a singularity index equal to . In general, the strength of the field PEC edges, will vary with distance along the edge so (1) can be written more generally as follows:

or, since

as

as

(2)

as

(3)

,

This suggests at once an approach to the construction of the new basis: find a scalar basis that can represent functions like and take its gradient. However, this is not sufficient because a trial built in such a way would have a zero curl and we know that its curl must be able to represent the magnetic field . Like , the magnetic field has the asymptotic form as

(4)

We need additional basis functions for that have a curl able to represent . Rotational functions of the form will do this.

0018-9480/$25.00 © 2007 IEEE

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Fig. 2. Line and point tetrahedra.

Fig. 4. Coordinates ,  ,  , and  for the point tetrahedron. The thick arrows show the direction in which each coordinate is increasing. The shaded regions are surfaces on which the corresponding coordinate is constant.

Fig. 3. Coordinates ,  , and  for the line tetrahedron. The thick arrows show the direction in which each coordinate is increasing; the shaded regions are surfaces on which the corresponding coordinate is constant.

III. LOCAL COORDINATE SYSTEMS The element shapes considered are tetrahedra. While prismshaped elements are a more natural fit to line singularities (e.g., [8] and [9]), automatic tetrahedral mesh generators are widely used and requiring special element shapes along the sharp edges would impose extra complexity at the mesh generation stage. There are two types of tetrahedral elements that require singular basis functions: those that have an edge along the singularity, which we will call line tetrahedra, and those that have just one node on the singular edge, which we will call point tetrahedra (Fig. 2). Note that it is necessary to consider both, even though we are not providing special basis functions for reentrant corners. For each tetrahedron type, we need a local coordinate system in which one of the coordinates is approximately the distance away from the singular edge. For the line tetrahedron, the three coordinates , , and are as defined in Fig. 3 in terms of the simplex coordinates , , , and of the element. It can be seen that , , and correspond roughly to , , and , respectively. In particular, we have

(5) so that, as tends to zero along a radial line ( ) within the tetrahedron

,

(6) for any function . This allows us to construct functions in , , and that have the correct asymptotic behavior as tends to zero.

Also indicated in Fig. 3 are three vectors , , and . These are in the direction of the gradients of the three coordinates, but are scaled to make them bounded everywhere in the tetrahedron. They will be used later to construct basis functions. are defined for the The four coordinates , , , and coordinates corpoint tetrahedron (Fig. 4). The , , and respond to the three simplex coordinates of the vertices that are are not all not on the singular edge. Note that , , and independent, (7) Only and two of the other three coordinates are needed to define a point in the tetrahedron, but it is sometimes convenient to use all four coordinates. Also indicated in Fig. 4 are four vectors , , , and . These are in the direction of the gradients of the four coordinates, but as before, they are scaled to make them bounded everywhere in the tetrahedron. The line and point coordinates match very simply. Consider a point tetrahedron that has a singular vertex in common with a line tetrahedron, and assume the numbering shown in Figs. 2 and 3. 1) Along all shared edges and faces, of the line tetrahedron matches of the point tetrahedron. 2) If vertex 3 of the line tetrahedron is coincident with vertex of the point tetrahedron ( or ), then along the shared edge from vertex to the common singular vertex, matches . 3) If vertex 4 of the line tetrahedron is coincident with vertex of the point tetrahedron ( or ), then along the shared edge from vertex to the common singular matches . vertex, 4) If vertices 3 and 4 of the line tetrahedron are coincident with vertices and of the point tetrahedron, respectively, then over the shared face formed by vertices and and the common singular vertex and match and , respectively. Also on this face, vanishes, where or , but different from and .

WEBB: SINGULAR TETRAHEDRAL FINITE ELEMENTS FOR VECTOR ELECTROMAGNETICS

IV. GRADIENT BASIS To build the gradient basis we start with functions that can . In the line tetrahedron, the following approximate four edge functions are chosen:

Each function vanishes tangentially on all edges, except the singular edge, and on every face, except its own. The first two of the face functions have to match corresponding functions in the point tetrahedron. The following three face functions provide that continuity:

Edge 13: Edge 14: Edge 23:

Face 132: Face 143:

Edge 24: where

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Face 124:

(8)

is this linear function of , (9)

Each function vanishes on every edge, except its own and, on the singular edge, has the desired asymptotic dependence on . The function will be explained later (towards the end of Section V). In the point tetrahedron, there are three edge functions as follows: Edge 1 :

(10)

It may readily be confirmed that these functions are continuous between line and point tetrahedra. The corresponding and are just the gradients of these vector functions scalars. V. ROTATIONAL BASIS As discussed above, rotational functions should be of the . In the line tetrahedron, the four scalar functions form given in (8) can be used to generate four “radial” edge functions of the right form as follows:

function becomes simply (Note that, on its own face, the .) All of the functions described thus far are tangentially zero on edge 34 of the line tetrahedron and edges 23, 34, and 42 of the point tetrahedron. Yet, along these edges, we want the singular elements to be compatible with the standard Whitney tetrahedron, which has one tangential basis function per edge, of the form Edge

Edge 34:

Edge 23: (11)

Each function vanishes tangentially on every edge, except its in the above own and the singular edge. (Note that the might seem to be unnecessary since is perpendicular to edge 34 . However, without the the curl of the functions .) is singular at Matching these are the following three edge functions of the point tetrahedron based on (10): (12)

The functions in (11) and (12) provide an approximately radial field. They must be complemented with azimuthal and axial basis functions. In the line tetrahedron, we add the following four face functions: Face 143: Face 234: Face 132: Face 124:

(13)

(15)

(16)

and for edges 23, 34, and 42 of the point tetrahedron, we have the following: Edge 23: Edge 34: Edge 42:

Edge 14:

Edge 1 :

:

It is necessary, therefore, to introduce a final set of edge functions to provide this compatibility. Standard Whitney functions could be used, but it seems more appropriate to use functions factor. For edge 34 of the line tetrahedron, that include the we have the following basis function:

Edge 13:

Edge 24:

(14)

(17)

The rotational basis given above [(11), (13), and (16)] for the line tetrahedron is not, in fact, purely rotational (though, for convenience, it will still be referred to by that name). Examination of the curls of the nine functions reveals that the space spanned by them has a two-dimensional subspace of gradient functions. This is analogous to the Whitney element, in which the space spanned by the six edge functions has a three-dimensional gradient subspace. To further characterize the two-dimensional gradient subspace, consider the following scalar vertex functions: Vertex 3: Vertex 4:

(18)

which have gradients Vertex 3: Vertex 4:

(19)

part of these can be built from the edge functions (11) The and the part from linear combinations of (13) and (17). The gradients of (18), then, form a basis for the two-dimensional gradient subspace. Note that the gradients of (18) are independent of (8) and, thus, there is no overlap between the rotational and gradient spaces as defined.

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From (8) and (18), it is evident that the gradient subspace and . Consequently, contains the radial factors it contains the function , which is the radial variation of the simplest solution in the neighborhood of the singular edge [7]. This function would have been absent from the trial function space if the factor had been omitted from (8). In the same way, the rotational basis [(12), (14), and (17)] for the point tetrahedron is not purely rotational. Consider the following three scalar vertex functions: Vertex :

(20)

which have gradients Vertex :

(21)

The part of these can be built from the edge functions (12) and the part from combinations of (14) and (17). The vertex functions form a basis for a three-dimensional gradient subspace. Note that (20) is independent of (10) and, thus, there is no overlap between the rotational and gradient spaces as defined. VI. IMPROVED BASIS FOR ITERATIVE SOLVERS Iterative methods for solving the global matrix equation are widely used in finite-element analysis because of their low memory requirements and, particularly for very large problems, low computation times compared to direct methods. However, iterative solvers often do not perform well when applied to time–harmonic microwave problems of the sort considered here. The reasons are discussed in [10]. The solution is to ensure that there is, within the set of basis functions used, a subset whose span is the entire gradient subspace of the discrete function space [10]–[12]. In general, this requires adding to the basis additional gradient functions that are linearly dependent on the existing ones. Though this makes the global matrix singular, paradoxically it dramatically improves convergence. The simplest example is the Whitney basis. On its own, it leads to slow convergence; when one gradient function per vertex is added, fast convergence is restored. In the case of the line tetrahedron, the required set of gradient basis functions almost exists, but not quite. As explained above, there is a two-dimensional gradient subspace that is part of the “rotational” space and is, therefore, not spanned by any gradient basis functions. However, the two-dimensional gradient subspace is spanned by gradients of the scalars in (18) so adding one function of this kind to each nonsingular vertex of a line tetrahedron solves the problem. We now have 15 basis functions per element [(8), (11), (13), (16), and (18)] instead of 13, but much faster convergence. For the point tetrahedron, the same problem arises and the remedy is the same: we add one gradient function, the gradient of (20), to each nonsingular vertex. There are now 15 basis functions per element [(10), (12), (14), (17), and (20)] instead of 12. Note that the new vertex functions (18) for the line tetrahedron match those of the point tetrahedron (20) at common vertices, edges, and faces. They also match the vertex functions that are added to the Whitney tetrahedra for the same reason. The effect of the vertex functions is shown in Table I for a . The second column geometry with singularity index

TABLE I EFFECT OF  ON THE CONVERGENCE OF THE CONJUGATE GRADIENT METHOD

shows the number of conjugate gradient steps needed to solve the global matrix equation when all the tetrahedra are Whitney elements, i.e., no singular elements are used. The corresponding number of degrees of freedom is shown in the first column. When the tetrahedra along the singular edges are replaced with singular elements, including vertex functions and using the correct value of according to (9), the number of conjugate gradient steps barely changes (column 3). However, if the wrong value of is used for all the gradient basis functions [(8), (10), (19), and (20)], the irrotational part of the “rotational” space is no longer duplicated by the gradient functions and many more steps are needed (column 4). VII. MATRIX ASSEMBLY For finite-element analysis, it is necessary to compute the enand , whose tries of the local mass and stiffness matrices entries for tetrahedron are given by

(22) where is if is a line tetrahedron and if is a point tetrahedron. When the vertex functions are included, the local matrices have dimension 15. Each of the basis functions for the line tetrahedron can be written in the form (23) where

,

, and

are polynomials in , , and . Since

(24) the general basis function can then be rewritten as

(25) are again polynomials in , , and . Taking the curl where of both sides, after some algebra we get

(26)

WEBB: SINGULAR TETRAHEDRAL FINITE ELEMENTS FOR VECTOR ELECTROMAGNETICS

where

are polynomials in , , , and (with cyclic permutation of the subscripts in the range

of 1–3). Now substituting (25) and (26) into (22) and following the procedure described in [13], we get

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Matrix assembly for point tetrahedra proceeds in the same and way, but with some small differences. The matrices are

(27) where and are a set of 12 matrices, each 15 15, and is the volume of the tetrahedron. Equation (27) equally holds for the polynomial elements described in [13], but in that and are universal matrices that can be encase, tirely precomputed and stored with the program. In the current and are somewhat different as follows: case,

(31) The inner two integrals are over a triangular region so, in this case, we have

(32) , , and are the abscissas of the three-point 2-D Gaussian quadrature formula for the unit right-angled triangle [14]. where

(28) VIII. RESULTS is 1 if and are unequal, 1/2 if they are equal. where Evaluation of these integrals using a series of universal matrices is possible, but unwieldy because and are embedded in and and these are only known at run time. An alternative is and are to use numerical integration. It turns out that first-order functions of , and first-order functions of . Consequently, we have

(29) where and are the abscissas of the two-point 1-D Gaussian quadrature formula on . Expression (29) is exact. After evaluation at the interval and , the term is a quadratic in with numerical coefficients and the rest of the integral can be handled symbolically at run time using the identity

(30) and are independent of the size and shape of the tetrahedron and, once computed, can be used for all line tetrahedra that have the same singularity index.

The new elements were used to compute the port parameters of a number of waveguide discontinuities. For the most part, the is reported. This was obtained using reflection coefficient the weighted residual method described in [15] with the electric field as the solution variable in every case. For each device, three curves are presented, all showing the versus the computation time for solving the global error in matrix problem (using the conjugate gradient method with symmetric successive over-relaxation (SSOR) preconditioning [12]). The meshes used were approximately uniform, and successive points along the curves correspond to a halving of , the maximum tetrahedron edge length in the mesh. The error , where is a reference solution value plotted is generated using a highly refined mesh of third-order tetrahedral elements of [13]). Note that this edge elements (the error measure captures both magnitude and phase information. The first curve (labeled “Order 1”) is obtained when every ] element. The tetrahedron in the mesh is a Whitney [or other two curves show the effect of replacing the Whitney elements touching a reentrant PEC edge by either second-order elements (“Order 1 2”) or the new singular elements (“Order 1 Singular”). - and -Plane Discontinuities in A. Parallel-Plate Waveguide The upper graphic in Fig. 5 shows a parallel-plate waveguide with an -plane discontinuity; the lower graphic is an -plane

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Fig. 5. H - and E -plane discontinuities in parallel-plate waveguide. Dimensions are in meters.

discontinuity. In each case, the dominant TEM mode is incident and the sidewalls are modeled as perfect magnetic conductors. The advantage of these simple, essentially 2-D, problems is that we know something about the nature of the field near the singular edge. In -plane problems, the electric field throughout is parallel to the singular edge and is nonsingular. The magnetic field, on the other hand, is perpendicular to the edge and is singular. Since the solution variable is , -plane problems highlight the rotational subspace of the singular elements, which is the only part of the trial function space that affects the accuracy of . In -plane problems, on the other hand, the magnetic field is parallel to the singular edge and is nonsingular; the electric field is perpendicular to the edge and is singular. -plane problems highlight the gradient part of the element function space. Two examples of each type of problem are considered. The insets in Figs. 6 and 7 are discontinuities consisting of an infinitely thin sheet of a PEC half-blocking the waveguide. The . Figs. 8 and 9 singularity index of the reentrant edge is show a step change in waveguide width and height, respectively. . The reentrant edge in this case is “less singular” since

Fig. 6. Error in S versus computation time for the inset H -plane problem. Dimensions are in meters. The analysis is at a free-space wavenumber k = 2 rad 1 m .

Fig. 7. Error in S versus computation time for the inset E -plane problem. Dimensions are in meters. The analysis is at a free-space wavenumber k = 2 rad 1 m .

B. Rectangular-Aperture Iris in Rectangular Waveguide The infinitely-thin PEC iris shown inset in Fig. 10 has singular edges oriented both parallel and perpendicular to the electric field of the incident wave and, to handle this, the singular element must be able to model both electric and magnetic field singularities adequately. This example is taken from [16], where normalized susceptance values are presented. The norusing the formula malized susceptance is obtained from (33) is referred to as the plane of the iris. At the analysis where frequency, i.e., 10 GHz, the measured value of is 0.674 [16]. This is within 1.5% of the reference value of 0.684, which is obtained using a fine mesh of third-order elements.

Fig. 8. Error in S versus computation time for the inset H -plane problem. Dimensions are in meters. The analysis is at a free-space wavenumber k = 2 rad 1 m .

WEBB: SINGULAR TETRAHEDRAL FINITE ELEMENTS FOR VECTOR ELECTROMAGNETICS

Fig. 9. Error in S versus computation time for the inset E -plane problem. Dimensions are in meters. The analysis is at a free-space wavenumber k = 2 rad 1 m .

Fig. 10. Error in normalized susceptance B versus computation time for the inset rectangular waveguide problem. a = 22:86 mm, b = 10:16 mm, a = 7:7 mm, and b = 4:8 mm. The analysis frequency is 10 GHz.

C. Ridge in a Rectangular Waveguide The final problem consists of a rectangular PEC block placed against the broad wall of a rectangular waveguide, forming a ridge (Fig. 11). In fact, this can be considered to be a transition from a rectangular waveguide to ridge waveguide and back . There again. The metal edges have a singularity index are also four singular corners for which no special allowance was made: was set to 2/3 in all the singular elements, even the ones touching the singular corners. Measured and computed results for this geometry are presented in [17]. At 13 GHz, the measured return loss of 4.86 dB agrees well with the reference value used for this study, i.e., 4.82 dB. IX. DISCUSSION OF RESULTS In nearly every case, replacing the first-order elements along the singular edge by the new singular elements significantly reduces the error. The only exceptions occur when the mesh is at its coarsest; presumably in these cases, the discretization error from other regions of the problem is so high that it masks the

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Fig. 11. Error in S versus computation time for the inset rectangular waveguide problem. a = 19:05 mm, b = 9:525 mm, L = 20 mm, u = 4:5 mm, w = 6 mm, and d = 10 mm. The analysis frequency is 13 GHz.

error coming from the singular edge. In most cases, the reduction in error is greater than is obtained by halving the element size, i.e., going to the next point along the “Order 1” curve. Exceptions are seen in Figs. 9 and 11, which are both “less sin) cases. gular” ( The significant reduction in error shows what a large effect singular edges have on accuracy of port parameters, and demonstrates the effectiveness of the new elements for both electric and magnetic field singularities. However, it could be argued that the reduction in error has been achieved simply by using elements with more degrees of freedom (e.g., two basis functions per edge rather than 1), and that this could be done just as effectively by using a second-order element. In fact, the second-order element has more degrees of freedom than the singular element because it has two basis functions per face instead of 1 so if the only issue were the number of degrees of freedom, the second-order element would always outperform the singular element. It does cases (Figs. 6, 7, and 10), the singular elenot. In the ment gives much less error than the second-order element, decases, spite having fewer degrees of freedom. In the there is a more mixed picture. In the -plane problem (Fig. 8), the singular element does better, but in the -plane problem (Fig. 9), the accuracy is about the same, whereas in the ridge problem (Fig. 11), second order does slightly better. In all the cases, polynomials can do a better job of representing the singular fields than in the more singular cases, which might explain the better performance of second-order elements. Also, in the ridge problem, there are singular corners present, for which the current method makes no allowance. For the finest mesh (rightmost point) of Figs. 6, the error obtained with singular elements is 3.4 times smaller than the error obtained with second-order elements for about the same computational cost. The corresponding factors for Figs. 7–11 are 2.2, 1.9, 1.2, 2.1, and 0.8; the average of these is 1.9. X. CONCLUSIONS How well the fields are modeled near singular edges has a large effect on the accuracy of global parameters, particularly in

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the case. Adding degrees of freedom to the elements touching these edges can greatly improve the accuracy. While order 2 will do, the results show that singular elements are more effective, i.e., they give greater accuracy at lower cost. It is likely that even greater accuracy can be achieved by the use of higher order singular elements. These might be generated by adding the singular basis functions described in this paper to the higher order polynomials in [13], or by constructing new singular basis functions capable of approximating more terms of the series expansion of the field near the singularity, as in [6]. Investigation of such elements is underway. REFERENCES [1] J. P. Webb, “Finite element analysis of dispersion in waveguides with sharp metal edges,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1819–1824, Dec. 1988. [2] J. M. Gil and J. P. Webb, “A new edge element for the modeling of field singularities in transmission lines and waveguides,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2125–2130, Dec. 1997. [3] Z. Pantic-Tanner, J. S. Savage, D. R. Tanner, and A. F. Peterson, “Two dimensional singular vector elements for finite-element analysis,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 2, pp. 178–184, Feb. 1998. [4] R. D. Graglia and G. Lombardi, “Singular higher order complete vector bases for finite methods,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1672–1685, Jul. 2004. [5] J. Masoni, G. Pelosi, and S. Selleri, “Substitutive divergent bases for FEM modelling of field singularities near a wedge,” Microw. Opt. Technol. Lett., vol. 44, no. 4, pp. 327–328, Feb. 2005. [6] D.-K. Sun, L. Vardapetyan, and Z. Cendes, “Two-dimensional curlconforming singular elements for FEM solutions of dielectric waveguiding structures,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 984–992, Mar. 2005. [7] J. Van Bladel, “Field singularities at metal-dielectric wedges,” IEEE Trans. Antennas Propag., vol. AP-33, no. 4, pp. 450–455, Apr. 1985. [8] M. Stern and E. B. Becker, “A conforming crack tip element with quadratic variation in the singular field,” Int. J. Numer. Methods Eng., vol. 12, pp. 279–288, 1978.

[9] M. Stern, “Families of consistent conforming elements with singular derivative fields,” Int. J. Numer. Methods Eng., vol. 14, pp. 409–421, 1979. [10] H. Igarashi and T. Honma, “Convergence of preconditioned conjugate gradient method applied to driven microwave problems,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1705–1708, May 2003. [11] R. Dyczij-Edlinger, G. Peng, and J.-F. Lee, “A fast vector-potential method using tangentially continuous vector finite elements,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 6, pp. 863–868, Jun. 1998. [12] J. P. Webb, “Combined direct-iterative matrix solvers for hierarchal vector finite elements,” IEEE Trans. Magn., vol. 38, no. 2, pp. 345–348, Mar. 2002. [13] ——, “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, Aug. 1999. [14] D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng., vol. 21, pp. 1129–1148, 1985. [15] H. Akel and J. P. Webb, “Design sensitivities for scattering-matrix calculation with tetrahedral edge elements,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1043–1046, Jul. 2000. [16] R. Yang and A. S. Omar, “Investigation of multiple rectangular aperture irises in rectangular waveguide using TE -modes,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 8, pp. 1369–1374, Aug. 1993. [17] D. Arena, M. Ludovico, G. Manara, and A. Monorchio, “Analysis of waveguide discontinuities using edge elements in a hybrid mode matching/finite elements approach,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 9, pp. 379–381, Sep. 2001.

Jon P. Webb (M’83) received the Ph.D. degree from Cambridge University, Cambridge, U.K., in 1981. Since 1982, he has been a Professor with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC, Canada. His research interest is computer methods in electromagnetics, especially the application of the finite-element method.

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Space-Mapping Optimization With Adaptive Surrogate Model Slawomir Koziel, Member, IEEE, and John W. Bandler, Life Fellow, IEEE

Abstract—The proper choice of mapping used in space-mapping optimization algorithms is typically problem dependent. The number of parameters of the space-mapping surrogate model must be adjusted so that the model is flexible enough to reflect the features of the fine model, but at the same time is not over flexible. Its extrapolation capability should allow the prediction of the fine model response in the neighborhood of the current iteration point. A wrong choice of space-mapping type may lead to poor performance of the space-mapping optimization algorithm. In this paper, we consider a space-mapping optimization algorithm with an adaptive surrogate model. This allows us to adjust the type of space-mapping surrogate model used in a given iteration based on the approximation/extrapolation capability of the model. The technique does not require any additional fine model evaluations. Index Terms—Adaptive surrogate model, engineering optimization, microwave design, space mapping, space-mapping optimization.

I. INTRODUCTION

S

PACE MAPPING is a recognized engineering optimization methodology [1]–[5]. It shifts the optimization burden from an expensive “fine” (or high fidelity) model to a cheap “coarse” (or low fidelity) model by iterative optimization and updating of the surrogate model, which is built using the coarse model and available fine model data. A similar idea is exploited by other surrogate-based methods [6]–[12], although many of them construct a surrogate model by direct approximation of the fine model data with no underlying coarse model. Space mapping was originally applied to the optimization of microwave devices [1], where fine models are often based on full-wave electromagnetic simulators, whereas coarse models are physically based circuit models. Recently, space-mapping techniques have been applied to design problems in a growing number of areas (see, e.g., [13]–[15]). A review of advances in space-mapping technology is contained in [4]. Recent efforts have focused in several areas, which are: 1) the development of new algorithms that use different space-mapping techniques such as implicit space mapping [2] and output Manuscript received August 31, 2006; revised November 25, 2006. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007239, Grant STGP269889, and Grant STGP269760, 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.2006.890524

space mapping [3]; 2) the development of new space-mappingbased models [16]; 3) theoretical justification of space mapping and convergence theory for space-mapping optimization algorithms [17], [18]; 4) neuro-space mapping [19]–[22]; and 5) applications of space mapping (e.g., [23]–[26]). The common problem in space-mapping-based optimization is the proper choice of type of mapping. Space-mapping techniques available include input, implicit, and different variations of output space mapping, as well as customized mappings such as frequency space mapping [3]. By combining these mappings in different configurations, one can adjust the flexibility of the space-mapping surrogate model, which is correlated with the number and type of space-mapping parameters. The space-mapping surrogate model cannot be too simple, otherwise it will not properly reflect the features of the fine model. The surrogate model cannot be over-flexible because its extrapolation properties would then be too poor to allow accurate prediction of the fine model response in the neighborhood of the current iteration point. Unfortunately, it is difficult to tell beforehand which combination of mappings may be optimal for a given problem. A wrong choice of space-mapping type may lead to poor performance of the space-mapping optimization algorithm. Another issue is that surrogate models that are flexible and theoretically suitable for a given problem may exhibit poor performance due to difficulties in the extraction of the model parameters. In this paper, we present a space-mapping-based optimization algorithm with an adaptive surrogate model. Our technique allows us to adjust the type of space-mapping surrogate model used in a given iteration based on the approximation/extrapolation capability of the model. This capability is estimated by comparing properly chosen quality factors that measure the ability of the surrogate model to match the fine model and to extrapolate its response at points not used in parameter extraction. The technique does not require any additional fine model evaluations because the quality factor calculation is based on already available data.

II. BASICS OF SPACE-MAPPING OPTIMIZATION Let , , denote the response vector of a fine model of the device of interest. Our goal is to solve (1) where is a given objective function. To solve (1), we use an optimization algorithm that generates a sequence

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of points models

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, ,

, and a family of surrogate , so that (2)

where denotes the trust region radius at iteration . We use a trust region method [27], [28] to improve the convergence properties of the algorithm. , , denote the response vectors Let are constructed of the coarse model. The surrogate models from the coarse model so that proper matching conditions are satisfied. A variety of space-mapping-based surrogate models are available [1]–[5], [17], [18]. Here, we use a surrogate model that incorporates both input [1] and output [3] space mapping. and We define (3) for

of space mapping both with respect to approximation and extrapolation quality. be a set of candidate surrogate Let models considered at iteration . Each of is a special case of (3). For simplicity, we assume the following compact way of , where writing the surrogate models: is a parameter domain of the model. We shall denote by the set of initial values of the parameters of candidate model . The model is set up by proper , which are determined using choice of its parameter values the parameter-extraction procedure (6)

is a subset of , the where set of all previous iteration points. Let , such that . Now, let us define the following two quantities:

, where (7) (4) and (5)

Apart from model (3)–(5), there is an optional frequency scaling that works in such a way that the coarse model is evaluated at a different frequency than the fine model using the transformation: , where is obtained in a parameter-extraction process similar to (4). Flexibility of the surrogate model can be adjusted by disabling some of the parameters, i.e., constraining them to their initial values. We shall use the following naming convention for the surrogate models: the presence of any of the letters , , , and is equivalent to enabling (not constraining) the corredenotes sponding model component, e.g., surrogate model the model that uses nontrivial components and . III. SPACE MAPPING WITH ADAPTIVE SURROGATE MODEL The general space-mapping model (3) allows us to use different combinations of space mapping. However, an optimal choice of mapping is usually problem dependent and may also be iteration dependent. We do not want the surrogate model to be too simple because, in that case, it cannot properly reflect the features of the fine model. We do not want the surrogate to be over flexible because its extrapolation properties, i.e., its capability to properly model the fine model response in the neighborhood of the current iteration point, may be lost. In general, a suitable choice requires both knowledge of the problem and engineering experience. Here, we describe a simple algorithm that makes the process of choosing a good space mapping automatic. The algorithm is adaptive in the sense that it can change the space mapping used from iteration to iteration based on the estimated performance

(8)

If is empty, we set . The first factor, i.e., , measures the quality of the apbecause it is the ratio of proximation properties of model the matching error before and after parameter extraction, calculated for the points which were used in parameter extraction. , measures the quality of the exThe second factor, i.e., trapolation properties of model because it is the ratio of the matching error before and after parameter extraction, calculated for the points which were not used in extraction. At iteration , we select the surrogate model based on the combined quality factor (9) In particular, we set (10) where (11) A good surrogate model exhibits high values for both and ; however, we consider extrapolation properties as even more important than approximation properties because indicates the capability of modeling the fine model

KOZIEL AND BANDLER: SPACE-MAPPING OPTIMIZATION WITH ADAPTIVE SURROGATE MODEL

outside the points at which the surrogate was created. This factor also indicates potential over flexibility of the surrogate model. Therefore, in practice, we use small values of (e.g., ). The space-mapping algorithm with an adaptive surrogate model selection scheme can be summarized as follows. Step 0: Set ; Choose the candidate model set ; Step 1: Given , set and ; Step 2: Perform parameter extraction and calculate quality and ; Choose the current surfactors ; rogate model and obtain using (2); Step 3: Optimize Step 4: Update ; is accepted set , ; Step 5: If Step 6: If the termination condition is not satisfied go to Step 1; else terminate the algorithm. The proposed adaptive scheme does not require any extra fine model evaluations because the surrogate model assessment is based on already existing fine model data. Additional computational effort concerns the coarse model only, and because we assume that the coarse model evaluation is significantly cheaper than the fine model evaluation, this additional effort does not substantially affect the total execution time of the optimization algorithm. In order to further reduce the execution time, unsuccessful candidate models can be gradually eliminated from (based, for example, on the ranking of ). In our nufixed throughout the merical experiment, however, we keep algorithm. The proposed adaptive scheme alleviates certain problems in parameter extraction. In particular, the space-mapping surro, which is potentially better than another model, gate model , may appear worse at a given iteration because of pasay, rameter-extraction problems, e.g., a large number of parameters or the wrong starting point may prevent the parameter-extraction process from finding optimal values of the space-mapping parameters. A possible extension of the proposed method is to exploit more than one coarse model so that the candidate surrogate models are combinations of both different space-mapping types and different coarse models. In particular, our method can be used for the adaptive selection of the coarse model.

Fig. 1. Six-section

Fig. 2. Six-section model [29].

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H -plane waveguide filter: the 3-D view [29].

H -plane waveguide filter: the equivalent empirical circuit

Fig. 3. Geometry of the microstrip bandpass filter [31].

Fig. 4. Coarse model of microstrip bandpass filter (Agilent ADS).

IV. EXAMPLES A. Test Problem Description -plane waveguide filter [29] Problem 1: Six-section (Fig. 1). The fine model is simulated using MEFiSTo [30] in a 2-D mode. The MATLAB coarse model (Fig. 2) has lumped inductances and dispersive transmission line sections. We simplify formulas due to Marcuvitz for the inductive susceptances corresponding to the -plane septa. Design parameters . The design speciare for GHz GHz, fications are for GHz GHz, and for GHz GHz. The starting point is mm (coarse model optimal solution).

Problem 2: Microstrip bandpass filter [31] (Fig. 3). The design parameters are . The fine model is simulated in FEKO [32], the coarse model is the circuit model implemented in Agilent ADS [33] (Fig. 4). The design specidB for GHz GHz, fications are dB for GHz GHz, and dB for GHz GHz. The initial design is the coarse model optimal solution mm. Problem 3: Microstrip bandpass filter with double-coupled resonators [31] (Fig. 5). The design parameters are . The fine model is simulated in FEKO [32], the coarse model is the circuit model implemented

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TABLE I DESCRIPTION OF CANDIDATE MODEL SETS

the naming convention for surrogate models (e.g., Bcd ) is explained in Section II. I denotes implicit space mapping with preassigned parameters being electrical permittivities (initial value 9) and heights (initial value 0.66 mm) of microstrip elements MLIN, MCORN, and MTEE (all these elements are grouped into five groups; there is a separate electrical permittivity and height for each group, which makes a total of ten preassigned parameters).

Fig. 5. Geometry of the microstrip bandpass filter with double-coupled resonators [31].

TABLE II OPTIMIZATION RESULTS

Fig. 6. Coarse model of microstrip bandpass filter with double-coupled resonators (Agilent ADS).

in Agilent ADS [33] (Fig. 6). The design specifications dB for GHz GHz, are dB for GHz GHz, and dB for GHz GHz. The initial mm. design is B. Experimental Setup For each of the test problems, we performed space-mapping optimization using the adaptive surrogate model selection scheme and the algorithm described in Section III. Table I shows the candidate model sets for each of the problems. For the sake of comparison, we also solved our problems using each of the candidate surrogate models separately. For all the problems, we have set , where , and ( is empty if , i.e., ). Thus, roughly 2/3 of the available points are used to assess approximation quality, as well as to determine the next surrogate model, while 1/3 of the points are used to assess extrapolation capability of the model. Obviously, there is a number of other choices available; the above is just one of many reasonable ones. Among other benefits, it allows reuse of space-mapping parameters for the winning candidate model without performing additional parameter extraction.

the naming convention for surrogate models (e.g., Bcd ) is explained in Section II. the results marked bold are acceptable solutions; the rest of the results are considered not acceptable with respect to the given specification.

C. Experimental Results Table II shows the results of our experiments, i.e., the objective function value (minimax error) and the number of fine model evaluations necessary to obtain the solution for problems 1–3. As an illustration, Table III shows, for test problem 3, the actual sequence of surrogate models used in subsequent iterations of the algorithm, as well as the corresponding combined quality factors compared to quality factors averaged over the whole set of candidate models. Note that the number of iterations does not correspond to the number of fine model evaluations shown in Table II because we use a trust region approach and there may be more than one fine model evaluation per iteration. Figs. 7–9 show the fine model responses at the initial and final solution obtained using our space-mapsolution ping algorithm with adaptive model selection for problems 1–3, respectively.

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TABLE III SURROGATE MODEL EVOLUTION FOR TEST PROBLEM 3

the naming convention for surrogate models (e.g., Bcd ) is explained in Section II. F K F is the average combined quality factor (i.e., the combined quality factor averaged over all candidate surrogate models).

=

Fig. 9. Optimization results for the microstrip bandpass filter with double-coupled resonators problem using adaptive surrogate model selection: initial solution (dashed line) and final solution (solid line).

TABLE IV PERFORMANCE COMPARISON: ADAPTIVE SPACE MAPPING VERSUS FIXED SURROGATE MODEL

Values averaged over the whole candidate set for a given test problem. Fig. 7. Optimization results for the six-section H -plane waveguide filter problem using adaptive surrogate model selection: initial solution (dashed line) and final solution (solid line).

Fig. 8. Optimization results for the microstrip bandpass filter problem using adaptive surrogate model selection: initial solution (dashed line) and final solution (solid line).

model evaluations than the best fixed-model algorithms to obtain a solution of similar quality. Note that some of the space-mapping algorithms with a fixed model fail to find an acceptable solution. In some cases (problem 2), this applies to the most flexible surrogate model . In some cases (problems 2 and 3), most of the fixed-model algorithms give results that are not acceptable. This means that choosing the surrogate model type “at random” may lead to inadequate performance of the algorithm. On the other hand, because choosing the surrogate model without prior knowledge and experience is almost the same as a random choice, it is fair to make a comparison between the adaptive space-mapping approach and the average performance of the fixed-model algorithm. Table IV provides this kind of comparison. It shows, in particular, that the average performance of the fixed-model space-mapping algorithm is much worse than the performance of the space-mapping algorithm with adaptive surrogate model selection for all considered test problems. V. CONCLUSION

D. Discussion The results shown in Table II indicate that the algorithm working with adaptive model selection performs either better than the algorithm using any fixed surrogate model (problem 1) or almost as good as the best algorithms using a fixed surrogate model (problems 2 and 3). In the latter case, our space-mapping algorithm with adaptive model selection requires fewer fine

A novel adaptive surrogate model selection procedure has been presented. The proposed technique allows us to adjust the type of space-mapping surrogate model used in a given iteration based on the approximation/extrapolation capability of the model. The technique does not require any extra fine model evaluations. Examples verifying the performance of our approach are provided. It follows that our adaptive surrogate

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model selection improves the performance of the space-mapping optimization algorithm. It prevents a bad choice of the space-mapping type. The algorithm working with our adaptive space mapping never failed to find a solution close to the optimal one. Failures happened to the algorithms working with a fixed space-mapping type, and this is exactly what may occur when the space-mapping type is wrongly chosen. The optimization results obtained with adaptive surrogate model selection are comparable to or better than the results obtained with a fixed space-mapping type.

ACKNOWLEDGMENT The authors thank Agilent Technologies, Santa Rosa, CA, for making ADS available, and Dr. W. J. R. Hoefer, Faustus Scientific Corporation, Victoria, BC, Canada, for making the Faustus MEFiSTo software available. The authors acknowledge discussions with N. K. Nikolova, McMaster University, Hamilton, ON, Canada, and thank J. Zhu, McMaster University, for setting up the microstrip filter examples.

REFERENCES [1] J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R. H. Hemmers, “Space mapping technique for electromagnetic optimization,” IEEE Trans. Microw. Theory Tech., vol. 4, no. 12, pp. 536–544, Dec. 1994. [2] J. W. Bandler, Q. S. Cheng, N. K. Nikolova, and M. A. Ismail, “Implicit space mapping optimization exploiting preassigned parameters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 378–385, Jan. 2004. [3] J. W. Bandler, Q. S. Cheng, D. H. Gebre-Mariam, K. Madsen, F. Pedersen, and J. Søndergaard, “EM-based surrogate modeling and design exploiting implicit, frequency and output space mappings,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1003–1006. [4] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H. Bakr, K. Madsen, and J. Sondergaard, “Space mapping: The state of the art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 337–361, Jan. 2004. [5] D. Echeverria and P. W. Hemker, “Space mapping and defect correction,” Int. Math. J. Comput. Methods Appl. Math., vol. 5, no. 2, pp. 107–136, 2005. [6] N. M. Alexandrov and R. M. Lewis, “An overview of first-order model management for engineering optimization,” Optim. Eng., vol. 2, no. 4, pp. 413–430, Dec. 2001. [7] A. J. Booker, J. E. Dennis, Jr., P. D. Frank, D. B. Serafini, V. Torczon, and M. W. Trosset, “A rigorous framework for optimization of expensive functions by surrogates,” Structural Optim., vol. 17, no. 1, pp. 1–13, Feb. 1999. [8] J. E. Dennis and V. Torczon, “Managing approximation models in optimization,” in Multidisciplinary Design Optimization, N. M. Alexandrov and M. Y. Hussaini, Eds. Philadelphia, PA: SIAM, 1997, pp. 330–374. [9] S. J. Leary, A. Bhaskar, and A. J. Keane, “A knowledge-based approach to response surface modeling in multifidelity optimization,” Global Optim., vol. 26, no. 3, pp. 297–319, Jul. 2003. [10] S. E. Gano, J. E. Renaud, and B. Sanders, “Variable fidelity optimization using a Kriging based scaling function,” presented at the 10th AIAA/ISSMO Multidisciplinary Anal. Optim. Conf., Albany, NY, 2004, Paper AIAA-2004-4460. [11] T. W. Simpson, J. Peplinski, P. N. Koch, and J. K. Allen, “Metamodels for computer-based engineering design: Survey and recommendations,” Eng. with Comput., vol. 17, no. 2, pp. 129–150, Jul. 2001.

[12] N. V. Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidynathan, and P. K. Tucker, “Surrogate-based analysis and optimization,” Progr. Aerosp. Sci., vol. 41, no. 1, pp. 1–28, Jan. 2005. [13] S. J. Leary, A. Bhaskar, and A. J. Keane, “A constraint mapping approach to the structural optimization of an expensive model using surrogates,” Optim. Eng., vol. 2, no. 4, pp. 385–398, Dec. 2001. [14] M. Redhe and L. Nilsson, “Using space mapping and surrogate models to optimize vehicle crashworthiness design,” presented at the 9th AIAA/ISSMO Multidisciplinary Anal. Optim. Symp., Atlanta, GA, Sep. 2002, Paper AIAA-2002-5536. [15] 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. [16] 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. [17] S. Koziel, J. W. Bandler, and K. Madsen, “Towards a rigorous formulation of the space mapping technique for engineering design,” in Proc. Int. Circuits Syst. Symp., Kobe, Japan, May 2005, pp. 5605–5608. [18] ——, “Space mapping optimization algorithms for engineering design,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 1601–1604. [19] 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. [20] J. E. Rayas-Sanchez, “EM-based optimization of microwave circuits using artificial neural networks: The state-of-the-art,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 420–435, Jan. 2004. [21] J. E. Rayas-Sanchez, F. Lara-Rojo, and E. Martinez-Guerrero, “A linear inverse space-mapping (LISM) algorithm to design linear and nonlinear RF and microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 960–968, Mar. 2005. [22] 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. [23] 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. [24] K.-L. Wu, Y.-J. Zhao, J. Wang, and M. K. K. Cheng, “An effective dynamic coarse model for optimization design of LTCC RF circuits with aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 393–402, Jan. 2004. [25] 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. [26] M. Dorica and D. D. Giannacopoulos, “Response surface space mapping for electromagnetic optimization,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1123–1126, Apr. 2006. [27] A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust Region Methods, ser. MPS–SIAM Optim. Philadelphia, PA: SIAM, 2000. [28] S. Koziel, J. W. Bandler, and K. Madsen, “Space-mapping based interpolation for engineering optimization,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2410–2421, Jun. 2006. [29] M. H. Bakr, J. W. Bandler, N. Georgieva, and K. Madsen, “A hybrid aggressive space mapping algorithm for EM optimization,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2440–2449, Dec. 1999. [30] “MEFiSTo-3D Pro,” ver. 4.0, Faustus Sci. Corporation, Victoria, BC, Canada, 2004. [31] A. Hennings, E. Semouchkina, A. Baker, and G. Semouchkin, “Design optimization and implementation of bandpass filters with normally fed microstrip resonators loaded by high-permittivity dielectric,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1253–1261, Mar. 2006. [32] “FEKO User’s Manual, Suite 4.2” EM Softw. Syst. S.A. (Pty) Ltd., Stellenbosch, South Africa, 2004 [Online]. Available: http://www.feko.info [33] Agilent ADS. ver. 2003C, Agilent Technol., Santa Rosa, CA, 2003.

KOZIEL AND BANDLER: SPACE-MAPPING OPTIMIZATION WITH ADAPTIVE SURROGATE MODEL

Slawomir Koziel (M’03) received the M.Sc. and Ph.D. degrees in electronic engineering from Gdansk University of Technology, Gdansk, Poland, in 1995 and 2000, respectively, and the M.Sc. degree in theoretical physics and mathematics and Ph.D. degree in mathematics (with honors) from the University of Gdansk, Gdansk, Poland, in 2000, 2002, and 2003, respectively. He is currently a Research Associate with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada. He has authored or coauthored over 90 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, London, U.K. 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 currently Professor Emeritus. Until November 20, 1997 (the date of acquisition by the Hewlett-Packard Company), he was President of Optimization Systems Associates Inc., which he founded in 1983. He is President of Bandler Corporation, Dundas, ON, Canada, which he founded in 1997. Dr. Bandler is a Fellow of several societies including the Royal Society of Canada. He was the recipient of the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Microwave Application Award.

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Ceramic Layer-By-Layer Stereolithography for the Manufacturing of 3-D Millimeter-Wave Filters Nicolas Delhote, Dominique Baillargeat, Member, IEEE, Serge Verdeyme, Member, IEEE, Cyrille Delage, and Christophe Chaput

Abstract—Three-dimensional (3-D) ceramic stereolithography has been developed and used to manufacture RF devices using periodic structures. To our knowledge, it is the first time that zirconia and high-performance Ba3 ZnTa2 O9 ceramics are used with that process to manufacture high unloaded quality factor resonant structures and bandpass filters working in the -band. A theoretical study on the performances of periodic arrangements of high-permittivity structures has been carried out and has led to the manufacture of a high unloaded quality factor cavity and narrow three-pole filter (1% bandwidth) measured at a working frequency of 33 GHz. Index Terms—Ba3 ZnTa2 O9 (BZT), Bragg reflector, ceramic stereolithography process, high unloaded quality factor, narrow bandpass filter, periodic structure, 3-D ceramic technology, zirconia.

I. INTRODUCTION

T

HREE-DIMENSIONAL (3-D) ceramic stereolithography is usually applied to the biomedical and mechanical domains with well-known zirconia and alumina ceramics. The aim of this study is to validate the use of such a process, with its current performances, for the manufacturing of high unstructures and RF filtering loaded quality factor resonant devices. In order to make them as compact and efficient as possible, high-permittivity and low-loss ceramics are required. High-frequency dielectric resonators are usually made out of such high-performance dielectrics such as Ba ZnTa O (BZT), but to our knowledge, it is currently used for the manufacturing of simple cylinder-shaped resonators by standard processes such as pressing or molding. It is the first time that 3-D ceramic stereolithography has been used to manufacture complex innovative filtering devices based on periodic arrangements [1] with high-performance ceramics. An air cavity surrounded by such spatial arrangement can be used as a resonant structure [2]. Careful cavity dimensioning allows to work at the chosen frequency [3], [4] and a high uncan potentially be achieved using the loaded quality factor proper dielectric because that parameter mainly depends on the dielectric losses [5].

Manuscript received October 17, 2006. N. Delhote, D. Baillargeat, and S. Verdeyme are with the XLIM Laboratory, Unité Mixte de Recherche, Centre National de la Recherche Scientifique 6172, University of Limoges, 87060 Limoges, France (e-mail: nicolas.delhote@ xlim.fr; [email protected]). C. Delage and C. Chaput are with the Centre de Transfert de Technologies, 87068 Limoges, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.891690

Previous works in the millimeter-wave domain [5]–[7] have shown that the number of periodic cells must be carefully chosen in order to avoid leakage through the structure. The use of a high dielectric constant ratio can be useful to limit the number of cells needed to avoid that kind of trouble. The study described in this paper is performed on compact structures using periodic dielectric plates. The main motivation is to integrate them in microwave devices while using as little metal as possible to achieve a high unloaded quality factor in the millimeter wavelength range. The first objective of this study is to design a high quality multipole bandpass filters factor resonant cavity and narrow using periodic structures. The second objective is to validate the use of the 3-D ceramic stereolithography process with its current accuracy in order to manufacture such devices first with zirconia and then with BZT. This paper is an extension of [8], previously presented at the 2006 European Microwave Conference (EuMW). More measurements have been performed on a single cavity resonant structure and multipole filters made of zirconia. Further studies on the enhancement of the unloaded quality factor have been carried out and has led to the use of the low-loss and high-permittivity BZT. Its compatibility with the 3-D ceramic stereolithography process has been studied and tested. To our knowledge, it is the first time this ceramic has been used with stereolithography for the design of RF filtering structures. Section II describes the manufacturing technique used. In Section III, periodic structures are studied and designed in order to create high quality factor resonant cavities while using highpermittivity ceramics (zirconia and BZT). In Section IV, a threepole narrow filter is designed, manufactured and tested. A conclusion is then presented in Section V. All theoretical studies are performed with 3-D finite-element method simulations applying a homemade software developed at the XLIM Laboratory, Unité Mixte de Recherche, Centre National de la Recherche Scientifique, University of Limoges, Limoges, France. All structures described in this paper are designed at the XLIM Laboratory and manufactured at the Centre de Transfert de Technologies (CTTC), Limoges, France.

II. STEREOLITHOGRAPHY PROCESS The presented manufacturing process appears to be innovative because of its ability to use ceramic powders, which present interesting RF performances, and to build complex structures. This kind of process has already been applied in the RF domain, but more often with resins [9].

0018-9480/$25.00 © 2007 IEEE

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Fig. 1. Principle of stereolithography. From [8] .

It is based on layer-by-layer ceramic stereolithography. This method is a fast prototyping technique [10], [11], which allows the fabrication of accurate 3-D structures, which cannot be made by standard shaping means such as pressing or molding. First of all, a digital 3-D file containing the dimensions of the structure is created. This digital image of the structure is sliced, each slice having a thickness from 25 to 200 m. The physical manufacturing then begins, consisting of the fabrication of the slice-by-slice (layer-by-layer) structure. A paste compound with ceramic powder, binders, monomers, and photosensitive resin [12] is deposited on the working surface (see Fig. 1) with a blade to give a good layer thickness (25 m for the presented structures). An Argon ionized UV laser, having a 100- m spot size, then draws on the layer, the pattern corresponding to the first slice of the 3-D structure by polymerizing the paste on its way. Another layer of paste is deposited above the previous one and the second layer pattern is drawn by the laser and so on until the whole structure is completed. The monomer surplus (i.e., the paste that was not exposed to the UV laser) is then removed and the resulting piece is debinded and sintered through several baking cycles [13], [14]. The average manufacturing accuracy is approximately 50 m with the current process and stays in this range for every manufactured piece, confirming the good reproducibility of this process. This parameter clearly is laser spot size dependant. Another laser having a 40- m-diameter spot is going to be used with the presented technology and will obviously improve the manufacturing accuracy, making it more likely close to 15 m. Considering the cost of such technology, it appears very interesting, especially when manufacturing complex 3-D structures. Indeed such devices would required extra manipulations with a standard manufacturing process and obviously extra costs when applying, for example, laser machining or other machining technics. The presented technology permits to make, in only one process, a monolithic part, which should have required different standard technologies to build equivalent structures. This process, developed for biomedical and mechanical applications using airconia and alumina ceramics, has to be adapted to the requirements of the RF domain. This study has been done by taking advantage of the partnership between the CTTC and XLIM laboratories. Moreover, in order to improve the performances of the presented structures, the development

Fig. 2. (a) Reflection coefficient of an normally incident EM wave on one-, two-, and four-period Bragg reflectors made out of Zirconia depicted in (b). Data from [8] in the table (c) sum up the reflection coefficient at 33 GHz.

of this process has been carried out with high-performance BZT ceramic. This study is a follow-up to what has already been presented in [8]. III. PRELIMINARY STUDIES A. Bragg Reflector One of our main objectives has been to validate the presented process for RF applications. A complex periodic structure has, therefore, been chosen in order to enhance the performances of the usual devices. The chosen ceramic for that purpose has been yttria-stabilized zirconia ZrO . It has been characterized at the XLIM Laboratory using the method described in [15] and presents a permittivity of 31.2 and a loss tangent of 1.8 10 at 30 GHz. This ceramic is a good candidate for periodic structures because of the high dielectric ratio given by this material compared to the air and because of its very good compatibility with ceramic stereolithography. Standard Bragg reflectors are studied with that ceramic: they consist of 1-D periodic arrangement of dielectric plates, each of them having a thickness equal to the quarter of the working and separated by a distance equal to the wavelength quarter of the working wavelength in free space . As it is well-known [16], a forbidden frequency band is created by this mean around the working frequency. Computations were performed with our homemade software based on the finite-element method. They were carried out considering a TM polarized wave normally incident to the reflector. Fig. 2 shows the reflection coefficient as a function of the frequency for a one-, two-, and four-period reflector dimensioned to provide a forbidden gap around 30 GHz. Thanks to the high-permittivity ceramic, a four-period reflector provides nearly 100% reflection for a TM polarized normally incident wave.

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Fig. 3. (a) Manufactured single-cavity central ceramic part on its brass metallic support (bottom metal plate). (b) Front/rear view of the whole structure.

B. Resonant-Cavity Design and Manufacturing Such reflectors are then used to make a resonant-cavity-based structure. Instead of using a standard and common air cavity shielded with metallic walls, we propose a hybrid structure composed of a central ceramic part made out of zirconia. This part consists of a resonant air cavity surrounding by a Bragg reflector, as shown in Fig. 3(a). Along the -axis, four-period reflectors are placed to strongly contain the field in the cavity. The thick fourth exterior wall of these four-period reflectors is to provide an extra mechanical robustness for this wall, which remains in free space. The air cavity is excited by a standard WR28 waveguide placed on each side of the central part along the -axis, and only two period reflectors are placed on this axis to excite the resonant mode of the cavity. The other parts of the structure are metallic plates put on the bottom and on the top [see Fig. 3(b)] of the ceramic central part. The whole structure can be easily connected to standard WR 28 waveguides on its front and rear faces for its measure [see Fig. 3(b)]. The main interest of this structure is that its performances not only depend on the metallic losses, but also on the dielectric ones, making it possible to reach a high unloaded quality factor with the proper dielectric. The cavity is dimensioned to work at 33 GHz on its fundamode and, therefore, has the following dimenmental 5.82 mm 3.56 mm. The reflectors prosions: 7.11 mm vide a forbidden frequency band around 30 GHz by having a thickness of 0.448 mm and by being separated by 2.5 mm. The whole ceramic part is 3.56-mm thick. The outside dimensions are 27.48 mm 26.5 mm 3.56 mm or (1) dB Fig. 4 presents the manufactured main part out of zirconia with its outside dimensions and the computed and measured transmission parameter. Brass is used for the plates S/m and the cavity was weakly excited in order to measure the unloaded quality factor. The manufactured dimensions are as follow: the cavity is 7 mm 5.86 mm 3.5 mm. Dielectric walls are 0.47-mm thick

Fig. 4. (a) Manufactured zirconia central ceramic part. (b) Computed (thin line) and measured (thick line) transmission parameter of the hybrid single cavity. (c) Close view of the measured (meas.) S -parameter.

and separated by 2.45 mm, making the average manufacturing accuracy close to 50 m. The outside dimensions are 27.2 mm 26.4 mm 3.5 mm. It represents a 0.95% manufacturing discrepancy compared to the theoretical values. The structure, measured with a Hewlett-Packard 85107A network analyzer, exhibits a working frequency of 33.564 GHz (1.7% shift compared to the theoretical value of 32.99 GHz) and a 3–dB bandwidth dB of 42 MHz. The scattering dB, parameters at 33.564 GHz are measured as dB, and dB. By using (1), the unloaded quality factor appears to be approximately 2400, whereas the theoretical value is 2800. However, this value is less than the given by a standard brass shielded air cavity having the 3000 same dimensions as the air cavity of the central ceramic part. Therefore, it is not satisfactory enough. Section IV of this study is focused on the enhancement of the for the presented structure. C. Enhancement of the Unloaded Quality Factor With this hybrid resonant structure, it has been deduced from the finite-element method analysis that roughly 60% of the total losses come from the dielectric and 40% from the metal. By providing a better conductive metal such as silver instead of brass for the top and bottom metal plates and keeping the same permittivity, it appears that the unloaded quality factor greatly

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Fig. 5. Evolution of the unloaded quality factor for the hybrid resonant structure as a function of the dielectric loss tangent. The top and bottom plates are chosen made out of silver and the dielectric permittivity is kept at 31.2.

Fig. 7. Cross section of BZT parts manufactured by stereolithography and sintered at T0 50 C (Temex Ceramics standard value) and BZT Temex Ceramics dielectric resonator.

6

Fig. 6. Manufactured pieces out of BZT made to verify its compatibility with the ceramic stereolithography process. (a) Three different parts are presented: 100-mm-thick plates, parallelepipeds, and complex structure composed of crossed bars. (b) Manufactured single-cavity resonant structure out of BZT having the same dimensions as the device presented in Section III-B.

rises as the dielectric loss tangent decreases (see Fig. 5). For relative permittivity of 31.2 and a loss tangent of 2 10 , the factor raises to 7000. It is very interesting because a silver shielded air cavity having the same dimensions exhibits only a 4500 quality factor. Such ceramic having the required characterexists and is known as BZT Ba ZnTa O . istics for a 7000 In [17], a commercially available dielectric resonator is characterized by a relative permittivity of 30.2 and a loss tangent of GHz , that actually means a loss tangent of 1.9 10 at 33 GHz. The obvious next step of this study is to determine if the presented structure can be manufactured by stereolithography out of BZT. D. Compatibility of BZT With Stereolithography Compatibility of that ceramic has been performed with Temex Ceramics’ BZT. The first step has been to manufacture various structures out of BZT in order to check if the versatility and accuracy of stereolithography is still valid with this new ceramic. Fig. 6 presents the manufactured pieces showing that simple structures (parallelepipeds or 100- m-thick plates), as well as more complex ones [the piece in the upper right corner in Fig. 6(a)] can be made with about the same accuracy as the structures made out of zirconia. Fig. 6(b) shows the manufactured single-cavity resonant structure presented in Section III-B, but this time out of BZT. This last device is much more complex than a BZT cylinder-shaped dielectric resonator usually made by pressing and has successfully been fabricated by ceramic stereolithography. Its dimensions are as follows: the

cavity is 7.01 mm 5.70 mm 3.51 mm. Dielectric walls are 0.49-mm thick and separated by 2.46 mm, making the average manufacturing accuracy close to 78 m. The outside dimensions are 27.33 mm 26.37 mm 3.51 mm. It represents a 0.75% manufacturing discrepancy compared to the theoretical values. This accuracy is currently being improved in order to be close to 50 m, which is the usual accuracy observed with zirconia structures. Unfortunately, it has not been measured in time to be presented in this paper. It appears that the microstructure strongly depends on that temperature. The BZT part fabricated by stereolithography and sintered at T0 C, T0 being the high temperature defined by the Temex Ceramics standard sintering procedure, exhibits approximately the same microstructure as the Temex Ceramics dielectric resonators (see Fig. 7). Based on this observation, a standard cylinder-shaped dielectric resonator was dimensioned and manufactured by 3-D ceramic stereolithography with a diameter of 8 mm and a height of 3.51 mm [see Fig. 8(a)]. This ceramic, while not being as mechanically resistant as the Zirconia, can easily be manipulated and inserted in a support without critical inconveniences. However, this mechnical parameter is clearly one of our concern and all the process is going to be optimized to maximize the ceramic resistance. The resonator made by stereolithography has been characterized at the XLIM Laboratory with a permittivity of 28.8. The mode resonator technique is used for the measure of the loss tangent with the dielectric resonator inserted in a copper shielded air cylinder-shaped cavity having a height and a diameter of 20 mm. Fig. 8(b) and (c) presents the measured transmission parameter with weak external coupling. mode of the resonator is 7.5382 GHz The and ( ), the 3–dB bandwidth is 782 kHz. dB. The is, therefore, approximately GHz . Actually, this mean a of 3.9 10 at 30 GHz, which is better than the loss tangent of the zirconia at 30 GHz), while having a permittivity of ( approximately 25.5. However, the Temex Ceramics’ resonator of GHz . This difference is due to exhibits a

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TABLE I EVOLUTION OF THE UNLOADED QUALITY FACTOR FOR TWO-, THREE-, FOUR-, AND FIVE-PERIOD REFLECTORS PLACED ALONG THE x-AXIS. COMPUTATIONS ARE PERFORMED FOR A SINGLE-CAVITY STRUCTURE MADE OUT OF BZT AND MADE OUT OF ZIRCONIA WORKING AT 33 GHz

Fig. 8. (a) Dielectric resonator made by ceramic stereolithography out of BZT (diameter: 8 mm, height: 3.51 mm). (b) Measured (meas.) transmission parammode is at 7.5382 GHz. The resonant mode at 9 GHz eter. The resonator TE is a cavity mode. (c) Close view of the transmission parameter at 7.5382 GHz.

carbonaceous residuals still existing after the debinding cycle and lowering the dielectric performances. Further investigations are currently performed in order to gain access to a loss tangent , which mean a of of approximately GHz 2.75 10 at 33 GHz. achievable Computations were then performed on the at with the potentially available BZT ( 33 GHz), while keeping the same dimensions as those used in Section III-B and the same brass support. Studying more as a function of the number of precisely the evolution of periods within the reflectors along the -axis [see Fig. 4(a)], the optimum unloaded quality factor achievable appears to be approximately 4500 for a four-period reflector (see Table I). obtained with This is approximately 58% higher than the the same structure made out of zirconia and is due to the lower loss tangent of BZT at 33 GHz. This result can be fully understood when referring to the reflection coefficient given by Bragg reflectors in Fig. 2. For three and less periods, they do not provide enough reflection and the leakage is, therefore, too important to correctly contain the TM polarized electrical field in the central air cavity. Five or more periods for these reflectors . It is interesting do not grant a significant enhancement on to note that a brass shielded air cavity having the same size as of only this surrounded by the Bragg reflectors provides a approximately 3000. The presented resonant structure appears to be bigger than this standard cavity, but its unloaded quality factor is 50% higher.

Fig. 9. (a) Manufactured three-pole central ceramic part in its brass support (i.e., bottom metallic plate). (b) Measured (thick line) and theoretical (thin line) responses of the three pole filter. From [8].

IV. NARROW BANDWIDTH FILTERS Using the presented resonant cavity, a three-pole filter is then designed. For that purpose, zirconia has been chosen because of its better current manufacturing accuracy by stereolithography and its sufficient characteristics to provide a narrow bandwidth -band. in the The three cavities required for the three-pole filter are placed along the -axis [8]. The coupling distance between them were dimensioned to provide approximately 1% bandwidth at 33 GHz and a ripple of 0.1 dB. Thus, cavities have the same dimensions as the previous ones and the spacing between the cavities is 2.85 mm. The outside dimensions are 27.48 mm 39.75 mm 3.56 mm. Brass tuning screws are placed above each cavity and each coupling spacing to refine the different coupling coefficient between the cavities [see Fig. 9(a)]. The manufactured dimensions 5.86 mm 3.45 mm. are as follow: the cavity is 7.1 mm Dielectric walls are 0.41-mm thick and separated by 2.52 mm.

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TABLE II COMPUTED AND MEASURED PERFORMANCES FOR THE THREE-POLE FILTER. DATA FROM [8]

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TABLE III COMPUTED AND MEASURED PERFORMANCES FOR THE TUNED THREE-POLE FILTER

V. CONCLUSION

Fig. 10. Measured (thick line) and theoretical (thin line) responses of the threepole filter for a greater tuning.

Finally, the spacing value is 2.9 mm, making the average manufacturing accuracy close to 45 m. The outside dimensions are 27.21 mm 39.61 mm 3.45 mm, which means approximately 1.4% manufacturing discrepancy. A 1.5% central frequency shift and modified couplings are observed on the manufactured structure. We present in Fig. 9(b) and Table II experimental results with minimal tuning provided by the tuning screws. As we can note, after tuning, the return loss in the band is approximately 17 dB and the bandwidth is around 1%. The measured insertion losses and ripple equal 1.7 and 1 dB, respectively, and are higher than the theoretical values (1 and 0.1 dB, respectively) (see Table II). These differences come from radiation leakage due to a gap between the ceramic piece and the upper and lower metallic plates. This has to be improved for future experimental structures. The low conductivity of the brass tuning screws is also involved. The screws can easily be used to compensate the discrepancy and shift the working frequency back to its initial value of 32.94 GHz. Fig. 10 and Table III show that the required frequency and bandwidth can easily be reached. However, the major insertion of the screws in this case brings much more insertion losses (3 dB instead of 1 dB), increases the ripple (0.5 dB instead of 0.1 dB), and limits the return loss to 10 dB instead of 16 dB. It is interesting to note that the input/output coupling can also be tuned by adding screws just before/after the first/last cavity The improvement of the manufacturing accuracy, currently under study, will, of course, reduce the use of the screws, and finally suppress them and, therefore, greatly improve the insertion losses and return loss.

In this study, two objectives have been completed. The first is the validation of the 3-D ceramic stereolithography for the fabrication of RF devices out of zirconia, but also out of the high-performance BZT. The manufactured accuracy is as low as 45 m and, therefore, the presented process is fully able to provide high unloaded quality factor resonant structures and narrow filtering devices for the millimeter-wave domain. It is the first time it has been brought to our knowledge that BZT has been successfully used with the ceramic stereolithography. The second objective is the efficient application of periodic structures for the design of high-frequency filtering devices. Four period Bragg reflectors made out of high-permittivity zirconia or BZT provide enough reflection for the creation of the high quality factor resonant structure needed for narrow filters with high electrical performances. The presented structures are compact, easily tunable, compatible with standard WR connections, and their working principle can be applied to integrated multiplexers. The study performed in this paper shows that, considering the innovative structures already available with zirconia, a 7000 unloaded quality factor can be achieved at approximately 33 GHz available with a with BZT. This value is 55% more than the silver shielded air cavity having the same size as this surrounded by the Bragg reflectors. With its manufacturing simplicity and versatility, the tested stereolithography process has proven its capabilities to manufacture complex innovative RF devices while being cheaper than building equivalent structures by common procedures such as laser machining. More developments are currently being performed to enhance the manufacturing accuracy and the electrical properties of structures made out of several ceramic materials such as zirconia, alumina, BZT, etc. ACKNOWLEDGMENT The authors thank P. Filhol and Temex Ceramics, Pessac, France, for their support in this study and for providing the BZT ceramic powder, as well as H. Jallageas and B. Casteignau, both with the XLIM Research Institute, Limoges, France, for their helpful contribution. REFERENCES [1] E. Yablonovitch, “Inhibited spontaneous emission in solid statephysics and electronics,” Phys. Rev. Lett., vol. 58, no. 20, pp. 2059–2062, May 1987. [2] E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett., vol. 67, no. 24, pp. 3380–3383, Dec. 1991.

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[3] R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B, Condens. Matter, vol. 44, no. 24, pp. 13772–13774, Dec. 1991. [4] P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B, Condens. Matter, vol. 54, no. 11, pp. 7837–7842, Sep. 1996. [5] W. J. Chappell, M. P. Little, and L. P. B. Katehi, “High two dimensional defect resonators—Measured and simulated,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, vol. 3, pp. 1437–1440. [6] B. Lenoir, D. Baillargeat, S. Verdeyme, and P. Guillon, “Finite element method for rigorous design of microwave devices using photonic bandgap structures,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1998, vol. 2, pp. 1061–1064. [7] M. Schuster and N. Klein, “Excitation of line and point defect modes in two-dimensional electromagnetic bandgap structures for microwave frequencies,” J. Phys. D, Appl. Phys., vol. 34, no. 3, pp. 374–378, Feb. 2004. [8] N. Delhote, D. Baillargeat, S. Verdeyme, C. Delage, and C. Chaput, bandpass filters made of high permittivity ceramic by “Narrow layer-by-layer stereolithography,” in Eur. Microw. Symp. Dig., Sep. 2006, pp. 510–513. [9] B. Liu, X. Gong, and W. J. Chappell, “Applications of layer-by-layer polymer stereolithography for the three-dimensional high-frequency components,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2567–2575, Nov. 2004. [10] C. Hull, “Apparatus for production of three dimensional objects by stereolithography,” U.S. Patent 4575 330, Mar. 11, 1986. [11] T. Chartier, C. Chaput, and F. Doreau, “Ceramic paste composition and prototyping method,” W.O. Patent 0042471, Jan. 11, 2000. [12] M. L. Griffith and J. W. Halloran, “Freeform fabrication of ceramics via stereolithography,” J. Amer. Ceram. Soc., vol. 79, no. 10, pp. 2601–608, 1996. [13] T. Chartier, C. Chaput, F. Doreau, and M. Loiseau, “Stereolithography of structural complex ceramic parts,” J. Mater., vol. 37, pp. 3141–3147, 2002. [14] F. Doreau, C. Chaput, and T. Chartier, “Stereolithography for manufacturing ceramic parts,” Adv. Eng. Mater., vol. 2, no. 8, pp. 493–496, 2000. [15] D. Thompson, O. Tantot, H. Jallageas, G. E. Ponchak, E. Tentzeris, and J. Papapolymerou, “Characterization of liquid crystal polymer (LCP) material and transmission lines on LCP substrates from 30 to 110 GHz,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1343–1352, Apr. 2004. [16] P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media: General theory,” J. Opt. Soc. Amer., vol. 67, no. 4, pp. 423–438, Apr. 1977. [17] J. Krupka, W.-T. Huang, and M.-J. Tung, “Complex permittivity measurements of low-loss microwave ceramics employing higher order modes excited in a cylindrical dielectric sample,” Meas. quasi-TE Sci. Technol., vol. 16, pp. 1014–1020, Mar. 2005.

Q

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

Ka

Nicolas Delhote was born in Limoges, France, in 1981. He received the Master’s degree in high frequency and optical telecommunications from the University of Limoges, Limoges, France, and is currently working toward the Ph.D. degree with the XLIM Laboratory, University of Limoges. His research interests are dedicated to the design of original filters and resonant structures using 3-D manufacturing process, high performances ceramics and electromagnetic-bandgap (EBG) concept.

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

Cyrille Delage was born in La Rochefoucauld, France. In 1974, he became a Ceramic Engineer with the Ecole Nationale Superieure de Ceramique Industrielle (ENSCI). He is currently with the Centre de Transfert de Technologies (CTTC), Limoges, France, where he is in charge of research programs on stereolithography process applied on ceramic materials. He possesses seven years of professional experience.

Christophe Chaput was born in Paris, France, in 1967. He is a currently a Ceramic Engineer with the Ecole Nationale Superieure de Ceramique Industrielle’ (ENSCI), where he is in charge of the direction of the Centre de Transfert de Technologies (CTTC), Limoges, France. He possesses 13 years professional experience. He has been involved with innovation and development for over ten years and has been involved in many research projects with European companies.

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Heat Conduction in Microwave Devices With Orthotropic and Temperature-Dependent Thermal Conductivity John Ditri

Abstract—This paper deals with steady-state heat conduction in integrated circuits due to surface heating. Temperature-dependent and orthotropic thermal conductivities are assumed, and an exact formula is obtained for the temperature field in the integrated circuit due to arbitrary surface flux loading. These general results are then specialized to specific forms of surface flux representative of multifingered FETs of arbitrary gate length, gatewidth, and pitch. Numerical results are presented for a representative transistor on a gallium–arsenide substrate and comparisons are made, where possible, to existing approximate solutions, as well as to finite-element results. The examples are chosen to highlight the effects of orthotropy and temperature dependence of thermal conductivity on the junction temperature of devices on integrated circuits. Index Terms—Fourier series, orthotropic, temperature dependent, transistor.

I. INTRODUCTION

HE ROLE that temperature plays on both the performance and ultimate life of electronic components is well known. Excessive channel temperatures in FETs can cause near instantaneous burnout, whereas elevated temperatures can cause gradual degradation and reduction in the life of the device, as well as degraded performance during operation. Due to this, much work has been done on the development of analytical and numerical methods of predicting the operating temperatures of devices subject to specific power dissipations, as well as to experimental techniques to measure the actual resulting temperatures. Previous analytical treatments of heat conduction in semiconductor devices can be separated into many categories, but two main separations are those methods of an approximate nature, represented by [1]–[4], and others that are exact within the context of the theory of heat conduction. Some representative papers of this type are [5]–[7], [10]. There are, of course, numerous other works in both categories, as well as a third category of purely numerical techniques, such as finiteand boundary-element methods. Reference to much of this literature can be found in this paper’s Reference section and their references therein. A collection of analytical solutions to many

T

Fig. 1. Rectangular body subject to heat flux loading on its upper surface, adiabatic sidewalls, and a fixed temperature at its base.

fundamental heat conduction problems can be found in [11] and [12]. The current work falls into the class of exact solutions, and is distinguished from most of the previous literature [5]–[10] by its focus on the inclusion of both orthotropic (i.e., different values of thermal conductivity along three mutually perpendicular coordinate directions) and temperature-dependent thermal conductivity. These two factors have been shown in the past to individually influence the ultimate junction temperature of FETs. It is the goal of this paper to incorporate these two effects into an exact, relatively simple, and compact formula for the temperature distribution in a microwave chip subject to arbitrary surface flux. It is a further goal to reduce this general formula to the specific case of a multifingered FET. Such a formula can be used by monolithic microwave integrated circuit (MMIC) designers during the design phase of microwave devices to predict the effect of various FET parameters on junction temperature. By providing exact benchmark results, this study also aids with finite-element mesh convergence studies, which are usually performed as the first step in the solution of problems with very complex geometries by finite-element methods. The mesh of the numerical model is refined until its prediction for the single FET is close enough to that predicted by the exact formula. After this is achieved, the remainder of the model can be built and meshed. A. Problem Statement

Manuscript received June 6, 2006; revised October 17, 2006. The author is with MS2, Lockheed Martin, Moorestown, NJ 08057 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.2006.890526

The problem addressed in this paper is depicted in Fig. 1. A in the - rectangular parallelepiped of dimensions and -direction, respectively, is subjected to an arbitrary distribution of surface flux on its upper surface, while its lower

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surface is presumed to be kept at a specified temperature, i.e., . The sides of the solid are assumed to be perfectly insulated such that there is no heat flux passing through them. Finally, the thermal conductivity of the solid is assumed to be orthotropic and temperature dependent, in a form to be discussed later. This particular geometry is an idealization of RF/MMIC circuits, where the heat-producing elements are mostly resistors and various types of FETs, which are produced on the upper (or very near upper) surface of the die using photolithographic imaging and deposition techniques. Such devices are ubiquitous throughout the modern commercial electronics industry, as well as in military applications. A typical device may have many heat-producing regions on its upper surface, primarily in the form of transistors and resistors. The lower surface of such devices, especially for high-powered devices used in RF and microwave transmitters and receivers, are typically mounted to a cold plate, the temperature of which is controlled by external means. It should be noted that many devices are more complicated than this simple model in the sense that they may contain multiple layers of conductors/insulators including a thin layer of top surface metallization and air bridges at the FETs. In addition, MMICs are often placed as the top component in a material stackup with several other conductors beneath them before the temperature-controlled surface is reached. Inclusion of these effects would require solving a composite medium problem, which is tedious, but which presents no insurmountable theoretical difficulties, provided the thermal conductivities of the different media are constant. In fact, several fairly comprehensive treatments of the multiple layer problem are available [5]–[7]. Composite medium problems with temperature-dependent material properties cannot, however, be solved by the usual Kirchoff transformation [13] and, therefore, the available exact solutions are all for materials with temperature-independent thermal conductivity. The mathematical statement of the current problem can be written as follows: (1)

The problem described by (1)–(5) is nonlinear because of the dependence of the thermal conductivity on temperature. Note that this statement of the problem assumes that the exact distribution of heat sources on the top surface of the die are known. In general, the exact distribution of heat within the die can only be obtained via a solution of coupled differential equations (Poisson equation, continuity equation, electron/hole generation, recombination equations, etc.) describing the potential and field distributions throughout the high field regions of the transistors, coupled with assumptions on electron mobilities versus applied fields [8], [9]. However, given the somewhat inexact knowledge of some of the basic material properties of the die, such as its thermal conductivity (versus temperature, if temperature dependent), low field mobility, electron velocity-field dependence, etc., it seems that the current approach of assuming a priori that the heat is contained to within the transistor gate regions will provide sufficient accuracy for the purpose of designing thermal margin into the transistor devices. B. Formal Solution Before beginning with the solution, an important simplifying assumption must be made concerning the temperature dependence of the principal thermal conductivities. It is assumed that , , and all vary with temperature in the same fashion, i.e., that they can be written in the form (6) with . is at this point an unspecified reference function of temperature, and the ’s are constants. Use is now made of the following transformation [13]: (7) where is, in general, an arbitrary constant, chosen in this case to be the same temperature appearing in (2) because that will homogenize that boundary condition. Utilizing (6) and (7), (1) becomes (8)

with boundary conditions (2) (3) (4)

which is a linear partial differential equation in the new variable . Note that is an explicit function of temperature , but it is also implicitly dependent upon the spatial coordinates , , and because the temperature field varies spatially. Finally, in order to reduce (8) further, the following change of coordinates is introduced:

(5) In the above, , , and represent the , , and components of the temperature-dependent thermal con, ductivity tensor of the assumed orthotropic material ( ), and is an arbitrary function (subject to certain restrictions later) representing the distribution of heat flux at the . A posupper surface (energy per unit area, per unit time) represents heat flux into the solid. The itive value of and lateral dimensions of the die are and , respectively.

(9) Utilizing (9) reduces (8) to the standard Laplace equation (10) The boundary conditions must also be transformed by (7) and the change of variables (9) in order to complete the specification

DITRI: HEAT CONDUCTION IN MICROWAVE DEVICES

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The general solution of (8), which also satisfies the boundary conditions (14)–(16), can be written

TABLE I REFERENCE VALUES OF K AND  FOR SEVERAL COMMON SEMICONDUCTOR MATERIALS. T = 300 K

(18) of the boundary value problem governing the new dependent . variable Before considering how the boundary conditions transform, a specific form of the dependence of thermal conductivity on temperature will be given. To keep matters somewhat genwill only be assumed to be exeral, the dependence of pressible in the general form

are at this point arbitrary constants, and

where defined as

(19) Transforming back to physical variables, (18) becomes

(20)

(11) where and are arbitrary constants ( is not to be confused with the thermal diffusivity, which will not appear in this is an arbitrary referproblem because it is steady state), and ence temperature, and all temperatures are in an absolute scale. The form chosen in (11) has been shown [3] to closely approximate the actual thermal conductivities of several well-known semiconductor materials, some of which are given in Table I. Substituting (11) into (7) results in

(12) The inverse of (12), giving the actual temperature in terms of the transformed dependent variable can be written

are

What remains is to satisfy the remaining boundary condition (17). This is achieved in the usual manner of substituting (18) into (17) and recognizing that the resulting expression is the . double cosine expansion of the function This enables the calculation of the expansion coefficients in the form

(21) where (22) and represents the Kronecker delta, equal to 1 if and 0 otherwise. Finally, making use of the inverse transformation (13) results in the formal solution for the temperature in the form

(13)

Utilizing (12) and the change of variables (9), boundary conditions (2)–(5) are transformed into (23) (14)

for

or

(15) (16) (17)

(24) for

. II. PARTICULAR CASES

where

,

,

, and

. Equations (8), and (14)–(17) are now in a form readily solvable by the customary method of separation of variables.

A. Rectangular Area The simplest case of surface flux loading is a rectangular area , in the - and -direction, respectively, with of extent

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, width , and the center-toEach finger is of length center gate spacing is denoted . If the centroid of the first , then the centroid of the last finger is located at finger is located at . If is equal to for points within any gate region and zero outside of all gates, the resulting expansion coefficients are given by

Fig. 2. Variables defining a rectangular patch load.

(27) These expansion coefficients can be evaluated more explicitly by making use of the following closed-form expression for the finite sum over the index [14]

(28) which only needs to be slightly modified to put it in the form required in (27). Utilizing (28) in (27) reduces it to

Fig. 3. Variables defining an x-oriented FET.

centroid at , , and of constant magnitude (W/m in the meter, kilogram, seconds (MKS) system of units), as shown in Fig. 2. can be expressed as In this case, the function if otherwise.

(29) where the coefficients

are defined as

(25)

Substituting (25) into (21) yields the expansion coefficients for the rectangular area in the form

(26) Further substituting this result into (23) or (24) yields the temperature at any location within the MMIC. The final formula that results is the generalization to orthotropic and temperature-dependent materials of the results obtained in [10] for isotropic and temperature-independent materials. It should be noted that the expansion coefficients appearing in (26) are undefined as and/or approach 0. However, once term effectively (26) is substituted into (23) or (24), the removes this singular behavior, and the resulting expressions for temperature are regular for any values of and provided the appropriate limits are taken as and/or approach 0. B. Multifinger -Oriented FET The extension of the results (26) to the case of a multifingered FET structure is straightforward. Consider the case of rectangular gate fingers oriented in the coordinate direction (shown in Fig. 3).

(30) It is easily shown that if the number of fingers simplifies to , and (29) reduces to (26).

,

C. Multifinger -Oriented FET The case of a -oriented FET follows immediately from that of the -oriented FET by simple substitutions. If the centroid , then the of the first gate finger is again located at centroid of the last finger is located at . If is equal to for points within any gate region and zero outside of all gates, the resulting expansion coefficients are given by

(31)

DITRI: HEAT CONDUCTION IN MICROWAVE DEVICES

Fig. 4. Comparison of the current exact analytical approach with numerical and approximate analytical approaches for a single FET with nonlinear thermal conductivity.

III. NUMERICAL EXAMPLES Two specific examples are now presented to highlight the role that temperature-dependent and anisotropic material properties play in the conduction of heat in MMICs. For the first example, consider a GaAs FET with fingers and other dimensions given by gate length m, gatewidth m, gate pitch m, and total dissipated power W (therefore, W m ). The MMIC dimensions m, m. The FET were assumed to be was centered on the MMIC, and the base temperature of the MMIC, i.e., , was varied from 0 C to 100 C. Shown in Fig. 4 is the junction temperature predicted by the current analysis (at the center of the middle finger), as well as that predicted by ANSYS, a commercial finite-element analysis program, and the simplified approximate, method of [1]. Also plotted in Fig. 4 is the predicted temperature, using the current analysis, for temperature independent material properties , where a constant thermal conductivity equal to 57 W/m K (which is the value of the thermal conductivity of GaAs at 0 C) was assumed. Fig. 4 shows that the current exact approach matches the finite-element results over the entire range of , whereas the approximate formula predicts junction temperatures approximately 5% to 17% higher. It can also be seen that, due to the material nonlinearity, each 10 C rise in MMIC base temperature results in an approximately 16 C rise in junction temperature. For the linear thermal conductivity case, the 10 C rise in base temperature results in an equal 10 C rise in junction temperature, as is to be expected since the thermal resistance is, in this case, a constant. The second example examines the effect of anisotropy on junction temperature. The same basic FET structure on a GaAs die was modeled, and the thermal conductivities in the -, -, and -direction were varied from 100% of their nominal values down to 30% of their nominal values. This was simply achieved , , and from 1.0 down to 0.3 by varying the constants individually, while the other two conductivity multipliers were held fixed at 1.0.

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Fig. 5. Effect of material anisotropy on junction temperature. Solid curves: current theory. Dashed curves: from finite-element analysis.

It should be noted that GaAs is normally modeled as isotropic, but this example is presented to show the applicability of the current results to other potentially orthotropic substrates. The results are shown in Fig. 5, along with the junction temperatures obtained using ANSYS. The baseplate temperature was held constant at 50 C, and nonlinear material properties were used. It can be seen that agreement between the current theory and the finite-element analysis is good. For the case of varying , the increasing discrepancy between the current exact approach and the finite-element results are due to the fact that the mesh of the finite-element analysis was fixed as the conductivity was reduced. Reducing the -component of thermal conductivity significantly increases the thermal gradients near the gate regions to the point where there were not sufficient elements across the gate length to sufficiently resolve the temperature. It is interesting to note from Fig. 5 that the -direction conductivity has the largest effect on heat transfer away from the FET, which is understandable since it directly affects the path between the heated top surface of the MMIC and the fixed temperature sur. face at IV. CONCLUSION The solution derived in this paper for heat conduction in materials with orthotropic and temperature-dependent thermal conductivity can serve as a valuable tool in MMIC development where relatively quick, but highly accurate, junction temperatures are required in order to asses the suitability or optimality of a particular MMIC layout. As can be seen from Figs. 4 and 5, the current exact analytical approach yields results that are very close to those obtained by finite-element analysis. Which approach yields the more correct answer cannot be concluded from those figures because those results fundamentally depends upon the mesh used in the finite-element approach and on the number of terms retained in the summation for the analytical approach. Fig. 5 shows the importance of material anisotropy on channel temperature for transistors. One important conclusion from Fig. 5 is that, where possible, the FETs should be oriented with their widths (longer dimension) in the direction of lowest thermal conductivity. The gates themselves (shorter dimension) should

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be oriented in the direction with highest (best) thermal conductivity to ensure better spreading and reduction of the high heat gradients near the channels. The main advantages of the current exact analysis, relative to approximate analytical techniques, is that it is free of limiting assumptions about MMIC sizes (usually assumed infinite in approximate analyses), FET gate geometry (width/length is usually assumed to be very large), and number of fingers (many fingers usually required to justify adiabatic boundary conditions). This exact analysis is also readily extendable to more complex MMIC designs such as two or more nearby FETs or other heat sources, which may be close enough to affect each other thermally. One simply adds the expansion coefficients from each heat source to obtain the total solution. When compared to numerical techniques, such as the finiteelement method, the main advantages are the time savings from not having to build the detailed models, as well as not having to climb the steep learning curve for becoming proficient with such programs. Of course, the cost of the numerical programs is also considerable, whereas the exact solution can be programed on any PC. On the other hand, numerical techniques are much more versatile than purely analytical approaches, allowing for much more complex geometries, material properties, and loading scenarios than could possibly be solved analytically [15], [16]. For this reason, the current approach is meant to compliment the numerical approaches, perhaps as an aid during initial device development, to be followed by more a detailed numerical analysis of the proposed final design. It should be noted, however, that despite its advantages, there are some significant disadvantages to the Fourier series approach. It is well known that the convergence behavior of the resulting double Fourier series can be extremely slow, requiring many thousands to millions of terms to stabilize to the 0.5% range. This is especially true for evaluation points, which are located close to the heat-generating regions where thermal gradients are large. This characteristic of Fourier series solutions is well known and documented. For example, in [7], the convergence behavior of series resulting from a similar problem was studied and it was found that the number of terms required for suitable convergence was a function of ratio of MMIC size to heat source size. This was also found numerically in the current study. For the figures presented above, 8 million terms were included in the summations, 10 000 in ( -direction), and 800 in ( -direction). This gave convergence to the 0.4% level (i.e., adding additional terms changed the answer by less than 0.4%). For 1% convergence, typically on the order of 4 million terms were required. Summation of so many terms would have been impractical several years ago, but with modern PCs, this takes a matter of minutes. Typical run times were on the order of 7.3 min/point on a personal computer with a 3.6-GHz Xeon processor. The large discrepancy in the number of terms required for convergence in the and summations is due to the fact that the FET gatelengths are in the -direction and, hence, the largest thermal gradients are likewise in the -direction. This is one additional problematic issue with exact approaches, namely, that achieving convergence is not simply a matter of adding more terms, rather, it is important that sufficient terms in the high gradient directions be added.

When such a large number of terms are summed, not only is time of concern, but the control of roundoff errors also becomes an issue. Although no particular measures were taken to prevent them in the current study (except for working entirely in double precision), some cases were found where slight changes in inputs (such as die size) resulted in large changes in predicted junction temperatures. It is believed, however, that as serious as they are, both the run time and accuracy concerns of the exact approaches can be addressed through careful programming techniques and, perhaps, by applying convergence acceleration techniques. REFERENCES [1] H. F. Cooke, “Precise technique finds FET thermal resistance,” Microw. RF, pp. 85–87, 1986. [2] F. Masana, “A closed form solution of junction to substrate thermal resistance in semiconductor chips,” IEEE Trans. Compon., Packag., Manuf. Technol. A, vol. 19, no. 4, pp. 539–545, Dec. 1996. [3] J. C. Freeman, “Channel temperature model for microwave AlGaN/GaN HEMTs on SiC and sapphire MMICs in high power, high efficiency SSPAs,” Glenn Reseach Center, NASA, Cleveland, OH, Tech. Rep. NASA TM-2004-212900, 2004. [4] A. M. Darwish, A. J. Bayba, and H. A. Hung, “Accurate determination of thermal resistance of FETs,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 306–313, Jan. 2005. [5] W. Batty, C. E. Christoffersen, A. J. Panks, S. David, C. M. Snowden, and M. B. Steer, “Electrothermal CAD of power devices and circuits with fully physical time-dependent compact thermal modeling of complex nonlinear 3-D systems,” IEEE Trans. Compon. Packag., Manuf. Technol. A, vol. 24, no. 4, pp. 566–590, Dec. 2001. [6] A. G. Kokkas, “Thermal analysis of multiple-layer structures,” IEEE Trans. Electron Devices, vol. ED-21, no. 11, pp. 674–681, Nov. 1974. [7] C. C. Lee, A. L. Palisoc, and Y. J. Min, “Thermal analysis of integrated circuit devices and packages,” IEEE Trans. Compon. Packag., Manuf. Technol. A, vol. 12, no. 4, pp. 701–709, Dec. 1989. [8] C.-S. Chang and D.-Y. Day, “Analytic theory for current–voltage characteristics and field distribution of GaAs MESFET’s,” IEEE Trans. Electron Devices, vol. 36, no. 2, pp. 269–280, Feb. 1989. [9] D. Denis, C. M. Snowden, and I. C. Hunter, “Coupled electrothermal, electromagnetic, and physical modeling of microwave power FETs,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 2465–2470, Jun. 2006. [10] R. D. Lindsted and R. J. Surty, “Steady-state junction temperatures of semiconductor chips,” IEEE Trans. Electron Devices, vol. ED-19, no. 1, pp. 41–44, Jan. 1972. [11] J. C. Jaeger and H. S. Carslaw, Conduction of Heat in Solids, 2nd ed. Oxford, U.K.: Oxford Univ. Press, 1959. [12] I. N. Sneddon, Fourier Transforms. New York: McGraw-Hill, 1951. [13] M. N. Ozisik, Heat Conduction. New York: Wiley, 1980. [14] Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products, 5th ed. San Diego: Academic, 1994, p. 36. [15] J. N. Reddy and D. K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, 2nd ed. London, U.K.: CRC, 2000. [16] R. W. Lewis, K. Morgan, H. R. Thomas, and K. N. Seetharamu, The Finite Element Method in Heat Transfer Analysis. New York: Wiley, 1996. John Ditri received the B.S., M.S., and Ph.D. degrees in applied mechanics from Drexel University, Philadelphia, PA, in 1988, 1990 and 1992, respectively. He then spent one year as an Instructional Post-Doctoral Scholar with Pennsylvania State University, followed by 12 years in industry, where he was involved in the design and analysis of equipment for experimental stress analysis, as well as high-speed semiconductor assembly equipment. In 2004, he joined MS2, Lockeed Martin, Moorestown, NJ, where he has been involved with the analysis and design of RF and microwave packaging solutions for advanced radar applications. He has authored or coauthored over 50 publications. He holds seven U.S. patents.

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Low-Loss Patterned Ground Shield Interconnect Transmission Lines in Advanced IC Processes Luuk F. Tiemeijer, Ralf M. T. Pijper, Ramon J. Havens, and Olivier Hubert

Abstract—In this paper, we provide an extensive experimental and theoretical study of the benefits of patterned ground shield interconnect transmission lines over more conventional layouts in advanced integrated-circuit processes. As part of this experimental work, we present the first comparative study taken on truly differential transmission line test structures. Our experimental results obtained on transmission lines with patterned ground shields are compared against a predictive compact equivalent-circuit model. This model employs exact closed-form expressions for the inductances, and describes key performance figures such as characteristic impedance and attenuation loss with excellent accuracy. Index Terms—Calibration, deembedding, integrated circuits (ICs), on-chip interconnect transmission lines, on-wafer microwave measurements.

I. INTRODUCTION ITH THE continuous increase of the operating frequencies of very large scale integrated circuits (ICs) the behavior of global interconnects becomes more important and (inductive) transmission line effects need to be taken into account in digital applications [1], whereas for the analog and RF circuit blocks, accurate interconnect transmission line models are required [2] to be able to quantify signal delay and loss. In conventional CMOS processes, unshielded coplanar waveguide (CPW) transmission lines realized in the thick top metals available for global power routing show signal losses of 10–20 dB/cm at 10 GHz due to dielectric losses in the silicon substrate [3]. Inclusion of a solid metal shield realized in the bottom metal layer reduces the signal loss to approximately 3 dB/cm at 10 GHz [2], [4]. However, as shown in this paper, eddy-current losses in these solid metal shields can make up a sizeable fraction of this signal loss in typical IC processes. Lowering the resistance of the ground shield by connecting two bottom metal layers in parallel reduces the loss further to 2 dB/cm at 10 GHz [5]. In general, however, these stacked metal ground shields raise line capacitance and reduce line characteristic impedance to less desirable levels. Up to 10 GHz, however, similar low-signal losses have also been reported when the shield is fabricated in a highly doped 8- sq buried layer [6], provided it has been patterned to prevent eddy-current losses and only allow current flow in the direction perpendicular

W

Manuscript received October 17, 2006. L. F. Tiemeijer and R. M. T. Pijper are with the Research Department, NXP Semiconductors, 5656 AE Eindhoven, The Netherlands (e-mail: [email protected]). R. J. Havens is with the Innovation Center RF, NXP Semiconductors, 6534 AE Nijmegen, The Netherlands. O. Hubert is with the Process and Library Technology Department, NXP Semiconductors Caen, 14079 Caen, France. Digital Object Identifier 10.1109/TMTT.2007.891691

to the line. Blocking the eddy currents enhances the ground inductance [6] and, therefore, patterned ground shields are now primarily promoted for differential circuit design [7]. Despite all attention [6]–[9], the reported benefits of patterned ground shield transmission lines have not yet been supported by electromagnetic (EM) modeling. In the course of this study, we found that the complex geometry of patterned ground shields provides a major roadblock for a straightforward analysis using conventional EM simulation software. Therefore, in this paper, we compare our experimental results obtained on various CPW interconnect transmission lines with an orthogonal patterned polysilicon grid acting as a ground shield against an accurate compact model based on an analytical closed-form expression for the transmission line inductance. Using a proven concept [10] for including current crowding and skin resistance effects combined with conventional interconnect capacitance modeling techniques, an excellent agreement with our experimental observations is achieved. Section II describes the basic layout and inductance modeling of the patterned ground shield CPW transmission lines, while the high-frequency compact model is discussed in Section III. Common-mode radiation loss and the final equivalent-circuit model is the topic of Section IV. This model is verified against measured data taken on single test lines in Section V. In Section VI, we present the first experimental comparative study taken on truly differential test lines, confirming the benefits of patterned ground shields for this type of transmission lines and illustrating the wide applicability of our modeling approach. Our results are summarized in Section VII.

II. TRANSMISSION LINE LAYOUT A cross section of the patterned ground shield CPW transmission lines studied in this paper is presented in Fig. 1. The signal current flows in the central conductor ( ) in the top metal of the IC process. The return current flows in two equal ground ( and ) conductors, which exist in all available metal layers. To reduce losses due to capacitive coupling to the conductive silicon substrate, a polysilicon patterned ground shield consisting of 1- m-wide bars at 1- m space perpendicular to the direction of the current flow, and connected to the ground conductors, is used. Inductance is defined based on magnetic fields generated by currents flowing in closed conductor loops. The partial-element equivalent-circuit approach [11], where this inductance is decomposed into a sum of partial self-inductance and mutual inductance, is well suited to model the transmission line (loop)

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Fig. 1. Cross section of the patterned ground shield CPW line. (top) Layout view. (bottom) current model view in TEM mode. The signal current flows in the central line (s) of width W and thickness T , whereas the return current flows in two ground lines (g and g ) of width W and thickness T . There is no current flow in the patterned ground shield (grey).

Fig. 3. Resistance of a 4-m-diameter circular copper wire [(3), symbols] versus frequency. The solid line is the model [see (2)] used in this paper. Both the dc and high-frequency limits and the corner frequency for 25% impedance increase are the same. Over frequency, the difference is never larger than 7%. The dotted line represents the equivalent strip result (2 -m wide and 2-m thick) [14]. Here, for the same high-frequency limit, the corner frequency for 25% impedance increase is approximately 1.6 higher.

2

III. HIGH-FREQUENCY MODEL At gigahertz frequencies, the current flow in the signal and ground conductors is governed by a delicate balance of resistive and inductive effects, and is not uniform anymore. We will model this using our general expression for current crowding effects first introduced in [10] for spiral inductors Fig. 2. Low-frequency signal and ground path inductance versus ground linewidth when W = 15 m, S = 5 m T = 3 m, and T = 11 m.

(2) inductance as , where and denote the (partial) signal and ground inductances, respectively,

(1) is the (mutual) inductance (Appendix-A) and where where . Since the transmission line series inducwe used tance and shunt capacitance can be modeled independently, and interconnect capacitance modeling techniques are well known from the literature [12], in this paper, only the inductance modeling will be detailed. An example of the signal and ground inductances per unit length calculated using (1) is shown in Fig. 2. The inductance in the ground lines is smaller than in the signal line, but certainly not negligible. As shown in Fig. 2, the width of the ground tracks would need to be fairly large to minimize this ground inductance. This is not feasible because of the metal density rules imposed in advanced IC processes, and apart from that would lead to an undesirably large footprint of these transmission lines.

where and denote the signal and ground line resistances, and and are the corresponding transition corner frequencies. To corroborate the use of this expression for transmission line modeling, we consider the case of a cylindrical conducting wire, for which an analytical solution exists relating its surface potential to the current passing through the conductor body. This complex frequency-dependent impedance is [13] (3) where is the radius of the wire, and represent Bessel , , and is functions of the first kind, the conductivity of the wire. For the same dc and high-frequency limits, (2) shows almost the same transition corner frequency (Fig. 3), contrary to the equivalent rectangular strip approach introduced in [14], and never deviates more than 7% from the exact solution, which is accurate enough for our purpose. Both (2) and (3) are unsuitable for direct use. To get an equivalent-circuit model suitable for time-domain simulations, the imped(2) are approximated with fourth-order L/R (inances and

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Fig. 4. Schematic representation of a set of n inductively coupled current filaments of equal resistance. Their interaction is captured in the inductance matrix M.

ductor/resistor) networks using similar coefficients, as in [10]. and , the behavior of a set of inductively coupled To find current filaments (Fig. 4), as found in a single partial transmisrepresent the mutual sion line element, is analyzed. Letting inductance matrix between the identical filaments, we may relate their voltages and currents through a simple matrix relationship

Fig. 5. Filamentary inductance profile calculated for a CPW line with W = W = 15 m, S = 5 m T = 3 m, and T = 11 m versus lateral distance and from the transmission line symmetry axis. The solid lines represent L L and the dashed lines represent L and L , respectively.

(4) where represents the unity matrix and is the dc resistance of the partial element. The element’s ac impedance is found and calculating the resulting total imposing a voltage yielding current

which, apart from providing the dc resistance and low-frequency inductance, also shows that the exact frequency where the dc resistance starts to increase since the current flow is no longer of the uniform can be derived from the variance filamentary inductance. Expanding (2) for small gives

(5) (11) After writing this as (6) expanding for small

into (7)

and in Comparing (10) and (11) shows that we can find (2) from and . and are found in a similar fashion. To calculate the variance of the filamentary inductance of our CPW transmission line, a uniform lateral meshing is performed, as indicated in the bottom portion of Fig. 1, where the signal and ground conductors are split into 20 subsections. Current in adjacent conductors clearly has its impact on the shape of the filamentary inductance profile (Fig. 5). This proximity effect is included using the filamentary version of (1)

and some rearranging, we obtain (12) (8) As shown in Appendix-B, the above sums can be formulated in . terms of different averages over the filamentary inductance Doing so, we arrive at (9) A second expansion, again for small , then gives the desired result (10)

where the index is the subsection number. calculated versus Fig. 6 shows the ground impedance frequency using (2) for the transmission line of Fig. 5. As seen over the entire frequency range up to 50 GHz, this impedance remains predominantly resistive and approximately shows a square root increase with frequency. In fact, at low frequency, the ground impedance is dominated by resistive loss, whereas at high frequency, where inductive effects are important, the effective ground inductance reduces due to the current crowding near the center conductor, where the filamentary ground inductance changes sign. As a result, even at high frequencies, the impedance seen in the ground conductor when its width is taken equal to the center conductor remains within acceptable limits.

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Fig. 8. Simplified transmission line model including common-mode radiation loss. TABLE I OVERVIEW OF THE WIDTHS AND SPACINGS (IN MICROMETERS) AND A FEW SELECTED RATIOS OF THE LINES CHARACTERIZED IN THIS STUDY

Fig. 6. Calculated magnitude (left axis) and phase (right axis) of the ground impedance Z versus frequency for the transmission line depicted in Fig. 4.

measurement setup was characterized measuring a special thru pattern (see Fig. 7 inset) and calculated using

(13) To include common-mode radiation loss, the transmission line is modeled using the equivalent circuit depicted in Fig. 8 and , and connecting the measured in using (2) for parallel with . The capacitance between the signal and ground lines is calculated from geometrical parameters, and taken independent of frequency, with a small series resistance representing the shield resistance, and distributed along the length of the transmission line, to obtain proper behavior at high frequency. V. SINGLE LINES

Fig. 7. Measured common-mode radiation impedance versus frequency. Inset: GG-S thru pattern found on the Cascade ISS 005-016 general-purpose calibration substrate.

IV. RADIATION LOSS In typical two-port on-wafer -parameter measurements, the RF probe ground tips can no longer be assumed to be at the instrument ground at gigahertz frequencies. As long as a low resistance connection exist between all ground pads of the test structure, this has no adverse effect on the measurements. In our transmission line structures, however, the voltage drop across the ground lines of our transmission lines can excite commonmode currents in the cables connecting the probes to the network analyzer. These common-mode currents [15] are partially radiated into space, and partly absorbed in the RF probe. To allow a proper comparison between model and experimental data, the (Fig. 7) seen in the common-mode radiation impedance

To evaluate the performance of single patterned ground shield CPW interconnect transmission lines, on-wafer -parameters were measured from 50 MHz to 50 GHz on six different transmission line geometries with center conductor widths ranging from 15 to 30 m and signal to ground conductor spacings ranging from 5 to 34 m (Table I) realized in a high-performance RF Bi-CMOS process [16]. The width of the two ground conductors was always taken equal to the center conductor width. The signal conductor was made in a 3- m-thick fifth aluminum top metal, while the ground conductors were realized connecting all available metal layers in parallel. For each line, the impedance seen at 10 GHz in is compared to that in the signal the ground conductor . As seen in Table I, the impedance in the ground conductor conductor is not always negligible. When we compare the skin (Appendix-C), which can be deperimeter rived from the high-frequency limit of (2), to the full conductor perimeter, it is seen that the current uses the available conductor surface at high frequency effectively, as shown in Table I. This is important for achieving low loss [17]. For each transmission line, versions without shield, with patterned polysilicon ground

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shield, and with a solid metal 1 ground shield were fabricated. For the lines without shield, deep trench isolation was used to suppress currents in the highly doped upper layers of the cm silicon substrate. 20A. Deembedding For transmission line characterization, conventional open short deembedding [18] is less suitable since a short dummy with a sufficiently low impedance termination of the transmission line connection plane is difficult to realize. Therefore, we used the double delay deembedding procedure [19] where two transmission line structures are assumed to be embedded between identical adapters (14) (15) where represents the chain -parameter matrices of the twoport networks denoted in the subscript. As any passive two-port, the adapters can be modeled with one series impedance and two shunt impedances, one facing the RF probes and one facing the device-under-test (DUT). Their ground impedance is neglected. only Since a measurement of provides us with two independent -parameters, there is insufficient information to extract the adaptor equivalent circuit. Having come this far, as already mentioned in [19], conventional open and short dummy structures can at best provide redundant information, but not solve the issue at hand. Therefore, instead of neglecting one of the two shunt impedances [19], we assumed a realistic ratio between them, where the shunt impedance facing the DUT was taken as 1/3 of the shunt impedance facing the RF probes. The double delay deembedding method was applied both on measured, as well as on simulated transmission line data, to ensure that all power losses resulting from imperfect coupling of the RF power into the transmission line and all end effects that might hamper proper comparison are eliminated. After that, the transmission line characteristic impedance and propagation constant were extracted [20], which, in turn, were used to study line resistance, inductance, and capacitance. B. Results Fig. 9 shows the inductance measured on the three - m-wide and - m-long transmission lines versus frequency. Apart from the noisy data below 300 MHz, excellent agreement between measured data (symbols) and the model curves, is obtained. In particular, the reduction in inductance with frequency is accurately reproduced. Fig. 10 shows the resistance of all six 750- m-long polysilicon patterned ground shield transmission lines versus frequency. Essential features, such as the level of the dc resistance, the increase in resistance due to current crowding beyond approximately 1 GHz, and the excess resistance seen in some lines beyond 10 GHz due to common-mode radiation loss are modeled correctly. Fig. 11 compares the characteristic impedances measured on three m m line, without any versions of the shield, with the patterned polysilicon ground shield, and with a solid metal 1 ground shield. As seen the impact of the shield

Fig. 9. Inductance of the three W = 30-m-wide 750-m-long transmission lines with different values of the ground line spacing. Symbols denote measurements, lines represent the model.

Fig. 10. Resistance of the six 750-m-long transmission lines versus frequency. Symbols denote measurements, lines represent the model.

type on is considerable due to changes in line capacitance, and in the case of the solid metal 1 ground shield, also due to lowering of the line inductance due to eddy currents in this shield. In line with the model, the patterned polysilicon ground shield provides the least frequency-dependent 50characteristic impedance. Fig. 12 compares the attenuation loss in decibels/centimeter measured on the same set of lines. The best performance is found for the line with the patterned polysilicon ground shield. As can be seen, despite the measures taken to suppress substrate currents, without a shield, substrate dielectric losses cause the attenuation to double beyond 2 GHz. For the conventional microstrip configuration, eddy-current losses in the solid metal 1 ground shield cause the attenuation to double between 300 MHz and 10 GHz. Beyond 20 GHz, the common-mode radiation loss in our characterization setup obscures a proper comparison; however, when this radiation loss

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Fig. 11. Characteristic impedances of the W = 30-m S = 24-m lines versus frequency. Symbols denote measurements, the line represents the model described in this paper.

Fig. 13. Filamentary inductance profiles calculated for common- and differential-mode operation for a differential transmission with dimensions, as described in the text.

path for common-mode signals. For differential operation with current , at frequencies above approximately 1 GHz, mirror of strength currents (16) are induced in these ground conductors, which slightly reduce the inductances seen in the signal lines. Fig. 13 shows the filamentary inductance profiles calculated for both differentialoperation, where the current flows in mode opposite directions in both signal conductors, and for commonoperation where the current flows in the mode same direction in both signal conductors and returns through the ground inductors. A. Deembedding

Fig. 12. Attenuation loss of the W = 30-m S = 24-m lines versus frequency. Symbols denote measurements, the solid line include the commonmode radiation loss seen in the measurement setup, the dashed line represents the result obtained with the intrinsic model.

is removed from the patterned ground shield CPW simulation, both solid and patterned ground shields show comparable skin resistance limited losses. VI. DIFFERENTIAL LINES Grounding issues are avoided when signals are transported over differential transmission lines. To further investigate the benefits of the patterned polysilicon ground shield and verify our modeling approach, differential transmission lines where fabricated in the 0.9- m-thick copper top sixth metal layer of a 90-nm node CMOS process [21]. The two 3.5- m conductors were spaced 5 m apart, while two 12- m-wide ground conductors were added at 17- m spacing, providing a controlled return

Since for four-port -parameters taken on a set of differential transmission line test structures the double-delay approach [19] did not provide the desired accuracy, 750- m-long differential transmission lines were terminated at one end with either an open or a short termination, after which two-port -parameters were taken on the other end of the lines, as shown in Fig. 14. Open-short deembedding [18] was used to correct for the test-pad parasitics. After that, the differential- and commonmode impedances seen at the line end were extracted. For each and propapropagation mode, the characteristic impedance gation constant are now easily found from (17)

(18) where and denote the impedances measured on the open and short terminated transmission lines, respectively. Since the open and short terminations used on the end of the lines are drawn identical to those used for the open and short

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Fig. 16. Common-mode attenuation loss versus frequency. Symbols denote measurements, lines represent the model.

Fig. 14. Two-port test structures used to characterize open and short terminated differential transmission lines. The lower photograph details the connection to the differential line with the patterned ground shield.

Fig. 17. Differential-mode attenuation loss versus frequency. Symbols denote measurements, lines represent the model.

Fig. 15. Characteristic impedance versus frequency. Symbols denote measurements, lines represent the model.

dummy deembedding structures, imperfections in the terminations cancel each other, and good accuracy is expected at frequencies where the line length is not close to a multiple of a quarter-wavelength. Since the differential transmission line is only probed at one side, with a low-impedance connection between the ground pads of the two test-structure ports, even for common-mode operation, radiation loss is not expected. B. Results Fig. 15 shows the measured and calculated characteristic for differential- and common-mode operation. impedance

Again, (2), together with a frequency-independent line capacitance, calculated from geometrical parameters was used to model the transmission line. Fig. 16 shows the common-mode attenuation loss for the differential transmission line with a patterned polysilicon ground and a solid metal 1 ground shield, respectively. As seen, the losses measured on the line with the patterned polysilicon shield are in line with the model. Compared to this, eddy currents in the solid metal 1 shield cause a significant excess attenuation loss in the frequency range of 1–30 GHz. Fig. 17 shows the differential-mode attenuation loss for the same set of lines. Again, the losses measured on the line with the patterned polysilicon shield are in line with the model, whereas for the line with solid metal 1 ground, eddy currents cause an excess attenuation loss in the frequency range of 2–30 GHz. However, due to the more confined EM field of the differential mode, here the differences are less prominent. In

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fact, twofold loss reductions found for single line test structures [7] reduce to around 25% differences when verified on truly differential test structures.

and [22]

VII. SUMMARY AND CONCLUSIONS We have presented a thorough theoretical and experimental study confirming the scattered observations of the benefits of patterned polysilicon ground shields over solid metal grounds shields such as lower attenuation losses and higher characteristic impedance. We also found, however, that these benefits tend to disappear at frequencies beyond several tens of gigahertz, where the skin depth becomes less than the thickness of the solid metal ground shield. Beyond these frequencies, a solid metal ground shield is preferred since it shows less ground inductance and, thus, better approaches the ideal ground, which can sink or supply any current at will without changing its potential. A patterned ground shield decouples the transmission line specific capacitances and inductances, and the resulting slow wave effects allow compact quarter-wavelength stubs. However, this decoupling makes the propagation constant of the line more susceptible to process variations and should, thus, be used with care. In differential transmission lines, the opposite currents cancel each other and ground quality is not an issue. In this paper, the expected lower attenuation loss of differential transmission lines with a patterned ground shield has been verified, to the best of our knowledge, for the first time on truly differential test structures. Due to the more confined EM field of the differential mode, the differences are less pronounced than seen on single lines. Our experimental results obtained on transmission lines with patterned ground shields are supported by a predictive compact equivalent-circuit model, which we have described in sufficient detail to allow reproduction of our study. This model employs exact closed-form expressions for the inductances, and describes key performance figures such as characteristic impedance and attenuation loss with excellent accuracy. APPENDIX

(21) where

, ,

,

,

,

,

, , and is the line length. Close investigation shows that the partial self-inductance given by (21) does not increase proportional to the line length , but instead includes end effects. In this paper, inductance per using two unit length is, therefore, either calculated from sufficiently large values for or from double delay deembedding. Inductance Matrix Sums: Defining and for the first matrix sum in (8), it is trivial to show that (22) For the second matrix sum in (8), we may write

Mutual Inductance Between Straight Rectangular Lines: required in (1) can be calcuThe partial mutual inductances lated from a weighted average of 16 partial self-inductances [22] assuming a uniform current flow

(23) whereas (24)

(19) where

is symmetrical, i.e., Since the mutual inductance matrix , the above two matrix sums yield equal results. Unfortunately, however, when the power of the matrix is larger than 2, we find (25)

(20)

making further expansion of (8) unattractive.

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Minimizing Conductive Losses: For a coaxial transmission line of impedance

(26) , where and denote the signal conductor outer and ground conductor inner radius, operating in the high-frequency regime, where the skin depth is less than half the conductor thickness, the conductive losses , expressed in decibels/meter, are

(27) where

(28) is an average of the skin perimeters and in the signal and ground conductors, and where (29) represents the skin resistance. For fixed , we find that the loss , which for an air dielectric correis minimized for . sponds to For general rectangular transmission line geometries, similar considerations apply, but the current will tend to crowd at the edges at high frequencies, and the skin perimeters entering (28) will be less than the full conductor perimeters. However, remains essential selecting a geometry that maximizes to minimize the conductive losses. For microstrip transmission lines integrated in a typical IC process, the thickness of the ground metal layer is less than the skin depth defined in (29) up to several tens of gigahertz. This causes higher losses in this frequency range than in CPW lines where the ground current (predominantly) flows in the thick top metal layers. ACKNOWLEDGMENT The authors wish to acknowledge the Crolles2 alliance for providing part of the silicon used for this paper. The authors further wish to thank T. Kamgaing, now with the Intel Corporation, Chandler, AZ, for his assistance in realizing the differential transmission line test structures. REFERENCES [1] A. Deutsch, G. V. Kopcsay, P. Restle, H. H. Smith, G. Katopis, W. D. Becker, P. W. Coteus, C. W. Surovic, B. J. Rubin, R. P. Dunne, T. Gallo, K. A. Jenkins, L. M. Terman, R. H. Dennard, G. A. Sai-Halasz, B. L. Krauter, and D. R. Knebel, “When are transmission-line effects important for on-chip interconnections?,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1836–1846, Oct. 1997.

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[2] T. Zwick, Y. Tretiakov, and D. Goren, “On-chip SiGe transmission line measurements and model verification up to 110 GHz,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 65–567, Feb. 2005. [3] V. Milanovic, M. Ozgur, D. C. DeGroot, J. A. Jargon, M. Gaitan, and M. E. Zaghloul, “Characterization of broadband transmission for coplanar waveguides on CMOS silicon substrates,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 632–640, May 1998. [4] B. Kleveland, C. H. Diaz, D. Vook, L. Madden, T. H. Lee, and S. S. Wong, “Exploiting CMOS reverse interconnect scaling in multigigahertz amplifier and oscillator design,” IEEE J. Solid-State Circuits, vol. 36, no. 10, pp. 1480–1488, Oct. 2001. [5] A. M. Mangan, S. P. Voinigescu, M.-T. Yang, and M. Tazlauanu, “deembedding transmission line measurements for accurate modeling of IC designs,” IEEE Trans. Electron Devices, vol. 53, no. 2, pp. 235–241, Feb. 2006. [6] R. Lowther and S.-G. Lee, “On-chip interconnect lines with patterned ground shields,” IEEE Microw. Guided Wave Lett., vol. 10, no. 2, pp. 49–51, Feb. 2000. [7] T. S. D. Cheung and J. R. Long, “Shielded passive devices for siliconbased monolithic microwave and millimeter-wave integrated circuits,” IEEE J. Solid-State Circuits, vol. 41, no. 5, pp. 1183–1200, May 2006. [8] R. D. Lutz, V. K. Tripathi, and A. Weisshaar, “Enhanced transmission characteristics of on-chip interconnects with orthogonal gridded shield,” IEEE Trans. Adv. Packag., vol. 24, no. 3, pp. 288–293, Aug. 2001. [9] P. Wang and E. C.-C. Kan, “High-speed interconnects with underlayer orthogonal metal grids,” IEEE Trans. Adv. Packag., vol. 27, no. 3, pp. 497–507, Aug. 2004. [10] L. F. Tiemeijer, R. J. Havens, R. de Kort, Y. Bouttement, P. Deixler, and M. Ryczek, “Predictive spiral inductor compact model for frequency and time domain,” in Proc. Int. Electron Device Meeting, 2003, pp. 875–878. [11] A. E. Ruehli, “Inductance calculations in a complex integrated circuit environment,” IBM J. Res. Develop., pp. 470–481, 1972. [12] U. Choudhury and A. Sangiovanni-Vinticelli, “Automatic generation of analytical models for interconnect capacitances,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 14, no. 4, pp. 470–480, Apr. 1995. [13] S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics. New York: Wiley, 1965, pp. 291–297. [14] H.-Y. Lee and T. Itoh, “Phenomenological loss equivalence method for planar quasi-TEM transmission lines with thin normal conductor or superconductor,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1904–1909, Dec. 1989. [15] I. F. Chen, C. M. Peng, and C. W. Hsue, “Circuit-concept approach to radiated emissions of printed circuit boards,” Proc. Inst. Elect. Eng. —Sci. Meas. Technol., vol. 151, no. 3, pp. 205–210, May 2004. [16] P. Deixler, T. Letavic, T. Mahatdejkul, Y. Bouttement, R. Brock, P. C. T. V. Saikumar, A. Rodriguez, R. Colclaser, P. Kellowan, H. Sun, N. Bell, D. Bower, A. Yao, R. van Langevelde, T. Vanhoucke, W. D. van Noort, G. A. M. Hurkx, D. Crespo, C. Biard, S. Bardy, and J. W. Slotboom, “QUBiC4plus: A cost-effective BiCMOS manufacturing technology with elite passive enhancements optimized for ‘silicon-based’ RF-system-in-package environment,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Oct. 2005, pp. 272–275. [17] J. Kim, B. Jung, P. Cheung, and R. Harjani, “Novel CMOS low-loss transmission line structure,” in Proc. IEEE Radio Wireless Conf., 2004, pp. 235–238. [18] M. C. A. M. Koolen, J. A. M. Geelen, and M. P. J. G. Versleijen, “An improved deembedding technique for on-wafer high frequency characterization,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Sep. 1991, pp. 188–191. [19] J. Song, F. Ling, G. Flynn, W. Blood, and E. Demircan, “A deembedding technique for interconnects,” Elect. Performance Electron. Packag., pp. 129–132, Oct. 2001. [20] W. R. Eisenstadt and Y. Eo, “S -parameter-based IC interconnect transmission line characterization,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. 15, no. 4, pp. 483–489, Aug. 1992. [21] L. F. Tiemeijer, R. J. Havens, R. de Kort, A. J. Scholten, R. van Langevelde, D. B. M. Klaassen, G. T. Sasse, Y. Bouttement, C. Petot, S. Bardy, D. Gloria, P. Scheer, S. Boret, B. Van Haaren, C. Clement, J.-F. Larchanche, I.-S. Lim, A. Zlotnicka, and A. Duvallet, “Record RF performance of standard 90 nm CMOS technology,” in Proc. Int. Electron Device Meeting, 2004, pp. 441–444.

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[22] G. Zhong and C.-K. Koh, “Exact closed-form formula for partial mutual inductances of rectangular conductors,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 10, pp. 1349–1353, Oct. 2003. Luuk. F. Tiemeijer was born in Son en Breugel, The Netherlands, in 1961. He received the M.S. degree in experimental physics from the State University of Utrecht, Utrecht, The Netherlands, in 1986, and the Ph.D. degree in electronics from the Technical University of Delft, Delft, The Netherlands, in 1992. In 1986, he joined Philips Research Laboratories, Eindhoven, The Netherlands, where he has conducted research on InGaAsP semiconductor lasers and optical amplifiers. Since 1996 he has been involved in the RF characterization and modeling of advanced IC processes. In October 2006 he joined NXP Semiconductors, Eindhoven, The Netherlands.

Ralf M. T. Pijper was born in Holtum, The Netherlands, in 1977. He received the M.Sc. degree in applied physics from the Technical University of Eindhoven, Eindhoven, The Netherlands, in 2003. In 2005, he joined Philips Research Laboratories, Eindhoven, The Netherlands, where he is currently involved with RF characterization of advanced IC technologies. In October 2006, he joined NXP Semiconductors, Eindhoven, The Netherlands.

Ramon J. Havens was born in Nijmegen, The Netherlands, in 1972. He received the Bachelor’s degree from Eindhoven Polytechnic, Eindhoven, The Netherlands, in 1995. He subsequently joined Philips Research Laboratories, Eindhoven, The Netherlands, where he became involved in the on-wafer RF characterization of the various active and passive devices found in advanced IC processes. At the end of 2005, he joined the Innovation Center RF, NXP Semiconductors, Nijmegen, The Netherlands, where he is currently active in the characterization of RF microelectromechanical systems (MEMS).

Olivier Hubert was born in Caen, France, in 1967. He received the Ph.D. degree in sciences from the University of Caen, Caen, France, in 1995. His doctoral research concerned low-noise superconducting devices applications. From 1997 to 2000, he was a Lecturer in electronics with the Ecole Superieure d’Ingénieurs en Electrotechnique et Electronique (ESIEE), Amiens, France. In 2000, he joined the Process and Library Technology Department, NXP Semiconductors Caen, Caen, France, where he is currently involved in the characterization and modeling of RF IC processes.

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Compact Left-Handed Transmission Line as a Linear Phase–Voltage Modulator and Efficient Harmonic Generator Hongjoon Kim, Member, IEEE, Alexander B. Kozyrev, Abdolreza Karbassi, and Daniel W. van der Weide, Senior Member, IEEE

Abstract—We construct a synthetic left-handed transmission line with cascaded varactors and shunt inductors. By modulating dc bias, the capacitance of the varactors can be changed and modulation of the output phase state is possible. For frequencies from 4.7 to 6.4 GHz, a very linear phase variation versus voltages of over 200 phase variation with low insertion-loss variation ( 0.5 dB) is demonstrated. This circuit can also act as an efficient harmonic generator when a large signal is applied. Since the left-handed transmission line shows high-pass filter response, harmonics generated are not seriously attenuated. However, because this synthetic transmission line is a very dispersive medium, strong dispersions and instabilities may arise. The circuit size is determined by the diode size and lumped-element inductor, allowing it to be compact. Index Terms—Harmonic generator, left-handed transmission line, phase shifter, varactor.

I. INTRODUCTION

M

ANY microwave applications using left-handed transmission lines have been reported [1]–[6] since negative refractive index metamaterials were first described by Veselago in 1968 [7]. In a conventional nonlinear transmission line in which varactors are distributed along the transmission line, phase propagation and power flow are in the same direction, and this configuration can be called a right-handed nonlinear transmission line. By contrast, left-handed transmission lines use cascaded capacitors and shunt inductors. More recently [6], the cascaded capacitors were replaced with varactors. This configuration can be called a left-handed nonlinear transmission line. Due to the varactors’ nonlinear capacitance–voltage (C–V) relationship, the input wave experiences distortion. Furthermore, in this structure, phase velocity increases with frequency, resulting in anomalous dispersion. Thus, this nonlinear transmission line structure can be used as an effective

Manuscript received December 13, 2006. This work was supported by the Air Force Office of Scientific Research through the Multiuniversity Research Initiative Program under Grant F49620-03-1-0420. H. Kim is with the Electrical Engineering Department, City College of City University of New York, New York, NY 10031 USA (e-mail: [email protected]. edu). A. B. Kozyrev, A. Karbassi, and D. W. van der Weide are with the Electrical Engineering Department, University of Wisconsin–Madison, Madison, WI 53706 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.891692

harmonic (even parametric) generator [6]. However, due to anomalous dipersion, the harmonic generation process in the left-handed nonlinear transmission line could be instable. Right-handed nonlinear transmission lines can be used as broadband analog phase shifters. Analog phase shifters differ from digital phase shifters since they require only one control bias line for an infinite number of phase states, and they require no high-quality switches [8]. The two most famous configurations of analog phase shifters are reflection- and transmission-type phase shifters. Reflection-type phase shifters use 90 branch-line couplers and varactors [9], resulting in a narrow bandwidth and a small range of phase variation. At low frequencies, the circuit size can still be large due to the branch-line couplers. To overcome this drawback, lumped-element branch-line couplers are used to reduce the circuit size in [10] and [11]. In [11], the authors also increased the phase-variation ranges using the LC resonance and cascading as many cells as necessary. Many distributed-element transmission-type phase shifters have been reported [12]–[16]. These structures require long transmission lines to achieve sufficient inductance and, therefore, the circuit size also becomes large. However, these circuits exhibit low insertion loss and broadband performance. In [17], a very compact transmission-type phase shifter was achieved by replacing a long transmission line with a lumped-element inductor; however, the phase variation versus voltage was found to be nonlinear and the insertion-loss variation was large. An approach to minimize harmonic distortion inherent in the right-handed nonlinear transmission line has been reported [18]. With the technique suggested, large signals can be applied without serious harmonic distortion when using a right-handed nonlinear transmission line as a phase shifter. The dual of the right-handed nonlinear transmission line, the left-handed nonlinear transmission line, can be constructed and used both as a voltage-variable linear phase modulator and a harmonic generator. Due to the balanced relationship between phase constant and the voltage relationship dispersion in the left-handed nonlinear transmission line, the phase change is very linear with respect to applied voltage [19]. Also, while changing the output phase state, insertion-loss variation can be low in the left-handed nonlinear transmission line. However, because the left-handed nonlinear transmission line is an anomalous dispersion medium, we cannot use a broadband signal when we use it as a phase shifter. In this paper, we present the detailed design and analysis of a left-handed non-

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frequency can be determined. This parameter is the Bragg cutoff , which can be defined as frequency

(3) In the case of positive power flow (indicated by the Poynting vector), (1) becomes (4) For as

,

becomes very small, and we can approximate

Fig. 1. Unit cell of a left-handed nonlinear transmission line. The diode can be modeled as voltage variable capacitance (C (V )) and resistance (R (V )).

(5)

linear transmission line for linear phase–voltage modulation and efficient harmonic generation.

When we ignore the diode resistance, the characteristic can also be approximated as (6) when impedance as follows: (6)

II. FUNDAMENTAL THEORIES ON LEFT-HANDED NONLINEAR TRANSMISSION LINE Using (5), the phase velocity can be found as follows:

and the group velocity

A. Left-Handed Nonlinear Transmission Line Unit Cell (7) Fig. 1 shows a schematic of a single left-handed nonlinear transmission line unit cell. Two diodes are cascaded in series, and a shunt inductor is placed between them. By supplying a dc bias through a large resistance value, modulation of the diode capacitance is possible. By changing the dc bias, the diode series resistance is also changed. A reverse bias is applied in order to modulate the diode capacitance. In all of following equations, refers to the diode reverse dc bias. In the Appendix, this circuit is analyzed to find the dimensionless attenuation constant and the phase constant . The results are as follows:

(1) and (2)

and (8) Here, is the period of the unit cell length (see Fig. 4). As can be seen in (7), negative phase propagation occurs in the left-handed nonlinear transmission line and strong dispersion is expected to occur in the medium. B. Left-Handed Nonlinear Transmission Line as a Linear Phase–Voltage Modulator Since the structure of a left-handed nonlinear transmission line acts as a high-pass filter, its bandwidth is bounded on the lower end by the Bragg cutoff frequency. For this frequency per unit cell range, the controllable phase variation range can be expressed as [using (4)]

where

is the voltage variable diode capacitance and is the diode resistance. The phase-modulation range can be increased when several unit cells are cascaded. Furthermore, the nonlinearity of the line would increase as the input wave interacts with more nonlinear unit cells. However, because the input signal is attenuated more with the number of unit cells, there exists an optimized unit cell number in a left-handed nonlinear transmission line. is larger than 1, must become If the value of imaginary. By setting the left side of (1) equal to unity, the cutoff

(9) For

,

is simplified as (10)

KIM et al.: COMPACT LEFT-HANDED TRANSMISSION LINE AS LINEAR PHASE–VOLTAGE MODULATOR AND EFFICIENT HARMONIC GENERATOR

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where and are the maximum and minimum reverseis bias voltages, respectively. The total phase variation given as (11) where is the number of identical unit cells in a left-handed nonlinear transmission line. To achieve a full 0 –360 controlis lable phase change, the number of unit cells required given as (12) In order to minimize reflections, we set so the diode capacitance values satisfy the following equation ( large signal diode capacitance): (13) A desirable feature of the left-handed nonlinear transmission line phase shifter is that the dispersion relation (4) permits very linear phase variation control with respect to the bias voltage level. The general diode C–V relationship is (14)

Here, is the zero-bias diode capacitance, is a junction is the built-in diode voltage. doping grading parameter, and for abrupt and for hyper-abrupt From [20], diodes. Fig. 2(a) shows C–V curves for several diodes, i.e., an abrupt , and the diode used for diode, a hyper-abrupt diode was set as 0.7 V. the experiments described in this paper. As can be seen in Fig. 2, very nonlinear C–V relations for several diodes [see Fig. 2(a)] become quite linear –V relations [see Fig. 2(b)]. This can be easily seen by eye on the graph, which is and nH, the actual drawn using (9), where measured values used in constructing the left-handed nonlinear transmission line for the experiments. Although the graph depicts data obtained using a frequency of 5.2 GHz, a very linear phase–voltage variation curve can be obtained for other frequencies. This means that for most abrupt and hyper-abrupt diodes, the –V relation becomes very linear for any frequencies above the Bragg frequency, and the achievement of high-frequency very linear phase shifters can be realized using left-handed nonlinear transmission lines. C. Left-Handed Nonlinear Transmission Line as a Harmonic Generator When a large signal is applied, a left-handed nonlinear transmission line functions as a harmonic generator [6]. The nonlinearity occurs due to the nonlinear diode C–V relationship. Since the left-handed nonlinear transmission line is the high-pass filter structure, the harmonics generated are not significantly attenuated if they are larger than the Bragg frequency. This is different

Fig. 2. C–V curve and –V curve. (a) C–V curve for model abrupt, hyperabrupt, and measured diodes used in this study (MACOM MA46H120). (b) Section of the theoretical phase variation curve with voltage using (9). Phase state taken as 0 at 0 V. Values used for the case when C = 1:47 pF, f = 5:2 GHz, and L = 1:25 nH. A very linear phase variation with voltages is observed. For the MACOM varactor, maximum deviation from the linear response is just 3.1 .

6

from the harmonic generation process in the right-handed nonlinear transmission line, which shows a low-pass filter response. Harmonics generated in the right-handed nonlinear transmission line are significantly attenuated. Due to strong dispersion [see (5)], the phase mismatch among the harmonics at the output of the left-handed nonlinear transmission line is serious. In some cases, parametric generation occurs and harmonic generation in the left-handed nonlinear transmission line is very unstable [21]. This is different from the right-handed nonlinear transmission line case in which the input wave experiences shock-wave generation [22]. The theoretical background for the harmonic generation process in the left-handed nonlinear transmission line is explained in [6]. III. LEFT-HANDED NONLINEAR TRANSMISSION LINE FABRICATION We constructed a left-handed nonlinear transmission line on . MA/COM hyperRogers RT/Duroid 3010 board abrupt junction GaAs flip-chip varactor diodes (MA46H120) were attached using conductive silver epoxy. After thru-reflect-

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Fig. 3. Diode series-resistance variation (MACOM MA46H120). Series resistance is important for insertion-loss estimation.

line (TRL) calibration, we measured the diode capacitance variof the diode is ation at 5 GHz using a network analyzer. 1.1 pF, as described by the manufacturer’s data sheet. Measurefor the diode of 1.47 pF. The disment showed a similar crepancy may arise from the packaging capacitance, the parasitics of the silver epoxy, or process variation in assembly. The diode C–V curve was given in Fig. 2(a). From this curve, we pF. Since two diodes are in series in each calculated section, the total diode capacitance is halved. Thus, to achieve pF, the ina characteristic impedance of 50 with ductance must be 1.25 nH from (6). The diode series resistance, which is a function of applied dc bias, was also measured (see Fig. 3). This is an important parameter used to estimate the insertion loss of the phase shifter (2). Circuit inductances were implemented by connecting 0.12-mm-diameter copper wire to the ground plane on the back side of the board. This wire inductor approach works better than chip inductors or thin transmission line stubs since the parasitic capacitance inside the inductor can be minimized. DC-bias wires were connected using 2.2-k resistors between the diodes. Since our objective was to achieve a full 360 phase shifter at 5.2 GHz, seven identical unit cells were required [using (12)]. The final circuit is seen in Fig. 4. IV. MEASUREMENTS AND DISCUSSION A.

-Parameters as a Phase Modulator

Since the Bragg cutoff frequency is a function of the applied bias voltage (3), the cutoff frequency changes as we vary the voltage and also modulate the output phase state. Theoretically, the Bragg cutoff frequency is 2.63 and 4.46 GHz for 0and 4-V bias, respectively [using (3)]. However, the theoretical Bragg cutoff frequency (3) is defined when infinite numbers of unit cells are cascaded. Our circuit is constructed with seven unit cells. We assumed the Bragg cutoff frequency to be at 20 dB. The Bragg frequency in the actual measurement shows 2.65 GHz and 4.18 GHz for 0- and 4-V bias, respectively [see Fig. 5(a)], which agrees with theory. The circuit performance deviates from theory for frequencies larger than 6.4 GHz due to the stopband where insertion loss be-

Fig. 4. Fabricated seven-section left-handed nonlinear transmission line. Circuit size is 1.4 cm 0.3 cm ignoring connectors and bias circuit and is determined by diode and lumped-element inductor sizes. A unit length d is defined as shown.

2

comes very large. The insertion loss at the stopband frequency increases with bias voltage [see Fig. 5(a)]. The stopband is unavoidable since pads are required for attaching cascade diodes and inductor wires, and these pads work as shunt capacitors. As a result, the whole circuit is a bandpass filter. Furthermore, the silver epoxy used to attach devices also has parasitic electrical properties. The amount of silver epoxy used in each pad is also not uniform, making the parasitics caused by the silver epoxy difficult to model. Monolithic designs will reduce parasitic effects. When designing this left-handed phase shifter, one should consider the operating voltage range, the Bragg frequency, and parasitic effects in order to realize a desired passband frequency. Thus, a tradeoff exists between bandwidth and controllable phase states. For the passband (4.7–6.4 GHz), this phase shifter shows good performance with a maximum insertion loss of only 6.4 dB and minimum insertion loss of 4.6 dB and, therefore, a variation of only 0.9 dB. However, the maximum insertion loss in the actual measurements is almost 3 dB larger than that of the mathematical calculation using (2). This effect is also associated with the above-mentioned parasitics. For this passband, the phase variation curve with voltage is very linear [see Fig. 6(a)]. A theoretical phase variation curve using (4) is drawn in Fig. 6(b). Though some discrepancies are observed in the phase-variation quantity due to parasitics, the phase variation predicted by theory is similar to our measurement. This ensures that this passband is in the left-handed region. Fig. 7 shows return loss as a function of voltage. For the passband, the return loss is less than 10 dB for any dc bias.

KIM et al.: COMPACT LEFT-HANDED TRANSMISSION LINE AS LINEAR PHASE–VOLTAGE MODULATOR AND EFFICIENT HARMONIC GENERATOR

Fig. 5. Insertion loss according to voltage. (a) Insertion loss versus voltage for the whole bandwidth. Bragg frequency is a function of voltage. (b) Insertion loss versus voltage for passband only. Maximum is 6.4 dB and minimum is 4.6 dB. Insertion loss variation is very small ( 0.9 dB) for the whole passband.

6

This low reflection is possible due to impedance matching using (13). Our objective is to achieve full 360 phase change at 5.2 GHz (Fig. 8). We found an insertion loss of 5.45 dB 0.5 dB and a very linear phase variation for an almost full 360 controllable range (maximum deviation from a linear response is just 15 ), while maintaining the return loss at less than 14.5 dB. At this frequency, 1-dB input compression point is larger than 13 dBm for any bias. Larger power can be used without serious distortion for the left-handed nonlinear transmission line. Its performance at 5.2 GHz makes this phase shifter attractive for an indoor wireless local area network (WLAN), possibly for adaptive antenna applications. This phase shifter has a drawback due to the phase velocity and frequency relation defined in (7): strong dispersion is expected when a broadband input signal is applied. Thus, though the phase shifter is wideband tunable, narrow bandwidth signals must be applied in order to avoid signal distortion at the output.

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Fig. 6. Relative phase variation curve with several voltages. (a) Measured results. For any frequency in passband, phase state gap between voltages is almost the same. It means phase variation is very linear. (b) Theoretical phase variation using (9). Though the results are similar, some differences, especially in the phase-variation range, are observed between (a) and (b) due to the parasitics in the actual circuit explained in Section III. S 21 phase is 0 at 0 V for all frequencies in both cases.

B. Result of Left-Handed Nonlinear Transmission Line for Harmonic Generation The left-handed nonlinear transmission line can be used as an effective harmonic generator. However, because of anomalous dispersion, the magnitude of higher harmonics is very sensitive to frequency variations and to dc bias because of phase mismatch between the fundamental and its higher harmonics (4). Our measurements have shown that harmonic generation is most effective when the fundamental frequency is close to the Bragg frequency. Fig. 9(a) shows general harmonic generation in left-handed nonlinear transmission line. When the fundamental frequency is changed slightly (from 4.076 to 4.016 GHz), instabilities arise that affect harmonic generation [see Fig. 9(b)]. Thus, due to anomalous dispersion, the harmonic generation process can be

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Fig. 7. Measurement result of return loss. For the passband (4.7–6.4 GHz), return loss is less than 10 dB for the any dc bias.

Fig. 8. Performance of phase shifter at 5.2 GHz. Insertion loss is 5.45 dB 0.5 dB. Phase variation is very linear and return loss is less than 14.5 dB.

0

6

stable or unstable for a given bias and frequency. This qualitatively agrees with predictions in [6]. Detailed anomalous dispersion process in the left-handed nonlinear transmission line is explained in [21].

Fig. 9. Nonlinear behavior in left-handed nonlinear transmission line. (a) General harmonic generation. Fundamental frequency is 4.076 GHz. (b) Instability affects harmonic generation. Fundamental frequency is 4.016 GHz. Input power is 18 dBm for both cases.

+

V. CONCLUSION We have proposed a left-handed nonlinear transmission line structure. It performs both as a phase shifter and harmonic generator. As a phase shifter, very linear phase modulation versus voltage is possible with small insertion-loss variation over a wide range of frequencies. Due to its high-pass filter structure, a high-frequency phase shifter can also be realized. It consumes little power because we use reverse bias of the diodes to modulate phase state, and requires only a single bias control. For large-signal operation, the left-handed nonlinear transmission line becomes an effective harmonic generator. Though there exist instabilities due to anomalous dispersion, its compact size and effective harmonic generation properties will be useful for microwave system applications.

Fig. 10. Unit cell of a left-handed nonlinear transmission line equivalent to Fig. 1.

APPENDIX DERIVATION OF ATTENUATION CONSTANT AND PHASE CONSTANT IN LEFT-HANDED NONLINEAR TRANSMISSION LINE STRUCTURE The circuit in Fig. 1 is equivalent to the circuit in Fig. 10 (dc-bias circuit is not included).

KIM et al.: COMPACT LEFT-HANDED TRANSMISSION LINE AS LINEAR PHASE–VOLTAGE MODULATOR AND EFFICIENT HARMONIC GENERATOR

First, let us define a complex propagation constant

as (15)

Here, Then,

is an attenuation constant and

is a phase constant. (16)

and (17) By referring to Fig. 10, the following equations are derived using Kirchhoff’s law: (18) and (19) By substituting (19) into (18), we obtain

(20) is used. Thus, by simplification, we Here, can find the following equation: (21) We can simplify (21) further as follows: (22) At this point, we need an approximation to simplify the equation further. By referring to the insertion-loss result [see Fig. 5(b)], the maximum insertion loss is 6.4 dB. It translates per section, which is small enough for approxto and . Therefore, (21) imation. Thus, becomes (23) By equating the real and imaginary parts of (23), we can get (24) and (25) We obtain (26) from (24) (26) By considering positive power flow, Thus,

must be negative.

(27)

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is negative. Thus, must be positive. Since From (27), we ignored transmission line length, the above quantities are dimensionless. Strictly speaking, and are a section of attenuation and phase constant, respectively, and the unit must be 1/section. ACKNOWLEDGMENT The authors would like to thank C. Paulson, currently with the University of Wisconsin–Madison, for reviewing this paper’s manuscript. REFERENCES [1] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [2] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [3] S. Lim, C. Caloz, and T. Itoh, “Electronically scanned composite right/ left handed microstrip leaky-wave antenna,” IEEE Microw. Wireless Compon. Lett, vol. 14, no. 6, pp. 277–279, Jun. 2003. [4] ——, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 12, pp. 2678–2690, Dec. 2004. [5] M. A. Antoniades and G. V. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett., vol. 51, no. 2, pp. 103–106, Feb. 2003. [6] A. B. Kozyrev and D. W. van der Weide, “Nonlinear wave propagation phenomena in left-handed transmission line media,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 238–245, Jan. 2005. [7] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp, vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [8] I. D. Robertson and S. Lucyszyn, RFIC and MMIC Design and Technology. London, U.K.: IEE, 2001, ch. 9. [9] C.-L. Chen, W. E. Courtney, L. J. Mahoney, M. J. Manfra, A. Chu, and H. A. Atwater, “A low-loss Ku-band monolithic analog phase shifter,” IEEE Trans. Microw. Theory Tech., vol. MTT-35, no. 3, pp. 315–320, Mar. 1987. [10] F. Ellinger, R. Vogt, and W. Bächtold, “Compact reflective type phase shifter MMIC for C -band using a lumped element coupler,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 913–917, May 2001. [11] ——, “Ultracompact reflective-type phase shifter MMIC at C -band with 360 phase-control range for smart antenna combining,” IEEE J. Solid-State Circuits, vol. 37, no. 4, pp. 481–486, Apr. 2002. [12] W. M. Zhang, R. P. Hsia, C. Liang, G. Song, C. W. Domier, and N. C. Luhmann, Jr., “Novel low-loss delay line for broadband phase antenna array applications,” IEEE Microw. Guided Wave Lett, vol. 6, no. 11, pp. 395–397, Nov. 1996. [13] R. P. Hsia, W. M. Zhang, C. W. Domier, and N. C. Luhmann, Jr., “A hybrid nonlinear delay line-based broadband phased antenna array,” IEEE Microw. Guided Wave Lett, vol. 8, no. 5, pp. 182–184, May 1996. [14] P. Akkaraekthalin, S. Kee, and D. W. van der Weide, “Distributed broadband frequency translator and its use in a 1–3 GHz coherent reflectometer,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2244–2250, Dec. 1998. [15] A. S. Nagra and R. A. York, “Monolithic GaAs phase shifter with low insertion loss and continuous 0 –360 phase shift at 20 GHz,” IEEE Microw. Guided Wave Lett, vol. 9, no. 1, pp. 31–33, Jan. 1999. [16] N. S. Baker and G. M. Rebeiz, “Optimization of distributed MEMS transmission-line phase shifters—U -band and W -band designs,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1957–1966, Nov. 2000. [17] F. Ellinger, H. Jäckel, and W. Bächtold, “Varactor-loaded transmission-line phase shifter at C -band using lumped elements,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1135–1140, Apr. 2003. [18] H. Kim, S.-J. Ho, C.-C. Yen, K.-O. Sun, and D. W. van der Weide, “Balanced distributed-element phase shifter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 3, pp. 147–149, Mar. 2005.

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[19] H. Kim, A. B. Kozyrev, A. Karbassi, and D. W. van der Weide, “Linear tunable phase shifter using a left-handed transmission line,” IEEE Microw. Wireless Compon. Lett, vol. 15, no. 5, pp. 366–368, May 2005. [20] R. F. Pierret, Semiconductor Device Fundamentals. Reading, MA: Addison-Wesley, 1996, ch. 7. [21] A. B. Kozyrev, H. Kim, A. Karbassi, and D. W. van der Weide, “Wave propagation in nonlinear left-handed transmission line media,” Appl. Phys. Lett, vol. 87, no. 12, pp. 121109–121109, Sep. 2005. [22] D. W. van der Weide, “Delta-doped Schottky diode nonlinear transmission lines for 480-fs, 3.5 V transients,” Appl. Phys. Lett., vol. 65, pp. 881–883, 1994. Hongjoon Kim (S’05–M’05) was born in Taegu, Korea, in 1972. He received the B.S. degree in electrical/electronics engineering from Kyungpook National University, Taegu, Korea, in 1997, the M.S. degree in communication engineering from the University of Southern California, Los Angeles, in 1999, and the Ph.D. degree in electrical engineering from the University of Wisconsin–Madison, in 2006. In 2000, he was a Research Engineer with the Samsung Electronics Company. From August 2006, he joined City College, City University of New York, New York, as an Assistant Professor. His research focuses on microwave phase shifters and their applications, especially using right- and left-handed nonlinear transmission lines.

Alexander B. Kozyrev was born in Gorky (now Nizhny Novgorod), Russia, in 1971. He received the Diploma degree (with distinction) in radiophysics and electronics from Nizhny Novgorod State University, Nizhny Novgorod, Russia, in 1993, and the Ph.D. degree from the Institute for Physics of Microstructures, Russian Academy of Sciences (RAS), Nizhny Novgorod, Russia, in 2001. From 1993 to 1994, he was with the Institute of Applied Physics, RAS. In 1994, he joined the Institute for Physics of Microstructures, RAS. Since 2003, he has been a Research Associate with the University of Wisconsin–Madison. His research interests include wave-propagation phenomena in nonlinear transmission lines and their applications in microwave electronics, left-handed metamaterials, ultrafast phenomena in semiconductors, and semiconductor heterostructures.

Abdolreza Karbassi received the M.S. degree in electrical engineering from Ferdowsi University, Mashhad, Iran, in 2000, and is currently working toward the Ph.D. degree at the University of Wisconsin–Madison. His research interests include left-handed metamaterials and near-field probes.

Daniel W. van der Weide (S’86–M’86–SM’06) received the B.S.E.E. degree from the University of Iowa, Ames, in 1987, and the Master’s and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1989 and 1993, respectively. He has held summer positions with Lawrence-Livermore National Laboratory and the Hewlett-Packard Company, as well as full-time positions with Motorola as an Engineer and Watkins-Johnson Company as a Member of the Technical Staff. From 1993 to 1995, he was a Post-Doctoral Researcher with the Max-Planck-Institut für Festkörperforschung (Solid State Research), Stuttgart, Germany, after which he joined the Department of Electrical and Computer Engineering, University of Delaware, as an Assistant and Associate Professor and Director of the Center for Nanomachined Surfaces. In 1999, he joined the Department of Electrical and Computer Engineering, University of Wisconsin–Madison, as an Associate Professor, becoming a Full Professor in 2004. His current research involves ultrafast electronics, low-dimensional electron systems, and the application of high-frequency techniques in biotechnology. He was the Principal Investigator on a 2003 Air Force Office of Scientific Research (AFOSR) Multiuniversity Research Initiative (MURI) entitled “Nanoprobe Tools for Molecular Spectroscopy and Control.” From 2002 to 2004, he was a University of Wisconsin Vilas Associate. Dr. van der Weide was the recipient of the 1997 National Science Foundation (NSF) CAREER Award and PECASE Award and the 1998 Office of Naval Research (ONR) Young Investigator Program Award.

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An LTCC-Based Wireless Transceiver for Radio-Over-Fiber Applications Luca Pergola, Ralf Gindera, Student Member, IEEE, Dieter Jäger, Fellow, IEEE, and Rüdiger Vahldieck, Fellow, IEEE

Abstract—This paper describes the realization of a low-temperature co-fired ceramic (LTCC)-based wireless transceiver with optical interface for radio-over-fiber applications involving several standards. The RF front-end including an antenna is fabricated in LTCC technology, while the optical transceiver with a single-mode optical interface is built on a silicon motherboard. The front-end operates in the 5-6-GHz band, while the modulated optical carrier is transmitted at 1.55- m wavelength. The front-end module is an attractive solution for wireless local area network applications such as IEEE 802.11a or HIPERLAN2 requiring a direct link to an optical backbone. Index Terms—Low-temperature co-fired ceramic (LTCC), optically fed antennas, opto-electronics, radio-over-fiber, transceiver, wireless local area network (WLAN).

I. INTRODUCTION

R

ADIO-OVER-FIBER applications are of growing interest to a number of outdoor and indoor applications. The basic idea consists of employing several base stations connected to a single central unit by means of a fiber-optic backbone. Such a setup is also generally referred to as a distributed antenna system. The RF base stations and the central unit are either linked through baseband-modulated optical carriers or by subcarrier multiplexing the optical carrier with the RF signal. The benefits in using such a distributed antenna system are twofold: great versatility and potentially large cost effectiveness. Using subcarrier-multiplexed radio-over-fiber is versatile in several ways. First of all, changes in the transmission standards and protocols can easily be implemented by reprogramming or substituting the central unit instead of the transceivers. Secondly, the transceiver architecture can be kept simple and, thus, small and cost effective. The use of radio-over-fiber solutions has been investigated with respect to various wireless standards, ranging from wireless local area network (WLAN) (IEEE 802.11a/b/g) to global system for mobile communications (GSM) [1]. Application of this technology ranges from pico-cell interconnects in urban

Manuscript received April 1, 2006; revised October 11, 2006. L. Pergola and R. Vahldieck are with the Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology, Eidgenössische Technische Hochschule Zurich, CH-8092 Zurich, Switzerland (e-mail: [email protected]; [email protected]). R. Gindera and D. Jäger are with the Department of Optoelectronics, Universität Duisburg-Essen, D-47048 Duisburg, Germany (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890527

areas to in-house distribution of RF signals to intelligent transport systems [2]. In all these applications, the use of lightweight and compact transceivers is necessary in order to fully exploit the benefits of a distributed antennas system. To merge optical and RF functionality in one circuit environment, low-temperature co-fired ceramic (LTCC) technology has shown great potential due to its possibility to enable three-dimensional integration and interconnection, as well as packaging of active and passive RF components up to millimeter-wave frequencies. Passive alignment of a multimode optical fiber to a laser by using LTCC structures has already been successfully investigated. In [3], an edge-emitting laser diode situated in a cavity was passively aligned to a multimode fiber placed in a grooved area in the LTCC. In [4], a via-hole is used to position a fiber underneath the active region of a vertical cavity surface-emitting laser placed on top of an LTCC substrate. In both papers, the tolerances inherent in the LTCC process are fully compatible with the requirements for multimode fiber coupling. In order to enable transmission bandwidths of several gigahertz over large distances and also to easily interconnect remote base stations to the single-mode fiber-optic backbone, merging of LTCC with single-mode fibers (SMFs) can be quite attractive. Although the LTCC process is not capable of achieving the tight sub-micrometer tolerances required for passive alignment of single-mode optical fibers to lasers or photodiodes (PDs), silicon technology can be used instead, and subsequently integrated into LTCC. By using V-grooved regions on a silicon motherboard to fix the SMFs, high-precision alignment of the fibers to the laser and PD on the same substrate can be achieved. This paper describes the design, realization, and testing of an LTCC-based wireless transceiver with an integrated optical interface. The resulting optical module with 9/125 SMFs was placed on the LTCC chip and wire bonded to the RF circuitry of the LTCC transceiver [5]. The transceiver operates in time duplex using a switch to separate between TX and RX paths. Its operating band is 5–6 GHz for WLAN-oriented applications like IEEE 802.11a (5.15–5.825 GHz) or HIPERLAN2 (5.15-5.725 GHz). II. SYSTEM LAYOUT The block diagram of the optical transceiver is represented in Fig. 1. A patch antenna, designed for working in the 5–6-GHz frequency band, featuring a gain of 5.5 dBi, is directly connected to a bandpass filter. The filter is designed for a passband of 1 GHz centered at 5.5 GHz with an insertion loss of 3 dB and dB at 6.6 GHz. a selectivity of

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Fig. 1. Block diagram of the radio-over-fiber transceiver with integrated optical interfaces.

A commercial monolithic-microwave integrated-circuit (MMIC) switch, suited for WLAN applications with an insertion loss of 1 dB, allows for RX/TX time duplexing by connecting the filter to both the receiving and transmitting branches. The switch is a commercial surface mountable chip in GaAs technology. In the receiving mode, the filtered signal is delivered to a GaAs-based low-noise amplifier (LNA), featuring a gain of 20 dB and connected by wire bonds. The amplified signal nm. modulates a distributed feedback (DFB) laser at A passive matching network is used to match the standard 50- output of the receiver LNA to the small 10- signal impedance of the laser, limiting the insertion loss to 1 dB. The optical signal produced by the modulation of the laser current is injected into a single-mode 9/125- m fiber. The biasing of the laser is through a bias tee, using discrete microwave rated components. In the RF transmitting mode, a modulated optical signal enters the device through a single-mode 9/125- m fiber coupled to a microwave-rated p-i-n PD, which converts the optical signal into an RF current. The current from the photodetector is fed to a commercial SiGe transimpedance amplifier (TIA), featuring a 1-k transimpedance. The TIA is also used for biasing the PD. The TIA output signal is amplified by a commercial GaAs LNA chip with 15-dB gain. Both the TIA and transmitter LNA are interconnected by wire bonds. Power amplification was not considered a primary issue at this prototyping stage. The output of the transmitter LNA is fed to the transmitting branch of the switch, filtered, and finally delivered to the antenna feed line. III. LTCC MODULE The core of the transceiver is a ten-layer LTCC module including the antenna feed network. The antenna is designed as an aperture-coupled patch radiator fed by means of a stripline located inside the LTCC die. The slot for coupling to the radiator is located on the upper ground plane of the LTCC module. The actual radiator is placed on top of the module and held in position at a distance of 2.5 mm by means of plastic screws (Fig. 2). The patch radiator is a square with a 21-mm edge. The dimensions of the ground plane (30 30 mm) needed for satisfactory radiation characteristics of the antenna clearly exceeded the LTCC size. Therefore, the antenna ground plane was split into two parts involving both the LTCC module and motherboard. The LTCC module is connected to the ground plane side of an RT/Duroid motherboard by means of a ball grid array (BGA).

Fig. 2. LTCC module placed on the motherboard. The patch radiator is held in position by means of plastic screws.

Fig. 3. Different layers of the LTCC module.

Fig. 4. Laser matching network.

The size of the LTCC die is 14 14 1.3 mm. The fabrication of the module involved the use of a DuPont Greentape . The 951A2, with post-fired thickness of 135 m and metallization is realized by gold screen printing. Fig. 3 shows a more detailed sketch of the LTCC die features. An internal ground plane divides the module in two parts. The upper part consists of four layers and contains the antenna feed line; the lower part is composed of six layers and contains the laser matching network and the interdigital six resonator bandpass filter used both in the transmitting and receiving branch. The interdigital bandpass filter is realized with six quarterwavelength stripline resonators, which are at one side grounded to both ground planes. The laser matching network was designed, on the one hand, to match the laser impedance to the output of the receiver LNA and, on the other hand, to compensate for the parasitic inductance due to the bond wires connecting the laser. The structure of this network is sketched in Fig. 4; two quarter-wavelength lines are stacked and separated by a metal layer connected to

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Fig. 6. Simulated insertion loss for an LTCC-embedded vertical transition. Fig. 5. Connection of an MMIC pad to a stripline embedded into the LTCC module. The MMIC ground pads are wire bonded directly to the module ground plane.

ground by means of via fences. A stub is used in order to improve the matching. The outside areas of the top and bottom layers of the LTCC module serve also as a packaging structure for the MMICs and as interconnect layer for the different components. Moreover, all of the lines used for biasing the active components and controlling the switch are embedded inside the LTCC die and distributed over several layers, which are interconnected by means of vias. The LTCC bias lines are then connected to the microstrip side of the RT/Duroid board by means of a BGA and vias realized through the motherboard itself. The way of connecting an external MMIC to the internal LTCC circuitry is illustrated in Fig. 5. A wire bond connects the MMIC pad to a pad realized on the LTCC ground plane; the latter is connected by means of a metal via to a stripline realized between the middle and the bottom ground plane of the LTCC module. The transition embedded inside of the LTCC can be very critical in terms of insertion loss and matching. The behavior of such a vertical transition has been investigated in [6] and [7], where several broadband solutions were proposed in order to reduce its negative impact on the circuit performance. The structure has been simulated in the operating frequency range and the results are shown in Fig. 6 for the insertion loss and in Fig. 7 for the return loss at the stripline side. Both the simulated return and insertion loss have proven to be excellent over the whole band. The full-wave simulations have been carried out by using Ansoft’s High Frequency Structure Simulator (HFSS). The receiver LNA is located on the top ground plane of the LTCC die. Holes were drilled into the RT/Duroid motherboard to allow for the presence of integrated circuits (ICs) and discrete passive components also on the bottom LTCC ground plane. Fig. 8 shows a detailed view of the motherboard (microstrip side) with a portion of the LTCC visible. The optical silicon bench has been placed on the bottom ground plane of the LTCC; the laser and PD are connected to the laser bias tee and TIA, respectively, by means of wire

Fig. 7. Simulated return loss for an LTCC-embedded vertical transition.

Fig. 8. Backside of the RT/Duroid motherboard showing the bias lines.

bonds. The transmitter LNA and the switch are also located on the same ground-plane side. To avoid unwanted coupling between the bias lines, they were separated as much as possible. Air bridges have been used to connect lines across the cut for the optical bench.

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Fig. 11. Layer structure of the Si submount.

Fig. 9. Layout of the hybrid integrated SiOB. From [5].

Fig. 12. Image of the realized SiOB (without fibers).

Fig. 10. SEM picture of the: (a) BH-DFB laser diode and (b) p-i-n PD.

IV. SILICON OPTICAL BENCH (SiOB) To enable optical coupling of the LTCC wireless transceiver to standard SMFs, a hybrid integrated SiOB, acting as an interface between the electrical circuitry placed on the LTCC module and the optical domain, was realized. Due to the tolerances in LTCC processing, the coupling of standard SMF to active optical devices could not be implemented directly in LTCC. Using instead an SiOB, the aligning process can easily be realized. The fiber coupled SiOB can then be placed onto the LTCC transceiver and electrically contacted using wire-bonding techniques to realize the optically fed wireless transceiver. The SiOB (Fig. 9) consists of an 8 2 mm silicon submount with an integrated V-groove and high-frequency planar electrical circuitry. On top of the Si substrate, a laser diode (LD) and a PD are bonded. The directly modulated buried heterostructure distributed feedback (BH-DFB) laser diode [see Fig. 10(a)] is optimized for 10-Gb/s applications in metropolitan area networks. The laser emits light at a wavelength of 1550 nm with a maximum optical output power of 10 dBm. The waveguide laser allows 500 m horizontal coupling to a standard SMF. The 400 The LD chip comprises two LDs, with one being a spare. The high-speed top illuminated p-i-n PD [see Fig. 10(b)] based on InGaAsP/InP exhibits a 3-dB bandwidth of 10 GHz. The 550 550 m PD chip exhibits an active area with a diameter of 38 m.

The PD is designed to be mounted by flip-chip technology. The high-frequency planar electrical circuitry is used for electrically contacting the LD and PD and establishes the connection to the LTCC transceiver. silicon substrate covFor realizing the SiOB, we used a ered with a silicon–dioxide and silicon–nitride layer structure (Fig. 11) for the Si submount. V-grooves were etched into this submount to enable planar integration of the vertically illuminated PD. The silicon–nitride layer is used to mask the silicon substrate during an anisotropic wet chemical etching process of the V-groove. The silicon–dioxide layer acts as a buffer layer for lattice matching of silicon and silicon–nitride. The top thermal silicon dioxide layer was removed using a 5% solution of HF etchant. Subsequently the silicon–nitride layer and the silicon–dioxide buffer layer were etched using an 85% solution of phosphoric acid and a 5% solution of HF, respectively. The V-groove was etched into the silicon substrate using a 30% solution of alkali-hydroxide etchant (KOH). To realize the high-frequency electrical circuitry, a Ti/Au alloy was thermally deposited on top of the Si submount. Simulations were carried out for designing the coplanar ground–signal–ground (GSG) transmission lines so that they are impedance matched over a frequency range up to 10 GHz. Flip-chip technology was used to mount the high-speed PD onto the Si submount. Therefore, gold bumps with a diameter of 75 m were deposited onto the contacts of the electrical circuitry of the Si submount. The PD was fixed via thermo-compression onto the bumps using the flip-chip bonder. The LD carries a backside metallization for the n-contact. Therefore, a thermally curing epoxy adhesive with high electrical and thermal conductivity was used to mount the LD onto the Si submount. The p-contact of the LD was connected to the Si submount using wire-bonding technology. In Fig. 12, an image the realized SiOB is shown.

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Fig. 13. Cross section of the PD coupling scheme.

Fig. 15. Measured optical output power of the LD as a function of the applied current.

Fig. 14. (a) Measured dark current of the PD. (b) Measured current–voltage characteristic of the PD for 0-dBm optical input power at 1550 nm.

For coupling a standard SMF to the vertically illuminated PD, the fiber was aligned in the V-groove and pushed forward under the PD. The 54.7 angled front mirror of the V-groove, which arises during anisotropic etching, was used for reflecting the light to the PD (Fig. 13). For coupling the LD, a butt-coupling scheme was used. Therefore, a cleaved standard SMF was placed on the SiOB, left floating and actively aligned to the LD waveguide. No V-groove was used for aligning the fiber to the LD in order to avoid complications related to the consequent shift of the fiber core with respect to the laser waveguide. After aligning the fibers to the LD and PD, they were fixed to the Si submount by using a nonshrinking UV curing liquid adhesive. This adhesive exhibits a refractive index matched to the core of the standard SMF for low-loss optical coupling of the active optical devices. The measurements of the packaged PD characteristics were carried out by using a reference laser diode with tunable wavelength in the range from 1500 to 1580 nm. The laser wavelength was tuned to 1550 nm by using an optical spectrum analyzer and the emitted power was fixed to 0 dBm with the help of a reference photodetector. The reference laser was then used to inject the 0-dBm optical power into the PD fiber. As shown in Fig. 14, the realized optical transceiver exhibits a PD responsivity of 81 mA/W for the transmitter branch.

Fig. 16. Measured antenna.

E -plane radiation pattern for the aperture coupled patch

The optical output power of the packaged LD was measured by connecting the LD fiber to a reference photodetector. The laser showed a small-signal responsivity of 2.25 mW/A when biased at 15 mA. The maximum optical output power of the LD is 4 dBm at 45 mA (Fig. 15). V. ANTENNA AND FILTER MEASUREMENTS The measured radiation patterns for the aperture coupled patch antenna are illustrated in Fig. 16 for the -plane and in Fig. 17 for the -plane. Both diagrams show the measured main and cross polarization. Since the antenna is linearly polarized, the cross polarization level is very low. The gain was measured as 5.5 dBi over the operating frequency band. The use of plastic screws for holding the radiator on top of the LTCC module introduced tolerance problems. Several simulations with small variations in the distance between the radiator and the LTCC module were carried out in order to figure out the impact on the antenna return loss. The results of the simulations

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Fig. 19. Tolerance analysis with respect to the dielectric constant of the Duroid 5880 substrate carrying the patch radiator. The nominal value is " = 2:2. Fig. 17. Measured H -plane radiation pattern for the aperture coupled patch antenna.

Fig. 20. Effects of the tolerances on the antenna return loss. The simulation with h = 2:3 mm and " = 1:9 agrees well with the measured results. Fig. 18. Tolerance analysis with respect to the distance between the patch radiator and the LTCC module. The nominal value is h = 2:5 mm.

are shown in Fig. 18; changes of 0.2 mm with respect to the optimum distance (2.5 mm) already introduce sensible variations in the return loss. The return loss is, however, below 10 dB over most of the 5.15–5.825-GHz band (IEEE 802.11 a). The tolerances in the dielectric constant of the 20-mil-thick Duroid 5880 substrate carrying the radiator have also been investigated. The changes in the return loss, corresponding to , have small variations around the nominal value of been simulated and the results are illustrated in Fig. 19. Results of the tolerance investigation clearly explain the disagreement between the simulated and measured antenna return loss. Fig. 20 shows a comparison between the simulated return loss, corresponding to the nominal values for the antenna height mm and dielectric constant , and the simulated mm and ; the latter has been return loss for found to agree well with the measured return loss. This is also illustrated by a comparison on the Smith chart (Fig. 21). The interdigital bandpass filter embedded in the LTCC die was measured, and the -parameters are shown in Fig. 22. The passband was designed somewhat wider than required in order to accommodate possible tolerance problems due to the LTCC process.

Fig. 21. Comparison between simulated and measured return loss for h =

2:3 mm and " = 1:9.

VI. OPTICAL TRANSCEIVER MEASUREMENTS Preliminary measurements were targeted at the RF performance of the LTCC chip to verify the performance of both the transmitting and receiving branches separately without the optical interface. For this reason, several LTCC test modules were

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Fig. 24. Measured electrical performance of the transmitter. Fig. 22. Measured filter S -parameters.

Fig. 25. Transmitter measurement setup.

Fig. 23. Measured electrical performance of the receiver.

realized in the same production lot. In the LTCC test phase, the SiOB was replaced by a 50- microstrip line on a separate substrate attached to the bottom side of the LTCC to mimic the SiOB. The microstrip line was also connected to the laser bias tee and a gain measurement of the RF part was performed. The gain was expected to be 16.5 dB, which was calculated by taking into account the antenna gain (5.5 dBi), receiver LNA gain (20 dB), filter loss (3 dB), and simulated loss in the laser matching network (6 dB). The measured gain versus frequency is illustrated in Fig. 23. The maximum in-band gain is 16 dB, which compares well with the expected value. The high loss of the matching network is due to the mismatch caused by the 50microstrip load instead of the low impedance of the biased laser diode. A similar test was done on the transmitter LTCC module, again with a 50- microstrip line in place of the optical silicon bench. The microstrip line was connected to the TIA. The measured gain is shown in Fig. 24. The maximum level of the in-band gain compares well with the expected value of 29.5 dB. The latter was calculated taking into account the TIA gain (12 dB), transmitter LNA gain (15 dB), filter loss (3 dB), and antenna gain (5.5 dBi). The expected TIA gain has been

calculated taking into account the strong mismatch at its input due to the 50- microstrip load instead of the high impedance of a reverse-biased PD. It is likely that this mismatch is also the reason for the ripple of the gain (Fig. 24) inside of the operating band. Both gain measurements were performed in an anechoic chamber by comparison with a known antenna. Finally, the complete transceiver is measured with the optical interface fully integrated. To be able to interpret the measurement results correctly, the optical power used in the measurements must be known. It should be noted that the prototype measurements were done with amplitude modulated (sinusoidal) optical signals of varying frequency from 4.5 to 8 GHz in order to allow a performance analysis over the frequency range. Thus, the total , optical power can be written as where and are constant. The amplitude is related to the power of the signal traveling in the fibers. This power can be measured by a spectrum analyzer and is referred to as modulated optical power. Two different measurement setups have been used in order to test the performance of both the transmitting and receiving branches. The setup for the transmitter is sketched in Fig. 25. An HP85645A tracking generator, synchronized to an HP71400C spectrum analyzer, is used to drive the HP83402A laser. In this configuration, a modulated optical power of 9.2 dBm is injected into the transmitter fiber. The output of the PD is then amplified through the LTCC RF transmitter and

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Fig. 26. Measured received power at the sensing antenna in the transmitter measurement setup.

Fig. 27. Receiver measurement setup.

radiated by the antenna. A sensing antenna (R70 horn antenna, 14.5-dBi gain) is placed at a distance of 1 m away from the device-under-test and connected to a spectrum analyzer. The expected in-band power can be calculated and compared with measured values. The input modulated optical power is dBm mW and the PD responsivity is mA/W, resulting in a PD current amplitude of A. The PD current feeds the TIA, which produces an mV across the tranoutput voltage amplitude of k . This translates into an output power simpedance of delivered to the 50- input of the transmitter LNA of dBm. The calculated radiated power from the transceiver patch antenna amounts to a level of 13.7 dBm. Considering the free-space loss ( 48.5 dB) and the sensing antenna gain, an expected power level of 47.7 dBm at the sensing antenna output can be expected. The measured power level is shown in Fig. 26 for a frequency band ranging from 4.5 up to 8 GHz. A maximum measured power level of 48.9 dBm is detected, confirming the expected performance of the transmitter. The power level drops down to 80 dBm in the band from 7 to 8 GHz due to the filter selectivity. A similar measurement setup, sketched in Fig. 27, was used in order to test the performance of the receiver. The tracking generator, driven by the spectrum analyzer, is connected to the R70 antenna and the transceiver under test is again placed 1 m away. The power delivered by the tracking generator to the R70 antenna is 4 dBm, resulting in a power level of 30 dBm at the transceiver patch radiator. The small-signal power delivered to the biased laser can be calculated taking into account the patch

Fig. 28. Measured modulated optical power from the laser.

antenna gain of 5.5 dBi, receiver LNA gain of 20 dB, switch insertion loss (1 dB), filter insertion loss (3 dB), and loss due to the matching network (1 dB), resulting in an expected power level of 9.5 dBm. The laser is biased at 15 mA with a signal responsivity of 2.25 mW/A. A small signal power of 9.5 dBm delivered to the biased laser results in a small-signal modulation mA. This value multiof its current with amplitude plied by the responsivity gives a value of 19.7 dBm for the expected modulated optical power generated by the laser. The modulated optical power from the laser was measured by connecting the pigtailed laser to the optical plug-in port (HP70810A) of the spectrum analyzer. The measurement results are illustrated in Fig. 28; a maximum value of 20 dBm was measured, confirming the calculated performance of the receiver. VII. CONCLUSION For the first time, the design of an LTCC-based optical wireless transceiver for radio-over-fiber systems utilizing standard SMFs has been presented. This device is particularly attractive for indoor and outdoor distributed antenna systems. The front-end consists of an LTCC module including RF circuitry and antenna, as well as an SiOB including laser and PD for optically feeding the wireless transceiver using standard SMFs. The performance of the single passive components was tested and compared with simulations. The results of a tolerance analysis on the patch radiator have been used in order to explain the disagreement between the measured and simulated antenna return loss. Measurements on the complete transceiver, as well as on testing structures, have been carried out in order to compare the actual performance with the expected one. The maximum measured power levels in the 5–6-GHz operating frequency range agree well with the theoretical expectations for both the transmitting and receiving branch. ACKNOWLEDGMENT The authors would like to acknowledge the great support of H. Benedickter, M. Lanz, C. Maccio, and S. Wheeler, all members of technical staff of the Institut für Feldtheorie und Höchstfrequenztechnik (IFH), Zurich, Switzerland.

PERGOLA et al.: LTCC-BASED WIRELESS TRANSCEIVER FOR RADIO-OVER-FIBER APPLICATIONS

REFERENCES [1] P. Hartmann, X. Qian, R. V. Penty, and I. H. White, “Broadband multimode fibre (MMF) based IEEE 802.11a/b/g WLAN distribution system,” in IEEE Int. Microw. Photon. Top. Meeting, Oct. 2004, pp. 173–176. [2] H. B. Kim, M. Emmelmann, B. Rathke, and A. Wolisz, “A radio-overfiber network architecture for road vehicle communication systems,” in IEEE 61st Veh. Technol. Conf., May 2005, vol. 5, pp. 2920–2924. [3] J. A. Hiltunen, K. Kautio, J.-T. Mäkinen, and P. Karioja, “Passive multimode fiber-to-edge-emitting laser alignment based on a multilayer LTCC substrate,” in Electron. Compon. Technol. Conf., May 2002, pp. 815–820. [4] M. Karppinen, K. Kautio, M. Heikkinen, J. Häkkilä, and P. Karioja, “Passively aligned fiber-optic transmitter integrated into LTCC module,” in Electron. Compon. Technol. Conf., May 2001, pp. 20–25. [5] L. Pergola, R. Gindera, D. Jäger, and R. Vahldieck, “An LTCC-based wireless transceiver with integrated optical interface,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, [CD ROM]. [6] H.-H. Jhuang and T.-W. Huang, “Design for electrical performance of wideband multilayer LTCC microstrip-to-stripline transition,” in Proc. 6th Electron. Packag. Technol. Conf., Dec. 2004, pp. 506–509. [7] R. Valois, D. Baillargeat, S. Verdeyme, M. Lahti, and T. Jaakola, “High performances of shielded LTCC vertical transitions from DC up to 50 GHz,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 2026–2032, Jun. 2005.

Luca Pergola (S’01) was born in Cagliari, Italy, in 1974. He received the Master degree in electronic engineering from the University of Cagliari, Cagliari, Italy, in 2000, and is currently working toward the Ph.D. degree at the Eidgenössische Technische Hochschule (ETH), Zürich, Switzerland. In January 2001, he joined the Electromagnetic Field Theory Group, ETH. His current research fields are RF frontends, antenna arrays, and radio-over-fiber systems realized by using LTCC technology.

Ralf Gindera (S’04) was born in Oberhausen, Germany, in 1976. He received the Dipl.-Ing. degree in electrical engineering from the Universität Duisburg-Essen, Duisburg, Germany, in 2003. His thesis focused on the development of an optical 1.3 m DFB-GRINSCH laser with a monolithic integrated 40-Gb/s electroabsorption modulator. He is currently a Research Associate with the Department of Optoelectronics, Universität DuisburgEssen. His main research interests include opto-electronic devices and communication systems, as well as microwave photonic devices and their applications in microwave and millimeter-wave transmission systems.

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Dieter Jäger (F’01) received the Diplomphysiker, Dr.rer.nat., and Habilitation degrees in physics from the University of Münster, Münster, Germany, in 1969, 1974, and 1980, respectively. From 1974 to 1990, he was Head of a research group with the Institute for Applied Physics, University of Münster, where he became an Associate Professor of physics in 1985. From 1989 to 1990, he was a Visiting Professor with the Universität DuisburgEssen, Duisburg, Germany. Since 1990, he has been with the Faculty of Electrical Engineering, Universität Duisburg-Essen, where he is Head of the Department of Optoelectronics. From 1998 to 2001, he was Dean of the faculty. He is an Honorary Professor of Brasov University/Romania and Consultant Professor of the Huazhong University of Science and Technology, China. He is a consultant of the Institution of Electrical Engineers (IEE), U.K., Photonics Network, a member of the Photonics Competence Center, and founder of OpTech-Net, a German Network of Excellence on optical technologies. He has authored or coauthored over 300 papers in books, journals, and conference proceedings. Prof. Jäger is chair of the German IEEE Lasers and Electro-Optics Society (LEOS) Chapter, as well as a member of the IEEE Microwave Photonics Steering Committee.

Rüdiger Vahldieck (M’85–SM’86–F’99) received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the University of Bremen, Bremen, Germany, in 1980 and 1983, respectively. From 1984 to 1986, he was a Postdoctoral Fellow with the University of Ottawa, Ottawa, ON, Canada. In 1986, he joined the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, where he became a Full Professor in 1991. During Fall and Spring of 1992–1993 he was a Visiting Scientist with the Ferdinand-Braun-Institute für Hochfrequenztechnik, Berlin, Germany. In 1997, he was became a Professor of electromagnetic field theory with the Swiss Federal Institute of Technology, Zurich, Switzerland, and became Head of the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH) in 2003. In 2005, he became President of the Research Foundation for Mobile Communications and was elected Head of the Department of Information Technology and Electrical Engineering (D-ITET), Eidgenössische Technische Hochschule (ETH) Zurich. Since 1981, he has authored or coauthored over 300 technical papers in books, journals, and conferences. His research interests include computational electromagnetics in the general area of electromagnetic compatibility (EMC) and, in particular, for computer-aided design of microwave, millimeter-wave, and opto-electronic ICs. Prof. Vahldieck is the past president of the IEEE 2000 International Zurich Seminar on Broadband Communications (IZS’2000) and, since 2003, president and general chairman of the International Zurich Symposium on Electromagnetic Compatibility (EMC Zurich). He is a member of the Editorial Board of the IEEE TRANSACTION ON MICROWAVE THEORY AND TECHNIQUES. From 2000 to 2003, he was an associate editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, and from July 2003 to 2005, he was the editor-in-chief. Since 1992, he has been on the Technical Program Committee (TPC) of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), the IEEE MTT-S Technical Committee on Microwave Field Theory, and in 1999, on the TPC of the European Microwave Conference. From 1998 to 2003, he was the chapter chairman of the IEEE Swiss Joint Chapter on IEEE MTT-S, Antennas and Propagation (AP), and Electromagnetic Compatibility (EMC) societies. He was the recipient of the 1996 Institution of Electronics and Telecommunication Engineers (IETE) J. K. Mitra Award for the best research paper. He was corecipient of the 1983 Outstanding Publication Award of the Institution of Electronic and Radio Engineers and the 2004 ACES Outstanding Paper Award.

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Development of a 2.45-GHz Local Exposure System for In Vivo Study on Ocular Effects Kanako Wake, Member, IEEE, Hiroyuki Hongo, Soichi Watanabe, Member, IEEE, Masao Taki, Member, IEEE, Yoshitsugu Kamimura, Member, IEEE, Yukio Yamanaka, Toru Uno, Senior Member, IEEE, Masami Kojima, Ikuho Hata, and Kazuyuki Sasaki

Abstract—We developed a new exposure system to irradiate microwaves locally on a rabbit eye using a small coaxial-to-waveguide adapter filled with low-loss dielectric material as an antenna. A numerical rabbit model was also developed using X-ray computer tomography images, and the specific absorption rates (SARs) in the rabbit, especially in the eye, were analyzed with the finite-difference time-domain method. The temperature elevation in the exposed eye was also evaluated by solving a bioheat equation. Our exposure system can generate incident power density of 15 mW/cm2 at the surface of a rabbit eye with input power of 1 W. When the incident power density on the rabbit eye is 300 mW/cm2 , average SAR over the exposed eye and the whole body were approximately 108 and 1.8 W/kg, respectively. The exposure system can realize localized exposure to the eye with the ratio of exposed-eye averaged SAR to the whole-body averaged SAR was 60. The developed exposure system can achieve high-intensity exposure such as the threshold of cataracts, i.e., the eye-averaged SAR over 100 W/kg or the lens temperature over 41 C with the incident power density of 300mW/cm2 without significant whole-body thermal stresses. Index Terms—Cataract, eye, microwave, ocular effect, specific absorption rate (SAR), temperature.

I. INTRODUCTION

O

CULAR effects due to microwave exposure are one of the effects significantly related to the safety guidelines for limiting exposures to microwaves. Several experiments have been performed to investigate the effects of microwave irradiation on eyes [1]–[3]. A comprehensive review of ocular effects has been done [4]. Guy et al. irradiated 2.45-GHz continuous microwaves on an eye of a rabbit and reported that cataracts can be caused by microwave irradiation at an incident power density of

Manuscript received July 10, 2006. This work was supported in part by the Committee to Promote Research on the Possible Biological Effect of Electromagnetic Fields, Ministry of Internal Affairs and Communications. K. Wake, S. Watanabe, and Y. Yamanaka are with the National Institute of Information and Communications Technology, Tokyo 184-8795, Japan. H. Hongo and T. Uno are with the Department of Electrical and Electronic Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan. M. Taki is with the Department of Electrical and Electronic Engineering, Tokyo Metropolitan University, Minamiosawa 1-1, Hachioji-shi, Tokyo 1920397, Japan. Y. Kamimura is with the Department of Information Science, Utsunomiya University, Tochigi 321-8585, Japan. M. Kojima, I. Hata, and K. Sasaki are with the Department of Ophthalmology, Kanazawa Medical University, Ishikawa 920-0293, Japan. 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.2006.890531

150 mW/cm for 100 min and its threshold temperature is 41 C [1]. This incident power density is much higher than 1 mW/cm , which is the reference level for the general public in the International Commission on Non-Ionizing Radiation Protection (ICNIRP)’s guidelines [5]. Kues et al. reported that corneal endothelial abnormalities were caused by pulsed microwave of 10 mW/cm and continuous microwave of 20 mW/cm [2]. However, a replication study by Kamimura et al. showed no effects for continuous microwave irradiation, while they did not investigate the effect of the pulsed microwave [3]. There is a guideline in Canada to limit the eye exposure based on Kues et al.’s experimental data [6]. The absence of reproducibility among the previous studies suggests the need for further investigation on the effects of microwaves on eyes. We list the following three requirements for the exposure setup for the eye exposure experiment with high accuracy: • ability of high-intensity exposure enough to replicate wellunderstood effects; • localization of exposure of the eye to avoid thermal effects of whole-body power absorption; • well-established dosimetry. We have developed a new system to expose the rabbit eye to microwaves [7]. This exposure system was used in the experiment to investigate the ocular effects and temperature in rabbit eyes [8]. In this paper, the specific absorption rate (SAR) in the rabbit and the temperature elevation due to exposure are analyzed. II. EXPOSURE SETUP For our eye exposure experiment, we need to generate highintensity exposure enough to replicate well-understood effects such as cataracts without thermal effects of whole-body power absorption. A coaxial-to-waveguide adapter has been employed as an antenna in several eye exposure experiments [2], [3]. However, the aperture size of an R22 waveguide, the ordinary waveguide at 2.45 GHz, is 54.6 109.2 mm , and is comparable to the size of a rabbit’s head, which makes it difficult to expose the rabbit eye locally. Therefore, we developed an experimental setup to expose the rabbit eye locally to microwaves of 2.45 GHz [7]. A diagram and photograph of the exposure setup are shown in Fig. 1(a) and (b), respectively. Continuous or pulsed microwaves can be fed to an antenna. A signal generator (ESG-D3000A, Hewlett-Packard, Palo Alto, CA) and an amplifier (GRF5034, Ophir, New Orleans, LA) are employed for continuous microwave irradiation. Pulsed microwaves are generated by a pulse generator (MKN-156-3S2A, Nihon Koshuha,

0018-9480/$25.00 © 2007 IEEE

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Fig. 1. Exposure system. One eye of the rabbit is exposed to 2.45-GHz microwave radiated from a small coaxial-to-waveguide adapter. The rabbit is under constraint with a plastic holder. The distance between the antenna aperture and the surface of the rabbit eye is 40 mm. (a) Diagram of experimental setup. (b) Experimental view.

Kanagawa, Japan). Both forward and reverse powers were monitored with power meter (E4417A, Agilent Techology, Palo Alto, CA) through directional coupler to determine radiated power from the antenna. We developed a small waveguide-adapter by filling an R84 waveguide with a low-loss dielectric material, i.e., macerite, so that we could localize the microwave exposure on the eye and avoid whole-body thermal stresses due to whole-body power absorption. The aperture size of our small waveguide-adapter antenna is 12.6 28.5 mm . Solid macerite was inserted inside the waveguide adapter, except the feed line and antenna surface was covered with polyvinyl chloride. The relative permittivity of macerite is approximately 5.5 at 2.45 GHz. The developed waveguide adapter is shown in Fig. 2. The frequency characteristics of the input impedance of the small waveguide antenna in free space is shown in Fig. 3. The real part of the antenna input impedance has a peak around 2.46 GHz. The antenna input . impedance at 2.46 GHz is For our animal experiment, it is necessary to avoid significant changes in the antenna input impedance due to animal movement and to achieve high-intensity exposure for positive control conditions. Fig. 4 shows the dependence of the antenna input impedance on the distance between the antenna aperture and the surface of the rabbit eye. The antenna input impedance varies with the distance as the rabbit gets close to the antenna. Fig. 5 shows the incident power density distribution along the center axis of the waveguide. The incident power density was calculated by the measured electric fields and the wave impedance in free space. The electric fields were measured with a small isotropic electric-field probe (ER3DV5, SPEAG, Zurich, Switzerland) at a distance over 20 mm. The distance

Fig. 2. Coaxial-to-waveguide adapter at 2.45 GHz. Small waveguide adapter is filled with low-loss material (macerite). (a) Ordinary (R22) and small (R84) waveguide adapter. (b) Dimension of the small waveguide adapter.

Fig. 3. Antenna input impedance of the small waveguide antenna in free space.

over 20 mm satisfies far-field conditions are defined as follows for our waveguide antenna: (1) (2) where is the distance from the antenna, is the wavelength, and is the maximum dimension of the antenna.

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Fig. 4. Dependence of the antenna input impedance on the distance between the antenna and the surface of the rabbit eye.

Fig. 6. Measured electric fields. (a) 40 mm from small waveguide (R84). (b) 250 mm from ordinary waveguide (R22).

Fig. 5. Incident power density along the center axis of the waveguide antenna.

Our small waveguide antenna can generate relatively high incident power density in the vicinity of the antennas. To reduce the changes in the antenna input impedance due to animal movement, the distance between the antenna aperture and the rabbit should be large [see Fig. 4]. However, the incident power becomes smaller with larger distance [see Fig. 5]. The distance between the eye of the rabbit and the aperture of the waveguide was determined to be 40 mm considering both requirements. Fig. 6(a) shows the electric field distribution at 40 mm from the aperture of the small waveguide. Results are normalized against the maximum value. We also measured the electric-field distribution at 250 mm from the aperture of the ordinary waveguide R22 [see Fig. 6(b)]. The distance of 250 mm is the length that also satisfies the far-field conditions for the ordinary waveguide. Electric fields were well localized around the center with our small waveguide compared to the ordinary one. The developed exposure system with the small waveguide antenna generates the incident power density of approximately 15 mW/cm at 40 mm from the aperture with the antenna input power of 1 W, although that of the ordinary waveguide at 250 mm from the aperture is approximately 1 mW/cm for the same input power. In previous studies, animals were sometimes anesthetized and sometimes not, and the results were different [2], [3]. In our

experimental setup, rabbits are constrained with a plastic holder made of polycarbonate [see Fig. 1(b)], which makes it possible to accurately expose the eye with and without anesthesia. III. SAR EVALUATION A. Methods and Models There are two methods for evaluating the SAR in biological bodies: numerical analysis and experimental dosimetry. The finite-difference time-domain (FDTD) method [9] is used frequently for numerical analysis. In the FDTD analysis, the heterogeneous numerical model based on anatomical data is often used for the biological body with millimeter resolution. However, this millimeter resolution is sometimes insufficient for modeling the antenna structure and can result in substantial errors in calculation of the antenna input impedance. On the other hand, the actual antenna and exposure setup can be used for experimental dosimetry, e.g., the thermographic method [10]. In the thermographic method, exposure-induced temperature elevation in a phantom is measured with a thermographic camera, and the SAR is evaluated by analyzing the measured temperature increase using (3) as follows: (3)

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TABLE I TISSUES AND THEIR ELECTRICAL PROPERTIES USED FOR THE CALCULATION

TABLE II ELECTRICAL PROPERTIES OF THE PHANTOM

Fig. 7. Rabbit model. (a) Numerical model. (b) Experimental phantom.

where , , and denote the SAR (in watts/kilogram), specific heat capacity of the phantom (joule/gram/kelvin) and temperature (kelvin), respectively. Homogeneous phantoms are usually used for experimental dosimetry, and it is difficult to consider the heterogeneity of biological bodies. Our purpose is to evaluate SAR distributions in the exposed eye. We, therefore, used numerical analysis for the SAR evaluation taking anatomical structure into account. We performed experimental dosimetry to validate our numerical analysis by comparing results in a homogeneous model between two methods. For a detailed evaluation of the SAR inside a rabbit, a rabbit model having a realistic shape and heterogeneous structure is necessary. We developed a numerical rabbit model based on X-ray computer tomography (CT) images. X-ray CT images of an anesthetized rabbit constrained by the plastic holder were taken at the Kanazawa Medical University, Ishikawa, Japan. Resolution of each image was 2.13 pixel/mm and slices of the image were taken at 1-mm intervals. Conversion from the CT data to a numerical (voxel) model and discrimination of tissues was partly automated in a similar method to [12]. We modeled the rabbit eye in detail by hand segmentation because we are interested in that. This hand segmentation was checked by medical doctors on ophthalmology. The developed model consists of 11 types of tissues and the resolution of the model is 1 mm [see Fig. 7(a)]. Table I lists the tissues and their electrical properties used in our calculations. The electrical properties of the biological tissues are available online,1 which is based on the data by Gabriel [11]. We take average of bone cancellous and bone cortical for bone and substitute vitreous humour for anterior chamber. Fig. 7(b) shows a homogeneous phantom of the rabbit with the same shape as that of the numerical model used for experimental dosimetry. We referred to [10] for the ingredients of the 1[Online].

Available: http://www.fcc.gov/fcc-bin/dielec.sh

phantom material. The electrical properties of the solid phantoms were measured several times with a dielectric probe kit (85070C, Agilent Technology, Palo Alto, CA) and the results are listed in Table II. The specific heat capacity of phantoms was 3.77 J/g/K. The phantoms were exposed to 2.45-GHz microwave with the small waveguide antenna for 30 s. The antenna input power was 35 W. The temperature elevation in the phantom due to the exposure was measured using a thermographic camera (TVS-8100MK II, Nippon Avionics, Tokyo, Japan). The phantoms are divided into two parts at a frontal plane across the eyes. The antenna was also modeled for numerical dosimetry. The plastic holder for the rabbit was not taken into consideration for our analysis. We assumed effects of the plastic holder on the SAR especially inside the eye is not significant because the relative permittivity of the plastic is relatively low compared to biological tissues and the plastic holder dose covers only the body and the nose, but not around the eyes. The FDTD method was used for calculating the SAR distribution in the rabbit model. The calculation region was 240 335 241 voxels and the resolution of each voxel was 1 mm. We employed second-order approximations of Mur’s absorbing boundary conditions [13]. B. Comparison of Numerical Analysis to Experimental Dosimetry The calculated and measured reflection coefficients of the antenna in free space are shown in Fig. 8. The calculated result was approximately equal to the measured one. We compared the SARs calculated in the homogeneous numerical model with those evaluated by the experimental dosimetry using the homogeneous solid phantom. We first compared calculated and measured SARs in a cube model of 100 100 mm [see Fig. 9(a)]. The calculated and 100 measured SAR values agreed well for the cube model. From this result, we confirmed our FDTD code. Fig. 9(b) shows the SAR value along the line from one eye to the other eye in the homogeneous rabbit model. Measured SARs were smaller than calculated results in and around the exposed eye. The complex shape of the rabbit, especially around eyes, whose curvature

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Fig. 8. Calculated and measured reflection coefficient of the antenna. Fig. 10. SAR distribution in the heterogeneous rabbit model exposed to near field of the small waveguide antenna. SAR is normalized against maximum value. (a) Surface. (b) Frontal plane across eyes.

Fig. 11. SAR distribution in the heterogeneous rabbit model exposed to plane wave. SAR is normalized against maximum value. (a) Surface. (b) Frontal plane across eyes.

Therefore, the numerical analysis with the FDTD method was used to evaluate the SAR inside the rabbit for our exposure system. C. SAR Distribution in the Rabbit

Fig. 9. Calculated and measured SAR in homogeneous model with antenna input power of 1 W. (a) SAR values through center in the homogeneous cube model of 100 100 100 mm . (b) SAR values along the axis through one eye to another eye in the homogeneous rabbit model.

2

2

is large, could cause significant error in the measured SAR distribution because of the cooling effect of the external air. The setup of our thermal experiment allows us only the measurement of temperature with approximately 1-mm resolution. We also cannot rule out the possibility that capturing the topology of the phantom head with 1-mm resolution is not sufficient.

SAR distributions shown in Fig. 10 are calculated with the heterogeneous numerical model exposed to microwave from our small waveguide-adapter antenna. For comparison, we evaluated the SAR when the rabbit was exposed to a vertically polarized plane wave, which propagates in the direction of one eye to the other eye (Fig. 11). High SARs are found near the exposed eye of the rabbit for the exposure with the small antenna compared to the exposure with plane waves. The average SAR over the whole body, head, and exposed eye are summarized in Table III. SAR values were normalized against the incident power density at the surface of the exposed eye of 1 mW/cm . For the exposure with the small waveguide, the ratios of the exposed-eye averaged SAR to the whole-body

WAKE et al.: DEVELOPMENT OF 2.45-GHz LOCAL EXPOSURE SYSTEM FOR In Vivo STUDY ON OCULAR EFFECTS

TABLE III AVERAGE SAR (watts/kilogram) OVER THE WHOLE BODY, THE HEAD, AND THE EXPOSED EYE FOR THE EXPOSURE WITH INCIDENT POWER DENSITY AT THE EYE OF 10 W/m

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TABLE IV SAR VALUES (watts/kilogram) IN THE TISSUES OF THE EXPOSED EYE WITH INCIDENT POWER DENSITY AT THE EYE OF 10 W/m

a corner reflector and found high absorption in the vitreous humour behind the lens [1]. D. SAR Variation Due to Rabbit’s Movement

Fig. 12. SAR distribution near the eye of heterogeneous rabbit model. (a) Rabbit model. (b) SAR distribution.

averaged SAR and to the head averaged SAR were 60 and 7.6, respectively. For the plane-wave exposure, the ratios of the exposed-eye averaged SAR to the whole-body averaged SAR and to the head averaged SAR were 5.7 and 3.2, respectively. These results indicate that localized exposure of the rabbit eye was achieved by using the developed small waveguide antenna. With our new exposure system, when the incident power density to the rabbit eye was 300 mW/cm , the average SARs over the whole body and the exposed eye were approximately 1.8 and 108 W/kg, respectively. It is, therefore, suggested that our system can provide extremely high-strength exposure of the eye without significant whole-body thermal stress, which can be caused by the whole-body averaged SAR over 4 W/kg [5], [14]. Fig. 12 shows the SAR distribution near the eye of the heterogeneous rabbit model exposed with our exposure system. The SAR distribution in the exposed eye is highly complex. The maximum is not found at the surface of the eye, but in the rear part. Table IV lists the average SAR and the maximum SAR for 1 voxel in each tissue of the exposed eye. Maximum SARs are higher in the rear part of the eye, i.e., vitreous humour and sclera, than in other tissues. It is consistent with the results reported by Guy et al. though there are some differences in exposure condition. They used a near field of a 2.45-GHz dipole antenna with

Rabbits are constrained with plastic holders to limit their movement and to maintain exposure stability. However, there is still possibility that the geometrical relationship between the rabbit and antenna changes. Therefore, we evaluated effects on the SARs due to the changes in the distances between the antenna and the surface of the rabbit eye ( ), antenna height ( ), antenna position in the direction of the main body axis of . the rabbit ( ), and azimuth of the rabbit The average SARs over the whole body and the exposed eye vary with the changes in the geometrical relationship between the rabbit and antenna (Fig. 13). SARs are normalized against mm , mm , that of the standard condition ( mm , and ). SAR variation is most remarkable when the distance between the antenna and the surface of the rabbit eye ( ) is changed. This indicates that adjusting the distance between the antenna and the eye is crucial. When the antenna is 20 mm from the eye, the average SAR over the exposed eye became 200% higher than that for the standard posimm . We used a plastic holder to limit the rabbit’s tion movement and arranged the rabbit carefully at the desired position. The variations of the geometrical relationship is supposed to be within a few millimeters. The deviations of the SAR with 2 mm in the distance between the antenna and the rabbit eye ( ) were approximately 7%. IV. TEMPERATURE EVALUATION INSIDE THE RABBIT EYE DUE TO EXPOSURE It is also important to investigate mechanisms of the effect of microwaves on the eye, evaluating the temperature increase due to exposure. The temperature elevation due to exposure can be calculated using bioheat equations [15], [16]. In addressing the problem associated with the eye, the eye is often assumed to be thermally isolated [15], [16]. In this calculation, we also assumed thermal isolation of the eye and that the temperature in the rest of the body was 37 C. The thermal parameters of the eye used for our calculation are listed in Table V, where and denote the specific heat capacity, thermal conductivity, and density of tissues, respectively. These values are taken from [15] and [16]. Steady-state temperature when the SAR is 0 was taken as an initial condition. During the thermal analysis, an ambient temperature of 25 C was assumed. The heat transfer coefficients at the boundary with air and the rest of the body C, respectively. were 20 and 65 W/m Steady-state temperature distributions for incident power density of 10, 150, and 300 mW/cm are shown in Fig. 14.

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TABLE V THERMAL PARAMETERS OF THE EYE TEMPERATURE ANALYSIS

C; K; and  denote specific heat capacity, thermal conductivity, and density of tissues, respectively. These values are taken from [15] and [16].

Fig. 14. Temperature distribution in vertical section across the eye. (a) 10 mW/cm . (b) 150 mW/cm . (c) 300 mW/cm .

TABLE VI AVERAGE TEMPERATURE IN THE TISSUES OF THE EXPOSED EYE WITH THE INCIDENT POWER DENSITY AT THE EYE OF 10, 150, AND 300 mW/cm

Fig. 13. Relative deviation of SAR by the changes in the distance between the antenna and the rabbit eye (x), the height of the antenna (y ), the antenna position in the body axis of the rabbit (z ), and the rotation of the rabbit (). The relative deviation are normalized against the standard position (x = 40 mm, y = 0 mm, z = 0 mm,  = 0). (a) x-direction. (b) y -direction. (c) z -direction. (d)  -direction.

Table VI lists the average temperatures in the tissues of the exposed eye. With this exposure system, the average temperature of the exposed eye exceeds 41 C with the incident power density of 300 mW/cm . In this case, the difference in the average temperature between the lens and the vitreous humor is approximately 2 . This large difference may be due to the boundary condition, i.e., the thermal isolation between the eye and the rest of the body. Guy et al. reported that the threshold

temperature of cataracts is 41 C [1]. This indicates that the developed exposure setup can achieve high-intensity exposure that can cause cataracts known as a well-understood ocular effect. V. CONCLUSION We have developed an experimental setup to expose rabbit eyes locally to continuous and pulsed microwaves at 2.45 GHz. A coaxial-to-waveguide adapter filled with low-loss dielectric material was used as an antenna to focus the exposure area

WAKE et al.: DEVELOPMENT OF 2.45-GHz LOCAL EXPOSURE SYSTEM FOR In Vivo STUDY ON OCULAR EFFECTS

within the eye. With this system, the antenna input power of 1 W can generate the incident power density of approximately 15 mW/cm at the distance of 40 mm from the aperture. The SAR in the rabbit and the temperature in the exposed eye were evaluated. The SAR is effectively localized in and around the exposed eye of the rabbit with our exposure system. When the incident power density on the rabbit eye is 300 mW/cm , average SARs over the exposed eye and the whole body were approximately 108 and 1.8 W/kg, respectively. This exposure can elevate the temperature of the lens to over 41 C, the threshold of the cataract, without significant whole-body thermal stresses. ACKNOWLEDGMENT The authors thank Prof. I. Yamamoto, Kanazawa Medical University, Ishikawa, Japan, for obtaining CT images of the rabbit. The authors also acknowledge Dr. A. Hirata, Nagoya Institute of Technology, Nagoya, Japan, for providing helpful advice on the calculation of temperature elevation. REFERENCES [1] A. W. Guy, J. C. Lin, P. O. Kramar, and A. F. Emery, “Effects of 2450-MHz radiation on the rabbit eye,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 6, pp. 492–498, Jun. 1975. [2] H. A. Kues, L. W. Hirst, G. A. Lutty, S. A. D’Anna, and G. R. Dunkelberger, “Effects of 2450-MHz microwaves on primate corneal endothelium,” Bioelectromagnetics, vol. 6, pp. 177–188, 1985. [3] Y. Kamimura, K. Saito, T. Saiga, and Y. Amemiya, “Effects of 2.45 GHz microwave irradiation on monkey eyes,” IEICE Trans. Commun., vol. E77, pp. 762–765, 1994. [4] J. A. Elder, “Ocular effects of radiofrequency energy,” Bioelectromagn. Suppl., vol. 6, pp. 148–161, 2003. [5] ICNIRP, “Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 GHz),” Health Phys., vol. 74, no. 4, pp. 494–522, 1998. [6] “Limits of human exposure to radiofrequency electromagnetic fields in the frequency range from 3 kHz to 300 GHz—Safety code 6,” Minister Public Works and Govern. Service, CITY, PROVINCE, Canada, 1999. [7] K. Wake, H. Hongo, S. Watanabe, Y. Yamanaka, M. Taki, Y. Kamimura, T. Uno, M. Kojima, I. Hata, and K. Sasaki, “Localized exposure system of 2.45 GHz microwave to the rabbit eye,” presented at the URSI Gen. Assembly, 2002, KA.P.6. [8] M. Kojima, I. Hata, K. Wake, S. Watanabe, Y. Yamanaka, Y. Kamimura, M. Taki, and K. Sasaki, “Influence of anesthesia on ocular effects and temperature in rabbit eyes exposed to microwaves,” Bioelectromagnetics, vol. 25, pp. 228–233, 2004. [9] A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1998. [10] Y. Okano, K. Ito, I. Ida, and M. Takahashi, “The SAR evaluation method by a combination of thermographic experiments and biological tissue-equivalent phantoms,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 2094–2103, Nov. 2000. [11] C. Gabriel, “Compilation of the dielectric properties of body tissues at RF and microwave frequencies,” Brooks Air Force Base, Brooks AFB, TX, Tech. Rep. AL/OE-TR-1996-0037, 1996. [12] C. K. Chou, K. W. Chan, J. A. McDougall, and A. W. Guy, “Development of a rat head exposure system for simulating human exposure to RF fields from handheld wireless telephones,” Bioelectromagnetics, vol. 20, no. 4, pp. 75–92, 1999. [13] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 4, pp. 377–382, Nov. 1981. [14] IEEE Standard for Safety Levels with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 3 kHz to 300 GHz, IEEE Standard C95.1-1991, 1991.

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[15] P. Bernardi, M. Cavagnaro, S. Pisa, and E. Piuzzi, “SAR distribution and temperature increase in an anatomical model of the human eye exposed to the field radiated by the user antenna in wireless LAN,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2074–2082, Dec. 1998. [16] A. Hirata, S. Matsuyama, and T. Shiozawa, “Temperature rises in the human eye exposed to EM waves in the frequency range 0.6–6 GHz,” IEEE Trans. Electromagn. Compat., vol. 42, no. 4, pp. 386–393, Nov. 2000.

Kanako Wake (M’05) received the B.E., M.E., and D.E. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1995, 1997, and 2000, respectively. She is currently with the National Institute of Information and Communications Technology (NICT), Tokyo Japan, where she is involved with research on biomedical electromagnetic compatibility. Dr. Wake is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers (IEE), Japan, and the Bioelectromagnetics Society. She was the recipient of the 1999 International Scientific Radio Union (URSI) Young Scientist Award. Hiroyuki Hongo was born in Sendai, Miyagi Prefecture, Japan, on November 27, 1978. He received the Associate degree in information and communication technologies from the Sendai National College of Technology, Sendai, Japan, in 2000, and the Bachelor degree in electrical and electronic engineering from the Tokyo University of Agriculture and Technology, Tokyo, Japan, in 2003. While with the Tokyo University of Agriculture and Technology, he has engaged in development and evaluation of the RF irradiation device for a rabbit eye with the Communication Research Center (currently the National Institute of Information and Communications Technology) as an Intern. Soichi Watanabe (S’93–M’96) received the B.E., M.E., and D.E. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1991, 1993, and 1996, respectively. He is currently with the National Institute of Information and Communications Technology (NICT), Tokyo, Japan. His main interest is research on biomedical electromagnetic compatibility. Dr. Watanabe is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers (IEE), Japan, the Institute of Electrical and Electronics Engineering, and the Bioelectromagnetics Society. He was the recipient of several awards including the 1996 International Scientific Radio Union (URSI) Young Scientist Award and 1997 Best Paper Award presented by the IEICE. Masao Taki (M’02) was born in Tokyo, Japan, in 1953. He received the B.E., M.E., and Ph.D. degrees from the University of Tokyo, Tokyo, Japan, in 1976, 1978, and 1981, respectively. In 1981, he joined the Metropolitan University, Tokyo, Japan, where he is currently a Professor with the Department of Electrical and Electronic Engineering. He has been engaged in research on the compatibility of electromagnetic fields (EMFs) with the human body. Dr. Taki is a member of the Bioelectromagnetic Society, the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers (IEE), Japan, the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), and the Japan Health Physics Society. Since 1995, he has been a member of the International Commission on Non-Ionizing Radiation Protection (ICNIRP). He is the chairman of the Japanese National Committee for International Electrotechnical Commission (IEC) TC106 and the chairman of the Japanese National Committee for URSI-K.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 3, MARCH 2007

Yoshitsugu Kamimura (M’84) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Nagoya University, Nagoya, Japan, in 1980, 1982, and 1985, respectively. From 1985 to 1991, he was a Scientist with the Communications Research Laboratory, Tokyo, Japan, where he carried out research in electromagnetic compatibility. Since 1991, he has been an Associate Professor with the Department of Information Science, Utsunomiya University, Tochigi, Japan, where he has been involved with biological effects of electromagnetic fields and electromagnetic compatibility. Dr. Kamimura was the recipient of the 1989 Shinohara Memorial Young Investigators Award presented by the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, and the 1999 Best Paper Award presented by the Japanese Society of Medical Imaging Technology (JAMIT).

Yukio Yamanaka was born in Yamaguchi Prefecture, Japan, on March 25, 1958. He received the B.S. and the M.S. degrees in electrical engineering from Nagoya University, Nagoya, Japan, in 1980 and 1983, respectively. In 1983, he joined the Radio Research Laboratory, Ministry of Posts and Telecommunications [now the National Institute of Information and Communication Technology (NICT)], Tokyo, Japan, where he has been engaged in the study of statistical characteristics of man-made noise and electromagnetic compatibility (EMC) measurements. He is currently the Group Leader of the EMC Group, NICT.

Toru Uno (M’85–SM’02) received the B.S.E.E. degree from the Tokyo University of Agriculture and Technology (TUAT), Tokyo, Japan, in 1980, and the M.S. and Ph.D. degrees in electrical engineering from Tohoku University, Sendai, Japan, in 1982 and 1985, respectively. In 1985, he became a Research Associate with the Department of Electrical Engineering, Tohoku University, and then an Associate Professor from 1991 to 1994. He is currently a Professor with the Department of Electrical and Electronic Engineering, TUAT. From August 1998 to May 1999, he was on leave from the TUAT as a Visiting Scholar with the Electrical Engineering Department, Pennsylvania State University. He has authored two books regarding the FDTD method for electromagnetics and antennas. He was an Associate Editor for the IEICE Transactions on Communications (2000–2005). His research interests include the electromagnetic inverse problem, computational electromagnetics, medical and subsurface radar imagings, and EMC. Dr. Uno is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, American Geophysical Union (AGU), Applied Computational Electromagnetic Society (ACES), the Japan Society for Simulation Technology, and the Japan Society of Archaeological Prospection. He has served as a secretary of the Technical Group on Antennas and Propagation of the Institute of Electronics, IEICE (1999–2000). He also served as a vice-chair of the IEEE Antennas and Propagation Society (AP-S) Japan Chapter (2003–2004) and as a chair (2005–2006). He was the recipient of the Young Scientist Award and the Distinguished Contributions Award presented by the IEICE.

Masami Kojima received the Ph.D. degree from Kanazawa Medical University, Kanazawa, Japan, in 1991. He is currently a University Lecturer and Senior Scientist with Department of Ophthalmology, Kanazawa Medical University, Ishikawa, Japan. From 1988 to 1990, he was an Alexander von Humboldt Research Fellow with the Department of Experimental Ophthalmology, University of Bonn, Bonn, Germany. His research interests focus on the mechanisms of cataract development in the lens. He is currently involved in the field of nonionizing exposure such as ultraviolet, infrared, and microwave and millimeter-wave exposure-related ocular damages. Dr. Kojima was a consulting member of the International Commission on Non-Ionizing Radiation Protection (2001–2004). He was a member of the International Members Committee of the Association for Research in Vision and Ophthalmology (2001–2004). He has been the recipient of many academic honors such as the International Society of Ocular Toxicology (1990), The Japanese Society for Cataract Research Award (1992), the International Scheimpflug Club Meeting Award (1993), the U.S.–Japan Cooperative Cataract Research Group Meeting Award (1997), and the Kanazawa Medical University Article Award (1997).

Ikuho Hata was received the B. Pharm. degree from Kyoritsu University of Pharmacy, Tokyo, Japan, in 1997. From 1997 to 1999, she was engaged in studies on lens eye as a Member of Research Staff with the Department of Oral Molecular Biology, Oregon Health Sciences University, Portland. From 1999 to 2004, she was a Member of Research Staff with the Department of Ophthalmology, Kanazawa Medical University, Kanazawa, Japan.

Kazuyuki Sasaki received the M.D. and Ph.D. degrees in medicine from the School of Medicine, Tohoku University, Sendai, Japan in 1966. From 1966 to 1976, he was a Lecturer with the Department of Ophthalmology, School of Medicine, Tohoku University. From 1973 to 1975, he was an Invited Lecturer with the Eye Clinic, University of Bonn, Bonn, Germany. From 1976 to 2001, he was an Associate Professor with the Department of Ophthalmology, School of Medicine, Tohoku University. He was also a Professor and Chairman with the Department of Ophthalmology, Kanazawa Medical University, Uchinada, Japan. Since 2001, he has been a Professor Emeritus with Kanazawa Medical University and China Medical University, Shenyang, China. He is currently the Head of the Division of Vision Research for Environmental Health, Medical Research Institute, Kanazawa Medical University. Dr. Sasaki was the recipient numerous awards presented by the Scheimpflug Club Meeting (1985) and the International Association for Cataract and Related Research (1989). He was also the recipient of the 1995 U.S. Cataract Cooperative Research Group International Award and the 2002 International Society of Ocular Toxicology Scientific Award.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 3, MARCH 2007

597

Letters Comments on “TOA Estimation for IR-UWB Systems With Different Transceiver Types”

Using the fact

8(x) = 12 1 + erf px 2

Saralees Nadarajah

where erf(1) denotes the error function defined by

I. INTRODUCTION The above paper [1] analyzed performances of stored-reference, transmitted-reference (TR), and energy-detection (ED)-based time-of-arrival estimation techniques for impulse-radio ultra-wideband (IR-UWB) systems at sub-Nyquist sampling rates. A main part of the analysis was the derivation of the probability of accurate peak detection for each transceiver. However, these probabilities were presented in the form of integrals (see [1, eqs. (7)–(15)]). Each integral presented takes the form

1

=

P

1 0 c8

0

N 01

z 

(z)dz

erf(x) = p2

P

=

(1)

N 01

1

=

N 01 k=0

1

2

and x

=

dy:

FA (a; b1 ; . . . ; bn ; c1 ; . . . ; cn ; x1 ; . . . ; xn ) 1 1 (a)m +111+m (b1 ) 1 1 1 (bn ) m m 111 = ( c 1 )m 1 1 1 (cn )m m =0 m =0

N 01

1

1

torial. Numerical routines for the direct computation of (2) are widely available (see, e.g., Mathematica and [2]).

P

=

0

=

k=0

N 01 k=0

N 01 k

(0 c) 8

(0 c )k

1 0

8

z 

k

(z )dz

z 

k

(z )dz:

(z )dz

k

erf pz 2

k l

0 2c

k

k k l=0

l

(z )dz

k I (l ) l

(4)

where

1

I (l ) =

erf pz 2

0

l

(z )dz:

1 (01)m x2m+1 erf(x) = p2 (2m + 1)m! m=0 the integral I (l) in (4) can be expressed as

I (l ) =

1

p2

= p2

Using the binomial expansion, one can express (1) as k

k

0 2c

N 01 k

k=0

0

II. EXPLICIT EXPRESSION FOR (1)

N 01 k

k

Using the series expansion

m m 2 xm ! 11 11 11 xmnn ! (2) where (f )k = f (f + 1) 1 1 1 (f + k 0 1) denotes the ascending fac-

1 N 01

0 2c

N 01 k l=0

0

(n)

0

1 + erf pz 2

0

2

The above paper [1] did not attempt to derive an explicit expression for (1). In this letter, for the first time, we derive an expression for (1) that is a finite sum of a well-known special function—namely, the Lauricella function of type A [2] defined by

N 01 k

k=0

2

p1 exp 0 x2 2

2 8(x) = p1 exp 0 y2 2 01

exp(0t2 )dt

one can rewrite (3) as

where (1) and 8(1) denote the probability density function and the cumulative distribution function of the standard normal distribution defined, respectively, by

(x) =

x

(3)

l

1 (01)m (z=)2m+1 m=0

2m+1=2 (2m + 1)m! 2 (z)dz

1 1

0

m

l

=0

111

1 (01)m +111+m 2m +111+m +l=2

m

=0

)2(m +111+m )+l (z)dz 2 (2m(1z= + 1) 1 1 1 (2ml + 1)m1 ! 1 1 1 ml ! 1 (01)m +111+m l 1 = p2 111 2m +111+m +l=2 m =0 m =0 02(m +111+m )0l

Manuscript received October 5, 2006. The author is with the School of Mathematics, University of Manchester, Manchester M60 1QD, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.891469 0018-9480/$25.00 © 2007 IEEE

2 (2m + 1) 1 1 1 (2ml + 1)m ! 1 1 1 ml ! 2

1

0

1

z 2(m +111+m )+l (z )dz:

1

(5)

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 3, MARCH 2007

REFERENCES

An elementary integration shows that

1

z

2(m +

111+m )+l (z)dz = 01=2 2m +111+m +l=201

0

20

m1 + 1 1 1 + ml +

l+1

2

[1] I. Guvenc, Z. Sahinoglu, and P. V. Orlik, “TOA estimation for IR-UWB systems with different transceiver types,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 1876–1886, Jun. 2006. [2] H. Exton, Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. New York: Halsted, 1978.

and, thus, (5) reduces to

I (l) = 0(l+1)=2 201 0l

1

1 m

=0

111

Authors’ Reply

(01=2 )m +111+m (m1 + 1=2) 1 1 1 (ml + 1=2)m1! 1 1 1 ml ! m =0 2 0 m1 + 1 1 1 + ml + l +2 1 : Now, using the fact (f )k one can simplify (6) to

2F A

(6)

= 0(f + k)=0(f ) and the definition in (2),

I (l) = 0(l+1)=2 201 0l 0 (l)

Ismail Guvenc, Zafer Sahinoglu, and Philip V. Orlik

l+1

2 1 ; ; . . . ; ; 3 ; . . . ; 3 ; 0 12 ; . . . ; 0 12 : 2 2 2 2 2  

l+1 1

First of all, we would like to thank Nadarajah for his comments on the above paper [1]. In [1], the probability of accurate peak detection is given in an integral form. In their comments, Nadarajah derives explicit expressions for [1, eqs. 7–15] as a finite sum of special functions. We appreciate these derivations. At the time of the publication of [1], we did not know the existence of Louricella functions.

REFERENCES (7)

[1] I. Guvenc, Z. Sahinoglu, and P. V. Orlik, “TOA estimation in IR-UWB systems with different transceiver types,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 6, pp. 1876–1886, Jun. 2006.

Combining (4) and (7), we obtain the expression

P=

p1 2 

N 01 k=0

2FAl

( )

c k N 01 k 0 2 0l=2 0l 0 l +1 k 2 l=0 l 2 1 l +1 1 1 3 3 1 ; ; . . . ; ; ; . . . ; ; 0 2 ; . . . ; 0 2 : (8) 2 2 2 2 2   k

Note that (8) is a finite sum of the Lauricella function of type A.

Manuscript received October 10, 2006. I. Guvenc is with NTT DoCoMo USA Labs, Palo Alto, CA 94304 USA (e-mail: [email protected]). Z. Sahinoglu and P. V. Orlik are with Mitsubishi Electric Research Laboratories, Cambridge, MA 02139 USA (e-mail: [email protected]; porlik@merl. com). Digital Object Identifier 10.1109/TMTT.2007.891466

0018-9480/$25.00 © 2007 IEEE

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A. Deutsch V. Devabhaktuni Y. Deval T. Dhaene N. Dib C. Dietlein L. Ding A. Djordjevi J. Dobrowolski W. B. Dou P. Draxler R. Drayton A. Dreher J. Drewniak L. Dunleavy J. Dunsmore L. Dussopt M. Dvorak J. East K. W. Ecclestone M. L. Edwards R. Egri R. Ehlers N. Ehsan G. Eleftheriades F. Ellinger T. Ellis B. Elsharawy A. Elsherbeni N. Engheta T. Enoki M. Enqvist K. Entesari H. Eom K. Erickson N. Erickson C. Ernst D. Erricolo I. Eshrah M. Essaaidi H. Esteban M. C. Fabres C. Fager D. G. Fang S. J. Fang A. Faraone M. Farina W. Fathelbab A. E. Fathy A. P. Feresidis A. Fernandez A. Ferrero T. Fickenscher S. J. Fiedziuszko G. Fikioris I. Filanovsky F. Filicori D. Filipovic F. D. Flaviis B. Floyd P. Focardi N. H. Fong S. S. Fort K. Foster P. Foster C. C. Franco M. C. Francos J. C. Freire F. Frezza I. Frigyes J. Fu R. Fujimoto O. Fujiwara C. Fumeaux V. Fusco D. Gabbay T. Gaier J. D. Gallego B. Galwas A. Gameiro O. Gandhi S. Gao J. R. Garai H. Garbe J. A. Garcia K. Gard P. Gardner R. Garg J. L. Gautier B. Geelen F. Gekat B. Geller J. Gering F. German M. Geshiro S. Gevorgian H. Ghali M. Ghanevati F. Ghannouchi K. Gharaibeh G. Ghione K. Ghorbani O. Giacomo E. D. Giampaolo F. Giannini P. Gilabert A. Goacher M. Goano E. Godshalk M. Goldfarb P. Goldsmith M. Golosovsky R. Gonzalo S. Gopalsami D. Gope A. Gopinath R. Gordon A. Gorur M. Gottfried G. Goussetis W. Grabherr J. Graffeuil R. Graglia L. Gragnani J. Grahn G. Grau A. Grebennikov I. Gresham J. Grimm A. Griol E. Grossman Y. Guan J. L. Guiraud S. E. Gunnarsson L. Guo Y. X. Guo C. Gupta K. C. Gupta M. Gupta W. Gwarek J. Hacker S. Hadjiloucas S. H. Hagh S. Hagness A. Hajimiri D. Halchin P. Hale P. Hall D. Ham K. Hanamoto T. Hancock

A. Hanke E. Hankui G. Hanson Z. Hao H. Happy A. R. Harish L. Harle L. D. Haro F. J. Harris H. Harris M. Harris P. Harrison R. G. Harrison O. Hartin H. Hashemi K. Hashimoto O. Hashimoto J. Haslett S. Hay J. Hayashi L. Hayden T. Heath J. Heaton M. P. Heijden G. Heiter J. Helszajn R. Henderson D. Heo P. Herczfeld H. Hernandez J. J. Herren K. Herrick F. Herzel J. S. Hesthaven K. Hettak P. Heydari T. S. Hie M. Hieda A. Higgins A. Hirata J. Hirokawa T. Hirvonen J. P. Hof K. Hoffmann R. Hoffmann M. Hoft E. Holzman J. S. Hong S. Hong W. Hong A. Hoorfar K. Horiguchi Y. Horii T. S. Horng J. Horton J. Hoversten H. Howe H. M. Hsu H. T. Hsu J. P. Hsu P. Hsu C. W. Hsue M. Z. Hualiang C. W. Huang F. Huang G. W. Huang J. Huang T. W. Huang W. Huei M. Huemer H. T. Hui J. A. Huisman A. Hung C. M. Hung J. J. Hung I. Hunter M. Hussein E. Hutchcraft B. Huyart J. C. Hwang J. N. Hwang R. B. Hwang M. Hélier Y. Iida S. Iitaka P. Ikonen K. Ikossi M. M. Ilic A. Inoue T. Ishikawa T. Ishizaki S. Islam Y. Isota M. Ito N. Itoh T. Itoh Y. Itoh T. Ivanov D. Iverson M. Iwamoto Y. Iyama D. Jablonski D. Jachowski R. Jackson R. W. Jackson A. Jacob M. Jacob S. Jacobsen D. Jaeger B. Jagannathan V. Jamnejad V. Jandhyala M. Janezic M. Jankovic R. A. Jaoude B. Jarry P. Jarry J. B. Jarvis A. Jastrzebski B. Jemison W. Jemison S. K. Jeng A. Jenkins Y. H. Jeong A. Jerng T. Jerse P. Jia X. Jiang B. Jim J. G. Jiménez J. M. Jin J. Joe R. Johnk L. Jonathan J. Joubert E. J. Jr N. C. Jr R. Judaschke J. Juntunen D. Junxiong T. Kaho M. Kahrs T. Kaiser S. Kalenitchenko V. Kalinin T. Kalkur Y. Kamimura H. Kanai S. Kanamaluru H. Kanaya K. Kanaya

Digital Object Identifier 10.1109/TMTT.2007.894110

S. Kang P. Kangaslahtii V. S. Kaper B. Karasik N. Karmakar A. Karwowski T. Kashiwa L. Katehi H. Kato K. Katoh A. Katz R. Kaul R. Kaunisto T. Kawai K. Kawakami A. Kawalec T. Kawanishi S. Kawasaki H. Kayano M. Kazimierczuk R. Keam S. Kee L. C. Kempel P. Kenington A. Kerr A. Khalil A. Khanifar A. Khanna F. Kharabi R. Khazaka J. Kiang J. F. Kiang Y. W. Kiang B. Kim C. S. Kim D. I. Kim H. Kim H. T. Kim I. Kim J. H. Kim J. P. Kim M. Kim W. Kim S. Kimura N. Kinayman A. Kirilenko V. Kisel M. Kishihara A. Kishk T. Kitamura K. I. Kitayama T. Kitazawa T. Kitoh M. Kivikoski G. Kiziltas D. M. Klymyshyn R. Knochel L. Knockaert Y. Kogami T. Kolding B. Kolundzija J. Komiak G. Kompa A. Konczykowska H. Kondoh Y. Konishi B. Kopp K. Kornegay T. Kosmanis P. Kosmas Y. Kotsuka A. Kozyrev N. Kriplani K. Krishnamurthy V. Krishnamurthy C. Krowne V. Krozer J. Krupka W. Kruppa D. Kryger R. S. Kshetrimayum H. Ku H. Kubo A. Kucar A. Kucharski W. B. Kuhn T. Kuki A. Kumar M. Kumar C. Kuo J. T. Kuo H. Kurebayashi K. Kuroda D. Kuylenstierna M. Kuzuhara Y. Kwon G. Kyriacou P. Lampariello M. Lancaster L. Langley U. Langmann Z. Lao G. Lapin L. Larson J. Laskar M. Latrach C. L. Lau A. Lauer J. P. Laurent D. Lautru P. Lavrador G. Lazzi B. H. Lee C. H. Lee D. Y. Lee J. Lee J. F. Lee J. H. Lee J. W. Lee R. Lee S. Lee S. G. Lee S. T. Lee S. Y. Lee T. Lee T. C. Lee D. M. Leenaerts Z. Lei G. Leizerovich Y. C. Leong R. Leoni P. Leuchtmann G. Leuzzi A. Leven B. Levitas R. Levy G. I. Lewis H. J. Li L. W. Li X. Li Y. Li H. X. Lian C. K. Liao M. Liberti E. Lier L. Ligthart S. T. Lim E. Limiti C. Lin F. Lin H. H. Lin

J. Lin K. Y. Lin T. H. Lin W. Lin Y. S. Lin E. Lind L. Lind L. F. Lind D. Linkhart P. Linnér D. Linton A. Lipparini D. Lippens V. Litvinov A. S. Liu C. Liu J. Liu J. C. Liu Q. H. Liu S. I. Liu T. Liu T. P. Liu O. Llopis D. Lo J. LoVetri N. Lopez Z. Lou M. Lourdiane G. Lovat D. Lovelace H. C. Lu K. Lu L. H. Lu S. S. Lu Y. Lu V. Lubecke S. Lucyszyn R. Luebbers N. Luhmann A. Lukanen M. Lukic A. D. Lustrac J. F. Luy C. Lyons G. Lyons G. C. M H. Ma J. G. Ma Z. Ma P. Maagt S. Maas G. Macchiarella P. Macchiarella J. Machac M. Madihian A. Madjar V. Madrangeas A. Maestrini G. Magerl S. L. Mageur A. A. Mahmoud S. Mahmoud F. Maiwald A. H. Majedi M. Makimoto S. Makino J. Malherbe G. Manara R. Manas G. Manes T. Maniwa R. Mansour D. Manstretta J. Mao S. G. Mao A. Margomenos R. Marques G. Marrocco J. Martel J. Martens J. Marti G. Martin E. Martinez K. Maruhashi J. E. Marzo H. Masallaei N. Masatoshi D. Masotti G. D. Massa B. Matinpour T. Matsui A. Matsushima S. Matsuzawa H. Matt G. Matthaei L. Maurer J. Mayock J. Mazierska S. Mazumder G. Mazzarella K. McCarthy G. McDonald R. McMillan D. McNamara D. McQuiddy F. Medina C. Melanie A. Á. Melcon F. Mena C. C. Meng H. K. Meng W. Menzel F. Mesa A. C. Metaxas R. Metaxas P. Meyer E. Michielssen A. Mickelson D. Miller P. Miller B. W. Min R. Minasian J. D. Mingo J. Mink B. Minnis F. Miranda D. Mirshekar C. Mishra S. Mitilineos R. Mittra K. Miyaguchi M. Miyakawa H. Miyamoto R. Miyamoto M. Miyashita M. Miyazaki K. Mizuno S. Mizushina J. Modelski W. V. Moer S. Mohammadi H. Moheb J. Mondal M. Mongiardo P. Monteiro C. Monzon A. D. Morcillo J. Morente T. Morf D. R. Morgan M. Morgan

K. Mori A. Morini H. Morishita A. Morris J. Morsey H. Mosallaei H. Moyer M. Mrozowski C. H. Mueller J. E. Mueller B. Nabet P. Nadia A. S. Naeini Y. Nagano I. Naidionova K. Naishadham M. Nakajima M. Nakao Y. Nakasha M. Nakatsugawa A. Nakayama J. Nakayama M. Nakayama M. Nakhla J. C. Nallatamby S. Nam T. Namiki T. Narhi S. Naruhashi A. Nashashibi A. Natarajan J. Nath J. M. Nebus I. Nefedov D. Neikirk B. Nelson A. Neri H. S. Newman G. Ng E. Ngoya C. V. Nguyen T. Nichols K. Nickolas K. Niclas E. Nicol E. Niehenke S. Nightingale N. Nikita P. Nikitin A. M. Niknejad N. K. Nikolova K. Nishikawa T. Nishikawa T. Nishino F. Niu E. Niver D. Nobbe S. Nogi T. Nojima C. D. Nordquist Z. Nosal B. Notaros K. Noujeim D. Novak T. Nozokido E. Nyfors K. O M. O’Droma J. Obregon M. Odyniec K. Oh K. Ohata T. Ohira A. Ohta I. Ohta H. Okabe Y. Okano H. Okazaki V. Okhmatovski A. Oki G. Olbrich A. Ø. Olsen A. S. Omar M. Omiya K. Onodera B. L. Ooi A. Orlandi R. Orta B. Ortega S. Ortiz J. Osepchuk H. Ota S. Otaka J. Ou C. Oxley S. Pacheco M. Pagani G. W. Pan Y. H. Pang H. Y. Pao J. Papapolymerou A. Parfitt S. Parisi C. S. Park J. S. Park A. E. Parker D. Pasalic D. Pasquet M. Pastorino H. M. Pau M. Paul T. Pavio D. Pavlidis J. C. Pedro C. Peixeiro S. Pellerano G. Pelosi R. Pengelly C. Penney J. Pereda D. Peroulis L. Perregrini M. Petelin R. Petersen W. Petersen A. Peterson C. C. Peñalosa U. R. Pfeiffer A. V. Pham M. Pieraccini L. Pierce P. Pieters B. Pillans Z. Y. Ping A. Piovaccari M. Pirola E. Pistono C. Plett C. Pobanz A. Podell R. J. Pogorzelski J. L. Polleux J. Poltz G. Ponchak J. Pond J. Portilla M. Pospieszalski V. Postoyalko B. Potter D. Pozar L. Pradell

J. Prasad S. Prasad D. Prescott H. Pretl M. Prigent A. Priou S. Prosvirnin H. Qian Y. Qian D. Qiao J. X. Qiu T. Quach C. K. Queck C. Quendo R. Quéré F. Raab V. Radisic M. Raffetto T. Rahkonen R. Raich C. Railton S. Raman R. S. Rana P. Ratanadecho C. Rauscher J. Rautio B. Rawat T. Razban R. Reano G. M. Rebeiz J. Rebollar S. Remillard K. Remley L. Reynolds S. K. Reynolds A. Reynoso E. Rezek J. K. Rhee A. Riddle J. S. Rieh J. Ritter E. Rius J. Rizk R. Robert I. Robertson P. Roblin C. Rodenbeck M. Rodwell O. T. Rofougaran H. Rogier U. Rohde Y. Rolain J. Rolf N. Rolland R. Romanofsky S. Rondineau Y. Rong D. Ronnow M. J. Rosario L. Roselli A. Rosen U. Rosenberg M. Rosker E. Rothwell J. Roy L. Roy T. Rozzi J. Rubio A. Ruehli D. Rutledge T. Ruttan A. Rydberg D. Rytting D. Rönnow C. Saavedra R. Saedi A Safaai-Jazi M. Sagawa K. Saito K. Sakaguchi A. Samelis C. Samori L. Samoska A. Sanada M. Sanagi A. Sangster L. Sankey K. Sano K. Sarabandi T. Sarkar C. Sarris H. Sato M. Sato K. Sawaya H. Sayadian C. Schaffer H. Schantz I. Scherbatko G. Schettini M. J. Schindler E. Schlecht E. Schmidhammer D. Schmitt J. Schoukens D. Schreurs W. Schroeder A. Schuchinsky P. Schuh L. Schulwitz F. Schwering K. F. Schünemann J. B. Scott F. Sechi Y. Segawa E. M. Segura T. Seki E. Semouchkina H. Serizawa J. Sevic O. Sevimli F. Seyfert O. Shanaa Z. Shao I. Shapir M. Shapiro A. Sharma S. K. Sharma J. Sharp J. R. Shealy Z. X. Shen Y. Shestopalov H. Shigematsu Y. C. Shih M. Shimozawa T. Shimozuma H. Shin S. Shin N. Shinohara G. Shiroma W. Shiroma K. Shu C. N. Shuo D. Sievenpiper A. Sihvola J. M. Sill C. Silva M. G. Silveirinha K. Silvonen W. Simbuerger G. Simin R. N. Simons D. Simunic H. Singh

B. Sinha D. Sinnott Z. Sipus K. Sivalingam A. Skalare R. Sloan M. Slominski A. Smith P. Smith C. Snowden R. Snyder N. Sokal V. Sokolov K. Solbach J. Sombrin R. Sorrentino A. Soury N. Soveiko B. E. Spielman P. Staeker D. Staiculescu J. Stake A. Stancu S. P. Stapleton P. Starski J. Staudinger D. Steenson P. Steenson M. Steer J. Stenarson K. Steve M. Steyaert W. Steyn S. Stitzer B. Strassner E. Strid M. Stubbs M. Stuchly B. Stupfel A. Suarez G. Subramanyam N. Suematsu C. Sullivan S. Sun J. Svacina R. Svitek M. Swaminathan D. Swanson D. M. Syahkal M. Syahkal B. Szendrenyi A. Taflove M. Taghivand G. Tait Y. Tajima T. Takagi I. Takenaka K. Takizawa T. Takizawa S. Talisa S. G. Talocia N. A. Talwalkar A. A. Tamijani B. T. Tan C. Y. Tan J. Tan S. Tanaka C. W. Tang D. W. Tang W. C. Tang M. Taromaru A. Tasic P. Tasker J. J. Taub J. Tauritz D. Taylor R. Tayrani D. Teeter F. Teixeira M. Tentzeris V. Teppati M. Terrovitis J. P. Teyssier K. P. Thakur H. Thal W. Thiel B. Thompson M. Thorburn C. E. Thorn Z. Tian M. Tiebout R. Tielert L. Tiemeijer G. Tkachenko M. R. Tofighi P. Tognolatti T. Toifl T. Tokumitsu R. Tomar A. Tombak K. Tomiyasu A. Topa E. Topsakal G. Town I. Toyoda N. Tran S. Tretyakov R. Trew C. M. Tsai E. Tsai R. Tsai J. Tsalamengas T. Tsiboukis M. Tsuji T. Tsujiguchi T. Tsukahara M. Tsutsumi S. H. Tu W. H. Tu N. Tufillaro A. Turudic G. Twomey C. K. Tzuang H. Uchida S. Uebayashi M. Ugajin J. Uher F. H. Uhlmann Y. Umeda V. J. Urick T. Uwano N. Uzunoglu R. Vahldieck P. Vainikainen K. Vanhille G. Vannini J. C. Vardaxoglou K. Varian G. Vasilescu C. Vaucher J. Vaz J. Venkatesan F. Verbeyst A. Verma J. Verspecht P. Vial H. O. Vickes A. Vilcot F. Villegas C. Vittoria S. Vitusevich R. Voelker S. Voinigescu

V. Volman A. Vorobiev A. V. Vorst B. Vowinkel L. D. Vreede M. A. Vérez B. Z. Wang K. Wagner K. Wakino P. Waldow M. Wale A. Walker D. Walker C. Walsh C. Wan S. Wane C. Wang C. F. Wang H. Wang N. Wang S. Wang T. Wang X. Wang Y. Wang J. Ward K. Warnick P. Warr S. Watanabe Y. Watanabe R. Waugh J. Webb K. Webb R. Webster C. J. Wei D. V. Weide R. Weigel G. Weihs B. Weikle R. M. Weikle C. Weil T. Weiland A. Weily S. Weinreb T. Weller S. Wentworth D. D. Wentzloff R. Wenzel J. Whelehan J. Whitaker D. A. White J. Wiart W. Wiesbeck J. Wight D. Willems D. Williams J. Wiltse D. Wittwer I. Wolff K. Wong W. Woo J. Wood C. Woods G. Woods R. C. Woods D. Woolard M. Wren C. Wu H. Wu K. Wu K. L. Wu Q. Wu T. L. Wu C. Wwang T. Wysocki S. Z. Xiang G. Xiao Y. Xiaopeng C. Xie Z. Xing H. Xu J. Xu S. Xu W. Xu X. B. Xu Y. Xu Y. P. Xu Q. Xue M. Yagoub T. Yakabe A. Yakovlev K. Yamamoto K. Yamauchi F. Yang G. M. Yang H. Y. Yang J. Yang K. Yang L. Yang X. Yang H. Yano F. Yanovsky H. W. Yao J. Yao B. Yarman A. G. Yarovoy Y. Yashchyshyn K. Yashiro K. Yasumoto Y. Yasuoka S. Ye K. S. Yeo S. P. Yeo L. K. Yeung H. R. Yi W. Y. Yin D. Yongsheng J. G. Yook Y. Yoon Y. J. Yoon R. York J. L. Young H. K. Yu M. Yu P. Yu W. Yu Z. Yuanjin P. Yue S. W. Yun A. I. Zaghloul A. G. Zajic K. Zaki J. Zamanillo P. J. Zampardi J. Zapata L. Zappelli J. Zehentner H. Zhang L. Zhang Q. J. Zhang R. Zhang X. Zhang A. P. Zhao J. Zhao Y. Zhao F. Zhenghe W. Zhiguo W. Zhou A. Zhu L. Zhu N. H. Zhu H. Zirath S. Zouhdi A. J. Zozaya T. Zwick