CMOS switched transconductor mixer, IEEE J Solid-State Circuits 39(2004), 1231–1240.

7. H.H. Hsieh, K.S. Chung, and L.H. Lu, Ultra-low-voltage mixer andVCO in 0.18-�m CMOS, In IEEE Radio Frequency Integrated CircuitsSymposium Digest, June 2005, pp. 167–170.

8. C. Hermann, M. Tiebout, and H. Klar, A 0.6-V 1.6-mW transformer-based 2.5-GHz downconversion mixer with �5.4-dB gain and �2.8-dBm IIP3 in 0.13-�m CMOS, IEEE Trans Microwave Theory Tech 53(2005), 488–495.

9. A.Q. Safarian, A. Yazdi, and P. Heydari, Design and analysis of anultrawide-band distributed CMOS mixer, IEEE Trans Very Large ScaleIntegr 13 (2005), 618–629.

© 2007 Wiley Periodicals, Inc.

BROAD BAND BEHAVIORAL MODELINGOF ON-CHIP RF INDUCTORS ANDTRANSFORMERS

Rajarshi Bhattacharya, T.K. Bhattacharyya, and Asudeb DuttaElectronics and Electrical Communication Engineering Department,Indian Institute of Technology, Kharagpur 721302, India

Received 21 February 2007

ABSTRACT: The broad band behavioral models of on-chip inductorsand transformers are extracted from sampled frequency response data.A linearized least square based model parameter extraction algorithm isused to obtain ‘s’-domain rational function model. Model order is opti-mized to strike a balance between modeling error and time required fortransient simulation of circuits consisting of on-chip inductors and/ortransformers. A new cost function is proposed to improve optimal modelorder selection. Particle swarm optimization (PSO) is applied to maxi-mize the nonconvex fitness function, associated with model order optimi-zation. A modification of PSO algorithm is proposed to speed up opti-mal model order selection. While modeling on-chip inductors andtransformers, the proposed modeling technique extracts highly accurate,yet reduced order models, which are valid over a wide band width andare compatible with SPICE-like simulators. © 2007 Wiley Periodicals,Inc. Microwave Opt Technol Lett 49: 2212–2216, 2007; Published on-line in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22653

Key words: behavioral modeling; rational function model; on-chip in-ductor; on-chip transformer; particle swarm optimization

1. INTRODUCTION

In the domain of expanding wireless industry, the increasingdemand for low cost RF-ICs has developed a strong interest inon-chip passive components. It has been noted that, inclusion of apoorly modeled on-chip inductor or transformer in a circuit sim-ulation results in deviation of the simulated circuit performancefrom the experiments and as a result, uncertainty in circuit designincreases. Therefore, more and more accurate broad band devicemodels are demanded in RF integrated circuits.

The use of compact equivalent circuit-based scalable physicalmodel [1–3] is common trend in RF inductor and transformermodeling. These equivalent circuit models are inherently deficientdue to the approximations made during analytical modeling of thecomplex field distribution. Moreover, these approximations are notvalid for all frequencies of operation as well as different geome-tries of the structure.

Behavioral modeling or macromodeling is widely used inbroadband modeling of transmission line structures like high speedinterconnect networks [4]. The same idea has been recently ex-

tended to model more complicated structures such as on-chipplanar inductors [5, 6]. This work extends the use of behavioralmodeling to wideband modeling of monolithic transformers. Sincebehavioral modeling does not make any simplistic approximationof the complex field distribution, it can accurately model a broad-band passive RF-microwave component of any arbitrary shape.

Model parameter extraction of linear component from fre-quency sampled data involves a nonlinear least square (LS) opti-mization problem [7]. In [5], particle swarm optimization (PSO)algorithm was applied to solve the nonlinear optimization problem.It is known that evolutionary algorithms (EAs) like PSO arecapable of finding the global optimal solution of a nonlinearoptimization problem. However, EAs are time-consuming andrequire large memory space. Therefore, in this work, a fastermodel parameter extraction algorithm is used for model parameterextraction.

Here the nonlinear optimization problem is linearized usingLevy’s approach to form a linear LS problem [7–9]. It should benoted that, the linearized LS does not assure stability of the model.In this work, stability is enforced by reflecting unstable poles intothe left half of the s-plane [6]. Moreover, an attempt has been madeto enhance the accuracy of the stabilized model by Levenberg–Marquard (LM) nonlinear optimization, perhaps for the first timein RF/Microwave component modeling [6, 9].

It is true that higher order models may provide the requiredaccuracy, but its transient simulation becomes highly time-con-suming. Considering this fact, here model order is optimized tostrike a balance between modeling error and time required fortransient simulation of circuits consisting of on-chip inductors andtransformers. A new cost function is proposed to improve modelorder selection. A modified version of PSO is applied to maximizethe nonconvex optimization problem associated with model orderselection [10]. A further modification of PSO algorithm is pro-posed to speed-up model order optimization.

This letter is structured in the following manner. Model param-eter extraction of on-chip inductors and transformers is dealt within Section 2. Model order selection is discussed in Section 3.Subsection 3.1 describes the proposed cost function for modelorder optimization and Subsection 3.2 describes the PSO-basedalgorithm for model order optimization. The proposed modifica-tion of PSO for faster model order optimization is described inSubsection 3.3. Results are presented in Section 4. Conclusion ismade in Section 5.

2. MODEL PARAMETER EXTRACTION OF ON-CHIPINDUCTORS AND TRANSFORMERS

Inductor is a linear single port device. Hence, driving point im-pedance of an on-chip inductor can be modeled as a SISO transferfunction, whereas, transformer being a two port linear passivedevice, we propose to model the two port open circuit parameters(Z parameters) or short circuit parameters (Y parameters) of trans-former. Advantage of modeling Z or Y parameters of passivedevices like transformers is that since both Z and Y matrices of apassive device are symmetric, model parameter extraction is re-quired only for the upper or lower triangular matrix. A circuitconsisting of on-chip inductors and transformers can be simulatedin SPICE like simulators by replacing the inductor and transformerwith corresponding ‘s’ domain rational function model [11]. It is tobe noted that behavioral transfer function models computed by theproposed algorithm can either be directly used in HSPICE simu-lation or in SPICE simulation through net-list [11, 12].

This letter presents a general algorithm for model parameterextraction of N-port linear device behavioral model from sampledfrequency response data. Let G� j�f� be an experimental/simulated

2212 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007 DOI 10.1002/mop

observation of N-port device transfer function matrix, whereG � CN�N, �f � �, and � � ��1 �2, . . ., �NF� is the set of NF

frequency points where G� j�f� is evaluated. G� j�f� can be Z, Y, Sor any other N-port parameter matrix.

Let G�s,�� be the model of G�s� where s � j� and � is theunknown parameter vector to be estimated. Let Gk�s,�k� be the kthelement of G�s,��, where k � 1, 2, . . ., N2. Gk�s,�k� is consideredto be in the rational function form

Gk�s,�k� � � Ak�s���1Bk�s� � � �i0

Pk

aisi��1

�j0

Qk

bisj, (1)

with Pk � Qk. To obtain an identifiable parameterization, it isrequired to impose a constraint like a0 � 1 [7–9, 11, 13]. Theunknown parameter vector of Gk�s,�k� is defined as, �k

� �ao a1. . .aPk bo b1

. . .bQk�T, �k � R�Ok�2

, where Ok � �Pk

� Qk� is the model order of Gk�s,�k�. Let ek be the linearized andnormalized modeling error, where

ekf � �Gk� j�f� Ak� j�f,�k� � Bk� j�f,�k��/Gk� j�f� (2)

and ek � �ek1 ek2. . . ekNF�

T. In Eq. (2), linearization is doneusing Levy’s approach [8]. The model parameters are extracted bysolving the following linearized LS optimization problem (for theassigned model order):

arg min�k

E��k� � �ekHek�/NF (3)

where “H ” means conjugate transpose and ek � CNF. For detailedderivation, see [7].

The linear LS optimization problem shown in Eq. (3) oftenleads to an ill-conditioned linear system. So, Householder trans-

formation is used for obtaining the pseudo inverse solution [13].The LS solution does not assure stability of the model. Hence, theunstable poles are flipped into the left half s-plane to make thesystem stable. The stabilized model is further improved further byLM iteration [6, 9].

3. MODEL ORDER SELECTION

Model order selection plays a crucial role in modeling of on-chipinductors and transformers as discussed in Section 2. Generally,final prediction error (FPE) or Akai’s information criterion (AIC)are used for selection of optimal model order [14–16]. The draw-back of these conventional model order selection cost functions isthat these cost functions instruct to select the model with lowermodel order even when the modeling error is high.

3.1. Novel Cost Function for Model Order SelectionIn this letter, a new cost function is proposed for model orderselection. The optimal model order of Gk�s,�k� is selected byminimizing the following cost function:

Figure 1 EkMO is function of Ok and Ek

MP��k,Ok� with � 0.02 and Et

� 0.05

TABLE 1 Dimensions of the Inductor and the Transformer

Inductor Transformer

Outer diameter 160 �m 250 �mMetal width 13 �m 13 �mMetal thickness 2.055 �m 2.055 �mMetal spacing 1 �m 23 �mNo. of turns 4 3 Primary; 2.5 SecondaryMetal layer M3, M4, M5 M5Under pass M3 M4

Figure 2 Magnitude plot of three-stack inductor impedance (ZIND). Themodel extracted by the algorithm consists of fifth order numerator poly-nomial and seventh order denominator polynomial

Figure 3 Phase plot of three-stack inductor impedance (ZIND)

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007 2213

EkMO � f1�E

t�E1��k,Ok� � f2�Et�E2��k,Ok,E

t,� (5)

where E1 EkMP(�k,Ok)

1

NF¥f1

NF �(Gk(j�f) � Gk(j�f,�k))/

Gk( j�f)�2, E2 Et �(Et � EkMP(�k,Ok))e�Ok, f1(Et)

u(EkMP(�k,Ok)�Et), and f2(Et) u(Ek

MP(�k,Ok)) � f1(Et)..In Eq. (5), �k is the estimated model parameter vector of the

stabilized model Gk�s,�k�, Et is the maximum tolerable value of EkMP

(mean squared normalized modeling error) and u� � � stands forunit step function. is a positive number that determines the slopeof Ek

MO.The key idea is that, one need not care for the model order as

long as EkMP��k,Ok� is greater than Et. Hence, when Ek

MP��k,Ok� isgreater than Et, Ek

MO is set to EkMP��k,Ok� and when Ek

MP��k,Ok� is lessthan Et, Ek

MO exponentially increases with Ok (Fig. 1).

3.2. PSO-Based Model Order OptimizationPSO is a stochastic, population-based, self-adaptive searchingtechnique first introduced in [17]. Here, PSO is employed to solve

the following nonconvex maximization problem for model orderoptimization:

arg maxPk1,Qk1

fitkMO��k,Pk1,Qk1� � � Ek

MO (6)

The optimal model order is obtained by searching the N�2integer

parameter space.The original PSO algorithm, as introduced in [17], may not

always converge to the global optimal solution [10]. In this work,a modified version of PSO is used to ensure faster convergence.The modified version of PSO uses dynamic inertia weight anddynamic acceleration coefficients [10]. A further modification ofPSO is suggested in the following subsection to speed-up modelorder optimization.

3.3. Modification of PSO for Faster Model Order OptimizationEach evaluation of fitk

MO requires model parameter extraction.Hence, number of fitk

MO evaluations has to be reduced to lessen theoverall computational complexity. In this work, a modification ofPSO is proposed to reduce the number of fitk

MO evaluation byexploiting the fact that the search space for model order estimationis a space of real positive integers.

The existing PSO algorithm remembers only the current bestlocation (and the corresponding fitness value) of each particle ofthe swarm and the current best location (and the correspondingfitness value) of the entire swarm. In this work, PSO algorithm hasbeen modified to store the fitness of all the explored locations ofthe search space. This modification of PSO increases the memoryrequirement a little, but, at the cost of slight increase in memoryrequirement, the algorithm becomes much faster by avoiding un-necessary repeated model parameter extraction for same modelorder.

4. CASE STUDIES

The performance of the proposed algorithm is evaluated for be-havioral modeling of an on-chip three stack inductor and anon-chip planar transformer. The S-parameters of the inductor andthe transformer, fabricated in 0.18 �m CMOS process, are mea-sured over a band-width of 20 GHz. Subsequently, the inductor

Figure 4 Magnitude plot of transformer Z11. The model extracted by theproposed algorithm consists of fourth order numerator polynomial andfourth order denominator polynomial

Figure 5 Phase plot of transformer Z11.

Figure 6 Magnitude plot of transformer Z12. The model extracted by theproposed algorithm consists of third order polynomial and fifth orderdenominator polynomial

2214 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007 DOI 10.1002/mop

impedance and the Z-parameter of the transformer are extractedfrom the measured S-parameter. The dimensions of the three stackinductor and the planar transformer are summarized in Table 1.Result shows excellent match between the behavioral model andthe measured result (Figs. 2–9).

5. CONCLUSION

This letter employs behavioral modeling to characterize RF on-chip inductors and transformers. LS and LM optimization are usedfor model parameter extraction. Moreover, for faster transientsimulation of circuits consisting of on-chip inductors and/or trans-formers, this letter presents a novel model order selection criterionto select the optimal model order and modifies PSO algorithm tospeed-up model order optimization. One of the main advantages ofthe proposed algorithm is that unlike the conventional scalableequivalent circuit model-based technique, the proposed algorithmdoes not make any simplistic approximation of the complex fielddistribution and hence, it can be applied to model wide bandon-chip inductors or transformers with any arbitrary shape. This

proposed algorithm extracts accurate reduced order model of amonolithic inductor or transformers, which is of immense impor-tance for RF circuit design.

ACKNOWLEDGMENT

The authors wish to express their gratitude to Prof. RameshGarg, professor in the department of E&ECE of IIT Kharagpur,India, for his valuable technical comments. The authors wouldalso like to thank National Semiconductor for providing theexperimental data.

REFERENCES

1. A. Mohan, G. Boeck, and A. Biswas, Modeling of on-chip inductorsand transformers for GaAs MMICs, Microwave Opt Tech Lett 47(2005), 270–274.

2. W. Gao, C. Jiao, T. Liu, and Z. Yu, Scalable compact circuit model fordifferential spiral transformers in CMOS RFICs, IEEE Trans ElectronDev 53 (2006), 2187–2194.

3. K.-Y. Lee, S. Mohammadi, P.K. Bhattacharya, and L.P.B. Katehi, Awideband compact model for integrated inductors, IEEE MicrowaveWireless Compon Lett 16 (2006), 490–492.

4. E.-X. Liu, E.-P. Li, L.-W. Li, and Z. Shen, Finite-difference time-domain macromodel for simulation of electromagnetic interference athigh-speed interconnects, IEEE Trans Magn 41 (2005), 65–71.

5. R. Bhattacharya, A.P. Joshi, and T.K. Bhattacharya, PSO-based evo-lutionary optimization for black-box modeling of arbitrary shapedon-chip RF inductors, In IEEE Topical Meeting on Silicon monolithicIntegrated Circuits in RF Systems, MTTs, San Francisco, USA, 2006,pp. 103–106.

6. T. D’haene, R. Pintelon, and G. Vandersteen, An iterative method tostabilize a transfer function in the s- and z-domains, IEEE TransInstrum Measure 55 (2006), 1192–1196.

7. A.H. Whitfield, Transfer function synthesis using frequency responsedata, Int J Control 43 (1986), 1413–1426.

8. E.C. Levy, Complex-curve fitting, IRE Trans Auto Control 4 (1959),37–43.

9. P. Verboven, P. Guillaume, and B. Cauberghe, Multivirable frequen-cy-response curve fitting with application to modal parameter estima-tion, Automatica 41 (2005), 1773–1782.

10. A. Ratnaweera, S.K. Halgamuge, and H.C. Watson, Self-organizinghierarchical particle swarm optimizer with time-varying accelerationcoefficients, IEEE Trans Evolution Comput 8 (2004), 58–73.

11. M. Elzinga, K.L. Virga, and J.L. Prince, Improved global rationalapproximation macromodeling algorithm for networks characterizedby frequency sampled-data, IEEE Trans MTT 48 (2000), 1461–1468.

Figure 7 Phase plot of transformer Z12

Figure 8 Magnitude plot of transformer Z22. The model extracted by theproposed algorithm consists of fifth order numerator polynomial and sixthorder denominator polynomial

Figure 9 Phase plot of transformer Z22

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007 2215

12. J. Kolstad, C. Blevins, J.M. Dunn, and A. Weisshaar, A new circuitaugmentation method for modeling of interconnects and passive com-ponents, IEEE Trans Adv Packag 29 (2006), 67–77.

13. M. Elzinga, K. Virga, L. Zhao, and J.L. Prince, Pole-residue formu-lation for transient simulation of high-frequency interconnects usinghouseholder LS curve-fitting techniques, IEEE Trans Adv Packag 25(2000), 142–147.

14. T.G. Freeman, Selecting the best linear transfer function model, Au-tomatica 21 (1985), 361–370.

15. M. Iwasaki, M. Miwa, and N. Matsui, GA-based evolutionary identi-fication algorithm for unknown structured mechatronic systems, IEEETrans Ind Electron 52 (2005), 300–305.

16. C.-M. Huang, C.-J. Huang, and M.-L. Wang, A particle swarm opti-mization to identifying the ARMAX model for short-term load fore-casting, IEEE Trans Power Sys 20 (2005), 1126–1133.

17. J. Kennedy and R.C. Eberhart, Particle swarm optimization, In IEEEConference on Neural Networks IV, Perth, Australia, 1995, pp. 1942–1948.

© 2007 Wiley Periodicals, Inc.

RADIATION PERFORMANCE OFPURELY METALLIC WAVEGUIDE-FEDCOMPACT FABRY–PEROT ANTENNASFOR SPACE APPLICATIONS

O. Ronciere,1 B. A. Arcos,1 R. Sauleau,1 K. Mahdjoubi,1 andH. Legay2

1 IETR (Institut d’Electronique et de Telecommunications de Rennes),UMR CNRS 6164, Universite de Rennes 1, Campus de Beaulieu,Avenue du General Leclerc, 35042 Rennes Cedex, France2 Alcatel Alenia Space, 26 Avenue J.F. Champollion, B.P. 1187,31037 Toulouse Cedex 1, France

Received 26 February 2007

ABSTRACT: Metallic electromagnetic bandgap resonator antennas arehighly attractive to design very efficient primary feeds of focal array fedreflector antennas for space Tx applications. The antennas analayzed inthis work consists of purely metallic Fabry–Perot (FP) cavities with asingle partially reflecting surface (PRS) and four perfect electrical con-ductor (PEC) walls on the sides. They are excited by a standard metal-lic waveguide. In contrast to prior work, we focus our attention on com-pact FP resonators whose size is in the order of a few wavelengths infree space. Their radiation characteristics (namely, their resonant fre-quency, maximum directivity, aperture efficiency, and radiation band-width) are investigated numerically in X-band with the finite-differencetime-domain method. The extensive study of their radiation performanceas a function of their size and grid reflectivity of the PRS, as well as thecomparison with the well-known characteristics of large FP antennas,highlight three main features that must be taken into account for an op-timum design: (1) for a given grid reflectivity, the resonant frequencyincreases while reducing the size of the shielded resonator, (2) for agiven size of the antenna there exists an optimum reflectivity maxi-mizing the aperture efficiency, (3) at resonance, the beam radiatedby a square cavity is not circularly symmetric. In particular, we haveshown that, for a 2 � o cavity, it is possible to reach a high-aper-ture efficiency (s 80%) over a 2.8% (�1 dB radiation) band-width. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett49: 2216 –2221, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22703

Key words: Fabry–Perot cavity; aperture efficiency; EBG resonatorantenna

1. INTRODUCTION

Multilayer periodic structures like electromagnetic bandgap (EBG)defected resonators [1–6] and metamaterials [7–9] have beenwidely employed to design low-profile directive antennas withpencil beams [1–9], shaped beams [10, 11], or frequency-scannedbeams [12–14]. All these antenna configurations are based onFabry–Perot (FP) concepts [15], and the corresponding reflectingmirrors consist of 1-D, 2-D, 3-D dielectric, or metallodielectricstructures. In most cases, the proposed antennas have a highdirectivity (20 dBi), and moderate radiation [16] and aperture(�70%) efficiencies.

Owing to their very narrow radiation bandwidth (BW), veryhigh-gain FP cavities antennas (BW �0.5% for a 30 dBi gainantenna) are not suitable for single-beam satellite communications.An alternative solution consists in using one-feed-per-beam reflec-tor antennas, the so-called focal array fed reflector (FAFR) anten-nas. In this context, an interleaved feed design using a 1-D dielec-tric EBG resonator has been introduced recently [17]. Thispromising configuration enables one to reduce spillover loss whileenhancing crossover level between adjacent beams.

Another solution is based on a dense array of compact FP asprimary feeds of reflector antennas. In this case, the typical size ofthe radiating aperture is in the order [1.2 � o]2 to [2 � o]2). Forthis application, the ultimate objective is to maximize their aper-ture efficiency in order to minimize the intersource spacing andthus the beam crossover. Despite the abundant literature on EBGresonator antennas and FP cavities [1–7, 11–17], all prior worksdeal with large structures: their lateral dimensions typically varybetween five and 15 wavelengths at the operating frequency. Toour best knowledge, there is only one paper reporting on reduced-size EBG resonator antennas [17]; in this reference, the antenna iscircularly symmetric and is built from stacked finite dielectricdisks fed by a flanged waveguide. As dielectric materials are notsuitable for space applications requiring high power, we investi-gate here the radiation performance of purely metallic and compactEBG resonator antennas. The FP cavities operate in X-band andare excited by a standard WR-90 rectangular metallic waveguide(10.2 � 22.8 mm2). The metal parts are assumed ideal perfectelectrical conductor (PEC) and have a zero thickness. Our numer-ical results have been obtained with a 3-D in-house finite-differ-ence time-domain (FDTD) tool [18, 19].

This paper is organized as follows. In Section 2, we describeand discuss the radiation characteristics of compact FP antennas,with emphasis on the key differences with conventional (i.e. elec-trically large) configurations. Then, universal curves summarizingthe antenna performance in terms of resonant frequency, maximumdirectivity, aperture efficiency, and radiation BW are given inSection 3, as a function of the grid reflectivity and resonator size.Conclusions are finally drawn in Section 4.

2. PHENOMENOLOGICAL ANALYSIS AND COMPARISONWITH LARGE FP ANTENNAS

The standard geometry of a waveguide-fed metallic FP antenna isrepresented in Figures 1(a) and 1(b). The resonant cavity (L � L �D) is built from (i) the metal flange of the input waveguide, (ii) theupper partially reflecting surface (PRS), and (iii) the four lateralopen-ended or PEC walls. For linearly polarized antennas, the PRSusually consists of a 1-D inductive grid [Fig. 1(b)] whose spatialperiodicity, width of the metal strip, and power reflectivity [20] aredenoted by a, d, and R, respectively.

For the sake of consistency and forthcoming discussions, webriefly summarize the key characteristics (directivity and resonantfrequency) of large FP resonators. Their fundamental resonantfrequency fo is given by [21]:

2216 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007 DOI 10.1002/mop