An Energy-BasedCircuit Parameter Extraction Method

for CRLH Leaky Wave AntennasS. Otto∗, A. Rennings†, T. Liebig†, C. Caloz‡, K. Solbach∗

∗Hochfrequenztechnik (HFT) and †Allgemeine und Theoretische Elektrotechnik (ATE)University of Duisburg-Essen, Bismarck-Straße 81, 47057 Duisburg, Germany

simon.otto@uni-due.de‡Ecole Polytechnique de Montreal, ch. de Polytechnique, H3T 1J4, Quebec, Canada

Abstract—An energy based circuit parameter extractionmethod for composite right/left-handed (CRLH) leaky-wave an-tennas with a symmetric unit cell is introduced. This methodcomputes the CRLH LCRG equivalent circuit parameters interms of the stored energy and power loss obtained by afull-wave eigenmode analysis via even and odd current andvoltage considerations. In this fashion, the unit cell is accuratelycharacterized for the infinite periodic case, with full account ofelectromagnetic effects. Compared to previously reported drivenmode extraction techniques, the proposed eigenmode methoddoes not require the separation of the series and shunt partsof the cell along with de-embedding, it automatically accountsfor mutual coupling between the periodic cells, and accurately de-termines the radiation parameters (resistance and conductance)to assess the antenna efficiency and bandwidth. Furthermore, themethod provides physical insight into the radiation mechanismsof CRLH antennas.

I. INTRODUCTION

Due to their unprecedented flexibility and performance,composite right/left-handed (CRLH) leaky-wave and resonantantennas have spurred considerable interest throughout the lastdecade. A comprehensive overview on the topic is available in[1]. Starting then years ago from very speculative and exoticmetamaterial concepts, these antennas have now reached acertain level of maturity and even start to penetrate commercialmarkets.

However, CRLH antennas still require more accurate andefficient design procedures, as well as stronger theoreticalfoundations regarding their fundamental capabilities and lim-itations. As a contribution toward this demand, this paperpresents a novel energy based extraction method to computethe constitutive circuit parameters of the CRLH unit cell.

Previously reported circuit model parameter extraction ap-proaches typically employ driven mode analysis to determinethe unit cell LC parameters [2]. However, these approachessuffer from three major drawbacks. First, they require separatesimulations for the series and shunt elements, with delicate de-embedding of the lead lines required to avoid perturbation bythe higher order modes due to port discontinuities. Second,they fail to account for the strong electromagnetic couplings

occurring between the different cells and therefore produceinaccurate results. Third, they do not provide a direct andaccurate way to extract the dissipative parts of the structure,which are essential for antennas (radiation resistance andconductance). The proposed method, which is based on aneigenmode approach, overcomes all these limitations in asimple, fast and insightful manner.

II. CRLH STRUCTURE, MODEL AND PARAMETERS

Fig. 1 shows a typical CRLH microstrip unit cell. Thisparticular implementation uses metal-insulator-metal seriescapacitors and stub shunt inductors [3]. The figure also showsthe relevant layout and substrate parameters, whose valuesare listed in Tab. 1, and the equivalent circuit model whichincludes the six parameters to be extracted LR, CR, LL, CL,R and G.

1

2GW

1

2GW

Dvia

layer stack

εr = 3.55, tanδ = 0.004

h1 = 100µm

h2 = 250µmsheet

metalization

LH

x

y

z

PMC

wall

PMC wall

1

2LS

WStubLStub

Lp

G

1

2R 1

2LR 2CL 2CL

1

2R1

2LR

Fig. 1. Microstrip metal-insulator-metal CRLH unit cell setup and theequivalent circuit model.

The structure includes 3 metallization layers, the groundlayer, the stub layer and the trace layer. The dielectric substratehas a permittivity of εr = 3.55 and a loss tangent oftan δ = 0.004. The conductor loss of the metallizations is

modeled by a constant surface resistivity of Rs = 20 mΩ toaccount for the copper losses in a narrow band around 24 GHz.

In the full-wave analysis setup, perfect magnetic conductor(PMC) walls are placed at both sides of the structure, parallelto axis of the propagation direction (x). These walls modelthe quasi-TEM field distributions along the antenna.

TABLE IMIM CRLH UNIT CELL DIMENSIONS

Parameter Dimensions(µm)

Lp 1700LH 700LS 1425LStub 1300WStub 500GW 400Dvia 300

III. STORED ENERGY AND DISSIPATED/RADIATED POWER

The energy-based circuit parameter extraction method em-ploys an eigenmode solver with periodic boundaries. Insteadof sweeping the Bloch-Floquet phase angle parameter Φ, itrequires only a single simulation at Φ = 0 to compute thestored energy and dissipated/radiated power and derive theCRLH LCRG parameters.

Fig. 2 shows the HFSS (finite element method commercialsoftware) simulation setup used to compute the stored energyand dissipated/radiated power. These computations use thefollowing standard integral expressions [4]:

W = We +Wm = 2Wm (1)

=12µ0

∫V

~H · ~H∗ dv (2)

Pr =12

∫Sr

~E × ~H∗ · d~s (3)

Pd =12

∫Vd

~E · ~J∗ dv (4)

Pc =Rs

2

∫Sc

~Htan · ~H∗tan ds, (5)

where W and P represent energies and powers, respectively,the subscripts e and m refer to electric and magnetic quanti-ties, respectively, the subscript r, d and c refer to radiation,dielectric and conductor losses, respectively, and ~E, ~H and~J represent the electric field, the magnetic field and theconduction current, respectively. The integration domains V ,Sr, Vd and Sc are indicated in Fig. 2.

At the real angular eigenmode frequency ω0, the systemresonates with equal average electric and magnetic energiesWe and Wm. The integration over the volume V in (2) isan approximation of the stored electric and magnetic energiesin the unit cell. To accurately determine the reactive near-field stored energy, the energy flux density of the radiatingfield components must be subtracted from the total energyflux density, as mentioned in [5] and shown in [6] and

x

z

y

Fig. 2. HFSS simulation model of a symmetric CRLH MIM antenna unitcell with periodic boundaries in x-direction. The volumes V and surfaces Sare used for the field integrations in (1)-(5).

[7]. However, the direct V volume integration is sufficientlyaccurate if the volume extends up to the far-field zone of theantenna unit cell, and this simple approach is therefore usedhere. The radiated power Pr is accounted for by the integrationof the Poynting vector ~S = 1

2~E× ~H∗ on the surface Sr located

on top of the antenna structure. The power dissipation in thedielectric substrate Pd is calculated by Joule’s law over thevolume Vd of the substrate. The conductors are modeled assheets with surface resistivity Rs; the power loss Pc due to thefinite conductivity is computed from the tangential magneticfield ~Htan on the surface of the conductor sheets.

The overall quality factor Q can be expressed in terms ofthe above integral quantities

Q =ω0W

Pr + Pd + Pc=ω0W

P. (6)

where P represents to total power loss Pr + Pd + Pc.

IV. MODES AT THE ZERO PHASE PERIODIC CONDITION

Two resonant modes exist in the infinite wavelength λg (orzero propagation constant β = 2π/λg) regime of the CRLHunit cell, which is the regime used in this paper for parameterextraction, the series and shunt resonances, ωse = 1/

√LRCL

and ωsh = 1/√LLCR, respectively. In the case of a balanced

CRLH unit cell [2], these two resonant frequencies are equal,ωse = ωsh = ω0. The ωse resonance mode concentrates itsenergy in the short transmission line and the metal-insulator-metal capacitors, while the ωse resonance mode concentratesits energy in the stub length with the via connection to theground. The phase term for the periodic boundaries in the xdirection is set to Φ = 0 to compute ωse and ωsh. Both modescan be fully discriminated by examining the eigenmode fielddistribution, as will be shown below.

Figs. 3 and 4 show the electric field distribution on the topmetalization plane for the series resonance mode at ωse and theshunt resonance mode at ωsh, respectively. In the former case,

the electric field to the periodic boundary surfaces is normal,indicating a perfect electric conductor (PEC) condition, whileFig. 4 shows the latter case, where the electric field istangential to the same surfaces, indicating a PMC condition.

PEC wall behavior

PEC wall behavior

Fig. 3. Electric field distribution for the series mode ωse obtained by theΦ = 0 periodic boundary condition and showing the PEC wall behavior ofthis mode.

PMC wall behavior

PMC wall behavior

Fig. 4. Electric field distribution for the shunt mode ωsh obtained by theΦ = 0 periodic boundary condition and showing the PMC wall behavior ofthis mode.

Thus, it is seen that a y-symmetrical unit cell with periodicboundary phase condition Φ = 0 produces two fundamentalmodes, one with a PEC boundary and another one with a PMCboundary. In particular, a CRLH symmetric unit cell may beanalyzed without any periodic boundaries by just applyinga PEC and a PMC boundary conditions in an eigenmodesimulator.

V. CURRENT AND VOLTAGE OF THE SERIES AND SHUNTMODES

In terms of the CRLH circuit model of Fig. 1, the series-PEC and shunt-PMC models reduce to the circuits shown inFigs. 5 and 6, respectively.

Vse

=0

Vse

=0

Ise 6= 0 Ise 6= 0

Fig. 5. Equivalent circuit resonator for the series-PEC ωse mode.

Ish = 0 Ish = 0

Vsh6=

0

Vsh6=

0

Fig. 6. Equivalent circuit resonator for the shunt-PMC ωsh mode.

The corresponding current and voltage are obtained fromfield quantities as

Ise =∮

Li

~HPEC · d~l (Vse = 0), (7)

Vsh =∫

Lv

~EPMC · d~l (Ish = 0), (8)

where the integration contours are indicated in Fig. 7.

Lu

Li

Fig. 7. Definition of the voltage and current path integration lines for (7)and (8)

VI. CIRCUIT PARAMETER FORMULAS

Based on the results of the previous three sections, simpleformulas for the CRLH reactive and resistive parameters cannow be written. The reactive parameter values (LR and CR)for the circuit components of each resonator are obtained byrelating the stored magnetic or electric energy to the currentor voltage for the corresponding mode, while the resistiveparameter values (Rr,d,c and Gr,d,c) are computed by relatingthe the power loss for each type of loss to the terminal currentor voltage.

The formulas for the series resonator, corresponding to thecircuit model of Fig. 5 read

LR =4Wm

|Ise|2, (9)

CL =1

ω2seLR

, (10)

Rr =2Pr

|Ise|2, (11)

Rd =2Pd

|Ise|2, (12)

Rc =2Pc

|Ise|2, (13)

while the formulas for the shunt resonator, corresponding tothe circuit model of Fig. 6 read

CR =4We

|Vsh|2, (14)

LL =1

ω2shCR

, (15)

Gr =2Pr

|Vsh|2, (16)

Gd =2Pd

|Vsh|2, (17)

Gc =2Pc

|Vsh|2, (18)

Based on the parameter for the series resonator in (9)-(13),we define for later convenience the following series modequantities:

R = Rr +Rd +Rc (19)

and

Qse =ω0W

P=ωseLR

R(20)

Similarly, the following shunt mode quantities are definedfrom (14)-(18):

G = Gr +Gd +Gc (21)

Qsh =ω0W

P=ωshCR

G. (22)

The field calculator of HFSS provides an easy-to-use inter-face for the computation of the integral expression (1)-(5) andthe current and voltage path integration (7) and (8).

VII. FULL-WAVE VALIDATION

The circuit parameters extracted by the proposed method forthe CRLH structure of Fig. 2 are given in Tab. 2. These resultsshow that the radiation contribution from the shunt mode isnegligibly small in this case, since Gr ≈ 0.

TABLE IIEXTRACTED CRLH CIRCUIT PARAMETER

LR(pH) CL(fF) Rr(Ω) Rd(Ω) Rc(Ω)547.83 72.21 4.54 0.30 0.12CR(fF) LL(pH) Gr(mS) Gd(mS) Gc(mS)289.77 153.25 3.6e-5 0.17 0.075

In order to examine the validity of the proposed method, afull-wave phase sweep (0 < Φ < 30) analysis is performedwith HFSS and the corresponding results are compared withthe circuit model with the extracted parameter values. Forsmall phase angles 0 < Φ < 30, corresponding to anelectrically small (lumped) unit cell, the dispersion relationof the circuit model can be approximated by the effectivetransmission line relation [2]

γLp =√Zse(ω)Ysh(ω), (23)

where γ is the complex propagation constant, Lp the lengthof the unit cell and where Zse(ω) = R + j(ωLR − 1/ωCL)and Ysh(ω) = G+j(ωCR−1/ωLL) are the series impedanceand shunt admittance, respectively.

For the eigenmode solution, we can reformulate this ex-pression in terms of the complex angular frequency ω =ωre + jωim and a real phase Φ as

Φ =√Zse(ω)Ysh(ω), (24)

where approximate reactance and susceptance slope parame-ters (linearization) may be used at resonance

Zse(ω) ≈ R+ j2LR(ω − ωse), (25)Ysh(ω) ≈ G+ j2CR(ω − ωsh). (26)

Finally, the eigenmode solution with complex angular fre-quency ω is written as

Φ =√

(R+ j2LR(ω − ωse)) (G+ j2CR(ω − ωsh)). (27)

Equation (27) has been solved numerically for the complexangular frequency ω using the commercial software MATLAB.The phase dependent quality factor is

Q =|ω|

2ωim. (28)

Figs. 8 and 9 show the eigenmode frequency and qualityfactor, respectively, obtained from HFSS eigenmode full-wave simulation with periodic boundaries (Φ sweep) and bythe CRLH circuit model with extracted parameters by theproposed method. Excellent agreement is observed for smallangles Φ < 20, while some discrepancies appear at higherangles, as expected from the homogeneization of (23) and (24)and linearization of (25) and (26). as Here, the transitions toguided modes are not taken into account by the simple circuitmodel.

fsh = 23.88 GHz

fse = 25.30 GHz

Φ (degree)

f(G

Hz)

eigen mode solution

circuit model

0 5 10 15 20 25 3020

21

22

23

24

25

26

27

28

29

30

Fig. 8. Comparison of the eigenmode frequencies for the HFSS full waveanalysis and the circuit model (27).

fsh = 176.82

fse = 17.54

Φ (degree)

Qu

alit

yfa

cto

rQ

eigen mode solution

circuit model

0 5 10 15 20 25 300

25

50

75

100

125

150

175

200

Fig. 9. Comparison of the quality factors for the full wave analysis and thecircuit model (27).

VIII. CONCLUSION

An energy based circuit parameter extraction method forcomposite right/left-handed (CRLH) leaky-wave antennas witha symmetric unit cell has been introduced. This methodcomputes the CRLH LCRG equivalent circuit parameters interms of the stored energy and power loss obtained by a full-wave eigenmode analysis via even and odd current and voltageconsiderations. It solves the difficulties encountered in pre-viously reported extraction procedures and most importantlyprovides an exact estimation of the radiated power.

The proposed extraction procedure offers a powerful tool forthe design of CRLH leaky-wave and resonant antennas. Fur-thermore, it provides a deep insight into the electromagneticoperation of this structure. Finally, the authors are currentlyapplying it to determine fundamental bounds and solutions forthe radiation efficiency, in particular at broadside, for CRLHleaky-wave antennas.

REFERENCES

[1] C. Caloz, T. Itoh, and A. Rennings, “CRLH metamaterial leaky-waveand resonant antennas,” IEEE Antennas Propag. Mag., vol. 50, no. 5, pp.25–39, Oct. 2008.

[2] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission LineTheory and Microwave Applications. Wiley-IEEE Press, 2005.

[3] T. Liebig, A. Rennings, S. Otto, C. Caloz, and D. Erni, “Comparisonbetween CRLH zeroth-order antenna and series-fed microstrip patcharray antenna,” in Proc. 3rd European Conference on Antennas andPropagation EuCAP 2009, 2009, pp. 529–532.

[4] D. M. Pozar, Microwave Engineering. J.Wiley & Sons, New York, 2nded., 1998.

[5] D. Pozar, “New results for minimum Q, maximum gain, and polarizationproperties of electrically small arbitrary antennas,” in Proc. 3rd EuropeanConference on Antennas and Propagation EuCAP 2009, 2009, pp. 1993–1996.

[6] R. Collin and S. Rothschild, “Evaluation of antenna Q,” IEEE Trans.Antennas Propag., vol. 12, no. 1, pp. 23–27, Jan. 1964.

[7] R. Fante, “Quality factor of general ideal antennas,” IEEE Trans. AntennasPropag., vol. 17, no. 2, pp. 151–155, Mar. 1969.