CONCLUSION

Removal of multipath from analogue FM is problematic as it is ablind process. This paper used the CMA in association with amultiplier-less FIR filter to demonstrate how the equalizationprocess can be improved. The number of simulated multipathspikes was observed to be in agreement with published measure-ments for a similar environment. Simulation demonstrated that theerror introduced by the approximation of the logarithmic codec didnot compromise the multiplier-less filter’s ability to remove themultipath spikes. The software verified the efficiency of thismethod, which suggests that a viable hardware implementationshould reduce computational time and complexity. The work pre-sented in this paper can be combined with the signed data or signederror CMA to simplify the hardware even further. This systemcould be deployed for interference reduction in vehicles and higherfrequency mobile applications, and as the basis for fast-convergingadaptive beam steering systems.

ACKNOWLEDGEMENT

Mr. El-Eraki was supported by the Harada European TechnologyCentre, UK.

REFERENCES

1. J.R. Treichler, C.R. Johnson, and M.G. Larimore, Theroy and design ofadaptive filters, A wiley-interscience publication, 1987.

2. S.R. Nelatury and S.S. Rao, Increasing the speed of convergence of theconstant modulus algorithm for blind channel equalization, IEEE TransCommun 50 (2002), 872–876.

3. K. Fujimoto and J.R. James, Mobile antenna systems handbook, ArtechHouse, Boston, 2001.

4. P. Lee, An FPGA prototype for a multiplierless FIR filter built using thelogarithmic number system, Proc 5th Int Wkshp Field-ProgrammableLogic and Appl FPL, Berlin, 1995, pp. 303–310.

5. B. Hofflinger, Efficient VLSI digital logarithmic codecs, IEE ElectronLett 27 (1991), 1132–1134.

6. H. Combet, H. Van Zonneveld, and L. Verbeek, Computation of thebase two logarithm of binary numbers, IEEE Trans Electron Computers14 (1965), 863–867.

© 2003 Wiley Periodicals, Inc.

THEORY OF MAGNETOELECTRICMULTICONDUCTOR TRANSMISSIONLINES WITH APPLICATION TO CHIRALAND GYROTROPIC LINES

Ricardo Marques,1 Francisco Mesa,2 and Francisco Medina1

1 Dept. de Electronica y ElectromagnetisomoFacultad de FısicaUniversidad de SevilleAvenida Reina Mercedes s/n41012 Sevilla, Spain2 Dept. de Fısica Aplicada IE.T.S. de Ingenierıa InformaticaAvenida Reina Mercedes s/n41012 Sevilla, Spain

Received 20 December 2002

ABSTRACT: This paper presents a theory of multiconductor trans-mission lines that allows for dealing with possible magnetoelectriccouplings between the basic circuit quantities: voltage, current, andthe per-unit-length charge and flux linkage. The magnetoelectric cou-pling is accounted for by the introduction of two new characteristicmatrices of the line, the so-called charge and current memductance

matrices. The proposed theory is suitable for the description of linesloaded with gyrotropic, bi(iso/aniso)tropic chiral, pseudochiral, andTellegen media. © 2003 Wiley Periodicals, Inc. Microwave OptTechnol Lett 38: 3–9, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/mop.10955

Key words: magnetoelectric coupling; circuit model; multiconductorlines; memductance

1. INTRODUCTION

The standard theory of multiconductor transmission lines [1] isbased on the generalization of the two-conductor telegrapher equa-tions by using the per-unit-length (p.u.l.) capacitance [C] andinductance [L] matrices of the line. An inherent limitation of thistheory is that magnetoelectric couplings (such as those appearingin bi(iso/aniso)tropic media [2–5]) cannot be taken into account.Moreover, the standard telegrapher wave equations only involvesecond-order derivatives of voltages and/or currents, thus givingrise to pairs of propagation constants with equal magnitude andopposite sign. Hence, only bidirectional lines can be considered inthat formalism, thus precluding its application to study, for exam-ple, any kind of line loaded with ferrite media (except for axialmagnetization [6, 7]). Nevertheless, an important effort has beendevoted to generalize the transmission line theory to include bi(iso/aniso)tropic medial [2–5]. In this context, the authors [8–10]recently applied the concept of line memductance (firstly intro-duced by L.O. Chua [11]) to account for the nonreciprocal effectsraised by the coupling between the p.u.l. charge and the magneticflux linkage across ferrite-filled transmission lines. In the presentwork, the concept of memductance will be extended to the analysisof multiconductor transmission lines containing bi(iso/aniso)tropicmedia and/or gyrotropic media (media having a magneticallyinduced anisotropy, such as biased ferrites or magnetized plas-mas).

2. TRANSMISSION LINE THEORY

2.1. Definitions and Basic EquationsLet a uniform multiconductor transmission line be formed by asystem of N � 1 conductors embedded in an inhomogeneouslinear medium characterized by a 6 � 6 frequency-dependenttensor relating the polarization and magnetization of the mediumto the macroscopic electric E and magnetic B fields (exp( j�t)time-dependence and translational symmetry along the z-axis areimplicitly assumed throughout). Provided that some type of quasi-static approximation applies, the current Ii and the p.u.l. charge �i,at each conductor, can be properly defined. Similarly, a quasi-staticvoltage Vi between the ith conductor and the (N � 1)th groundedconductor can be also introduced. Finally, it is assumed that a p.u.l.flux linkage �i, across a curve joining the ith conductor andground, can be unambiguously described (the physical conditionsfor these assumptions will be further discussed in each example).

Faraday and charge conservation laws establish that

d

dzV � �j�� (1)

d

dzI � �j��, (2)

where V, �, I, and � are, respectively, the voltage, p.u.l. fluxlinkage, current, and p.u.l. charge vectors in the line: V � (V1,V2, . . . Vi, . . . VN), etc. If magnetoelectric couplings exist, theusual charge-voltage and flux-current linear relations have to be

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 1, July 5 2003 3

substituted by a set of more general linear relations. This can bedone through the following set of equations:

� � �C� � V � �WQ� � � (3)

I � �WI� � V � �L��1 � �, (4)

where the [C], [L], [WQ], and [WI] matrices depend on thegeometry of the line and on the frequency and, eventually, anexternal magnetostatic field, H0, through the constitutive parame-ters of the medium. The new [WQ] and [WI] quantities havedimensions of charge divided by magnetic flux (or, equivalently,conductance) and will be denoted as the charge memductancematrix and current memductance matrix, respectively. In thepresent analysis, it will be distinguished between [WQ] and [WI],although it will be shown that these quantities are not independent.It is interesting to note that any other linear combination of therelations (3) and (4) could have been used as the starting point ofthe formulation leading to a very similar formalism [3–5]. Any-how, it will be shown later that the present formulation will makeit possible to account also for the magnetoelectric effects ofnon-bi(iso/ani)sotropic lines (for example, nonreciprocal propaga-tion in ferrite-loaded structures).

Starting from the quasistatic expression for the variation of thep.u.l. total energy along the line

�U � V � �� � I � ��, (5)

and following the theory of the generalized susceptances in [14], itis found that (see Appendix 1)

�C��; H0, �� � � �C��; �H0, ��� �T (6)

�L��; H0, �� � � �L��; �H0, ��� �T (7)

�WQ��; H0, �� � � �WI��; �H0, ��� �T, (8)

where �� is the Tellegen tensor [15–17] (which changes of sign bytime inversion [17]) and superindex T indicates transposed. Fornon-gyrotropic non-Tellegen reciprocal media, it is apparent fromEq. (8) that [WQ] and [WI] are mutually transposed.

The generalized telegrapher equations are easily obtained fromEqs. (1)–(4) as

d

dz �VI � � �j��M� � �V

I �, (9)

where

�M� � � ��L��WI� �L��C� � �WQ��L��WI� �WQ��L��. (10)

Because of the exp(�jkzz) dependence imposed to the line volt-ages and currents, Eq. (9) turns into the following eigenvalueequation for the line propagation constant kz:

�kz

��1� � �M�� � �V

I � � 0, (11)

where [1] denotes the unit matrix. Note that Eq. (11) does notproduce, in general, solutions of the type kz, magnetoelectrictransmission lines being, in general, non-bidirectional [2–5].

2.2. Reciprocity and Mode OrthogonalityIntroducing relations (6)–(8) into (9), it follows after some alge-braic manipulations that

d

dz�V � I� � V� � I � 0, (12)

where [V, I] denotes an arbitrary solution along the line and [V�,I�] an arbitrary solution along the so-called complementary trans-mission line [12, 13]; namely, that line having the same geometryand media as the original one but with the magnetostatic bias fieldreversed: H�0 � �H0 (or, in the case of lines with Tellegen media,with the Tellegen tensor reversed: �� � � ��� ). Eq. (12) can beconsidered as the particular form of the general reciprocity theo-rem [13] for the source-free solutions in the line.

The solution to Eq. (11) has the from V � V0exp(�jkzz), I �I0exp(�jkzz), which defines a line mode with propagation con-stant kz. The line modes will be denoted as [V0, I0; kz], where V0

and I0 are the modal eigenvector amplitudes associated with the kz

modal propagation constant. Similarly, a mode of the complemen-tary line will be denoted as [V�0, I�0; k�z]. Taking into account theexponential nature of the solutions, Eq. (12) leads to

�kz � k�z�V0 � I�0 � V�0 � I0 � 0, (13)

which can be considered as the most general orthogonality relationbetween the modes of a magnetoelectric multiconductor transmis-sion line.

For lines having reflection symmetry at the z-plane, the exis-tence of the [V�0, I�0; k�z] mode implies the existence of the [V�0,�I�0; �k�z] mode in the same line. Substituting this latter mode in(12) and using (13), it is obtained that

�kz � k�zV0 � I�0 � 0, (14)

which substitutes to Eq. (13) as the orthogonality relation betweenthe modes in this type of lines. Finally, for the particular case oflines with non-gyrotropic reciprocal media, the prime and non-prime quantities will correspond to modes of the same line.

2.3. Lossless LinesFrom energy conservation, any source-free solution in a losslessline must satisfy that

�� d

dz�V � I*� �

1

2

d

dz�V � I* � V* � I � 0. (15)

Substitution of (9) into (15) establishes, after some algebra, thatcondition (15) is fulfilled, provided that

�C�T � �C�* (16)

�L�T � �L�* (17)

�WQ�T � ��WI�*. (18)

Thus, the capacitance and inductance matrices must be hermitianand the charge and current memductance matrices must be mutu-ally anti-hermitian. For the particular case of reciprocal media, thecombination of relations (6)–(8) with (16)–(18) also establishesthat the capacitance and inductance matrices are real and symmet-ric, whereas the memductance matrices must be imaginary and oneis the transposed form of the other.

4 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 1, July 5 2003

Taking now the complex conjugate of Eq. (11) and substitutingEqs. (16)–(18) into the resulting equation, the following importantresult is reached:

If �V0, I0; kz� is a mode in a lossless line with reciprocal media,

then �V*0, �I*0; �k*z� is also a mode of the same line.

For purely propagative modes having real values of kz, this resultimplies bidirectionality. (Note that the same conclusion cannot bedirectly applied to lossy lines with anisotropic media which, fol-lowing [18], might be non-bidirectional despite of the reciprocityof the media.) Therefore, all purely propagative modes in losslessreciprocal transmission lines must be bidirectional, in accordancewith the general statement in [18].

Since the [WQ] and [WI] matrices in lossless reciprocal linesare imaginary (assuming they are not null), it is deduced from(9)–(11) that modal impedances (that is, the ratio between thevoltage Vi and current Ii at each conductor) for purely propagativemodes in this type of lines are, in general, complex quantities;namely, currents and voltages are, in general, out of phase. Thisstriking property of magnetoelectric lossless transmission lines is aconsequence of the imaginary nature of the memductances, themodal impedances being real only in the case of vanishing mem-ductances.

The following general orthogonality modal relation is alsoobtained after substituting (11) into (15):

�k*z � k�z�V*0 � I�0 � V�0 � I*0 � 0. (19)

The above equation together with Eq. (13) can be considered as thetransmission line theory version of Eqs. (21) and (23) in [19].

2.4. Two-Conductor LinesTwo-conductor magnetoelectric transmission lines are an impor-tant particular case of the structures under study. In this case, sincethe line capacitance, inductance, and memductances—C, L, WQ,and WI—are scalar quantities, the telegrapher equations (11) arereadily solvable, giving the following propagation constants andcharacteristic impedances:

kz

��

1

2�L�WQ � WI � �L2�WQ � WI

2 � 4LC� (20)

and

Z �V

I � 2�WQ � WI � �L2�WQ � WI2 � 4CL�1��1. (21)

If only reciprocal media are involved, it is deduced from Eq. (8)that WQ � WI, which, in light of Eqs. (20) and (21), allows us toestablish that any reciprocal two-conductor transmission line mustbe bidirectional and non-symmetrical [2], provided WQ, WI do notvanish.

3. APPLICATIONS

3.1. Design Formulas for the Quasi-TEM Edge-Mode Parallel-Plate Ferrite Phase ShifterAs an example of the application of the present theory, the non-reciprocal parallel-plate phase shifter shown in the inset of Figure1 is studied. This structure is the generalization of the symmetricalphase shifter analyzed in [8, 9], which has been recently used as ananalytical model for a microstrip edge-mode phase shifter [10].

The resulting design formulas could be useful as an analyticalmodel for more general edge-mode phase shifters. Neglecting thefield variations along y, and assuming Hz � 0 (that is, quasi-TEMfields), a quasi-magnetostatic solution for the fields can be con-structed under the assumption of constant magnetic flux density,Bx � B0, across the line and zero magnetic fields along they-direction (Hy � 0, By � 0). The remaining magnetic fieldcomponents in the ferrite, Bz and Hx, can be deduced from B0

using the constitutive relation B � �� � H, where �� is the magneticpermeability tensor of the ferrite. Outside the ferrite Bz � 0 andHx � �0

�1B0. Once the magnetic field has been obtained, the totalcurrent is computed in terms of the total flux linkage � � B0d,and the line inductance L is finally obtained as

L ��

I� d �c � a

�0�

2a

��

b � a

�0��1

, (22)

where � is the �xx � �yy diagonal element of the ferrite perme-ability tensor.

According to Faraday’s law, the presence of a nonvanishingmagnetic flux density Bz in the ferrite produces a nonzero trans-verse electric field Ey in the line cross section, even in the absenceof a quasi-static voltage between the plates. This electric fieldwould induce a nonvanishing p.u.l. charge � on the metallic platesgiven by

� � 1 �c

�a

Ey� xdx � f �a

a

Ey� xdx � 2 a

b

Ey� xdx. (23)

Once Ey has been calculated as a function of B0, the above p.u.l.charge density can be related to the flux linkage to give the chargememductance, WQ � �/�, of the line.

However, the application of Faraday’s law to calculate Ey

leaves this quantity undetermined by certain additive constant.Nevertheless, this constant can be determined if it is consideredthat the line has to be bidirectional when 1 � f � 2 (that is, kz

�

� kz�). From Eqs. (20) and (8) this condition is satisfied by taking

WQ � WI � 0; that is, � in (23) must vanish for 1 � f � 2.

Figure 1 Quasi-TEM and full-wave dispersion curves of the normalizedphase constants of the forward and backward quasi-TEM modes (/k0)propagating in the asymmetric parallel-plate ferrite loaded transmissionline shown in the inset. Line parameters are: a � 0.1 mm, b � 1 mm, c �0.5 mm, 1/0 � 2.5, f/0 � 15, 2/0 � 45, 4�Ms � 1800 G, H0 �100 Oe

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 1, July 5 2003 5

Imposing this condition, Ey is determined, giving the followingresult for the charge memductance:

WQ ���

�

2a

d�b � c �b�c � a1 � a�c � bf � c�b � a2�,

(24)

where j� is the off-diagonal element �xy of the permeability tensorof the ferrite. Since a change of sign in the magnetizing field H0

results in a change of sign of � in Eq. (24), with � remaining thesame, then WQ(�H0) � �WQ(H0). Taking (8) into account, thislatter relationship implies that WI � �WQ.

Finally, the p.u.l. capacitance can be computed by means of thequasi-static formula

C �1

d �c � a1 � 2af � �b � a2�. (25)

In order to test the validity of the proposed analysis, Eqs. (22) and(24)–(25) are introduced in (20) to compute the dispersion curvesof this particular structure. These quasi-TEM results are shown inFigure 1 in comparison to the corresponding full-wave data (ob-tained by using the method proposed in [20]). A good agreementcan be observed between both approaches, with better agreementat low frequencies, as expected from the nature of the quasi-TEMapproximation. Table 1 shows results for the same structure at afixed frequency and varying the biasing magnetic field. Theseresults also confirm the soundness of the present approach (thedisagreement at 250 GHz can be explained because of the close-ness between the corresponding resonance and operation frequen-cies).

3.2. Quasi-TEM Modes in Multiconductor Planar TransmissionLines on Isotropic SubstratesOur theory will be now applied to the analysis of a uniform planarmulticonductor transmission line loaded with a bi-isotropic chiralmaterial characterized by the following constitutive relations:

D � 0rE � �0�0� � j�H (26)

B � �0�rH � �0�0� � j�E, (27)

which have been written in the Tellegen notation [15] with and� denoting the dimensionless Tellegen and chirality parameters,respectively. Quasi-TEM modes, characterized by Ez 0 andHz 0, are expected in the line at those frequencies where thetransverse dimensions of the line are much smaller than the free-space wavelength [3, 4]. Since a full-wave analysis of this type oflines is available, for instance, in [21], the present approach will bechecked by comparing the results of both analysis.

The quasi-TEM approximation imposes that Ez � 0 and Hz �0, Dz � 0 and Bz � 0 [3]. Introducing this fact into thelongitudinal part of Maxwell’s equations, both E and H fields canbe obtained from the scalar electric potential �, and the z-compo-nent of the vector potential Az. Both potentials satisfy the trans-verse Laplace’s equation subject to the appropriate boundary con-ditions at the interface:

E � �t�; �t2� � 0 (28)

B � �z � �tAz; �t2Az � 0, (29)

where �t is the transverse part of the � operator. The modalvoltages Vi and the p.u.l. flux linkages �i are determined from thevalues of � and Az, respectively, at the ith conductor. The linemodal currents Ii and the p.u.l. charges �i are given, respectively,by the circulation of H and by the p.u.l. flux integral of D aroundeach conductor (both contour integrals do not depend on theintegration path since �t � H � 0 and �t � D � 0). Once Vi, Ii,�i, and �i have been defined without ambiguity, the application ofthe present theory entirely lies on the appropriate computation ofthe line capacitance, inductance, and memductance matrices.

Looking for a case where the concept of memductance isrelevant, the quasi-TEM modes of the symmetric coupled micros-trip line printed on a bi-isotropic substrate [see inset of Fig. 2(b)]

TABLE 1 Normalized Phase Constants and Differential Phase Shift When Varying the Magnetizing Field H0 for the StructureShown in the Inset of Fig. 1. Line Parameters: a � 0.1 mm, b � 1 mm, c � 0.5 mm, �1/�0 � 2.5, �f/�0 � 15, �2/�0 � 45, 4�Ms �

1800 G, Freq � 2 GHz

H0(Oe)

Quasi-TEM Full-Wave

�/k0 �/k0 �/k0 �/k0 �/k0 �/k0

0 5.28 5.62 0.34 5.27 5.60 0.3350 5.18 5.58 0.40 5.15 5.56 0.41

100 5.02 5.51 0.49 4.98 5.49 0.51150 4.75 5.41 0.66 4.67 5.38 0.71200 4.14 5.17 1.03 3.96 5.14 1.18250 1.22 3.72 2.50 1.05 4.01 2.96

TABLE 2 Normalized Phase Constants of a Coupled Microstrip Line with h � 0.5 mm, w1 � w2 � 0.3 mm, s � 0.2 mm, �r � 10,� � 0

� � 0.3 � � 0.6 � � 1.0

q-even q-odd q-even q-odd q-even q-odd

Quasi-TEM 2.6307 2.3625 2.6142 2.3176 2.5684 2.21450.1 GHz 2.6307 2.3625 2.6142 2.3176 2.5684 2.21451.0 GHz 2.6322 2.3626 2.6157 2.3177 2.5697 2.214710 GHz 2.6875 2.3717 2.6693 2.3285 2.6208 2.227950 GHz 2.9413 2.5305 2.9650 2.5092 3.0138 2.4678

6 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 1, July 5 2003

will be now analyzed using the spectral domain Galerkin method.As a first step, this analysis requires the computation of theGreen’s function of the structure. In the present case and assumingthe following spatial dependence for the potentials: � � �0( x,y)exp(�jkzz) and Az � A0,z( x, y)exp(�jkzz), the spectralGreen’s function matrix, [G], is given by the following relation-ship in the spectral domain:

� �Az� � �G�kx� � � �s

Js,z�, (30)

where kx is the spectral variable and �( y, kx), Az( y, kx), �s( y,kx), and Jz,s( y, kx) are the Fourier transforms of the scalar electricpotential, the z-component of the magnetic vector potential, thesurface charge density and the surface current density at the stripsinterface, respectively. Once the spectral Green’s function is ob-tained (see Appendix 2), the line capacitance, inductance, andmemductance matrices are computed after solving the integralequation for the appropriate line excitations by means of theGalerkin method. Weighted Chebyshev polynomials are used asbasis functions for expanding the surface charge and current den-sities on the strips.

To check the validity of the proposed quasi-TEM analysis.Table 1 shows a comparison between the quasi-TEM and full-wave results for a chiral structure that fulfils the conditions for thevalidity of the quasi-TEM analysis at frequencies below 10 GHz.Specifically, Table 2 shows the values of the normalized (tofree-space wavenumber) phase constants for the quasi-odd (q-odd)and quasi-even (q-even) modes of a coupled microstrip line fordifferent values of the chirality parameter and frequencies. Aperfect agreement is found at low frequencies for both modes,holding this good agreement at frequencies up to 10 GHz for theq-odd mode (as expected due to the higher quasi-TEM nature ofthis mode). Clearly, important discrepancies appear at higher fre-quencies as the quasi-TEM approximation is not suitable.

Once the validity of the proposed theory has been shown, thequasi-TEM approach will be applied to a more practical structurein order to show its usefulness as a first approximation for thestudy of those bi-isotropic lines that can be made with the currenttechnology. Thus, Figure 2(a) and (b) show the normalized phaseand attenuation constants of the two fundamental modes of acoupled microstrip line printed on a bi-isotropic substrate com-posed of helices embedded in a lossy host medium. Its chiralityparameter shows a dispersive behavior here characterized follow-ing the model proposed by Bahr et al. [22]. It can be seen that ourtheory accounts satisfactorily for the dispersive and resonant be-havior of the propagation constants, showing a very good agree-ment at low frequencies.

Figure 2 Normalized (a) phase and (b) attenuation constants of a cou-pled microstrip line with h � 5 mm, w1 � w2 � 3 mm, s � 2 mm. Hostmedium: r � 2.95 � j0.07, �r � 1. Helices: three-turn helices of radius0.625 mm and 0.667-mm-pitch made with cooper-coated wire of diameter0.1524 mm and surface resistance Rs � 2.52 � 107�f ( f � frequencyin Hz). Solid lines: quasi-TEM data; dashed lines: full-wave data

Figure 3 (a) Real and (b) imaginary parts of the modal impedances ofthe coupled microstrip line studied in Fig. 2

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 1, July 5 2003 7

The quasi-TEM values for the real and imaginary parts of thefour modal impedances of the above structure are shown in Figure3(a) and (b). Although these results are only approximate at higherfrequencies, their values can be very helpful to understand thequalitative behavior of the structure.

4. CONCLUSIONS

A novel proposal has been developed to account for magnetoelec-tric couplings between the basic circuit magnitudes of multicon-ductor transmission lines. The magnetoelectric couplings are in-corporated by means of new line parameters with dimensions ofelectric charge divided by magnetic flux: the line memductances.These new line parameters can model the magnetoelectric cou-plings present in transmission lines loaded with gyrotropic mediaand/or bi(iso/aniso)tropic media. The noticeable physical effectscaused by these couplings (for example nonreciprocity, non-sym-metry, non-bidirectionality, and complex impedances in losslesslines) are properly dealt with by the proposed theory.

The application of the fundamental Onsager principle of sym-metry to magnetoelectric transmission line has yielded all thefundamental symmetries to be satisfied by the characteristic pa-rameters of any physical transmission line. The orthogonality andother fundamental properties of the line modes are also establishedand interpreted in relation to previous results involving moregeneral guiding structures. Two important particular cases (loss-less lines and two-conductors lines) have been analyzed in moredetail to highlight their main physical properties.

Our proposed quasi-TEM approximation has been applied tothe study of two particular examples of magnetoelectric transmis-sion lines: a parallel-plate ferrite phase shifter and a pair ofcoupled microstrip lines printed on a bi-isotropic substrate. Goodagreement was found when the quasi-TEM results were comparedwith full-wave data (within that frequency range where the quasi-TEM approximation is satisfied). As an additional result of theabove quasi-TEM analysis, analytical design formulas have beenprovided for the parallel-plate ferrite phase shifter.

APPENDIX 1: ONSAGER SYMMETRY RELATIONS

The relations (3) and (4) can be written in the following slightlydifferent form:

V � �a� � � � �b� � � (31)

I � �c� � � � �d� � �. (32)

This new form is easily found to be equivalent to Eqs. (3) and (4),provided that

�C� � �a��1 (33)

�WQ� � ��a��1�b� (34)

�WI� � �c��a��1 (35)

�L��1 � ��c��a��1�b� � �d�. (36)

As far as expression (5) for the energy is valid, the application ofthe Onsager symmetry principle for the kinetic coefficients (takinginto account the different symmetry transformation properties of Vand I by time reversal) leads to the following relationships for thegeneralized susceptances [a], [b], [c], and [d] [14]:

�a��; H0, �� � � �a��; �H0, ��� �T (37)

�d��; H0, �� � � �d��; �H0, ��� �T (38)

�b��; H0, �� � � ��c��; �H0, ��� �T. (39)

From these relationships and from Eqs. (34)–(36), Eqs. (6)–(8) arereadily obtained.

APPENDIX 2: QUASI-TEM GREEN’S DYAD FOR AGROUNDED BI-ISOTROPIC LAYER

For the bi-isotropic microstrip line under study, the spectralGreen’s function matrix of the present problem is found to be

�G�kx� �1

kx��g� �

� g�, (40)

where

g��kx �coth�kxh1

�0�r�

coth�kxh2

�0, (41)

g�kx � pcoth�kxh1 � 0coth�kxh2, (42)

� �� � j

�0�r, (43)

� �� � j

�0�r, (44)

��kx � g�g � ��, (45)

and

p � 0r ��2 � 2

�0�r. (46)

ACKNOWLEDGEMENTS

This work has been supported by the Spanish Comision Intermin-isterial de Ciencia y Tecnologıa (CICYT) and European UnionFEDER funds under project no. TIC2001-3163.

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© 2003 Wiley Periodicals, Inc.

A DUAL-FREQUENCY DIELECTRICRESONATOR ANTENNA

C. Nannini, J. M. Ribero, J. Y. Dauvignac, and Ch. PichotLaboratoire d’Electronique, Antennes et TelecommunicationsUniversite de Nice-Sophia AntipolisCNRS UMR 6071250 rue Albert Einstein06560 Valbonne, France

Received 19 December 2002

ABSTRACT: A dual-frequency cylindrical dielectric resonator antennais presented. This paper proposes a configuration in which the shapeand volume of the radiating structure remain unchanged. The behaviourof this antenna after altering some of its attributes is investigated.© 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 9–10,2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10956

Key words: dielectric resonator; antenna; bi-frequency

INTRODUCTION

Dielectric resonator antennas have received much attention fromnumerous researchers for their use as radiating elements in spatial,mobile, and wireless communication systems. Indeed, they presenta better alternative to microstrip patch antennas because they offergreater coupling ability for a commonly used feeding scheme andpotential to obtain different radiation patterns by using variousmodes. Furthermore, they present some advantages, such as rela-tively large bandwidth and compact size. Another attractive featureis their high radiation due to the fact of having no inherentconductor loss, since there is no metallic part.

Recently, substantial efforts have been devoted to DRA band-width enhancement, and many investigations have been reportedon multi-frequency operations using an arrangement of resonatorswith different shapes [1–4].

This paper presents a structure to improve some antenna char-acteristics while operating at the HEM11�. The proposed antennaexhibits dual-frequency behaviour which maintains the shape andvolume of the radiating structure.

ANTENNA STRUCTURE

The study of this antenna has been carried out with a high-frequency structure simulator (HFSS) which characterizes 3D de-signs, to determine the bandwidth and the radiation patterns of theantenna.

The survey consists in changing the permittivity in some partsof the structure, thus the design investigated is an arrangement in

Figure 1 Antenna configuration

Figure 2 Return loss

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 38, No. 1, July 5 2003 9