ANALYSIS OF THE COUPLING BETWEEN RECTANGULAR IVIICROSTRIP PATCHES ON UNIAXIAL ANISOTROPIC SUBSTRATES

The effect of the dielectric anisotropy in the behavior of the resonant frequencies of coupled microstrip patches is investigated. In the analysis, the Hertz vector potentials technique, in the spectral domain, is used to determine the field components expressions, while the moment ( Galerkin ) method is used to solve the matrix equation obtained by imposing the boundary conditions of the structure under consideration. The microstrip patches substrates are composed by two uniaxial anisotropic dielectric materials. The optical axes in these dielectric layers are oriented perpendicularly to the ground plane. Numerical results are presented for the resonant frequency for the geometry under consideration. Excellent agreement was observed with results available in the literature for coupled microstrip patches on isotropic substrates.


INTRODUCTION
In the last two decades, a growing interest in the analysis of microstrip resonators has been observed because of their application in the development of microstrip patch antennas [1].The main advantages of the microstrip resonators, compared to the conventional ones, are: low weight and volume, and because they are easy to build and mount on plane and curved surfaces [2]- [3].
The first reported works in the literature were dedicated to the analysis of microstrip patches and resonators on isotropic dielectric substrates.Nevertheless, because of the fact that most of dielectric materials used in microwaves are anisotropic ones, the effect of the dielectric anisotropy has been considered by several authors [2]- [6].
By the way, results for microstrip patches on planar structures with magnetized ferrimagnetic substrates are available in the literature [3], [7], [8].
Furthermore, the study of planar coupled resonators on anisotropic substrates has been considered by several authors for both suspended stripline [9] and microstrip [10] resonators.to determine the structure resonant frequency.
In this work.results for the resonant frequency of coupled microstrip patches,mounted over two different anisotropic dielectric layers are presented, as well as the effect of the dielectric anisotropy in the resonant structure performance ( Fig, I).
In the analysis for the resonant frequency of a single microstrip patch, the Hertz potentials theory, in the spectral domain, in combination with the moment method is used, For the case of coupled microstrip patches, the analysis is performed by considering the even and odd modes theory, taking advantage of the symmetry of the structure considered.

THEORY
The structure considered in this work is shown in Fig. 1.Regions I and 2 are anisotropic dielectric ones, being characterized by the electric permittivity tensor E ( j = 1, 2 ), Region 3 is air.The patches and the ground plane are j perfectly conducting, The conducting patch thickness is neglected.
In this analysis.the determination of the resonant frequency of a single microstrip patch over two anisotropic layers is required, In order to do that.the Hertz vector potentials are defined to be along the optical direction ( y in Fig.I ) according to Ill.' =Il(fiy Then. the wave equations for TIe and TI h are obtained from Maxwell's equations, in the spectral domain, for each dielectric region in Fig, 1, The Fourier transformation is defined by [2] tp(CX,~)=f= f= 'P('-_),)(cxx+~.:-),t -oc -oc ., ~ -t (/_\.(/~(4) 1 f~ f~ --)"(a.Y + r:t~) (5) where 'P(x, z) is a generic function.
The electric and magnetic fields components, for dielectric region j ( j = 1, 2, 3 ) are determined, in the spectral domain, from Maxwell's equations and are expressed by ani" ani" The solutions of the wave equations ( 6) and ( 7) are expressed, in a general form, as ii "=B(ex,~)cosh(rv)+l/(ex,~)senh(rl) respectively, for dielectric region j ( j = 1,2 ).In dielectric region 3, which is an open one, the expressions for TI e3 and TI The unknown coefficients A j , Bj, Ai' and B/ are determined by imposing the boundary conditions at the interfaces, for the electric and magnetic field components, which are: where the subscript "t" means tangential components and Ix and I z are the transformed current density components on the conducting patch.
After determining the expressions for A j ,B j ,A; and B;, in dielectric region j U=1,2,3), the field components, in the Fourier domain, are obtained by using (10) to (15).
At the conducting patch plane, y == d, + d 2 , ( Fig. 1 ), the transformed electric field components, Ex and E z' are expressed as function of the transformed current density components, I x and I ' as shown below Once the impedance matrix [Z] was obtained, the moment method (Galerkin ) is used to determine the complex resonant frequency for the microstrip patch [2], [3].
In the moment method, the current density components Ix and I z are expanded as a combination of basis functions, Ixm and I zm ' respectively, as [ j __ -I

_(j
Non-trivial solutions for (30) and (31) are obtained by imposing det[K] = 0, where the matrix [K] components are given by (32 to (35).The solutions for this equation are the resonant frequencies for the single microstrip patch.
To determine the resonant frequency for coupled microstrip patches (Fig. 1 ), the electric current density components are expressed as functions of those considered in the analysis of a single microstrip patch, according to [10] .Jcc(a, ~) = [~e-jU(S, +11) + "IU(I, +11) pc(a, ~) where ~ = 1 for the even mode and ~ =-1 for the odd mode.As it is shown in Fig. 2, for suspended coupled microstrip patches, the resonant frequencies for the even and odd modes decrease when the patch length, L, is increased.Note that an equalization is obtained at L = 6 mm, or nearby.The structural parameters considered in the determination of the resonant frequencies are shown in Fig. 2. regions 1 and 3 are air-filled, while region 2 is filled with pyrolytic boron nitride (p.b.n) [2], for which £xx2 = 5.12 and £yy2 = 3.4.

RESULTS
In Fig. 3, the numerical results obtained for suspended coupled microstrip patches on sapphire ( £xx2 = 9.4 e £yy2 = 11.6 ) are shown [2].Lower values for the resonant frequencies, for a given L, are obtained in this structure compared to the one using p.b.n (shown in Fig. 2 ), as expected.Note that there is no equalization between the even and odd mode results.Nevertheless, for low values of L, from 4.0 to 5.0 ern, the differences between the even and odd mode results are smaller.
Fig. 4 shows the effect of the dielectric anisotropy ( in region 1 ) on the resonant frequency for single and coupled microstrip patches on a double layer.Dielectric region 2 is alumina filled ( £r2 =9.6 ).The anisotropy ratio, nxlny (for region 1 in Fig. 1 ), is given by As it is shown in Fig. 4, as nxlny increases, the resonant frequency increases for both single and coupled microstrip patches.It was observed that the results for both even and odd mode resonant frequencies approach the results for the resonant frequency of a single microstrip patch, when Sx F r (GHz) 8 , -----, -------, --------, 7.5 ~ 00, as expected.
Fig. 6 shows the behavior of the resonant frequency as function of the normalized spacing, s/W, (around 100), which is F; = 5.61 + jO.OOll Ghz.Fig. 7 shows the resonant frequency versus the anisotropy ratio for coupled patchess on two A comparison between the results of different techniques is shown in Tables I to IV, for single microstrip patch on isotropic and anisotropic substrates, respectively.A good agreement is observed.

CONCLUSION
The analysis and numerical results for the resonant frequencies of parallel coupled microstrip patches over anisotropic substrates was presented.
:>; the determination of the resonant frequency for a single microstrip patch ( particular case ), the Hertz vector potential and the moment methods were used.A determinantal equation was obtained, which solutions are the complex resonant frequencies of the structure considered.
The resonant frequencies of parallel coupled microstrip patches was performed by using the even and odd mode theory.The electric current density components for these modes were expressed as functions of that used in the analysis of a sing lc patch.
A ~ooJ agreement between the results of this work and those available in the literature for the particular case of coupled patchc-on a single isotropic dielectric layer was observed.
Fig, I -Coupled patches.Since the optical axis of the anisotropic dielectric regions I and 2 are oriented along the y direction ( Fig, I ), their electric perrnittivities, E. (j = 1.2), are given by .I