A Volume-Surface Integral Equation Solver for Radiation from Microstrip Antenna on Anisotropic Substrate

A volume-surface integral equation (VSIE) solver is presented for the calculation of electromagnetic radiation from arbitrary shaped microstrip antenna on anisotropic substrate. The method of moments (MoM) is used to convert the integral equation into a matrix equation, where the equivalent volume current and surface current are expanded into a finite series of SWG and RWG basis function, respectively. A simple strip model is incorporated in the VSIE to simplify the analysis of the probe-fed microstrip antenna. The present approach is sufficiently versatile in handling microstrip antenna with arbitrary shaped anisotropic dielectric substrate. Numerical results indicate the reliability and accuracy of the proposed method.


Introduction
During recent years, great interests have been shown in using microstrip antenna deposited on anisotropic substrate since the substrate anisotropy could have important applications on the operation of microstrip antennas [1][2][3][4].With the increasing complexity of geometry and material property, designing these antennas requires more and more dedicated and sophisticated computer-aided-design (CAD) tools to predict the characteristics.The method of moments (MoM) has been proven to be one of the most powerful CAD tools for solving this class of problems.By now, a number of microstrip antennas with anisotropic substrate have been investigated using the MoM-based spectral domain analysis method [1][2][3].However, in the MoM-based approaches for the analysis of microstrip antenna, the volume-surface integral equation (VSIE) [5][6][7][8][9] solver is more suitable to analyze complex structure targets and has a number of advantages over the other MoM-based approaches [2,10], such as the applicability to various inhomogeneous materials, the same simple form regardless of the complexity of the materials and no special treatment for problems with junctions.It should be noted that the VSIE solver has been applied to calculate the electromagnetic scattering by the composite anisotropic material and metal targets in previous work [11,12].
In this paper, the VSIE solver is extended and applied for the computation of electromagnetic radiation from the microstrip antenna on an anisotropic substrate.A simple strip model is incorporated in the VSIE to simplify the analysis of the probe-fed microstrip antenna.Radiation properties of both the plane and the cylindrical conformal microstrip antenna with uniaxial substrate are calculated using the VSIE solver.

Theory
The microstrip antenna is discretized into triangular patches for metallic surfaces and tetrahedral cells for dielectric volume, respectively.The unknown surface currents on the metallic surface and the volume currents in the anisotropic dielectric substrate can be obtained by solving the following coupled volume-surface integral equation [12]: where S and V denote the metallic surface and the dielectric body, respectively.D v (r) is the electric flux density in V , while E i (r) and E s (r) are the electric field due to the applied source and the scattered field from the induced currents on S and in V , respectively.The metallic surface and dielectric body are meshed into triangular patches and tetrahedral cells, and then RWG and SWG basis and test functions are applied to expand the unknown surface current J s (r) and International Journal of Antennas and Propagation electric flux density D v (r), respectively.ε r is the permittivity tensor of the dielectric material and given as ( The volume current J v is related to the total electric flux density D v (r) by where κ(r) is the contrast ratio tensor defined as The hybrid integral equation ( 1), togther, together with ( 2)-( 4), can be solved by using Galerkin's MoM, and then RWG and SWG basis and test functions are applied to construct the MoM matrix, respectively.The detail derivation of the MoM matrix equation can be found in [11].
A simple equivalent strip model [5] is applied in the treatment of the feeding probe, where the wire-surface junction [13,14], as shown in Figure 1(a), is changed to a surface-surface junction, as shown in Figure 1(b).The feeding probe of diameter d is replaced by its equivalent strip structure with the width w = 2d, as shown in Figure 1.The excitation source locates at the junction, and the E i in equation can be established by using the delta-gap voltage model: where V is the voltage across the gap and n l is normal to the junction l n .

Numerical Results
In this section, two numerical results will be shown.For the first example, We consider a planar microstrip antenna, as shown in Figure 2. The geometric parameters are taken from [4] and are as follows: the dimension of the patch is a × b = 12.45 mm × 16 mm, the substrate of the microstrip antenna is 28.1 mm × 32 mm × 0.794 mm, and the permittivity tensor is chosen as ε r = [2.310 0,0 2.31 0, 0 0 2.19].(Refer to [4], in Figure 5, when θ = 0.) The planar microstrip antenna is discretized into 1384 tetrahedral cells and 568 triangular patches, respectively, yielding a total of 4091 unknowns.
Figure 3 shows the S-parameter of the microstrip patch antenna solved by the presented method and compared with results in [4].It is observed that good agreements are observed between the two methods.Moreover, the E-plane radiation pattern of antenna at the frequency of 7.5 GHz is obtained from the MoM and compared with the fine finite-element method-(FEM)-based solution, as shown in Figure 4. Again, good agreements are exhibited.In addition, the CPU time for both the MoM and the FEM-based solution is tested using the same computer, and the test results are 757 seconds and 771 seconds, respectively.
The second example considers a cylindrical conformal microstrip antenna and discusses the effects of the uniaxial substrate on the input impedance.The geometric parameters are taken from [5], as shown in Figure 5.The cylindricalrectangular patch has a dimension of L × W = 30 mm × 40 mm with a curvature radius of 50.795 mm, and the cylindrical-rectangular substrate is L g × W g × h = 50 mm × 60 mm × 0.795 mm.The feeding point is on the centerline of the curved wide side and is 10 mm from the center of the patch, where the size of the rectangular strip feedline is 2 mm×0.795mm.The curved microstrip antenna is modeled with 600 triangular patches and 1340 tetrahedral cells, resulting in 4035 unknowns.The substrate is chosen the uniaxial anisotropic medium and the diagonalized permittivity tensor is taken in the form ε r = [ε 1 0 0, 0 ε 1 0, 0 0 ε 2 ].
Figure 6 shows the input impedance of the antenna versus different permittivity, where ε 1 is 2.32, and ε 2 is chosen as 1.8, 2.32, 2.8, respectively.It can be seen that the permittivity ε 2 has a significant influence on the impedance versus frequency variation, which implies that increasing the ε 2 decreases the resonant frequency of the antenna.

Conclusion
The VSIE solver has been applied to analyze the electromagnetic radiation from the microstrip antenna with arbitrary shaped anisotropic substrate.The targets are modeled using tetrahedral volume elements for substrate and triangle face elements for metal patches.The coupled volume-surface integral equations are derived by introducing tensor permittivity in conventional MoM.A simple strip model is used to simplify the analysis of the probe-fed microstrip antenna.Compared with the conventional MoMbased spectral domain analysis method, the VSIE solver approach offers good flexibility to model arbitrarily shaped microstrip antenna structures while keeping a good accuracy.The presented approach is formulated using the free-space Green's function.This feature makes it easy to apply the MoM-based fast algorithms to reduce the computational complexity of microstrip antenna with arbitrary shaped anisotropic substrate.

Figure 1 :Figure 2 :Figure 3 :
Figure 1: A strip model in the excitation source of antenna.(a) Wire-surface junction.(b) Surface-surface junction model.