## 1. Introduction

[2] The study of microstrip structures is a subject that is attracting much attention in microwave engineering, specially for their mechanical advantages, and for the integration capabilities with other radio frequency circuits offered by these configurations. For the analysis of microstrip structures, the integral equation (IE) technique in combination with the multilayered media Green's functions formulated both in the spectral [*Mesa and Marques*, 1995] and in the spatial [*Kinayman and Aksun*, 1997] domain, has grown in popularity. Using this approach, the information of the dielectric layers is included in the Green's functions, so that the numerical treatment of the problem is reduced to the metallic areas printed on the substrates, thus considerably increasing the efficiency of the technique. A big limitation of this approach, however, is that the dielectric layers are considered to be of infinite transverse dimensions [*Bunger and Arndt*, 1997]. In addition, the infinite size analysis can not extract other important information, such as the backside radiation of printed microstrip antennas [*Mosig and Gardiol*, 1982]. A third limitation of this technique is that, while the analysis of strictly planar geometries is relatively easy [*Mosig and Gardiol*, 1982], the same is generally not true when vertical or arbitrarily oriented metalizations are included inside the dielectric layers [*Michalski and Zheng*, 1990; *Kinayman and Aksun*, 1997; *Gay-Balmaz and Mosig*, 1997]. Also the practical circuits must be of finite size. In order to obtain accurate results with respect to the infinite model, the size of the substrates must contain several wavelengths of the circuit, in order to approach to the ideal infinite size situation. A drawback of this design procedure is that the size and volume of the total circuit can not be conveniently optimized, which might not be acceptable for many satellite and space applications.

[3] The techniques that are readily available for the analysis of full 3-D finite microstrip structures are the finite elements, finite differences, or the method of lines. These techniques, however, require very intensive numerical computations, specially because the whole volume around the structure must be discretized, often in combination of some sort of absorbing boundary condition [*Aksun and Dural*, 1996]. As regards the integral equation technique, some previous attempts for the analysis of finite microstrip structures have been reported, both using volume and surface equivalent formulations [*Sarkar and Arvas*, 1990; *Sarkar et al.*, 1990; *Chew et al.*, 2001]. Also, in the work of *Lu and Chew* [2000], metallic areas coated with dielectric regions were analyzed with a volume integral equation formulation, in the context of the calculation of the scattering properties of such objects. However, only very preliminary results of very simple microstrip antenna structures have been presented in the work of *Sarkar et al.* [1990]. Also, in the work of *Sarkar et al.* [1990], the radiation patterns obtained with volume and surface formulations appear not to agree well, thus indicating limited accuracy in the presented results. A different but very interesting approach for extracting the radiation patterns is addressed in the work of *Bokhari et al.* [1992], where the induced currents on the metallic areas are first calculated assuming an infinite dielectric slab. The finite dimensions are later taken into account during the computation of the far field, by considering only vertical polarization currents. In addition, there are not many precise input impedance results of finite size microstrip structures reported in the literature [*Tavakkol-Hamedani et al.*, 2002; *Lu and Yu*, 2002].

[4] In this contribution we present two efficient numerical implementations of the volume equivalent integral equation technique applied to the analysis of finite microstrip structures. The technique combines rooftop basis and testing functions for the conductor areas, with two different choices for the finite dielectric objects. The first is based on combining pulse basis functions and deltas for testing, in a similar way as presented in the work of *Sarkar et al.* [1990]. Unlike the technique in the work of *Sarkar et al.* [1990], however, the use of rooftop basis and testing functions for the conductor areas increases the numerical stability of the method, and accurate results are shown to be obtained, both for the input impedance and for the radiation patterns of microstrip antennas. The second choice to treat the finite dielectric objects is a full Galerkin approach, which employs rooftop basis and testing functions defined in the volume of the brick cells used in the discretization of the dielectrics. This type of basis functions is similar to the one introduced in the work of *Catedra et al.* [1989] in the context of the radar cross-section computation of pure dielectric objects. In that work, however, one dimensional pulses were used for testing, and uniform cells had to be used to employ the conjugate gradient-fast Fourier transform method. The advantage of using rooftop basis functions is that the derivatives on the scalar potential Green's functions are avoided, therefore reducing considerably the level of singularity in the subsequent numerical treatment.

[5] In addition, a novel excitation model, which can be used with finite size ground planes, has also been derived. The advantage of this model is that the ground plane can be meshed independently from the input feeding line. Also, a novel numerical procedure has been developed to allow for the independent meshing of the dielectric bodies with respect to touching metallic areas of the circuit. All these novel numerical techniques add flexibility to the volume equivalent IE method. Theoretical results are compared with measurements indicating that the new numerical models are indeed accurate, and can be used for the analysis of real life microstrip antennas and printed circuits.