## 1. Introduction

[2] The use of microwave ferrite materials is well known to provide the nonreciprocal characteristics required in some microwave devices as well as tuning capabilities through the application of an external magnetic field [*Baden Fuller*, 1987; *Schuster and Luebbers*, 1996; *Xie and Davis*, 2001]. The inclusion of ferrite layers in planar transmission lines, planar circuits and planar antennas has been object of attention by a number of researchers [*Pozar and Sanchez*, 1988; *Pozar*, 1992; *Yang*, 1994; *Fukusako and Tsutsumi*, 1997; *Tsang and Langley*, 1998; *Oates and Dionne*, 1999; *How et al.*, 2000; *Nurgaliev et al.*, 2001; *León et al.*, 2001, 2002]. Unfortunately, most of the common computer tools currently employed for the analysis and design of planar printed circuits and antennas cannot be applied to structures whose layered substrate includes nonisotropic materials. Nevertheless, a spectral domain implementation of the electric field integral equation (EFIE) [see, e.g., *Pozar*, 1992; *León et al.*, 2002] is available to deal with planar structures loaded with ferrite layers. Indeed, the inclusion of nonisotropic layers is relatively straightforward in the spectral domain frame since spectral domain Green's functions have been developed for general linear media, including ferrites. However, a clear disadvantage of the spectral domain approach lies on its inability to handle efficiently with nonrectangular shape conductors. This limitation can be very important in practice and strongly reduces the versatility of the numerical tools based on that approach.

[3] The incorporation of nonrectangular shaped conductors requires to use space domain formulations, which are suitable for using basis functions that can match any geometry. Thus a possible solution of the aforementioned problem could be the implementation of the corresponding EFIE in the space domain after performing the necessary inverse Fourier transformations to obtain the space domain counterpart of the spectral domain Green's dyadic. However, the space domain Green's dyadic required to solve the EFIE (for both isotropic and/or anisotropic structures) presents hypersingularities [*Bressan and Conciauro*, 1985; *Tai*, 1971], which are further transferred to the reaction integrals appearing after application of the method of moments (MOM) to solve the integral equation [*Arcioni et al.*, 1997]. The presence of these hypersingularities in the reaction integrals clearly degrades the numerical performance of the method and makes it necessary a lot of previous analytic preprocessing. This preprocessing has been already carried out in the case of using only isotropic and/or some kind of nonisotropic substrates (for example, uniaxial dielectrics). In such situations the above hypersingularities have been conveniently treated by the authors, thus making the space domain EFIE as competitive numerical tool as the alternative mixed potential integral equation (MPIE) in those circumstances [*Plaza et al.*, 2002; *Mesa and Medina*, 2002]. Unfortunately, the techniques reported by *Plaza et al.* [2002] and *Mesa and Medina* [2002] cannot be easily extended to deal with more general types of anisotropy. In particular, it has been the considerable difficulty to find a closed-form expression for the quasi-static part of the spectral domain Green's dyadic in the case of general anisotropy what has precluded the obtaining of explicit and closed-form expressions for the hypersingular terms of the corresponding EFIE space domain Green's dyadic [*Plaza et al.*, 2002].

[4] Nevertheless, there is still another possibility. Indeed, a convenient solution to the problem under discussion would be the implementation in the space domain of a MPIE (which is free of hypersingularities) that could also deal with complex nonisotropic layers. This purpose seems to be feasible, at least for planar structures whose layered substrate presents any type of magnetic anisotropy, once a numerical method to compute the required space domain Green's functions associated with the MPIE has been reported [*Mesa and Medina*, 2004]. Thus, starting from the Green's functions reported by *Mesa and Medina* [2004], the present paper will extend the work of *Mesa and Medina* [2005] presenting the details of the explicit implementation and numerical solution of the MPIE for planar structures having metallizations of arbitrary shape and layers of isotropic/uniaxially anisotropic dielectrics and/or ferrites magnetized by an external biasing field arbitrarily oriented. The power of the method is illustrated by means of the simulation of planar filters printed on magnetized ferrite substrates.