J Comput Electron DOI 10.1007/s10825-017-1048-x Spectral domain analysis of rectangular stacked patches printed on a substrate characterized by dielectric and magnetic anisotropy Youssouf Bentrcia 1 · Sarah Bouttout 1 · Siham Benkouda 2 · Tarek Fortaki 1 © Springer Science+Business Media, LLC 2017 Abstract Spectral domain formulation is provided for the analysis of rectangular stacked patches printed on a substrate characterized by dielectric and magnetic uniaxial anisotropy. Detailed analytical expressions of the dyadic Green’s func- tions are derived. Galerkin’s procedure is applied to solve the electric field integral equations, and the resonance character- istics are determined by solving the characteristic equation. Numerical results show that the influence of the magnetic anisotropy on the resonant frequencies is highly dependent on the permeability of the medium, where for a non-magnetic medium, the impact of the existing anisotropy was found negligible. However, for a magnetic medium, the anisotropy has a large impact on the resonant frequencies. Moreover, the influence of each of the components of the permeability tensor has been also reported. Keywords Spectral domain analysis · Stacked microstrip patches · Anisotropic materials 1 Introduction Anisotropic materials have been considered in the analy- sis and the design of many microwave devices, where the anisotropy may occur naturally or it may be deliberately implanted during the fabrication process [1, 2]. The analy- sis of microstrip patch antennas with uniaxial anisotropic substrates has been reported for a single patch [1, 3–6] and B Youssouf Bentrcia anoirnaser@gmail.com 1 Electronics Department, University of Batna 2, 05000 Batna, Algeria 2 Electronics Department, University of Constantine 1, 25000 Constantine, Algeria for stacked patches [7, 8] configurations. Most of the pub- lished works in this regard, except those reported in [8–11], were directed toward non-magnetic substrates; thus, the per- meability and the magnetic anisotropy consequently were not considered in the analysis process. Rectangular stacked patches antenna characterized by both permittivity and per- meability tensors has not been treated in any of the previously published works. This paper provides a complete formulation of such a structure. The treatment this subject is organized as follows: First, the general process allowing the derivation of the analytical expressions of the dyadic Green’s functions and the formulation of the integral equations of the structure is provided. The Galerkin’s procedure is used to transform the integral equations into a matrix equation allowing the computation of the resonant frequencies and the determina- tion of the current densities on the two patches. Second, the effect of the dielectric and the magnetic anisotropy has been investigated; moreover, the influence of the anisotropy ratio and the permeability tensor’s components is also presented. And finally, some important notes and results are listed. 2 Theory The microstrip antenna structure under study illustrated in Fig. 1 consists of stacked rectangular patches printed on two- layer uniaxial anisotropic substrate with permittivity ¯ ¯ ε and permeability ¯ ¯ μ tensors having the form ¯ ¯ ε = ε 0 ⎡ ⎣ ε x 0 0 0 ε x 0 0 0 ε z ⎤ ⎦ (1) 123