This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS 1 Efficient Analysis of Multiple Microstrip Transmission Lines With Anisotropic Substrates Veenu Kamra and Achim Dreher , Senior Member, IEEE Abstract—Our aim with this letter is to exhibit the exten- sion of discrete mode-matching method to analyze multilayered microstrip transmission lines with anisotropic dielectric layers. The mathematical formulation is presented to deal with mul- tilayered structures with metallizations in the interfaces. The application is demonstrated by computing propagation constant and characteristic impedance for multilayered microstrip and two-layer coplanar waveguide with uniaxial anisotropic dielectric. The validation of the results has been done by comparing with simulations with ANSYS HFSS and some of them also with open literature. A very good agreement has been observed. Index Terms— Anisotropic media, discrete mode- matching (DMM), multilayered microwave structures, numerical procedures. I. I NTRODUCTION M ICROSTRIP transmission lines are widely used in the navigation and communication systems. Substrates with uniaxial media are often integrated for various microwave and optical applications. The anisotropy influences the dispersion characteristics of the line structures. Thus, it is important to characterize them with anisotropic substrate accurately. Several numerical procedures to predict the characteristics of the microtrip lines have already been described, such as potential theory in [1] and [2] and the calculation of equiv- alent isotropic substrate in [3]. Commercial softwares using numerical techniques like finite difference time domain and finite element method are also available to analyze microwave structures. But they are very time-consuming and need lots of storage when dealing with multilayers. Here, an efficient full-wave analysis method, i.e., the dis- crete mode-matching (DMM), is used for analyzing the struc- tures. The advantages and validation of the DMM theory have already been presented in [4]. It uses exact eigenvalues of the waveguide modes. It is considered that the transmission lines are infinite in propagation direction, therefore, it uses sampling of the wave equation in one dimension and analytical solution in the remaining direction perpendicular to the layers. For this reason, only two lateral boundaries are required. It has been successfully applied to the analysis of planar, quasi-planar, and cylindrical multilayered structures [4]–[6]. In this letter, we extend the DMM method to analyze mul- tilayered striplines on planar dielectric layers having uniaxial Manuscript received April 26, 2018; accepted June 10, 2018. This work was supported by a DLR/DAAD Research Fellowship. (Corresponding author: Veenu Kamra.) The authors are with the German Aerospace Center, Institute of Communications and Navigation, D-82234 Wessling, Germany (e-mail: veenu.kamra@dlr.de; achim.dreher@dlr.de). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LMWC.2018.2847032 Fig. 1. Discretized multilayered microwave structure with metallization in the interfaces. anisotropy. The full-wave equivalent circuit (FWEC) [7] is used to characterize the structure with anisotropic dielec- tric layers and to determine the system equation in spectral domain. II. ANALYSIS The general cross-sectional view of the multilayered microwave structure is depicted in Fig. 1. Here, we consider several microstrip lines in the interfaces of the anisotropic dielectric layers (optical axis in the z -direction). For the present analysis, the structure is stratified in the z -direction and the wave propagation [exp(- jk y y )] is assumed in the y - direction. The analysis begins with Helmholtz’s wave equa- tion, normalized by the free space wavenumber k 0 , to obtain the full-wave solution  ∂ 2 ∂ x 2 + ∂ ∂ z 2 + ε d  ψ k = 0, ε d = K 2 - k 2 y (1) where ψ represents two independent field components E z or H z and K is the propagation constant in any arbitrary layer k . The remaining field components related to these can be calculated as  ∂ 2 ∂ z 2 + ε xx μ xx  ⎡ ⎢ ⎣ E x H x E y H y ⎤ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂ ∂ x ∂ ∂ z - j μ xx ∂ ∂ y j ε xx ∂ ∂ y ∂ ∂ x ∂ ∂ z ∂ ∂ y ∂ ∂ z j μ xx ∂ ∂ x - j ε xx ∂ ∂ x ∂ ∂ y ∂ ∂ z ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  E z H z  . (2) The structure is assumed to be infinite in the propagation direction, so just 1-D discretization along x is used for the analysis of multilayered microstrip lines as shown in Fig. 1. 1531-1309 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.