## 1. Introduction

[2] In the nineties, postwall waveguides (PWWGs) were proposed as an alternative to classical transmission lines at microwave frequencies [*Furuyama*, 1994; *Hirokawa and Ando*, 1998; *Uchimura et al.*, 1998; *Deslandes and Wu*, 2001]. Such waveguides are generally embedded in substrates, e.g., printed circuit boards (PCBs), and belong to the broader class of substrate-integrated waveguides (SIWs). The side walls of a PWWG are composed of cylindrical posts, which together with optional metallic top and bottom plates enclose a rectangular region similar to the rectangular (metallic) waveguide. Consistent with classical planar transmission-line structures, PWWGs are relatively easy and inexpensive to manufacture, offer integration with other circuits of a microwave front-end, and are less bulky than classical (metallic) waveguides. Furthermore, thanks to their waveguide-like characteristics PWWGs have in general lower losses than other PCB transmission-line structures, in particular above 30 GHz [*Coenen*, 2010, section 1.2]. For these reasons PWWGs have attracted substantial interest during the last decade.

[3] In our paper we focus particularly on the electromagnetic analysis of PWWGs. Not surprisingly the initial analysis methods found their origin in the optical domain with dielectric-post PWWGs evolving from photonic bandgap materials [*Yablonovitch*, 1987] and where one utilizes the Bloch-Floquet theorem and the plane wave expansion method to study periodic guiding structures [*Benisty*, 1996]. Such an approach is adopted in early PWWG work [*Hirokawa and Ando*, 1998] and by *Pissoort and Olyslager* [2004]. In the microwave community, however, lumped-element circuit models are most often used in the design of microwave components. Therefore, efforts were undertaken to identify suitable layouts for a circuit model of PWWGs [*Bray and Roy*, 2003; *Bozzi et al.*, 2003, 2006]. Parallel to this development, early PWWG studies stressed the determination of transmission characteristics, in particular the propagation constant, the loss factor, and the equivalent waveguide width [*Coenen*, 2010, chapter 3]. Beside the periodic-structure approach, several full-wave approaches were employed [*Cassivi et al.*, 2002; *Xu et al.*, 2003; *Deslandes and Wu*, 2006; *Bozzi et al.*, 2008] as well as discretization and averaging techniques applied to the permittivity and conductivity [*Ranjkesh et al.*, 2005; *Patrovsky and Wu*, 2005; *Cassivi and Wu*, 2005].

[4] More recently, the development of PWWG analysis techniques has more and more conformed to the practice of microwave component design, in which large structures are decomposed into smaller building blocks whose separately computed scattering parameters at predefined ports are cascaded to obtain the overall performance. The BI-RME method [*Cassivi et al.*, 2002; *Bozzi et al.*, 2008] offers such a procedure based on Generalized Admittance Matrices that relate modal currents and voltages at the PWWG ports. Outside these ports, the structure is laterally enclosed by fictitious metal walls, thus assuming negligible radiation loss. The approach described by *Arnieri and Amendola* [2008] and *Arnieri et al.* [2009] also considers the responses at designated ports, which include coaxial ports, waveguide ports, and slots. Instead of enclosing the PWWG by fictitious walls, the admittance matrix corresponding to equivalent magnetic current sources is calculated by expanding the Green's function of the parallel-plate waveguide in cylindrical waves and by adding separately the contribution of the posts to the corresponding integral equation. Finally using the approach described by *Van de Water et al.* [2005] the equivalent sources describing a specific building block are transferred to its entire boundary by linear embedding of the related Green's operator. The blocks are cascaded in a 2-D or 3-D lattice to obtain the field distribution of the overall structure.

[5] Following *Coenen* [2010] and *Coenen et al.* [2010], we present in this paper a model for PWWG building blocks that is based on the same principles as those of *Van de Water et al.* [2005], in particular Lorentz's reciprocity theorem. However we characterize the PWWG building block entirely in terms of electric and magnetic input and output surface currents at predefined port interfaces, as in the BI-RME and *Arnieri and Amendola* [2008]. Hereby we wish to facilitate the future integration of PWWG components in an existing circuit simulator. In section 2 we describe the details of our model, in particular how we relate the input and output currents using a current matrix, and in section 3 we derive expressions for the scattering parameters of the dominant TE_{10} mode at the ports. To validate our model, we compare in section 4 the propagation characteristics obtained by our model with those obtained by a commercial tool (Ansoft HFSS, version 11.1) and those obtained from measurements of uniform PWWGs of both metallic and dielectric posts. Finally, in section 5, we present the conclusions of our work.