## 1. Introduction

[2] In multilayer microwave integrated circuits such as low-temperature cofired ceramics (LTCCs) or multilayer printed circuit boards (PCBs); waveguide-like structures can be fabricated in planar form by using periodic metallic via holes called substrate-integrated waveguides (SIW) [*Deslandes and Wu*, 2001a; *Hirokawa and Ando*, 1998]. The SIW structures largely preserve the well-known advantages of conventional rectangular waveguides, viz., high Q and high power capacity, and include the advantages of microstrip lines, such as low profile, small volume and light weight etc. The SIW structure is convenient for the design of millimeter-wave circuits such as filters, resonators, and antennas etc., [*Zhang et al.*, 2005; *Deslandes and Wu*, 2001b, 2003; *D'Orazio et al.*, 2004; *Cassivi and Wu*, 2003; *Cassivi et al.*, 2002a].

[3] In addition, the SIW structure can easily be connected to microstrip or coplanar circuit using simple transitions [*Zhang et al.*, 2005; *Deslandes and Wu*, 2001a, 2001b, 2003; *D'Orazio et al.*, 2004; *Cassivi and Wu*, 2003; *Cassivi et al.*, 2002a], which may lead to the design and development of compact low-loss millimeter-wave integrated circuits and systems. Such developments should enhance manufacturing repeatability, reliability, and cost reduction significantly especially with the advent of LTCC and multilayer printed circuit board. For this reason, it is important to understand and analyze with simplicity the loss characteristics of SIW structures. It has been found that the substrate-integrated waveguide (SIW) has nearly the same propagation and cutoff characteristics with the conventional rectangular waveguide. In fact, the SIW can be considered a rectangular waveguide structure with an equivalent width [*Xu and Wu*, 2005; *Che et al.*, 2006; *Cassivi et al.*, 2002b]. Because of the periodic cylinders forming the sidewalls of the SIW, a SIW structure is subject to possible loss of leakage and ohmic attenuation. Studies of such losses have been carried out numerically or modally in several references [*Xu and Wu*, 2004, 2005; *Xu et al.*, 2003]. In this paper, for clearer physical insights, the losses of the SIW through the cylinder walls are studied analytically.

[4] Section 2 derives the formula of leakage loss from the surface impedance at the sidewalls of SIW, through an understanding of the physical significance of the self-term in the MoM (method of moments) matrix of *Harrington* [1993]. The approach may be called the “analytical MoM” and has been used to find formulas with good accuracy in capacitance from a finite and perforated ground plane [*Chow et al.*, 2002] and its extension, the accurate formula of the equivalent width on SIW [*Che et al.*, 2005, 2007], corresponding to the numerical and modal solutions mentioned above [*Xu and Wu*, 2004, 2005; *Xu et al.*, 2003].

[5] Section 3 converts the formula of ohmic loss of a regular waveguide of solid sidewalls to that of an SIW of cylinder walls through the ratio of their surface areas. Section 4 compares the loss formulas of the SIW prototype with the numerical results from HFSS, and from, experimental measurements. Good agreements between all three are observed.