**2. Ordinary SIW and SIW periodically loaded with cylindrical air holes or with metallic cylinders**

SIWs are planar structures that emulate dielectric-filled rectangular waveguides (see **Figure 1**). The two ground planes represent the top and bottom metal walls of the rectangular waveguide, and the rows of metal vias replace the sidewalls of the waveguide. The ordinary SIW made with a homogeneous substrate basically has the same guided-wave characteristics as the conventional rectangular waveguide [3], and its fundamental mode is similar to the TE10 mode of the rectangular waveguide. However, since the electric current density on the metal vias can only flow in the vertical direction, only TE*n*<sup>0</sup> modes are supported by SIWs [4]. The metallic vias are characterized by their separation *s* and diameter *d*. Their values must be appropriately chosen [5] to avoid radiation losses, so they must fulfill the following conditions:

$$d < \lambda\_{\mathfrak{g}}/5, \quad s \le 2d \tag{1}$$

where *λ<sup>g</sup>* is the guided wavelength. The propagation constant of the SIW fundamental mode is mainly determined by the width *aSIW* of the SIW (see **Figure 1**). A previous study [3] demonstrates that a SIW can be analyzed as an equivalent rectangular waveguide of effective width *a* given by

$$a = a\_{SW} - \frac{d^2}{0.95s}.\tag{2}$$

Therefore, all the presented results in this section have been obtained using the equivalent waveguide of width *a*, related to the cutoff frequency *f <sup>c</sup>*<sup>10</sup> of the TE10 mode and the relative permittivity of the substrate material by:

$$a = \frac{c}{2f\_{c10}\sqrt{\varepsilon\_r}}.\tag{3}$$

Going a step further, some propagation regions of the SIW can be conveniently modified so that it can behave as if it is loaded by a different dielectric permittivity in such regions with respect to that of its substrate, which may be of practical interest in *Novel Filtering Applications in Substrate-Integrated Waveguide Technology DOI: http://dx.doi.org/10.5772/intechopen.105481*

**Figure 2.**

*Scheme of a SIW with periodic air holes with a rectangular pattern.*

filtering applications, as shown in the following sections. A simple way of it can be achieved when the propagation region of the SIW is periodically loaded by air holes (see **Figure 2**), in which case a considerable reduction of the effective permittivity can be obtained (as long as the perforated substrate is shielded in the top and bottom walls, so the electric field distributes transversely through both the substrate and the air holes regions following the TE10 mode profile). Additionally, it is expected a decrease in dielectric losses due to the removal of substrate material, which may be of special interest in high-frequency bands. Effective permittivity of the periodically perforated SIW can be obtained to be used in filtering design by analyzing a unit cell of the perforated structure, by using the eigenmode solver of the commercial software tool Ansys HFSS [6], so that it is possible to relate the effective permittivity of the waveguide with the cutoff frequency of the TE10 mode through the following expression:

$$
\varepsilon\_r = \frac{c^2}{4a^2 f\_{c10}^2} \tag{4}
$$

A parametric study of the effective permittivity obtained in a SIW periodically loaded with cylindrical air holes following a rectangular pattern can be found in Ref. [7], where it has been analyzed the effect of the air holes parameters (the diameter *da* and separation *sa*) in the resultant effective relative permittivity of the waveguide, achieving a reduction of more than a 60% of the substrate relative permittivity.

Alternatively, the use of high effective permittivity structures, which behave as slow-wave structures, are also of special interest for device miniaturization in filtering applications. With this regard, the increase of the effective permittivity of a SIW can also be obtained by inserting in the dielectric an array of metallic inclusions, as already demonstrated in Refs. [8, 9]. A simple implementation of a SIW with a high effective permittivity can be achieved by inserting an array of metallic cylinders (see **Figure 3**), whose height must be lower than but not far from the waveguide height, to achieve a high effective permittivity in the waveguide. In Ref. [10], a parametric study of the effective permittivity in a SIW has been done, in which an array of metallic cylinders with a triangular pattern has been inserted to synthesize a higher effective permittivity, obtaining an effective permittivity that is more than twice the value of the substrate permittivity with the proper selection of the cylinder parameters (the diameter *dc* and separation *sc*).

**Figure 3.** *Scheme of a SIW with an array of periodic cylinders with a triangular pattern.*

Both types of periodically loaded SIWs with reduced or increased effective permittivity can be employed in novel filter solutions, as shown in the following section.
