**3. Band-pass step impedance filters in SIW technology**

In this section, the design procedure of novel band-pass filters in SIW technology using the impedance inverter model has been described [11], where the quarter-wave sections constituting the resonators are coupled through evanescent mode sections, which can be implemented by classical iris waveguides, or alternatively, by reduced permittivity SIW sections.

#### **3.1 Band-pass iris filter in SIW technology**

We begin this section by giving the basic guidelines for the practical design of a waveguide iris filter in SIW technology using the well-known impedance inverter model [11], which will be used as the basis for more sophisticated stepped-impedance configurations of SIW filters (with high and low dielectric constant sections). The filter design is made using the equivalent rectangular waveguide of effective width *a* given in Eq. (2), and the final design of the equivalent filter in SIW technology is accomplished using such equivalence in each of the respective waveguide sections. For the design of the waveguide-based filter, the equivalent circuit model of impedance inverters of an inductive waveguide iris through a *T* network can be employed [11], as can be seen in **Figure 4(a)**. The filter consists of half-wave resonators separated by the inductive iris. Using an electromagnetic simulator, the iris scattering matrix can be obtained and therefore its equivalent *T* network. Each iris is represented by two series reactances denoted by *Xs* and a shunt reactance denoted by *Xp*. The equivalent circuital rectangular iris filter is shown in **Figure 4(a)**. To transform it into the impedance inverter model, we use the impedance inverter circuit consisting of an inductive *T* network and two sections of length *φ=*2 on each side. The inverter is created by adding a length *φ=*2 and �*φ=*2 on each side of the discontinuity, as shown in **Figure 4(b)**. In this case, the resonators are transmission lines of length *Ln* connected to two transmission lines of artificial lengths �*φn=*2 and �*φ<sup>n</sup>*þ<sup>1</sup>*=*2. These

*Novel Filtering Applications in Substrate-Integrated Waveguide Technology DOI: http://dx.doi.org/10.5772/intechopen.105481*

**Figure 4.**

*(a) Equivalent circuit model of an inductive waveguide iris through a* T *network. (b) Equivalent impedance inverters model of an inductive waveguide iris through a* T *network.*

lengths represent the load of the resonator from the adjacent coupling inverters (**Figure 4(b)**).

As an example, we show the design of a five-pole Chebyshev filter, as illustrated in **Figure 5**, consisting of several sections of rectangular waveguide coupled with an inductive iris. The filter is designed with a center frequency *f* <sup>0</sup> ¼ 4 GHz, a bandwidth of 600 MHz, and return loss RL = 15 dB. The rectangular waveguide dimensions are *a* ¼ 15*:*8 mm and *b* = 0.63 mm, *b* equal to the thickness *h* of the employed substrate. We have used a Taconic CER-10 substrate with *ε<sup>r</sup>* ¼ 10 and tan ð Þ¼ *δ* 0*:*0035. Irises in all cases have a thickness *t* = 3 mm.

The filter center frequency *f* <sup>0</sup> and bandwidth BW are expressed by:

$$f\_0 = \sqrt{f\_1 f\_2}, \quad \text{BW} = f\_1 - f\_2,\tag{5}$$

which give *f* <sup>1</sup> ¼ 3.7 GHz, *f* <sup>2</sup> ¼ 4.3 GHz, and the filter relative bandwidth is:

$$
\Delta = \frac{\lambda\_{\rm g1} - \lambda\_{\rm g2}}{\lambda\_{\rm g0}} = 0.3636\tag{6}
$$

**Figure 5.** *Fifth order classical iris-type rectangular waveguide filter.* Then, the values obtained for the impedance inverter factors are:

$$\frac{K\_{01}}{Z\_0} = \frac{K\_{56}}{Z\_0} = \sqrt{\frac{\pi \Delta}{2 \text{g}\_0 \text{g}\_1}} = 0.7051\tag{7}$$

$$\frac{K\_{12}}{Z\_0} = \frac{K\_{45}}{Z\_0} = \frac{\pi \Delta}{2\sqrt{\mathfrak{g}\_1 \mathfrak{g}\_2}} = 0.45546\tag{8}$$

$$\frac{K\_{23}}{Z\_0} = \frac{K\_{34}}{Z\_0} = \frac{\pi \Delta}{2\sqrt{\mathcal{g}\_2 \mathcal{g}\_3}} = 0.34706\tag{9}$$

Using an electromagnetic simulator, the scattering parameters of a rectangular iris (referred to the discontinuity planes) can be obtained, which are related to the *T* network elements shown in **Figure 4(a)**, *Xs* and *Xp*, by the following Eqs. (10) and (11):

$$j\frac{X\_s}{Z\_0} = \frac{\mathbf{1} - \mathbf{S}\_{12} + \mathbf{S}\_{11}}{\mathbf{1} - \mathbf{S}\_{11} + \mathbf{S}\_{12}}\tag{10}$$

$$j\frac{X\_p}{Z\_0} = \frac{\mathfrak{L}\mathfrak{S}\_{12}}{\left(\mathfrak{1} - \mathfrak{S}\_{11}\right)^2 - \mathfrak{S}\_{12}^2} \tag{11}$$

where *S*11, *S*21, and *S*<sup>12</sup> are the scattering parameters of the *TE*<sup>10</sup> fundamental mode of the input waveguide at the filter center frequency *f* <sup>0</sup>. For the impedance inverter shown in **Figure 4(b)**, *Xs* and *Xp* are related to *K=Z*<sup>0</sup> and *φ* by:

$$\frac{K}{Z\_0} = \left| \tan \left( \frac{\rho}{2} \text{atan} \frac{X\_s}{Z\_0} \right) \right| \tag{12}$$

$$\varphi = -\text{atan}\left(2\frac{X\_p}{Z\_0} + \frac{X\_s}{Z\_0}\right) - \text{atan}\frac{X\_s}{Z\_0} \tag{13}$$

The scattering parameters of several iris of different widths have been obtained, so the values of the iris widths for this filter are *W*<sup>1</sup> = *W*<sup>6</sup> = 12.4 mm, *W*<sup>2</sup> = *W*<sup>5</sup> = 10.65 mm, and *W*<sup>3</sup> = *W*<sup>4</sup> = 9.85 mm. For these values of iris widths, the values of the phases provided by Eq. (13) are: *φ*<sup>1</sup> ¼ *φ*<sup>6</sup> ¼ �1.7 rad, *φ*<sup>2</sup> ¼ *φ*<sup>5</sup> ¼ �1.28 rad, and *φ*<sup>3</sup> ¼ *φ*<sup>4</sup> ¼ �1.07 rad. Finally, the resonator lengths are obtained as:

$$L\_n = \frac{\lambda\_{\rm g0}}{2\pi} \left[ \pi + \frac{1}{2} \left( \rho\_n + \rho\_{n+1} \right) \right], \qquad n = 1, \ldots, N \tag{14}$$

so the values of the resonator lengths are *L*<sup>1</sup> = *L*<sup>5</sup> = 9.4 mm, *L*<sup>2</sup> = *L*<sup>4</sup> = 11.2 mm, and *L*<sup>3</sup> = 11.8 mm. **Figure 6** shows the simulated response of the designed rectangular waveguide filter.

The final step is to obtain the equivalent waveguide and iris widths in SIW technology with the equivalence given by Eq. (2), considering via holes diameter of *d* = 0.7 mm and separation of *s* = 0.95 mm, and also the design of microstrip to SIW transitions. For the microstrip to SIW transition, a microstrip taper has been implemented [5]. Finally, an optimization process of the designed filter response has been performed, providing the following final filter parameters: *a*1*SIW* ¼ *a*6*SIW* ¼ 11.86 mm, *a*2*SIW* ¼ *a*5*SIW* ¼ 10.48 mm, *a*3*SIW* ¼ *a*4*SIW* ¼ 9.90 mm, *L*<sup>1</sup> ¼ *L*<sup>5</sup> ¼9.4 mm, *L*<sup>2</sup> ¼ *L*<sup>4</sup> ¼ 11.2 mm, *L*<sup>3</sup> ¼ 11.8 mm, *Wt* ¼ 2.60 mm, *Lt* ¼ 7.08 mm, and *Wm* ¼ 0.6 mm. *Novel Filtering Applications in Substrate-Integrated Waveguide Technology DOI: http://dx.doi.org/10.5772/intechopen.105481*

**Figure 6.** *Electrical response of the designed rectangular waveguide iris filter.*

The designed iris SIW filter with its final dimensions is shown in **Figure 7(a)**, while its simulated and measured response is shown in **Figure 7(b)** with solid and dashed lines, respectively, showing a good impedance matching in the passband (better than 12.5 dB), and also a good out of band rejection performance (better than 20 dB).

### **3.2 SIW filters based on high and low dielectric constant sections**

By combining the filter design procedure detailed in the previous section with the obtained results in Section 2, the same concept of band-pass filter in SIW technology

*(a) Scheme of the designed iris filter in SIW technology. (b) Simulated and measured response of the iris filter in SIW technology.*

**Figure 8.**

*SIW filter with periodic perforations. (a) Physical geometry of the filter. (b) Equivalent structure based on the homogeneous permittivity of the perforated area.*

can also be implemented by exploiting SIW sections with reduced (perforated) or increased (periodically loaded with metallic cylinders) effective permittivity with ordinary SIW sections. For instance, the SIW may be perforated in some regions (see **Figure 8(a)**) to synthesize evanescent mode sections, as it has been proposed in Ref. [7], where the perforations in the dielectric substrate allow for reduction of the local effective permittivity, thus creating waveguide sections below cutoff (see **Figure 8(b)**). The lengths of the evanescent perforated waveguide sections, which are related to the impedance inverter factors in the impedance inverter model (see **Figure 8(a)**), are related to the number of hole columns. An example of SIW filter implementation with periodic perforations is shown in **Figure 9(a)** along with its simulated and measured response (**Figure 9(b)**), whose waveguide width, vias parameters, and employed substrate are the same as in Subsection 3.1. It is worth mentioning that the depth of the upper rejection band of the filter observed around 5 GHz (see **Figure 9(b)**) is directly related to the value of the reduced effective permittivity obtained in the perforated regions.

On the other hand, the combination of lower-permittivity (perforated) SIW sections with higher-permittivity (loaded with metallic cylinders) SIW sections can yield a better performance of this filter topology in terms of the rejection band, due to a higher contrast of permittivities, along with a reduction of the transversal dimension of the waveguide. An example of this phenomenon can be observed in the band-pass filter design shown in **Figure 10(a)**, constituted by the combination of rectangular perforations of the dielectric substrate and the insertion of metallic cylinders, where the SIW width has been reduced by a factor of 2 with respect to the SIW width

*Novel Filtering Applications in Substrate-Integrated Waveguide Technology DOI: http://dx.doi.org/10.5772/intechopen.105481*

#### **Figure 9.**

*(a) Example of SIW filter implementation with periodic perforations. Parameters of the air holes: da* ¼ *1.7 mm and sa* ¼ *1.95 mm. Filter parameters: L*<sup>1</sup> ¼ *L*<sup>5</sup> ¼ 10*:*12 *mm, L*<sup>2</sup> ¼ *L*<sup>4</sup> ¼ 6*:*79 *mm, L*<sup>3</sup> ¼ 6*:*42 *mm. Wt* ¼ 3*:*74 *mm, and Lt* ¼ 7*:*37 *mm. (b) Simulated and measured response of this filter.*

employed in the filter of **Figure 9** for a similar center frequency, and with the same dielectric permittivity (being the employed substrate in the filter of **Figure 10** of thickness *b* ¼ 1*:*5 mm). In this case, the perforations of the dielectric substrate have been done with a rectangular cross-section for a better selection of their widths. The simulated response of this filter is represented in **Figure 10(b)**, which reveals a deeper and wider rejection band in this case.

The perforated SIW filters described above have proven to show a good performance, exhibiting lower sensitivity to fabrication inaccuracies compared to iris-type filters with analogous frequency response. However, a limitation of such structures is that the length of the evanescent waveguide sections depends on the number of hole columns, and only discrete values are possible. To overcome it, with the aim to add flexibility to the design, a gap between the central hole rows in the perforated evanescent waveguide sections can be inserted, as it has been proposed in Ref. [12] (see **Figure 11(a)**), so a wide range of coupling coefficients can be achieved with them by changing the number of hole columns and the central gap *a*. This allows to design filters with the desired passband—narrow band filters, which require small couplings, can be obtained by increasing the length of the waveguide sections below the cutoff (i.e., the number of hole columns), and reducing the central gaps, and vice versa. An example of

#### **Figure 10.**

*(a) Example of SIW filter implementation with the combination of rectangular perforations of the dielectric substrate and the insertion of metallic cylinders. aSIW = 9.4 mm and b = 1.5 mm. Parameters of the metallic cylinders: dc= 1.1 mm, sc = 1.6 mm, and thickness of 1.25 mm. Parameters of the rectangular perforations: L*<sup>1</sup> ¼ *L*<sup>5</sup> ¼ 0*:*7 *mm, L*<sup>2</sup> ¼ *L*<sup>4</sup> ¼ 1*:*7 *mm, L*<sup>3</sup> ¼ 2*:*5 *mm. Wt* ¼ 8*:*05 *mm, Lt* ¼ 8*:*1 *mm, and Wm* ¼ *1.45 mm. (b) Simulated response of this filter.*

SIW filter implementation employing Taconic CER-10 substrate with these guidelines can be seen in **Figure 11**, with the geometrical dimensions of the filter and a photograph of the fabricated prototype, along with its simulated and measured response.

Finally, further developments of SIW filters with perforations of the dielectric substrate extending this concept to half-mode SIW structures can be done [12], with the aim to reduce the size of the filter. As an example, the perforated SIW filter in **Figure 11** has been applied to the half-mode SIW configuration by removing half of the top metal layer (where an HFSS reoptimization has been done). **Figure 12** shows the geometry of the filter with the geometrical dimensions and a photograph of the top layer, along with the comparison of the simulated and measured response. In this *Novel Filtering Applications in Substrate-Integrated Waveguide Technology DOI: http://dx.doi.org/10.5772/intechopen.105481*

#### **Figure 11.**

*Example of a four-pole perforated SIW filter incorporating gaps between the central hole rows in the perforated evanescent waveguide sections. (a) Geometry of the filter (dimensions in millimeter: v = 0.6, b = 2.6, d*<sup>1</sup> *= 10, d*<sup>2</sup> *= 7.95, a*<sup>1</sup> *= 4, a*<sup>2</sup> *= 0.55, a*<sup>3</sup> *= 0.25, w = 17.8, c = 7, and l = 81). (b) Photograph of the prototype. (c) Scattering parameters of the four-pole filter (HFSS simulation compared with measured data). Reprinted with permission from Ref. [12]; copyright 2017 IEEE.*

#### **Figure 12.**

*Example of a four-pole filter based on perforated half-mode SIW structure. (a) Geometry of the filter (dimensions in millimeter: v = 0.6,* b *= 1.6, d*<sup>1</sup> *= 9.6, d*<sup>2</sup> *= 8.35, a*<sup>1</sup> *= 1, a*<sup>2</sup> *= 0.2, a*<sup>3</sup> *= 0.4, w = 8.3,* c *= 7, and l = 83). (b) Photograph of the prototype. (c) Electrical response of the half-mode filter (HFSS simulation compared with measured data). Reprinted with permission from Ref. [12]; copyright 2017 IEEE.*

*Novel Filtering Applications in Substrate-Integrated Waveguide Technology DOI: http://dx.doi.org/10.5772/intechopen.105481*

#### **Figure 13.**

*Example of a three-pole half-mode SIW filter in folded configuration. (a) Geometry of the filter (dimensions in millimeter: v = 0.6,* b *= 2.6, d*<sup>1</sup> *= 1.65, d*<sup>2</sup> *= 10.56, d*<sup>3</sup> *= 3.9, d*<sup>4</sup> *= 6.7, a*<sup>1</sup> *= 2.45, a*<sup>2</sup> *= 1.95, a*<sup>3</sup> *= 5, a*<sup>4</sup> *= 9.45, w = 9.2,* c *= 7,* g *= 1, and l = 25.1). (b) Photograph of the prototype. (c) Electrical response of the folded filter (HFSS simulation compared with measured data). Reprinted with permission from Ref. [12]; copyright 2017 IEEE.*


#### **Table 1.**

*Comparison of performance of the SIW filter topology of this work (filter shown in Figure 9) with other filter topologies.*

case, a significant size reduction of the circuit has been achieved, although the structure is affected by radiation leakage (which is directly related to the observed higher insertion loss of the filter), due to the field distribution along the open boundary of the half-mode SIW. A folded filter configuration can be adopted to mitigate radiation losses of the half-mode SIW filter (see **Figure 13**), so the open boundaries of the halfmode SIW structure are located face-to-face, which reduces the radiation loss, and at the same time introduces a transmission zero in the frequency response through the direct input–output coupling. As an example, **Figure 13(a)** shows a three-pole halfmode SIW filter in folded configuration (geometry of the filter and final dimensions), while a photograph of the filter and its electrical response is shown in **Figure 13(b)** and **(c)**. The effect of the absence of radiation leakage is the flat insertion loss observed in **Figure 13(c)**.

A comparison between the obtained results of one of the filters in SIW technology presented in this chapter is based on stepped-impedance configurations with high and low dielectric constant sections (filter shown in **Figure 9**), and some other band-pass filters in SIW technology reported in the technical literature are presented in **Table 1**. As can be seen from **Table 1**, the proposed filters in this work show a clear improvement in bandwidth and insertion losses with respect to similar band-pass SIW filter topologies reported in the technical literature.
