Novel Filtering Applications in Substrate-Integrated Waveguide Technology

Abstract

The SIW technology combines complete shielding and fairly low losses with simple and cost-effective manufacturing, thus representing the ideal platform for the development of the next generation of wireless systems, including the band-pass filters among them. In this chapter, a number of novel SIW filter configurations will be presented to improve the filter performance, reduce losses, and minimize the filter footprint. To this end, different topologies of band-pass filters in SIW technology will be described based on stepped-impedance configurations (with high and low dielectric constant sections) making use of the impedance inverter model, extending this concept to half-mode SIW structures, with the aim to reduce the size of the filters.

1. Introduction

Substrate-integrated waveguides (SIWs) are planar structures that emulate a dielectric-filled rectangular waveguide (RWG) in a single circuit board, in which the lateral metallic walls are replaced with a periodic array of metallic vias (see Figure 1) [1, 2]. Thus, SIWs are good candidates to be used as building blocks for the implementation of microwave waveguide filters with different topologies, benefiting from the advantages of such technology (mainly low cost and easy integration), combined with the well-known advantages of conventional rectangular waveguides (complete shielding and high-power-handling capability).

Figure 1.

The geometry of the classical SIW structure.

In the following sections, we begin analyzing the main properties of ordinary SIWs with the homogeneous substrate, and those whose substrate is periodically loaded with either cylindrical air holes or with metallic cylinders, thus achieving a reduced/higher effective permittivity, respectively. After that, different topologies of band-pass filters in SIW technology are briefly described, starting from classical iris-type SIW filters and moving to more novel topologies, consisting of step impedance filters based on high and low dielectric constant sections, extending this concept to half-mode SIW structures, with the aim to reduce the size of the filters, showing in all cases good performances in terms of insertion and return losses in their passbands, along with deep and wide rejection bands.

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_n0 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:

$$d<{\lambda }_g/5,s\le 2d$$E1

where ${\lambda }_g$ is the guided wavelength. The propagation constant of the SIW fundamental mode is mainly determined by the width $a_{\mathit{SIW}}$ 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

$$a=a_{\mathit{SIW}}-\frac{d^2}{0.95s}.$$E2

Therefore, all the presented results in this section have been obtained using the equivalent waveguide of width a, related to the cutoff frequency $f_{c10}$ of the TE₁₀ mode and the relative permittivity of the substrate material by:

$$a=\frac{c}{2f_{c10}\sqrt{{\epsilon }_r}}.$$E3

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 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:

Figure 2.

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

$${\epsilon }_r=\frac{c^2}{4a^2{f}_{c10}^2}$$E4

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 $d_a$ and separation $s_a$) 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 $d_c$ and separation $s_c$).

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.

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 $X_s$ and a shunt reactance denoted by $X_p$. 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 $\phi /2$ on each side. The inverter is created by adding a length $\phi /2$ and $-\phi /2$ on each side of the discontinuity, as shown in Figure 4(b). In this case, the resonators are transmission lines of length $L_n$ connected to two transmission lines of artificial lengths $-{\phi }_n/2$ and $-{\phi }_{n+1}/2$. These lengths represent the load of the resonator from the adjacent coupling inverters (Figure 4(b)).

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.

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 $f_0=4$ GHz, a bandwidth of 600 MHz, and return loss RL = 15 dB. The rectangular waveguide dimensions are $a=15.8$ 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 ${\epsilon }_r=10$ and tan $\left(\delta \right)=0.0035$. Irises in all cases have a thickness t = 3 mm.

The filter center frequency $f_0$ and bandwidth BW are expressed by:

Figure 5.

Fifth order classical iris-type rectangular waveguide filter.

$$f_0=\sqrt{f_1{f}_2},\mathit{BW}=f_1-f_2,$$E5

which give $f_1=$ 3.7 GHz, $f_2=$ 4.3 GHz, and the filter relative bandwidth is:

$$\mathrm{\Delta }=\frac{{\lambda }_{g1}-{\lambda }_{g2}}{{\lambda }_{g0}}=0.3636$$E6

Then, the values obtained for the impedance inverter factors are:

$$\frac{K_{01}}{Z_0}=\frac{K_{56}}{Z_0}=\sqrt{\frac{\pi \mathrm{\Delta }}{2g_0{g}_1}}=0.7051$$E7

$$\frac{K_{12}}{Z_0}=\frac{K_{45}}{Z_0}=\frac{\pi \mathrm{\Delta }}{2\sqrt{g_1{g}_2}}=0.45546$$E8

$$\frac{K_{23}}{Z_0}=\frac{K_{34}}{Z_0}=\frac{\pi \mathrm{\Delta }}{2\sqrt{g_2{g}_3}}=0.34706$$E9

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), $X_s$ and $X_p$, by the following Eqs. (10) and (11):

$$j\frac{X_s}{Z_0}=\frac{1-S_{12}+S_{11}}{1-S_{11}+S_{12}}$$E10

$$j\frac{X_p}{Z_0}=\frac{2S_{12}}{{\left(1-S_{11}\right)}^2-S_{12}^2}$$E11

where $S_{11}$, $S_{21}$, and $S_{12}$ are the scattering parameters of the ${\mathit{TE}}_{10}$ fundamental mode of the input waveguide at the filter center frequency $f_0$. For the impedance inverter shown in Figure 4(b), $X_s$ and $X_p$ are related to $K/Z_0$ and $\phi$ by:

$$\frac{K}{Z_0}=\left|tan\left(\frac{\phi }{2}\text{atan}\frac{X_s}{Z_0}\right)\right|$$E12

$$\phi =-\text{atan}\left(2\frac{X_p}{Z_0}+\frac{X_s}{Z_0}\right)-\text{atan}\frac{X_s}{Z_0}$$E13

The scattering parameters of several iris of different widths have been obtained, so the values of the iris widths for this filter are W₁ = W₆ = 12.4 mm, W₂ = W₅ = 10.65 mm, and W₃ = W₄ = 9.85 mm. For these values of iris widths, the values of the phases provided by Eq. (13) are: ${\phi }_1={\phi }_6=$ −1.7 rad, ${\phi }_2={\phi }_5=$ −1.28 rad, and ${\phi }_3={\phi }_4=$ −1.07 rad. Finally, the resonator lengths are obtained as:

$$L_n=\frac{{\lambda }_{g0}}{2\pi }\left[\pi +\frac{1}{2}\left({\phi }_n+{\phi }_{n+1}\right)\right],n=1,\dots ,N$$E14

so the values of the resonator lengths are L₁ = L₅ = 9.4 mm, L₂ = L₄ = 11.2 mm, and L₃ = 11.8 mm. Figure 6 shows the simulated response of the designed rectangular waveguide filter.

Figure 6.

Electrical response of the designed rectangular waveguide iris 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: $a_{1\mathit{SIW}}=a_{6\mathit{SIW}}=$ 11.86 mm, $a_{2\mathit{SIW}}=a_{5\mathit{SIW}}=$ 10.48 mm, $a_{3\mathit{SIW}}=a_{4\mathit{SIW}}=$ 9.90 mm, $L_1=L_5=$9.4 mm, $L_2=L_4=$ 11.2 mm, $L_3=$ 11.8 mm, $W_t=$ 2.60 mm, $L_t=$ 7.08 mm, and $W_m=$ 0.6 mm. 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).

Figure 7.

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

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 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.

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.

Figure 9.

(a) Example of SIW filter implementation with periodic perforations. Parameters of the air holes: $d_a=$ 1.7 mm and $s_a=$ 1.95 mm. Filter parameters: $L_1=L_5=10.12$ mm, $L_2=L_4=6.79$ mm, $L_3=6.42$ mm. $W_t=3.74$ mm, and $L_t=7.37$ mm. (b) Simulated and measured response of this filter.

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 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 $b=1.5$ 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.

Figure 10.

(a) Example of SIW filter implementation with the combination of rectangular perforations of the dielectric substrate and the insertion of metallic cylinders. $a_{\mathit{SIW}}$ = 9.4 mm and b = 1.5 mm. Parameters of the metallic cylinders: $d_c$= 1.1 mm, $s_c$ = 1.6 mm, and thickness of 1.25 mm. Parameters of the rectangular perforations: $L_1=L_5=0.7$ mm, $L_2=L_4=1.7$ mm, $L_3=2.5$ mm. $W_t=8.05$ mm, $L_t=8.1$ mm, and $W_m=$ 1.45 mm. (b) Simulated response of this filter.

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 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.

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, $d_1$ = 10, $d_2$ = 7.95, $a_1$ = 4, $a_2$ = 0.55, $a_3$ = 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.

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 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 half-mode 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 half-mode 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).

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, $d_1$ = 9.6, $d_2$ = 8.35, $a_1$ = 1, $a_2$ = 0.2, $a_3$ = 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.

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, $d_1$ = 1.65, $d_2$ = 10.56, $d_3$ = 3.9, $d_4$ = 6.7, $a_1$ = 2.45, $a_2$ = 1.95, $a_3$ = 5, $a_4$ = 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.

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.

	Number of resonators	Center freq. (GHz)	FBW* (%)	IL (dB)	Minimum RL (dB)	Size (mm^2)
[13]	2	4.75	2.5	1.4	13	1085
[14]	1	7.4	16.9	2.0	18	66.6
[15]	2	10.0	0.02	1.0	20	401
This work	5	4.0	15	1.6	12	327.4

Table 1.

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

^*

FBW: fractional bandwidth.

4. Conclusions

In this chapter, we provide guidelines for the design of novel SIW filters based on the alternation of SIW sections with reduced (perforated) or increased (periodically loaded with metallic cylinders) effective permittivity with ordinary SIW sections, making use of the impedance inverter model. Examples of several SIW filter configurations show that this solution can lead to the design of compact filters with good performance and low sensitivity to fabrication inaccuracies.

Acknowledgments

A. Coves acknowledges grant PID2019-103982RB-C43 funded by Ministerio de Ciencia e Innovación (MCIN)/Agencia Estatal de Investigación (AEI)/10.13039/501100011033.