Introduction

The microwave bandpass filters are one of the most important building blocks of microwaves and millimeter wave communication systems. Waveguide and printed filters are known as the most common technologies, with their advantages and disadvantages well-discussed1. Several technologies, such as low temperature co-fired ceramic (LTCC)2,3,4 ,microstrips5,6,7, waveguides8,9, and micro-electromechanical systems (MEMS)10,11, are used to develop bandpass filters with specific characteristics such as low cost, high selectivity, low insertion loss, broad bandwidth, compact size, and power handeling capacity. Due to the trade-off between these parameters, sometimes we have to prioritize a number of characteristics that are more important for our design. Several other technologies have been introduced for the specific reason of integrating different technologies in order to use their advantages together. Substrate integrated waveguide (SIW) is one the most familiar ones, which has found a lot of applications such as low production costs, high quality factor, and ability of integration specially, specially in millimeter-waves bands12,13,14,15. Gap waveguides introduced in16 is another new technology with some advantages comparing with classic waveguides. The gap waveguide can be a low-loss building block with advantages in manufacturig waveguide components such as antennas, filters, couplers, and also integration of active components like amplifiers and microwave monolithic integrated circuits (MMIC’s) at millimeter-waves and THz bands17,18. These structures are based on two parallel plates, with a bed of pins structure in one of the plates (see Fig. 1). This bed of pins structure introduces a high impedance condition over this plate that avoids the electrical contact requirement between the top plate and the pins of the bottom plate. The direction of the wave propagation relative to the direction of corrugation determines whether it enables or prevents the wave propagation at the surface19.

Recently, manufacturing problems in mm-wave and THz designs motivated consideration of the gap waveguide technology20,21,22. Common types of gap waveguide technologies are groove gap waveguide (GGW)23, microstrip gap waveguide (MGW)24 ridge gap waveguide (RGW)25 and substrate integrated groove gap waveguide (SIGGW)26,27. SIGGW is used in this research to design a band-pass filter at Ka band with a transmission zero.

In section 2, we briefly explain the SIGGW design for Ka-Band. In Sect. 3, we analytically design the BPF using dual-post configuration. We also explain how to use the out of band resonance for the SIGGW posts to realize a transmission zero in lower rejection band. Section 4 is used for simulation of structure and verifying the results and compare with calculated response at third section. Finally, we present a design method, followed by some concluding remarks.

Substrate integrated groove gap waveguide (SIGGW) structure

The geometry of the SIGGW is shown in Fig. 1. Unlike classic waveguides, the SIGGW doesn’t require electrical contact between the upper and lower planes. The dimensions and positions of metallic bars are designed to create a metamaterial gap at desired frequency band to prevent propagation in the x-direction and virtually act as a wall in that band. As a result, EM wave propagates through the space between top plane, metal pins, and lower plane which is filled with a dielectric. Here, we use RO4003 with \(\epsilon _r\)=3.55 and \(tan\delta \)=0.002 as dielectric. The dimensions of metal pins are designed to achieve a band gap around 28 GHz25. The designed parameters are depicted in Table 1.

Figure 1
figure 1

The geometry of SIGGW. (a) Front view of structure with RO4003 (b) Perspective view of SIGGW without dielectric.

Full size image
Table 1 Desired parameter for design SIGGW. \(T_x\) and \(T_y\) are distance center to center of metal pins and D is diameter of metal pins.
Full size table

The simulated results for values in Table 1 are shown in Fig. 2. The dispersion curve indicates that the first bandgap is between 20.5GHz to 42GHz. In Fig. 3, the s-parameters of the designed SIGGW are displayed, revealing excellent return loss and transmission coefficients in the range of 24GHz-32GHz.

Figure 2
figure 2

The bandgap between first and second mode of excitation.

Full size image

Dual post filter design

One of candidates for band-pass waveguide filter is inductive dual-post structure28. In section 3, we designed the SIGGW, and in this section, we use those results to design our filter. The filter is designed at center frequency of 28GHz and fractional bandwidth of about 5 We use posts at distance about \(\frac{\lambda _g}{2}\) from each other, where \(\lambda _g\) is guided wavelength at center frequency, taking into consideration the \(\epsilon _r\) of structure. For designing dual-post filters in classic rectangular waveguide, we use posts connecting bottom to top of the waveguide, but in this case the posts are shorter than the height of SIGGW. The post lengths, denoted as L from the bottom, as shown in Fig. 1. Since the L is shorter than waveguide height, there will be a gap between the top of the posts and the upper plate. A capacitance is formed between the posts and upper conductor. This can be so useful to generate a transmission zero in rejection band of filter. The posts are inductive and having a capacitance in series cause a resonance frequency which short circuit the SIGGW and current is maximum in this case. Nevertheless, we need inductances for our filter topology and fortunately, as the frequency increases, the behavior of series inductance and capacitance becomes more inductive, so in this case we expect that if the frequency increases this design behaves perfectly as an inductive post. This means we have a resonant frequency at lower rejection band that could be useful if we can control it at proper frequency. So here, we introduce an algorithm to control this transmission zero. The equivalent circuit of waveguide filter is shown in Fig. 4.

Figure 3
figure 3

Return loss and transmission coefficient of designed SIGGW.

Full size image
Figure 4
figure 4

Equivalent circuit of filter.

Full size image

The ABCD matrix and input impedance of transmission line is given by Eqs. (1) and (2) respectively.

$$\begin{aligned}{}[ABCD]= & {} \begin{bmatrix} cos\beta l &{} jZ_0 sin\beta l \\ jY_0 sin\beta l &{} cos\beta l \\ \end{bmatrix} \end{aligned}$$
(1)
$$\begin{aligned} Z_{in}= & {} Z_0\frac{Z_l+jZ_0 tan\beta l}{Z_0+jZ_l tan\beta l} \end{aligned}$$
(2)

If \(Z_l\rightarrow \infty \) the Eq. (2) becomes \(Z_{in}=-jZ_{01} cot\beta l_{11}\). So, the input impedance of structure is Eq. (3) (see Fig. 5).

$$\begin{aligned} Z_{in}=Z_{02}\frac{-jZ_{01} cot\beta l_{11}+jZ_{02} tan\beta l_1}{Z_{02}+j(-jZ_{01} cot\beta l_1)tan\beta l_1}=jZ_{02}\frac{x tan\beta l_1-cot\beta l_{11}}{x+cot\beta l_{11} tan\beta l_1} \end{aligned}$$
(3)

where, \(x=\frac{Z_{02}}{Z_{01}}\) so: \(Y_{in}=jY_{02}\frac{x+cot\beta l_{11} tan\beta l_1}{x tan\beta l_1-cot\beta l_{11}}\)

We mentioned that distance between posts is about \(\frac{\lambda _g}{2}\) so the ABCD matrix of total structure is Eq. (4).

$$\begin{aligned} {[}T] = \begin{bmatrix} cos\beta l &{} jZ_0 sin\beta l \\ jY_0 sin\beta l &{} cos\beta l \\ \end{bmatrix} \times \begin{bmatrix} 1 &{} 0 \\ jY_{02}\frac{x+cot\beta l_{11} tan\beta l_1}{x tan\beta l_1-cot\beta l_{11}} &{} 1 \\ \end{bmatrix} \times \begin{bmatrix} cos\beta l &{} jZ_0 sin\beta l \\ jY_0 sin\beta l &{} cos\beta l \\ \end{bmatrix} \end{aligned}$$
(4)

where \(Z_0\) and \(\beta l\), is characteristic impedance and electrical length of resonators respectively. If \(l_{11}<\lambda /4\) from taylor series we have \(cot \beta l_{11}\approx 1\) due to Fig. 5 it is a reasonable assumption. As a result, with \(\beta l=\pi \):

$$\begin{aligned} {[}T] = \begin{bmatrix} -1 &{} 0 \\ 0 &{} -1 \\ \end{bmatrix} \times \begin{bmatrix} 1 &{} 0 \\ jY_{02}\frac{x+\frac{tan \beta l_1}{\beta l_{11}}}{x tan\beta l_1-\frac{1}{\beta l_{1_1}}} &{} 1 \\ \end{bmatrix} \times \begin{bmatrix} -1 &{} 0\\ 0 &{} -1\\ \end{bmatrix} = \begin{bmatrix} 1 &{} 0 \\ jY_{02}\frac{x\beta l_{11}+tan \beta l_1}{x\beta l_{11} tan \beta l_{11}} &{} 1 \\ \end{bmatrix} \end{aligned}$$

If we convert ABCD matrix to S-matrix due to29:

$$\begin{aligned}{} & {} S_{11}=\frac{A+\frac{B}{Z_0}-C Z_0-D}{A+\frac{B}{Z_0}+C Z_0+D}\qquad \qquad S_{12}=\frac{2}{A+\frac{B}{Z_0}+C Z_0+D}\\{} & {} S_{11}=S_{22}=\frac{(x \beta l_{11} tan \beta l_1 -1)}{2(x \beta l_{11} tan \beta l_1 - 1)+j(x \beta l_{11} + tan \beta l_1)} \end{aligned}$$

At resonance frequency we choose \(|S_{11}|=0\) and \(|S_{21}|=1\).

$$\begin{aligned} x \beta l_{11} + tan\beta l_1&=0 \end{aligned}$$
(5a)
$$\begin{aligned} \frac{(x \beta l_{11} tan \beta l_1 -1)}{\sqrt{{4(x \beta l_{11} tan \beta l_1 - 1)^2+j(x \beta l_{11} + tan \beta l_1)^2}}}&=1 \end{aligned}$$
(5b)

In this case we have Quasi-TEM mode so we compute inductance and capacitance of posts. It is assumed that the inductance of \(l_1\) and capacitance of \(l_{11}\) are dominant. First, we compute inductance, form Ampere law we have:

$$\begin{aligned} \oint _{C} H.\,dl= & {} I {\rightarrow } H=\frac{I}{2 \pi r} \\ \frac{\mu }{2} \iiint \ |H^2|\ dv= & {} \frac{\mu }{2} \iiint \ \frac{I^2}{4\pi ^2 r^2}\,rdrd\phi dz = \frac{\mu I^2}{4 \pi } \int \limits _{R_1}^{R_{11}} \frac{1}{r}. \, dr \int \limits _{0}^{l_{1}} \, dz = \frac{\mu I^2}{4\pi } l_1 \ln {\frac{R_{11}}{R_1}} \end{aligned}$$

On the other hand, we know the energy of inductance is W=\(\frac{1}{2}LI^2\) so: \(L=\frac{\mu }{2 \pi }l_1 \ln {\frac{R_{11}}{R_1}}\)

Second, we compute capacitance. \(C=\epsilon _r \frac{A}{G}\) \(\rightarrow \) \(C=\frac{3.55}{36 \pi \times 10^9}\frac{\pi R_{11}^2}{G}\)

So, the resonance frequency due to the post is:

$$\begin{aligned} \omega _c=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{{\frac{\mu }{2 \pi } l_1 \ln {\frac{R_{11}}{R_1}}}\times \frac{3.55}{36\pi \times 10^9}\frac{\pi R_{11}^2}{G}}} \end{aligned}$$
(6)

Electric field and magnetic field are minimum and maximum respectively on posts, so assuming that the waveguide is strip line the radius of \(R_1\) and \(R_2\) related to \(Z_{01}\) and \(Z_{02}\). The impedance \(Z_0\) of a transmission line consisting of a circular signal conductor having a diameter D and centered between parallel ground planes separated by distance T is given by Eq. (7)30.

$$\begin{aligned} Z_0=\frac{\eta _0}{2\pi \sqrt{\epsilon _r}} \ln {\frac{4T}{\pi D}} \end{aligned}$$
(7)

Algorithm of design:

  1. (1)

    Choose the frequency that want to have transmission zero out of band and compute \(\omega _c\)

  2. (2)

    From Eq. (7) compute \(R_{11}\) and \(R_1\) according to \(Z_{01}\) and \(Z_{02}\).

  3. (3)

    Now we have 3 equations and 4 unknowns. (3 equations are Eq. 5a,  b and Eqs. (6) and (4) unknowns are \(Z_{01}\), \(Z_{02}\), \(l_{11}\), \(l_1\))

  4. (4)

    By assuming that one parameter is constant, compute other parameters. (For example, initially set \(l_1\) around \(\frac{\lambda _g}{4}\) and compute \(Z_{01}\), \(Z_{02}\), \(l_{11}\)).

The final design parameters are depicted in Table 2 We must mention that there are limitations to this structure. At lower frequencies, designing and building this structure will become difficult due to the increase in substrate thickness. Similarly, at higher frequencies, it will also be challenging to design and build this structure because of the shrinking dimensions of the filter.The final structure and results are shown in Figs. 5 and 6 respectively.

Table 2 Design parameters for posts. Index 1 and 11 is for the first post, index 2,22 and 3,33 is for second and third posts respectively.
Full size table
Figure 5
figure 5

Final structure of filter. (a) Perspective view of SIGGW with cuting plane (b) Enlarged view of posts.

Full size image
Figure 6
figure 6

Final result of filter.

Full size image

Simulation of SIGGW filter

For verifying our results, we use sweep parameters in CST. If we reduce the radius of capacitors from Eq. (6), we see that resonance frequency is increased. It can be seen from Fig. 7 that if radius of each capacitor is decreased about 0.02 mm the resonance frequency increased about 100MHz.

Figure 7
figure 7

Effect of changing the area of capacitors in the location of transmission zero.

Full size image
Figure 8
figure 8

Flow chart of the main steps in the SIGGW filter design with finite transmission zero in lower band.

Full size image

Figure 8 illustrates the primary objective of this paper, which is to present the design of the SIGGW filter in the Ka band with finite transmission zero in the lower band. This analysis method can also be employed for designing other resonant filters that incorporate finite transmission zero.

Discussion

In this research, a dual-post band-pass filter in the 5G Ka-band has been designed and simulated using a three-layer substrate integrated gap groove waveguide (SIGGW). After interpreting the dual-post filters, we proceeded with the SIGGW filter design. The most crucial advantage of gap waveguide structures compared to conventional rectangular waveguides is that the posts are not connected to the top wall of the waveguide. In SIGGW, the multilayer PCB etching and plating technology forces us to provide some pads on both sides of the via holes. These pads play the role of capacitor plates when we realize the posts as blind vias. We have shown that this can be efficiently controlled and facilitates the utilization of the resonant frequency of the filter posts to synthesize a transmission zero at a lower reject band. The realized transmission zero improves the lower side rejection slope and can be used for suppressing the LO leakage, which is commonly employed for up-converting the 5G signal to the Ka-band. We have proposed an algorithm to control this transmission zero, verified using CST simulating software. This algorithm can create transmission zero in other gap waveguide resonant filters and obtain a suitable initial solution. The results agree with the introduced simple circuit model for the resonant frequency. Finally, Table 3 compares the BPFs that were recently reported and fabricated using different technologies (the mass of the construction was not included due to a lack of available data).

Table 3 Performance comparisons between proposed filter in this paper with other filters in different technologies.
Full size table