With the rapid development of wireless communication, the competition for a large number of various radio technologies and applications has become more and more fierce, and communication systems need to work in two different frequency bands, such as: Global Positioning System (GPS), 4G communication and 5G communication. Filters are an important part of RF/microwave systems, and in dual-band transceiver systems, using dual-band filters instead of two separate single-band filters can not only reduce the volume of the communication system but also save costs compared to designing a separate filter in each corresponding frequency band. In addition, higher requirements are put forward for the miniaturization, high selectivity, low loss, and multi-channel transmission capability of dual-passband filters. To meet the needs of modern communication scenarios, a new type of transmission line structure, the substrate integrated waveguide (SIW) structure, has gradually been developed and utilized. The SIW structure not only has the advantages of flattening the microstrip structure and low cost, but also has the advantages of low loss and high performance of the traditional metal waveguide structure, which is very suitable for multi-channel RF front-end systems in the field of wireless communication. In recent years, many high-quality substrate-integrated filters with dual-passband or multi-pass band have been researched and designed1–15,20−23.

In order to achieve miniaturized and high-performance dual-passband filters, the most common methods used by scholars are: (1) loading metamaterial structures such as Complementary Split Ring Resonators (CSRR) and Composite Right/Left-Hand (CRLH) on the cavity surface. The dual-passband is achieved by loading this electromagnetic structure in the SIW structure to generate a resonant frequency lower than the basic mode of the SIW structure2,13−16; (2) Load slow-wave affect structures such as Defected Ground Structure (DGS) and Electromagnetic Bandgap (EBG) on the cavity surface. By etching the surface of the SIW structure to destroy the surface standard current distribution, an obvious band-stop or resonant effect is formed, to achieve dual-passband 3–4; (3) Add disturbed metal holes in the standard SIW cavity to change the pattern distribution in the cavity and realize the separation of different resonant modes to form a dual-passband 5; (4) Cascade coupling of resonators, etc. Different resonant cavities provide different resonant frequencies, and under the action of cascade coupling, a dual-passband is formed 6–7.

In this paper, a filter design method for loading back-to-back CSRR and C-slot on HMSIW cavity is proposed. The back-to-back CSRR structure is composed of two identical and reverse-arranged CSRR resonators, which are used to provide the first passband lower than the cutoff frequency of the HMSIW structure, and the resonance frequency and coupling coefficient of the back-to-back CSRR resonator are calculated by parity-mode analysis method, and the corresponding coupling effect transmission curve is obtained and two transmission zeros are formed in the lower stop band of the first passband, which realizes the degree of miniaturization and increases the frequency selectivity of the filter. The second passband is provided by the coupling of the first three modes in the HMSIW cavity, and in order to improve the out-of-band characteristics of the second passband, the resonator C-slot is etched on the top layer of the filter, and the C-slot is a single-loop structure of CSRR with band-reject characteristics, which can add an additional transmission zero point at the edge of the second passband. The test results show that the proposed design method not only reduces the size of the filter, simplifies the structural complexity, but also has high selectivity, which is very suitable for modern multi-channel communication applications.

## Methods

**Theoretical design and the principle of operation.** *A HMSIW resonator analysis.* Cutting along the midline on the SIW structure can obtain HMSIW, which has the same working mode as the SIW but the size can be reduced by half. Therefore, HMSIW has the advantages of small size, convenient production, low cost, and easy integration in microwave millimeter wave circuits.

SIW is synthesized on a dielectric substrate using a periodic array of metallized vias, only TE mode exists, and since the wave propagation in the SIW and rectangular metal waveguides is similar, the resonant frequency of the HMSIW cavity is:

$${f_c}=\frac{c}{{2\pi \sqrt {\mu \varepsilon } }}\sqrt {{{\left( {\frac{{m\pi }}{{2{L_{eff}}}}} \right)}^2}+{{\left( {\frac{{n\pi }}{{2{W_{eff}}}}} \right)}^2}} .$$

1

where (1): ,=1,2,3...;is the propagation speed of electromagnetic waves in a vacuum, andthe dielectric constant and permeability of the dielectric substrate, respectively; andthe equivalent length and equivalent width of HMSIW, respectively, are:

$${L_{eff}}=L - \frac{{{d^2}}}{{0.95p}}.$$

2

$${W_{eff}}=W - \frac{{{d^2}}}{{0.95p}}.$$

3

In equations (2) and (3):,and,respectively, the spacing, diameter and length, and width of the SIW structure of the metalized vias. When,, satisfied, the magnetic wall composed of metal vias hardly occurs electromagnetic leakage.

Figure 1 shows the structure diagram of the initial half-mode substrate integrated waveguide (HMSIW). To study the transmission characteristics of the HMSIW structure, the eigenmode simulation of the structure by the simulation software HFSS is used, and the electric field distribution of the first three resonant modes in the HMSIW cavity is given in Fig. 2. Figure 2(a) The resonant frequency corresponding to the main mode TE101 of the HMSIW cavity is 5.97GHz, and it is also the cutoff frequency of the HMSIW structure. Figure 2(b) and (c) are the two higher-order modes TE201 and TE103, corresponding to 7.68GHz and 9.12GHz, respectively, at this time, the three mode resonance points in the HMSIW cavity are close to each other and can be used to provide a passband of the filter.

*B Back-to-back CSRR resonator structure.*Using the low-frequency bandpass transmission characteristics of conventional CSRR, it is possible to add a passband at a low frequency without increasing the filter volume, and realize a miniaturized design. Compared with the traditional coupled CSRR, the coupling strength of the back-to-back CSRR structure is more controllable, which can greatly affect the electric field distribution in the resonant cavity, which is convenient for the design and adjustment of the passband.

As shown in Fig. 3(a), back-to-back CSRR is formed by a pair of identical CSRR structures arranged side by side in reverse and CSRR can be achieved by nesting two open rings of different sizes, the equivalent circuit of which is shown in Fig. 3(b). Since there is no gap between back-to-back CSRRs, they can be tightly coupled to each other, and the coupling between two independent CSRRs is electromagnetic coupling, and electrical coupling and magnetic coupling work together as a pair of electric dipoles under axial electric field excitation and work near the resonant frequency. In Fig. 3(b), the self-inductance and self-capacitance equivalent to two independent CSRR structures are equivalent to the mutual inductance and mutual capacitance between back-to-back CSRRs 13–16.

*C Back-to-back CSRR-HMSIW transmission response.* The back-to-back CSRR is loaded onto the upper metal surface of the HMSIW cavity to form a back-to-back CSRR-HMSIW structure, as shown in Fig. 4.

The diameter of HMSIW waveguide metal vias is 0.8mm, and the distance between adjacent vias is 1.2mm. The electromagnetic field simulation software HFSS 15.0 was used to analyze the back-to-back CSRR-HMSIW resonator structure and the transmission characteristics of HMSIW structure, and the corresponding transmission characteristic curve is shown in Fig. 5.

As can be seen from Fig. 5, the HMSIW cavity is loaded with back-to-back CSRRs, forming another passband below the HMSIW cavity cutoff frequency. The back-to-back CSRR structure produces a passband center frequency of 4.25GHz, a relative bandwidth of 18%, and an in-passband insertion loss of less than 0.3dB.

Since there are no gaps between the two CSRR structures, this configuration can result in transmission zero points on each side of the passband. It is clear that there are two pathways from the input to the output through the two CSRRs, which act as two resonators in this filter. Two transmission zeros emerge from the two pathways' opposing phase shifts. At the same time, there is a higher level of out-of-band inhibition. Since the waveguide has a cutoff frequency of 5.97GHz and the resonant frequency shifts to a lower frequency after adopting a back-to-back CSRR structure, the conventional HMSIW filter is miniaturized by 42.6%.

SIW metal vias are equivalent to inductors by array modeling. Input coupling includes electrical coupling and magnetic coupling, which are expressed as and. The equivalent circuit model diagram of the resulting back-to-back CSRR-HMSIW filter is shown in Fig. 6(a).The resonance characteristics of odd mode and even mode analysis were studied, and the odd mode and even-mode equivalent circuit diagrams shown in Fig. 6 (b) and (c) were used to derive odd mode and even-mode input admittance , as shown in Eq. (4) and Eq. (5).

$${Y_{{\text{ino }}}}= - \frac{j}{{\omega {L_d}}}+\frac{{\left( { - \frac{j}{{\omega {L_r}}}+j\omega {C_r}} \right)\left( { - \frac{j}{{\omega {L_i}}}+j\omega {C_i}} \right)}}{{ - \frac{j}{\omega }\left( {\frac{1}{{{L_r}}}+\frac{1}{{{L_i}}}} \right)+j\omega \left( {{C_r}+{C_i}} \right)}}.$$

4

$${Y_{{\text{ine }}}}= - \frac{j}{{\omega {L_d}}}+\frac{{\left( { - \frac{j}{{\omega {L_r}}}+j\omega {C_r}} \right)\left[ { - \frac{j}{\omega }\left( {\frac{1}{{{L_i}}}+\frac{2}{{{L_c}}}} \right)+j\omega \left( {{C_i}+2{C_c}} \right)} \right]}}{{ - \frac{j}{\omega }\left( {\frac{1}{{{L_r}}}+\frac{1}{{{L_i}}}+\frac{2}{{{L_c}}}} \right)+j\omega \left( {{C_r}+{C_i}+2{C_c}} \right)}}.$$

5

When the odd mode and even mode transmission poles are close to infinity, the resonant frequenciesandof odd mode and even mode are shown in equations (6) and (7):

$${f_{{\text{odd }}}}=\frac{{\sqrt {{L_i}+{L_r}} }}{{2\pi \sqrt {{L_r}{L_i}\left( {{C_r}+{C_i}} \right)} }}.$$

6

$${f_{{\text{even }}}}=\frac{{\sqrt {{L_i}{L_c}+{L_r}{L_c}+2{L_r}{L_i}} }}{{2\pi \sqrt {{L_r}{L_i}{L_c}\left( {{C_r}+{C_i}+2{C_c}} \right)} }}.$$

7

From equations (6) and (7), it can be seen that only the even-mode resonant frequency is affected by the intercoupling elementsand. The electromagnetic simulation method is used to study, and the coupling coefficient is extracted as shown in Eq. (8):

$$K=\frac{{\left| {f_{{odd}}^{2} - f_{{even}}^{2}} \right|}}{{f_{{odd}}^{2} - f_{{even}}^{2}}}.$$

8

According to the strength of the coupling coefficient between the two resonant elements, the resulting coupling effect can strengthen or weaken the energy storage of the resonant ring at the operating frequency. From Eq. (7) and Eq. (8), it can be seen that the size of the coupling coefficientbetween back-to-back CSRRs is mainly affected by the mutual capacitancebetween the two CSRRs. As can be seen from Fig. 6(a), the larger the width of the gapbetween the back-to-back CSRRs, the greater the mutual capacitanceof the equivalent circuit.

Figure 7 shows the influence of adjusting the width of the gapbetween back-to-back CSRRs on the coupling coefficientand odd-mode and even-mode resonant frequencies. As shown in Fig. 7, as the gapwidth between back-to-back CSRRs increases, the coupling coefficientgradually decreases, and the even-mode resonant frequency gradually increases. It is verified that the even-mode resonant frequency is more sensitive to the change of mutual capacitance,while the change of odd-mode resonant frequency is negligible.

When, transmitting the zero point, as shown in equations (9) and (10):

$${Z_{{\text{T1}}}}=\frac{1}{{2\pi \sqrt {LcCc} }}.$$

9

$${Z_{{\text{T2}}}}=\frac{1}{{2\pi \sqrt {LrCr} }}.$$

10

*D Load the C-slot.* In order to improve the lower stopband suppression characteristics of the second passband, two symmetrical C-slots are loaded on the back-to-back CSRR-HMSIW structure. Its structure is shown in Fig. 8.

CSRR as a duality argument for an open resonant ring (SRR) has been shown to have a negative permittivity constant. Since the signal cannot propagate near its resonant frequency, CSRR has stopband performance. The C-slot designed in this section adopts a CSRR single-ring structure, and the induced current generated under the action of the electromagnetic field will also change the original magnetic field distribution, to suppress a certain frequency, so the C-slot also has the same stop-band performance.

In the simulation software, the opening length of the C-slot has a great influence on the stop-band suppression characteristics under the second pass-band, and the size of the C-slot opening length is changed, and the corresponding transmission characteristic curve is shown in Fig. 9.

When the C-slot opening length becomes larger, a parasitic pass band is created near the passband. Whenis small, the stopband performance of the C-slot generates a new transmission zero at the lower stopband of the second passband, which reduces the influence of the parasitic passband on the second passband of the filter. The final C-slot opening length is determined to be 0.8mm.