## 3 Design of the Wide Passband SIW-SSPP Filters with Multi-Sub-Bands
### The Wide Bandpass Filter with Multi-Sub-Bandpass
#### 3.2.1 Dual-Sub-Bandpass Filter Loaded with CSRRs  
As shown in Figure 6a, four identical CSRRs are symmetrically loaded on the ground plane: \(move_{x2}=16\) mm and \(move_{y2}=1.45\) mm represent the distances from the CSRRs to the central point, \(r_{1}=0.65\) mm and \(r_{2}=0.45\) mm denote the outer and inner radii of the CSRRs, \(g_{2}=0.1\) mm indicates the opening width, and \(d_{1}=0.1\) mm corresponds to the ring spacing. When operating near its resonant frequency, a CSRR can be equivalently modeled as an electric dipole. It exhibits negative permittivity around the resonant frequency, resulting in a band-stop effect and generating a resonant absorption peak.  
Figure 6b shows the equivalent circuit of the CSRR, which is a resonant loop. \(L_{CSRR}\) represents the equivalent inductance and \(C_{CSRR}\) represents capacitance of the CSRR, respectively. In the equivalent circuit of an ideal lossless transmission line composed of series inductors and shunt capacitors, the loop is connected in series with the shunt capacitor, forming the equivalent circuit of the CSRR and its MSL. Based on the duality principle, the values of \(L_{CSRR}\) and \(C_{CSRR}\) for the CSRR can be derived from the equivalent capacitance \(C_{SRR}\) and inductance \(L_{SRR}\) of its complementary structure SRR. The specific values of \(C_{SRR}\) can be calculated using Equation (7) [37].  
\[C_{SRR}=\frac{\pi(r_{1}+r_{2})C_{pull}}{4} \tag{7}\]  
where the \(C_{pull}\) represents the per-unit-length capacitance between adjacent rings, \(C_{pull}=2\pi r_{2}\varepsilon_{0}\varepsilon_{r}d_{1}/d\). \(\varepsilon_{0}\) denotes the permittivity of vacuum, and \(\varepsilon_{r}\) denotes the relative permittivity of the dielectric substrate. \(L_{SRR}\) is the equivalent inductance of a circular ring of radius \((r_{1}+r_{2})/2\) and ring width \((r_{1}-r_{2}-d_{1})\).  
Figure 6: Dual-band filter loaded with CSRRs: (**a**) schematic diagram; (**b**) equivalent circuit of the CSRRs; (**c**) surface current; (**d**) S-parameters.  
The relationship between the equivalent capacitance and inductance of CSRR and SRR can be expressed as follows:  
\[C_{CSRR}=4\frac{\epsilon_{0}}{\mu_{0}}L_{SRR} \tag{8}\]  
\[L_{CSRR}=\frac{\mu_{0}}{\epsilon_{0}}C_{SRR} \tag{9}\]  
where \(\mu_{0}\) denotes the permeability of vacuum.  
The resonant frequency is ultimately expressed as follows:  
\[f_{CSRR}=\frac{1}{2\pi\sqrt{C_{\text{CSRR}}L_{\text{CSRR}}}} \tag{10}\]  
The values of the locations and sizes for Figure 6a are obtained with these formulas and CST optimization. The S-parameters results are presented in Figure 6c. In the first passband (24-25.5 GHz), S\({}_{11}\) remains below \(-12\) dB with an insertion loss of 0.5 dB, while in the second passband (26.3-30 GHz), S\({}_{11}\) is maintained below \(-13.5\) dB with the same 0.5 dB insertion loss. By loading the CSRR structure on the back of the filter, effective passband segmentation is achieved. Figure 6c presents the surface current of the filter with the loaded CSRR. At 26 GHz, the CSRR clearly interrupts current flow, which contrasts markedly with the unimpeded current propagation observed at 27.5 GHz within the passband. This provides direct evidence of the CSRR's effect, correlating well with the S-parameter characteristics shown in Figure 6d. Ultimately, the desired dual-band performance has been successfully realized.