## 2 Structure and Design of The BPF
### Analysis of Upper Path I  
Figure 2 shows the equivalent circuit of upper path I. The coupling networks consisting of quarter- and half-wavelength resonators are marked as red (T1-T5), (T1+T2 and T4+T5 are quarter-wavelength, T2+T3+T4 is half-wavelength), which are excited by two shorted quarter-wavelength resonators marked as blue (T6) from port 1 and 2, respectively. T2 and T4 are two end-to-end coupled lines with even and odd impedance of \(Z_{e1}\) and \(Z_{v1}\), electric length of \(\theta_{2}\). Three Microstrip lines T1, T3 and T5 are connected to coupling networks T2 and T4. T1 and T5 are connected to the shorted lines in T2 and T4 with impedance of \(Z_{1}\) and electric length of \(\theta_{1}\) while T3 is used to connect T2 and T4 with impedance of \(Z_{2}\) but two times longer than T1 or T5. The total length of T1 and T2 or T4 and T5 is quarter-wavelength (\(\theta_{1}+\theta_{2}=\pi/2\)). To verify the analysis, we use transfer matrix ABCD which is defined as Equation (1) shows [16]:  
\[ \begin{pmatrix}V_{1}\\ I_{1}\end{pmatrix} = \begin{pmatrix}A&B\\ C&D\end{pmatrix}\begin{pmatrix}V_{2}\\ -I_{2}\end{pmatrix}  \tag{1}\]  
For transmission lines with character impedance \(Z_{0}\) and electrical length \(\theta\)[16]:  
\[ \begin{pmatrix}A&B\\ C&D\end{pmatrix} = \begin{pmatrix}cos\theta&jZ_{0}\sin\theta\\ jY_{0}\sin\theta&\cos\theta\end{pmatrix}  \tag{2}\]  
Thus the response of the coupling network can be calculated by multiplying the ABCD matrixes as Equation (3). Figure 1: Configuration of the proposed bandpass filter (BPF) (**a**) Top layer. (**b**) Middle layer. (**c**) Bottom layer. (**d**) 3D view. (**e**) Front view of the proposed BPF. (**f**) Bottom view of the proposed BPF. \[T=T_{1}\times T_{2}\times T_{3}\times T_{4}\times T_{5} \tag{3}\]  
where  
\[T_{1} = \begin{pmatrix}cos\theta_{1}&jZ_{1}\sin\theta_{1}\\ jY_{1}\sin\theta_{1}&\cos\theta_{1}\end{pmatrix} =T_{5} \tag{4a}\] \[T_{3} = \begin{pmatrix}\cos 2\theta_{1}&jZ_{2}\sin 2\theta_{1}\\ jY_{2}\sin 2\theta_{1}&\cos 2\theta_{1}\end{pmatrix} \] (4b) \[T_{2} =\frac{1}{ab-cf} \begin{pmatrix}a^{2}-df&ac-db\\ ea-bf&ec-b^{2}\end{pmatrix} \] (4c) \[T_{4} =\frac{1}{ab-de} \begin{pmatrix}b^{2}-ce&db-ac\\ fb-ae&fd-a^{2}\end{pmatrix}  \tag{4d}\]  
Figure 2: Equivalent circuit and MATLAB simulated results of resonators in upper path I. (**a**) Equivalent circuit of different sections in upper path I. (**b**) MATLAB simulated results of different sections in upper path I. Here a to f in Equation (4) represent formulas related to the dimensions of coupling network T2 and T4 [17] as Equation (5). \[a =\frac{1}{2}(\cos\theta_{2e}+\cos\theta_{2o}) \tag{5a}\] \[b =\frac{1}{2}(\cos\theta_{2e}-\cos\theta_{2o})\] (5b) \[c =j\frac{1}{2}(Z_{e1}\cos\theta_{2e}+Z_{o1}\cos\theta_{2o})\] (5c) \[d =j\frac{1}{2}(Z_{e1}\cos\theta_{2e}-Z_{o1}\cos\theta_{2o})\] (5d) \[e =j\frac{1}{2}(Y_{e1}\cos\theta_{2e}+Y_{o1}\cos\theta_{2o})\] (5e) \[f =j\frac{1}{2}(Y_{e1}\cos\theta_{2e}-Y_{o1}\cos\theta_{2o}) \tag{5f}\]  
\(\theta_{2e}\) and \(\theta_{2o}\) are the even and odd electric lengths of slot-coupled resonators while \(Z_{e1}\), \(Z_{o1}\) and \(Y_{e1}\), \(Y_{o1}\) are the impedances and admittances. All of these parameters can be obtained using quasi-static analysis [18]. It is noteworthy that if the length of the slots \(\theta_{2}\) is two-third of quarter-wavelength, discriminating coupling is formed. According to [19], a zero at 3f\({}_{0}\) will appear when discriminating coupling occurs, because the voltage distribution is odd on one line and even on the other, leading to null coupling coefficient. Figure 2 shows the simulated result using MATLAB to validate the performance of discriminating. Blue line in Figure 2 demonstrates the simulated result of coupling networks including the connecting microstrip lines (T1-T5) and it is seen that there are two modes and one zero. Thus this line validates that discriminating coupling is effective to introduce a transmission zero at 3f\({}_{0}\) in this design.