# A Filtering Switch Made by an Improved Coupled Microstrip Line
## 2 Materials and Methods  
We first perform a theoretical analysis of the coupled microstrip line given in Figure 2.  
Figure 2 shows the equivalent schematic diagram of the coupled microstrip lines required to compose the filtering switch, where the parallel-coupled feed lines exhibit bandpass filtering effects, and where the coupled-line stub-loaded resonators mainly produce transmission zeros to enhance the passband selectivity in the filtered state. First, the Z-parameter matrix of the parallel-coupled feed line is obtained primarily based on the coupled microstrip line's parity mode analysis principle:  
\[Z= \begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix} = \tag{1}\]  
\[ \begin{bmatrix}-\dfrac{j}{2}(Z_{o\text{e}1}+Z_{o\text{e}1})cot\theta_{1}&- \dfrac{j}{2}(Z_{o\text{e}1}-Z_{o\text{e}1})csc\theta_{1}\\ -\dfrac{j}{2}(Z_{o\text{e}1}-Z_{o\text{e}1})csc\theta_{1}&-\dfrac{j}{2}(Z_{o \text{e}1}+Z_{o\text{e}1})cot\theta_{1}\end{bmatrix}  \tag{2}\]  
It is easy to see that the enter impedance \(Z_{1}\) of this coupled microstrip line is:  
\[Z_{1}=\sqrt{Z_{11}^{2}-\dfrac{Z_{11}Z_{21}^{2}}{Z_{22}}}= \tag{3}\]  
\[\dfrac{\sqrt{(Z_{o\text{e}1}-Z_{o\text{e}1})^{2}-(Z_{o\text{e}1}+Z_{o\text{e} 1})^{2}cos\theta_{1}^{2}}}{2 sint\theta_{1}} \tag{4}\]  
Figure 2: Parallel-coupled feed lines (**left**); coupled-line stub-loaded resonators (**right**). The \(ABCD\) matrix \(M_{1}\) of the parallel-coupled feed lines can be obtained from the \(Z\)-parameter matrix:  
\[M_{1}= \begin{bmatrix}A_{1}&B_{1}\\ C_{1}&D_{1}\end{bmatrix} = \tag{5}\]  
\[ \begin{bmatrix}\frac{Z_{11}}{Z_{21}}&\frac{Z_{11}Z_{22}-Z_{12}Z_{21}}{Z_{21}}\\ 1/Z_{21}&Z_{22}/Z_{21}\end{bmatrix}  \tag{6}\]  
The \(ABCD\) matrix \(M_{2}\) of the coupled-line stub-load resonators (CLSs) is:  
\[M_{2}= \begin{bmatrix}A_{2}&B_{2}\\ C_{2}&D_{2}\end{bmatrix} = \tag{7}\]  
\[ \begin{bmatrix}\frac{Z_{o\sigma 2}+Z_{o\omega 2}}{Z_{o\sigma 2}-Z_{o\omega 2}} cos^{2}\theta_{2}&\frac{j\left(Z_{o\sigma 2}+Z_{o\omega 2}\right)^{2}}{Z_{ o\omega 2}-Z_{o\omega 2}}sin^{2}\theta_{2}-\frac{j2Z_{o\sigma 2}Z_{o\omega 2}}{(Z_{ o\sigma 2}-Z_{o\omega 2})sin\theta_{2}}\\ \frac{j2}{Z_{o\sigma 2}-Z_{o\omega 2}}&\frac{Z_{o\sigma 2}+Z_{o\omega 2}}{Z_{ o\omega 2}-Z_{o\omega 2}}cos^{2}\theta_{2}\end{bmatrix}  \tag{8}\]  
The equal impedance \(Z_{2}\) of the coupled-line stub-load resonators can be acquired from the \(M_{2}\) matrix:  
\[Z_{2}=\frac{B_{2}}{D_{2}}= \tag{9}\]  
\[\frac{j}{2}(Z_{o\sigma 2}+Z_{o\omega 2})tan\theta_{2}-\frac{j2Z_{o\sigma 2}Z_{ o\omega 2}}{(Z_{o\sigma 2}+Z_{o\omega 2})sin\theta_{2}cos\theta_{2}} \tag{10}\]  
According to [13], the transmission zeros (\(f_{a}\), \(f_{b}\)) of the bandpass filter in the skip state are acquired from the equal impedance at \(Z_{2}\) = 0. \[f_{a}=\frac{2}{\pi}arcsin\sqrt{1-K_{2}^{2}}<f_{0} \tag{11}\]  
\[f_{b}=2f_{0}-f_{a}>f_{0} \tag{12}\]  
where \(f_{0}\) is the center frequency at which the device operates, and \(K_{2}\) is the coupling coefficient of the coupled-line stub-load resonators. Compared with the traditional quarter-wavelength stub, the coupled-line stub-load resonators (CLSs) can produce two symmetric transmission zeros to improve the passband selectivity. Compared to stub-loaded resonators, coupled-line stub-load resonators (CLSs) can exhibit better passband effects at the same operating frequency. Figure 3 shows the equivalent schematic of the filter switch based on the above analysis of coupled microstrip lines. The physical structure of the proposed filtering switch consists of two coupled microstrip lines, a parallel-coupled feed line, and coupled-line stub-load resonators (CLSs), where the coupled microstrip line \(TL_{1}\) is connected to a PIN diode to achieve a good on/off control effect. The equivalent schematic diagram that makes up this filtering switch is given in Figure 3.  
Figure 4 depicts the equal circuit used for the filtering switch's PIN diode and the bias's specific states.