# A Balanced Symmetrical Branch-Line Microstrip Coupler for 5G Applications
## 2 Design of Resonator, Filters and Coupler  
Figure 1 shows a symmetrical stub-loaded transmission line with its equivalent LC circuit. The stub is a low-impedance cell. The lines with the physical lengths \(\mathrm{l_{a}}\) and \(\mathrm{l_{b}}\) are replaced by the inductors \(\mathrm{L_{a}}\) and \(\mathrm{L_{b}}\), respectively. Also, the open end of the shunt stub is presented by the \(\mathrm{C_{o}}\) capacitor. The equivalent circuit from \(\mathrm{T_{1}}\) to \(\mathrm{T_{2}}\) is presented by two inductors with two equivalent impedances of 0.5\(\mathrm{jx}\) and a capacitor with an equivalent impedance of jB. For a short-line \(\mathrm{l_{c}}\) (\(\mathrm{l_{c}}<(\lambda_{\mathrm{g}}/8)\)) with a characteristic impedance of \(Z_{c}\), the values of \(x\) and B can be approximated by [22]:  
\[x\approx Z_{c}(\frac{2\pi\,l_{c}}{\lambda_{g}})\ \&\ B\approx\frac{1}{Z_{c}}( \frac{2\pi\,l_{c}}{\lambda_{g}}) \tag{1}\]  
where \(\lambda_{\mathrm{g}}\) and \(\mathrm{l_{c}}\) are the guided wavelength and the physical length from \(\mathrm{T_{1}}\) to \(\mathrm{T_{2}}\), respectively. The equivalent impedance of the shunt stub (\(Z_{\mathrm{Sh}}\)) can be calculated as follows:  
\[ \begin{array}{ll}Z_{Sh}&=\frac{1}{\frac{1}{(\frac{1}{2\pi\mathrm{C_{o}}}+i \omega(L_{a}+0.5x))+j\omega(L_{a}+0.5x))}+j\omega(L_{a}+0.5x)}\Rightarrow\\ Z_{Sh}&=\frac{1-\omega^{2}\mathrm{C_{o}}(L_{a}+0.5x)+j\omega(L_{a}+0.5x)[j \omega C_{o}+j\omega B(1-\omega^{2}\mathrm{C_{o}}(L_{a}+0.5x))]}{[j\omega \mathrm{C_{o}}+j\omega B(1-\omega^{2}\mathrm{C_{o}}(L_{a}+0.5x))]}\\ \end{array}  \tag{2}\]  
where \(\omega\) is an angular frequency and \(\mathrm{L_{a}}\) and \(\mathrm{C_{o}}\) are defined before and presented in Figure 1. Substituting Equation (1) into Equation (2) results in:  
\[Z_{Sh}=\frac{1-\omega^{2}\mathrm{C_{o}}(L_{a}+Z_{c}(\frac{\pi\,l_{c}}{\lambda _{g}}))+j\omega(L_{a}+Z_{c}(\frac{\pi\,l_{c}}{\lambda_{g}}))[j\omega C_{o}+j \omega(\frac{2\pi\,l_{c}}{\lambda_{g}})(1-\omega^{2}\mathrm{C_{o}}(L_{a}+Z_{ c}(\frac{\pi\,l_{c}}{\lambda_{g}}))]}}{j\omega C_{o}+j\omega(\frac{2\pi\,l_{c}}{ \lambda_{g}})(1-\omega^{2}\mathrm{C_{o}}(L_{a}+Z_{c}(\frac{\pi\,l_{c}}{ \lambda_{g}}))]} \tag{3}\]  
The ABCD matrix of the proposed resonator (\(\mathrm{T_{R}}\)) can be obtained through [23]:  
\[T_{R}=\left[ \begin{array}{cc}A_{R}&B_{R}\\ C_{R}&D_{R}\end{array} \right]=\left[ \begin{array}{cc}1&j\omega L_{b}\\ 0&1\end{array} \right]\times\left[ \begin{array}{cc}1&0\\ \frac{1}{Z_{\mathrm{Sh}}}&1\end{array} \right]\times\left[ \begin{array}{cc}1&j \omega L_{b}\\ 0&1\end{array} \right]=\left[ \begin{array}{cc}1+\frac{j\omega L_{b}}{Z_{ \mathrm{Sh}}}&j\omega L_{b}(2+\frac{j\omega L_{b}}{Z_{\mathrm{Sh}}})\\ \frac{1}{Z_{\mathrm{Sh}}}&1+\frac{j\omega L_{b}}{Z_{\mathrm{Sh}}}\end{array} \right] \tag{4}\]  
Figure 1: Layout and LC circuit of the proposed resonator. Using the calculated \(\mathrm{T_{R}}\), we can obtain the scattering matrix of the proposed resonator (\(\mathrm{S_{R}}\)) as follows:  
\[ \begin{array}{c}S_{R}=\left[\begin{array}{cc}S_{11}&S_{12}\\ S_{21}&S_{22}\end{array}\right]=\left[\begin{array}{cc}\frac{A_{R}+B_{R}/Z_{0}- C_{R}Z_{0}-D_{R}}{A_{R}+B_{R}/Z_{0}+C_{R}Z_{0}+D_{R}}&\frac{2(A_{R}D_{R}-B_{R}C_{R})}{A _{R}+B_{R}/Z_{0}+C_{R}Z_{0}+D_{R}}\\ \end{array}\right]\Rightarrow\\ S_{11}=S_{22}=\frac{\frac{\frac{i\omega_{L}}{Z_{0}}(2+\frac{i\omega_{L}}{Z_{ 0}})-\frac{Z_{0}}{Z_{0}}}-\frac{Z_{0}}{Z_{0}}}{2+\frac{2i\omega_{L}^{2}+Z_{0}} {Z_{0}}+\frac{i\omega_{L}}{Z_{0}}(2+\frac{i\omega_{L}}{Z_{0}})}\\ S_{12}=S_{21}=\frac{2}{2+\frac{2i\omega_{L}^{2}+Z_{0}}{Z_{0}}+\frac{2}{Z_{0}} +\frac{i\omega_{L}}{Z_{0}}(2+\frac{i\omega_{L}}{Z_{0}})}\end{array} \right] \Rightarrow \tag{5}\]  
In Equation (5), \(Z_{0}\) is the impedance of the terminals and \(\mathrm{A_{R}}\), \(\mathrm{B_{R}}\), \(\mathrm{C_{R}}\) and \(\mathrm{D_{R}}\) are the transfer parameters. Also, \(\mathrm{S_{11}}\), \(\mathrm{S_{21}}\), \(\mathrm{S_{12}}\) and \(S_{22}\) are the scattering parameters.