# Compact High-Diectivity Contra-Directional Coupler
## 2 Materials and Methods
### Design of Directional Coupler
\[[T_{\epsilon,\rho}]=[T_{1\epsilon,\rho}][T_{3\epsilon,\rho}][T_{2\epsilon,\rho}] \tag{1}\]  
\[[T_{1\epsilon,\rho}]=\frac{1}{2}\sqrt{\frac{Z_{3\epsilon,\rho}}{Z_{01}}} \begin{bmatrix}1+Z_{01}\cdot(Y_{3\epsilon,\rho}+Y_{1\epsilon,\rho})&1-Z_{01} \cdot(Y_{3\epsilon,\rho}-Y_{1\epsilon,\rho})\\ 1-Z_{01}\cdot(Y_{3\epsilon,\rho}+Y_{1\epsilon,\rho})&1+Z_{01}\cdot(Y_{3 \epsilon,\rho}-Y_{1\epsilon,\rho})\end{bmatrix}  \tag{2}\]  
\[[T_{3\epsilon,\rho}]= \begin{bmatrix}exp(j\theta_{3\epsilon,\rho})&0\\ 0&exp(-j\theta_{3\epsilon,\rho})\end{bmatrix}  \tag{3}\]  
\[[T_{2\epsilon,\rho}]=\frac{1}{2}\sqrt{\frac{Z_{02}}{Z_{3\epsilon,\rho}}} \begin{bmatrix}1+Z_{3\epsilon,\rho}\cdot(Y_{02}+Y_{2\epsilon,\rho})&1-Z_{3 \epsilon,\rho}\cdot(Y_{02}-Y_{2\epsilon,\rho})\\ 1-Z_{3\epsilon,\rho}\cdot(Y_{02}+Y_{2\epsilon,\rho})&1+Z_{3 \epsilon,\rho}\cdot(Y_{02}-Y_{2\epsilon,\rho})\end{bmatrix}  \tag{4}\]  
\[\Re T_{11\epsilon,\rho}=\frac{1}{4\sqrt{n}}[(n+1)\cdot A_{\epsilon,\rho}-B_{ \epsilon,\rho}\cdot(b_{1\epsilon,\rho}+nb_{2\epsilon,\rho})] \tag{5}\]  
\[\Im T_{11\epsilon,\rho}=\frac{1}{4}[(b_{1\epsilon,\rho}+b_{2\epsilon,\rho}) \cdot A_{\epsilon,\rho}+B_{\epsilon,\rho}\cdot(1-b_{1\epsilon,\rho}b_{2 \epsilon,\rho})+C_{\epsilon,\rho}] \tag{6}\]  
\[\Re T_{12\epsilon,\rho}=\frac{1}{4\sqrt{n}}[(n-1)\cdot A_{\epsilon,\rho}+B_{ \epsilon,\rho}\cdot(b_{1\epsilon,\rho}-nb_{2\epsilon,\rho})] \tag{7}\]  
\[\Im T_{12\epsilon,\rho}=\frac{1}{4}[(b_{1\epsilon,\rho}+b_{2\epsilon,\rho}) \cdot A_{\epsilon,\rho}-B_{\epsilon,\rho}\cdot(1+b_{1\epsilon,\rho}b_{2 \epsilon,\rho})+C_{\epsilon,\rho}] \tag{8}\]  
\[A_{\epsilon,\rho}=2 cos\theta_{3\epsilon,\rho} \tag{9}\]  
\[B_{\epsilon,\rho}=2z_{3\epsilon,\rho}sin\theta_{3\epsilon,\rho}\,\ n=\frac{Z_{02}}{Z_{01}} \tag{10}\]  
\[C_{\epsilon,\rho}=\frac{2}{z_{3\epsilon,\rho}}sin\theta_{3\epsilon,\rho} \tag{11}\]  
With algebraic manipulation of Equations (5)-(11) and the contra-directional condition of Equations (12) and (13) [33], we can express the even/odd and input/output susceptances \(b_{ie,\rho}\) as in Equations (14)-(17), Appendix A. We assumed the input and output impedances are identical, where \(n=1\). \[T_{11\epsilon}=T_{11\epsilon} \tag{12}\]  
\[T_{12\epsilon}=-T_{12\epsilon},\ \Re T_{12\epsilon}=\Re T_{12\epsilon}=0 \tag{13}\]  
\[b_{1\epsilon}=\frac{A_{\epsilon}+\sqrt{A_{\epsilon}^{2}-B_{\epsilon}\cdot(B_{ \rho}-C_{\epsilon})}}{B_{\epsilon}},\ \ n=1 \tag{14}\]  
\[\Im T_{11\epsilon,\rho}=\frac{1}{4}[(b_{1\epsilon,\rho}+b_{2\epsilon,\rho}) \cdot A_{\epsilon,\rho}+B_{\epsilon,\rho}\cdot(1-b_{1\epsilon,\rho}b_{2 \epsilon,\rho})+C_{\epsilon,\rho}] \tag{6}\]0  
\[\Im T_{11\epsilon,\rho}=\frac{1}{4}[(b_{1\epsilon,\rho}+b_{2\epsilon,\rho}) \cdot A_{\epsilon,\rho}+B_{\epsilon,\rho}\cdot(1-b_{1\epsilon,\rho}b_{2 \epsilon,\rho})+C_{\epsilon,\rho}] \tag{6}\]1  
\[\Im T_{11\epsilon,\rho}=\frac{1}{4}[(b_{1\epsilon,\rho}+b_{2\epsilon,\rho}) \cdot A_{\epsilon,\rho}+B_{\epsilon,\rho}\cdot(1-b_{1\epsilon,\rho}b_{2 \epsilon,\rho})+C_{\epsilon,\rho}] \tag{6}\]2  
With the susceptances \(b_{ie,\rho}\), we can construct the lumped-distributed equivalent circuit and its even- and odd-mode equivalent circuits, as in Figure 3, where \(C_{ie}\) is the even-mode equivalent capacitance as in Equation (18), and \(C_{io}\prime\) is the internal coupling capacitance between two parallel coupled lines. The internal coupling capacitance is expressed with even- and odd-mode capacitances, as in Equation (20).