# Compact High-Diectivity Contra-Directional Coupler
## 2 Materials and Methods
### Implementation of Directional Coupler  
We implemented the coupler with RF35 HTC 20 mil 1 oz substrate with the described design method in the frequency covering 13 to 14.5 GHz, which is used in the uplink of Ku-band satellite communication. A width of 1 mm was selected for the four stubs, and the corresponding admittance \(Y_{s}\) was calculated with the transmission line calculation software from the open-source circuit simulator, Qucs [30]. For the initial design parameters of the parallel coupled line, we chose 25 dB coupling, and 60\({}^{\circ}\) electrical length. By Equation (32), we calculated the even- and odd-mode impedances and obtained the physical dimensions of the microstrip line coupler using the transmission line calculation software Qucs. \[Z_{e}=\sqrt{\frac{1+C}{1-C}}Z_{0},\;\;Z_{o}=\sqrt{\frac{1-C}{1+C}}Z_{0},\;\; where\;\;C=\frac{10^{C_{dB}}}{20}. \tag{32}\]  
The even- and odd-mode operations in Figure 1 are simulated in the circuit simulator, as in Figure 6. For the odd-mode excitation, out-of-phase ideal couplers (Coupler1 and Coupler2) in the upper circuit in Figure 6 are used at the input and output of the microstrip parallel coupled line under test. For the even-mode excitation, in-phase couplers are placed as in the lower circuit of Figure 6. \begin{table}
\begin{tabular}{l l} \hline \hline
1. & Start \\ \hline
2. & Determine the coupling factor and electrical length of CPL \\  & \(C=\frac{10^{C_{dB}}}{20}\theta_{3}<\frac{\lambda}{4}\) \\ \hline
3. & Calculate the even- and odd-mode impedance of CPL \\  & \(Z_{e}=\sqrt{\frac{1+C}{1-C}}Z_{0},\;\;Z_{o}=\sqrt{\frac{1-C}{1+C}}Z_{0}\) \\ \hline
4. & Simulate the even- and odd-mode phase of the CPL \\  & \(\theta_{3e}\), \(\theta_{3o}\) \\ \hline
5. & Calculate the normalized input/output, even- and odd-mode susceptances \\  & \(b_{1e}=b_{2e}=\frac{A_{e}+\sqrt{A_{e}^{2}-B_{e}-C_{e}}}{B_{e}}\), \(b_{1o}=b_{2o}=\frac{B_{e}}{B_{e}}b_{1e}-\frac{A_{e}-A_{e}}{B_{e}}\) \\ \hline
6. & Calculate the lumped capacitances \\  & \(C_{1e}=C_{2e}=\frac{b_{e}}{\omega_{1}\omega_{2}}\), \(C_{1o}^{\prime}=C_{2o}^{\prime}=\frac{C_{1e}-C_{2e}}{2}\) \\ \hline
7. & Calculate the electrical length of the distributed stubs, given Ys \\  & \(\theta_{1o}=\theta_{2o}=solve\Big{(}C_{ig}(\theta_{io})=C_{is}(\theta_{io}) \Big{)}\), \(\theta_{1e}=\theta_{2e}=atan\Big{(}\frac{\omega_{1}C_{io}-Y_{s}tan(\theta_{io}) }{Y_{s}}\Big{)}\) \\ \hline
8. & Calculate the physical dimensions of the stub lengths and the gap \\  & \(l_{1o}=l_{2o}=\frac{\theta_{3e}}{2\pi}\lambda,l_{1e}=l_{2e}=\frac{\theta_{3e}} {2\pi}\lambda,g_{i}=s_{cpl}-2l_{io}\) \\ \hline
9. & Stop \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The framework of the design process. The simulation results are shown in Figure 7. The even-mode electrical length is longer than the odd-mode length. The ratio between them is about 95.5% over the wide frequency range. With the calculated even- and odd-mode impedances, the simulated electrical lengths, and Equations (14)-(17), the susceptances, \(b_{ie}\) and \(b_{io}\), are calculated as Equation (33). With the calculated susceptances and Equation (18) through Equation (31), the initial dimensions of the physical coupler in Figure 5 (\(l_{ie}\), \(l_{io}\), and \(g_{i}\)) are calculated as Equation (34). \[b_{ie}=1.1142,\ b_{io}=1.3339 \tag{33}\]  
\[l_{ie}=0.291\ \text{mm},\ \ l_{io}=1.539\ \text{mm},\ \ g_{i}=0.6355\ \text{mm} \tag{34}\]  
Figure 6: The circuit simulation of the coupler’s even- and odd-mode electrical lengths. Figure 7: The simulated coupled line: (**a**) even- and odd-mode electrical lengths; (**b**) the ratio of even-mode and odd-mode electrical lengths. From the microstrip coupled-line coupler dimensions, and the initial physical dimensions in Equation (34), the coupler was laid out and simulated with a 3D electromagnetic simulator, MWS, from CST. The additional 50 Ohm lines were added for measurement, as in Figure 8a. There are additional discontinuities from the initial design, such as the additional microstrip around and across the physical layout.