# Compact High-Diectivity Contra-Directional Coupler
## 2 Materials and Methods  
### Design of Directional Coupler  
Microstrip parallel coupled lines are commonly used for directional couplers due to their simple structures and ease of use with other circuits. The parallel coupled-line coupler operation can be decomposed into even- and odd-mode operations, as in Figure 1. However, the microstrip transmission line has an inherently inhomogeneous structure, where dielectric material exists only between the signal line and the ground. Therefore, the even- and odd-mode operations have a different effective dielectric constant, resulting in different phase velocities and causing directivity degradation. Many works have been carried out to compensate for the difference. However, they usually have a structure that is vulnerable to high power or of a cumbersome size for integration with other circuits, as well as additional losses, as introduced before. Another way of implementing a directional coupler is using ring-type four-port structures, such as well-known branch line couplers or rat race couplers. However, these structures have limitations in implementing arbitrary coupling, especially weak coupling. In [33], the general ring-type four-port with lumped-distributed elements is well analyzed with even- and odd-modes of the T matrix for contra-, trans-, and co-directional coupler conditions. However, the lumped element is not readily available in microwave and  
Figure 1: Parallel coupled-line coupler operation: (**a**) even-mode operation; (**b**) odd-mode operation. millimeter waves. Moreover, there is coupling between transmission lines as the circuit size becomes smaller when the frequency increases. In this study, the original general ring-type four-port model in Figure 2a,b is modified by replacing the isolated four transmission lines (\(Z_{3},\theta_{3}\)) with a parallel coupled line (\(Z_{3e,\theta},\theta_{3e,\rho}\)) as in Figure 2c,d. Instead of compensating for the difference in even- and odd-mode phase velocities as in many other works, we used the different even- and odd-mode electrical lengths obtained by a circuit simulation of the coupled line. In Figure 2, \(Z_{i}\) and \(\theta_{i}\) are the characteristic impedances and electrical lengths of the transmission lines; \(X_{12}\), \(B_{i}\), and \(B_{pi}\) are the lumped reactance and susceptances; \(Z_{0i}\) is the port impedance; \(a_{i}\) and \(b_{i}\) are power waves for the T matrix analysis; \(Z_{ie,\rho}\)\((i=1,2)\) and \(\theta_{ie,\rho}\)\((i=1,2)\) are the impedances introduced by parallel coupled lumped elements. \(B_{i}\); and \(Z_{3e,\rho}\) are the even- and odd-mode impedances and electrical lengths of the replaced parallel coupled line. In the modified version of the general ring-type four-port model, the series reactances \(X_{12}\) are removed, and the original isolated transmission lines are replaced with a coupled line with different even- and odd-mode electrical lengths. The corresponding T matrix for the even- and odd-mode equivalent circuit in Figure 2d is modified as in Equations (1)-(4) from the original equations, where \(Y_{ie,\rho}\)\((i=1,2)\) are the admittance form of \(Z_{ie,\rho}\); \(Y_{3e,\rho}\) is the even- and odd-mode admittances of the parallel coupled line; and \(Y_{02}\) is the admittance form of \(Z_{02}\). Each element in the T matrix is updated in Equations (5)-(11), where \(b_{1e,\rho}\) is the normalized input even- or odd-mode susceptance,  
Figure 2: Equivalent circuit of: (**a**) the original general ring-type four-port with lumped-distributed elements [33]; (**b**) the original even- and odd-mode two-port equivalent circuit; (**c**) the modified general ring-type four-port with lumped-distributed elements [33]; (**d**) the modified even- and odd-mode two-port equivalent circuit. and \(b_{2\epsilon,\rho}\) is the normalized output even- or odd-mode susceptance; \(y_{ie,\rho}=jb_{ie,\rho}\), where \(y_{ie,\rho}\) is the normalization of \(Y_{ie,\rho}\); \(y_{ie,\rho}=Y_{ie,\rho}/\sqrt{Y_{01}Y_{02}}\).