Coupled transmission line

A coupled transmission line is described by two identical transmission line ABCD-matrices, one for the even mode (or common mode) and one for the odd mode (or differential mode). Because the coupled lines are a symmetrical 3-line system, the matrices are completely independent of each other. Therefore, its Y-parameters write as follows.

$$Y_{11} = Y_{22} = Y_{33} = Y_{44} = \frac{1}{2\cdot Z_{L,e} \cdot \tanh\left(\gamma_e\cdot l\right)} + \frac{1}{2\cdot Z_{L,o} \cdot \tanh\left(\gamma_o\cdot l\right)}$$ (9.216)

$$Y_{12} = Y_{21} = Y_{34} = Y_{43} = \frac{-1}{2\cdot Z_{L,e} \cdot \sinh\left(\gamma_e\cdot l\right)} + \frac{-1}{2\cdot Z_{L,o} \cdot \sinh\left(\gamma_o\cdot l\right)}$$ (9.217)

$$Y_{13} = Y_{31} = Y_{24} = Y_{42} = \frac{-1}{2\cdot Z_{L,e} \cdot \sinh\left(\gamma_e\cdot l\right)} + \frac{1}{2\cdot Z_{L,o} \cdot \sinh\left(\gamma_o\cdot l\right)}$$ (9.218)

$$Y_{14} = Y_{41} = Y_{23} = Y_{32} = \frac{1}{2\cdot Z_{L,e} \cdot \tanh\left(\gamma_e\cdot l\right)} + \frac{-1}{2\cdot Z_{L,o} \cdot \tanh\left(\gamma_o\cdot l\right)}$$ (9.219)

The S-parameters (according to the port numbering in fig. 9.12) are as followed [11].

reflection coefficients

$$S_{11} = S_{22} = S_{33} = S_{44} = X_e + X_o$$ (9.220)

through paths

$$S_{12} = S_{21} = S_{34} = S_{43} = Y_e + Y_o$$ (9.221)

coupled paths

$$S_{14} = S_{41} = S_{23} = S_{32} = X_e - X_o$$ (9.222)

isolated paths

$$S_{13} = S_{31} = S_{24} = S_{42} = Y_e - Y_o$$ (9.223)

with the denominator

$$D_{e,o} = 2\cdot Z_{L,e,o}\cdot Z_0\cdot \cosh(\gamma_{e,o}\cdot l) + \left(Z_{L,e,o}^2 + Z_0^2\right)\cdot \sinh\left(\gamma_{e,o}\cdot l\right)$$ (9.224)

and

$$X_{e,o} = \frac{\left(Z_{L,e,o}^2 - Z_0^2\right)\cdot \sinh\left(\gamma_{e,o}\cdot l\right)}{2\cdot D_{e,o}}$$ (9.225)

$$Y_{e,o} = \frac{Z_{L,e,o}\cdot Z_0}{D_{e,o}}$$ (9.226)

Figure 9.12: coupled transmission line