3.2.2 Classical [ABCD] matrix analysis method

The analysis of traditional transmission line filters with asymmetric structures is no longer within the application scope of classical odd-even-mode analysis method. To overcome this issue, the ABCD matrix method is employed to approach the overall transmission ABCD matrix; the Yin is then derived from the ABCD matrix of the overall structure. The ABCD matrix of several typical transmission line models and the ABCD matrix of several conventional circuit elements are depicted in Figure 5.

#### 3.2.3 Analysis of parallel-coupled lines

The analysis of parallel-coupled lines is more complicated than that of series/ shunt transmission lines. One of the reliable ways is to analyze the parallel-coupled lines as a four-port component, and parameters of parallel-coupled lines are shown in Figure 6.

Different paralleled coupling conditions and the position of the in/out port correspond to varied initial conditions. Therefore, the Z matrix of parallel-coupled lines can be solved according to this initial condition. The four-port impedance matrix is given as follows.

#### Figure 5.

Classical transmission line structure and their ABCD matrix.

Figure 6. Electrical diagram of parallel-coupled line.

$$
\begin{bmatrix} U\_1 \\ U\_2 \\ U\_3 \\ U\_4 \end{bmatrix} = \begin{bmatrix} Z\_{11} & Z\_{12} & Z\_{13} & Z\_{14} \\ Z\_{21} & Z\_{22} & Z\_{23} & Z\_{24} \\ Z\_{31} & Z\_{32} & Z\_{33} & Z\_{34} \\ Z\_{41} & Z\_{42} & Z\_{43} & Z\_{44} \end{bmatrix} \begin{bmatrix} I\_1 \\ I\_2 \\ I\_3 \\ I\_4 \end{bmatrix} \tag{14}$$

where

3.2.1 Classical even-odd-mode analysis method

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the symmetrical planes are considered to be open.

3.2.2 Classical [ABCD] matrix analysis method

3.2.3 Analysis of parallel-coupled lines

Classical transmission line structure and their ABCD matrix.

Figure 5.

in Figure 6.

Figure 5.

84

matrix is given as follows.

Since the odd-mode resonant frequencies of the symmetrical structure are orthogonal to the even-mode resonant frequencies, the whole transmission line model can be divided into odd-mode and even-mode circuits. Therefore, the resonant frequencies are then derived separately, which dramatically reduces the computation of resonant modes. It is worth noting that with odd-mode excitation, the symmetrical planes are considered to be grounded and with even-mode excitation,

Even-/odd-mode input admittance can be obtained from the even�/odd-mode

The analysis of traditional transmission line filters with asymmetric structures is no longer within the application scope of classical odd-even-mode analysis method. To overcome this issue, the ABCD matrix method is employed to approach the overall transmission ABCD matrix; the Yin is then derived from the ABCD matrix of the overall structure. The ABCD matrix of several typical transmission line models and the ABCD matrix of several conventional circuit elements are depicted in

The analysis of parallel-coupled lines is more complicated than that of series/ shunt transmission lines. One of the reliable ways is to analyze the parallel-coupled lines as a four-port component, and parameters of parallel-coupled lines are shown

Different paralleled coupling conditions and the position of the in/out port correspond to varied initial conditions. Therefore, the Z matrix of parallel-coupled lines can be solved according to this initial condition. The four-port impedance

Im ð Þ¼ Yine 0 (12) Im ð Þ¼ Yino 0 (13)

equivalent circuit, and Eq. (11) can be replaced by the following equations:

$$Z\_{11} = Z\_{22} = Z\_{33} = Z\_{44} = -\mathbf{j}\frac{Z\_{0\epsilon} + Z\_{0\sigma}}{2}\cot\theta\tag{15}$$

$$Z\_{12} = Z\_{21} = Z\_{34} = Z\_{43} = -\mathbf{j}\frac{Z\_{0\epsilon} - Z\_{0\sigma}}{2}\cot\theta\tag{16}$$

$$Z\_{13} = Z\_{31} = Z\_{24} = Z\_{42} = -\mathbf{j}\frac{Z\_{0\epsilon} - Z\_{0\epsilon}}{2}\csc\theta \tag{17}$$

$$Z\_{14} = Z\_{41} = Z\_{23} = Z\_{32} = -\mathbf{j}\frac{Z\_{0\epsilon} + Z\_{0\nu}}{2} \text{csc}\theta \tag{18}$$
