Cascading of two networks to get one equivalent network.
Finally, two cascaded networks can be replaced by one equivalent 2‐port network having equivalent [ABCD] parameter. It is given by the following expression:
Expression (3.1.22) can be extended to the cascading of N‐networks by multiplying the individual matrix of each network.
Example 3.5
Determine the [ABCD] parameters of the series impedance as shown in Fig (3.6).
Solution
The output port is open‐circuited, I2 = 0. Therefore, equation (3.1.17) provides V1 = A V2 and I1 = CV2. For the port 2 of Fig (3.6) open‐circuited, I2 = 0, V1 = V2 and I2 = I1 = 0. On comparing these equations, the computed parameters are A = 1 and C = 0.
For the output port is short‐circuited, V2 = 0. Therefore, equation (3.1.17) helps to get, V1 = BI2 and I1 = DI2. Using Fig (3.6) shows, V2 = 0, V1 = ZI2 and I1 = I2. The comparison of these equations provide B = Z and D = 1.
Thus, the [ABCD] matrix of series impedance is written as
Figure 3.6 Series impedance.
(3.1.23)
Example 3.6
Determine the [ABCD] parameters of a shunt admittance shown in Fig (3.7).
Solution
The output port‐2 is open‐circuited, I2 = 0. Therefore, from matrix equation (3.1.17): V1 = A V2 and I1 = CV2.At the open‐circuited output port 2: I2 = 0, V1 = V2 and I1 = Y V2. On comparing these equations: A = 1 and C = Y. At the short‐circuited output port 2: V2 = 0, V1 = BI2 and I1 = DI2.Using Fig (3.7), for V2 = 0, V1 = 0 and I1 = I2. On comparing these equations: B = 0 and D = 1. Finally, the [ABCD] matrix of shunt admittance can be written as
(3.1.24)
The [ABCD] matrix could be easily evaluated for the L, T, and π networks, shown in Fig (3.8). The [ABCD] matrix of each element is known and the complete circuit is a cascading of the elements.
Figure 3.7 Shunt admittance.
Figure 3.8 Basic networks.
Example 3.7
Determine the [ABCD] parameters of a section of transmission line shown in Fig (3.3).
Solution
Equations (2.1.79) of chapter 2 provide the voltage and current waves on a transmission line:
The V+ and V− are the amplitudes of the forward and reflected waves, respectively. For convenience, the distance x is measured from the port‐2. The voltage and current at the port‐2 are
The amplitudes of the forward and reflected voltages in terms of the port voltage and port current are
The voltage and current on a transmission line can be written as
The voltage and current at the input port‐1 are obtained for x = −ℓ:
Above equations can be written in the matrix form:
The [ABCD] parameters of the lossy and lossless transmission line sections are given by equation (3.1.25a) and equation (3.1.25b), respectively:
The above example can be further extended to a network of several cascaded transmission line sections having different ℓ, Z0, and γ. The overall [ABCD] parameter of the multisection transmission line can be obtained by a multiplication of the [ABCD] matrix of each line section. The line sections can be attached to the series and the shunt lumped elements. Even in such cases, one can find the overall [ABCD] parameter of a complete network. The input impedance, output impedance, Thevenin and Norton equivalent circuits,
