Introduction To Modern Planar Transmission Lines. Anand K. Verma

= I₃ and V₂ = V₃. Therefore, from the above equations, the following expression is obtained:

      (3.1.21)

Figure 3.5 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:

(3.1.22)

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:

(3.1.25)

      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