Anand K. Verma - Introduction To Modern Planar Transmission Lines

To obtain the transmission matrix description of a two‐port network, the output voltage and current are treated as the independent variables. The following expressions relate to the input and output voltage and current of the two‐port network shown in Fig (3.4):

(3.1.17)

Figure 3.4 Two‐port network for transmission parameter.

These expressions can be written in the matrix form,

(3.1.18)

In the case of the [Z] and [Y] parameters, the positive current I 2enters the port, while in the above network defining the [ABCD] parameter in Fig (3.4), the output current I 2 leaving the port is taken as positive [B.1, B.3]. It is an input to the next circuit block, as shown in Fig (3.5). However, like defining the [Z] and [Y] parameters, to define the [ABCD] parameter current, I 2could be taken as the current entering the output port. In this case, I 2in equation (3.1.18) is replaced by (−I 2) [B.1, B.4].

The matrix elements A, B, C, D can be determined from the open and short circuit conditions at the output port. When the output is open‐circuited, I 2= 0. Equation (3.1.17)provides the parameter‐A and C:

(3.1.19)

The parameter A is the voltage ratio that is a reciprocal of the voltage gain. The parameter C is the trans‐admittance of a network. It relates the output voltage of a network to its input current source.

When the output is short‐circuited, V 2= 0. Equation (3.1.17)again provides the parameters‐B and D:

(3.1.20)

The parameter B is the trans‐impedance of a network. It provides the output current when the input of a network is excited by the voltage source. The parameter D is the current ratio giving a reciprocal of the current gain of a network.

Fig (3.5)demonstrates the usefulness of the transmission parameters to obtain an equivalent [ABCD] parameter of the cascaded networks. The [ABCD] parameters for the first and the second network are written as

At the junction of two networks, I 2= I 3and V 2= V 3. 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.

Determine the [ABCD] parameters of the series impedance as shown in Fig (3.6).

The output port is open‐circuited, I 2= 0. Therefore, equation (3.1.17)provides V 1= A V 2and I 1= CV 2. For the port 2 of Fig (3.6)open‐circuited, I 2= 0, V 1= V 2and I 2= I 1= 0. On comparing these equations, the computed parameters are A = 1 and C = 0.

For the output port is short‐circuited, V 2= 0. Therefore, equation (3.1.17)helps to get, V 1= BI 2and I 1= DI 2. Using Fig (3.6)shows, V 2= 0, V 1= ZI 2and I 1= I 2. 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)

Determine the [ABCD] parameters of a shunt admittance shown in Fig (3.7).

The output port‐2 is open‐circuited, I 2= 0. Therefore, from matrix equation (3.1.17): V 1= A V 2and I 1= CV 2.At the open‐circuited output port 2: I 2= 0, V 1= V 2and I 1= Y V 2. On comparing these equations: A = 1 and C = Y. At the short‐circuited output port 2: V 2= 0, V 1= BI 2and I 1= DI 2.Using Fig (3.7), for V 2= 0, V 1= 0 and I 1= I 2. 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.

Determine the [ABCD] parameters of a section of transmission line shown in Fig (3.3).

Equations (2.1.79)of chapter 2provide 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