Anand K. Verma - Introduction To Modern Planar Transmission Lines

The impedance matrix could be obtained for a general linear two‐port network, shown in Fig (3.1). The wave entering the port is the incident voltage ( ) or the incident current ( ) wave. The reflected voltage ( ) and reflected current ( ) waves are also present at the ports. The total port voltage (V n) or current (I n) is the sum of the incident and reflected voltage or current:

(3.1.1)

In equation (3.1.1), n = 1, 2 is the port number, i.e. port‐1, port‐2. The power entering the network is taken as a positive quantity for the incident wave, so the power coming out of the network, i.e. the power of the reflected wave, is taken as a negative quantity. The reflected voltage ( ) is positive. To maintain the negative direction of power flow, the reflected current wave ( ) is taken as negative in the above equation. At each port, the current entering the port from outside is positive, whereas the current leaving the port is negative. For the linear networks, the voltage at any port is a combined response of the currents applied to all ports. On using the superposition of the voltage responses, the following set of equations is written:

Figure 3.1 Two‐port network to determine [Z] and [Y] parameters.

(3.1.2)

Equation (3.1.2)is written in a more compact matrix form:

(3.1.3)

where [V] and [I] are the column matrices. The two‐port impedance matrix is

(3.1.4)

The [Z] parameter can be easily extended to the N‐port networks [B.1, B.3–B.5]. The Z‐parameters are the open‐circuited parameters . The coefficient of the matrix can be defined in terms of the open circuit condition at the ports:

(3.1.5)

All the ports are open‐circuited, except the i thport at which the matrix element Z iiis defined. For instance, in the case of a two‐port network, Z 11is obtained when current I 1is applied to port‐1 and the voltage response is also obtained at the port‐1, while keeping the port‐2 open‐circuited, i.e. I 2= 0. The coefficient, Z 11, is known as the self‐impedance of the network. These are the diagonal elements of a [Z] matrix. The off‐diagonal elements of a [Z] matrix are defined as follows:

(3.1.6)

In this case, the current excitation is applied at the port‐j and the voltage response is obtained at the port‐i. All other ports are kept open‐circuited allowing I k= 0, except at the port‐j. For instance, in the case of a two‐port network to evaluate Z 12, the current source is applied at the port‐2, and the voltage response is obtained at the port‐1, while keeping the port‐1 open‐circuited. The coefficient Z 12is the mutual impedance that describes the coupling of port‐2 with the port‐1. A network can have Z 11= Z 22, i.e. both of the ports are electrically identical. Such a network is known as the symmetrical network. Furthermore, the voltage response of a network at the port‐1 due to the current at the port‐2 can be identical to the voltage response at the port‐2, due to the current at the port‐1. This kind of network is a reciprocal network. It has a Z 12= Z 21. If Z 12= Z 21= 0, the ports are isolated one.

Figure (3.2)shows lumped elements T‐network. Determine the [Z] parameter of the network.

For the port‐2 open‐circuited, I 2= 0. The voltage at the port‐1 is

Likewise, for the port‐1 open‐circuited, I 1= 0, and the parameters are Z 22= Z B+ Z C, Z 12= Z C.The [Z] matrix description of a T‐network is

(3.1.7)

Figure 3.2 Lumped T‐network.

The given circuit is asymmetrical. However, it is a reciprocal circuit. It becomes symmetrical for Z A= Z B.

Determine the [Z]‐parameter of a section of the transmission line of length ℓ shown in Fig (3.3).

Let the port‐2 be open‐circuited and an incident voltage is applied at the port‐1. The voltage wave reaches to the port‐2 and reflects from there. It reaches the port‐1 as the voltage wave . The maximum of the voltage wave occurs at the port‐2. The total port voltages are given below:

Thus, the [Z] parameters are obtained as follows:

The following [Z] matrix of a line section is obtained by keeping in view that the uniform transmission line is a symmetrical and reciprocal network: