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 Z12 = Z21. If Z12 = Z21 = 0, the ports are isolated one.
Example 3.1
Figure (3.2) shows lumped elements T‐network. Determine the [Z] parameter of the network.
Solution
For the port‐2 open‐circuited, I2 = 0. The voltage at the port‐1 is
Likewise, for the port‐1 open‐circuited, I1 = 0, and the parameters are Z22 = ZB + ZC, Z12 = ZC. The [Z] matrix description of a T‐network is
Figure 3.2 Lumped T‐network.
The given circuit is asymmetrical. However, it is a reciprocal circuit. It becomes symmetrical for ZA = ZB.
Example 3.2
Determine the [Z]‐parameter of a section of the transmission line of length ℓ shown in Fig (3.3).
Solution
Let the port‐2 be open‐circuited and an incident voltage
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:
For the lossless transmission line, γ = j β, α = 0 and [Z] is
(3.1.8)
Figure 3.3 A transmission line section.
3.1.2 Admittance Matrix
To define the [Y] parameters, the voltage is taken as an independent variable and current as of the dependent one for a two‐port network shown in Fig (3.1). In this case, the voltage is a source of excitation, and current at the port is the response. Thus, for a linear network, the total port current is a superposition of currents due to the voltages applied at both the ports:
where [V] and [I] are the voltage and current column matrices. The admittance matrix of the two‐port network is
(3.1.10)
The Y‐parameters are defined as the short‐circuited parameters. For the short‐circuited port‐2, V2= 0, and Y11 and Y21 are defined from equation (3.1.9):
(3.1.11)
Likewise, for the short‐circuited port‐1, the Y‐parameters are
(3.1.12)
The [Y] parameters are extended to a multiport network by defining its matrix elements as follows:
Equation (3.1.13) shows that to get Yii, i.e. the diagonal elements of the [Y] matrix, all the ports are short‐circuited, except the ith port. The current is evaluated at the ith port for the voltage applied at the ith port itself. To get Yij, i.e. the off‐diagonal elements of the [Y] matrix, the voltage is applied at the jth port. Yij is the mutual admittance describing the coupling between the jth port and the ith port. The current at the ith port is evaluated or measured, while all other ports are short‐circuited. The admittance element Yij is evaluated as
(3.1.14)
Example 3.3
Fig (3.2) shows the T‐network. Determine the [Y] parameter of the network.
Solution
The loop equations for the circuit are written as
For the short‐circuited port‐2, V2 = 0:
From the above equations:
