Impedance Parameters or Z Parameters

The input and output of a two port network can either be voltage or current. If the network is voltage driven, that can be represented as shown below.
voltage driven two port network
If the network is driven by current, that can be represented as shown below.
current driven two port network

From, both of the figures above, it is clear that, there are only four variables. One pair of voltage variables V1 and V2 and one pair of current variables I1 and I2. Thus, there are only four ratio of voltage to current, and those are,

These four ration are considered as parameters of the network. We all know,

This is why these parameters are called either impedance parameter or Z parameter.
The values of these Z parameters of a two port network, can be evaluated by making once

and another once

Let us explain in brief. For that, first, we make the output port of the network open circuited as shown below.
Two-port network with output open

In this case as the output is open, there will be no current in the output port. i.e.

In this condition, the ratio of input voltage to input current, is mathematically represented as,

This known as input impedance of the network, while output port is open. This is denoted by Z11
So, finally,


Now, Voltage source V2 is connected across port 2 that is output port, and the port 1 or input port is kept open as shown below
two port network with input open
Now, ratio of V2 and I2 at I1=0 is,

This is called open circuit output impedance. Similarly,


Since, all these above shown Z parameter have been obtained by open circuiting output port or input port, the parameters are also referred as open circuit impedance parameter. Now, we can relate all voltage and current variables of a two port network by these Z parameters.

These two equations can be represented in matrix form, as shown below,

In the equation (i), if we put I2 = 0, We get,

Similarly if we put I1=0, in the same equation, we get,

In same way, by putting I2 = 0 and I1 = 0 alternatively in equation (ii) We can prove,

Z11 and Z22 are also referred as driving point impedance.
Z21 and Z12 are also referred as transfer impedance. For better understanding, let us take the circuit below,
transfer impedance circuit
Let us put a voltage source V1 at input,
transfer impedance voltage source

Now, let us connect one voltage source V2 at output port and leave the input port as open as shown, below


So, Here,

When in a two port network, we get,

We can call it as symetrical network. Since, here,

As this ratio is same, the same voltage at any of the port results in same currents in the network. That means if we apply voltage V1 at output port then output current will be I1. That means the network will have a mirror-like symmetry between output and input ports, in respect of the imaginary central line.
When we get,


That means, if input exication and output response of the network are interchanged, the transfer impedance remain same.
Suppose, V is the input voltage and I is the output current in the network as shown below.
input voltage and output current
Now if we connect a current source of I at input port, so the voltage response of the network would be, V, at output port.

This is because the ratio of voltage to current between input and output remain same in both conditions. This is Reciprocity Theorem. The two port network behave like that is referred as reciprocal network.
For symetrical network,

For reciprocal network

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