Impedance Parameter (Z Parameter)

Impedance Parameter

The impedance parameter is used to describe the behavior of linear electrical networks and the small-signal response of a non-linear network in electrical engineering.

The impedance parameter is also known as Z-parameter or open-circuit impedance parameter. Because the values of Z parameters are calculated under open-circuit conditions.

In a two-port network, the input current is I1 and the output current is I2. Then the Z parameter calculate by choose port-1 is open (I1=0) in the first case and port-2 is open (I2=0) in the second case. The below figure shows an electrical two-port network.

Impedance Parameter
Impedance Parameter

As we know the impedance is a ratio of voltage and current. If we write the equation of impedance in terms of a matrix, it is given as;

    \[ [Z] = \frac{[V]}{[I]} \]

    \[ [V] = [Z][I] \]

In this equation, V, Z, and I is in the form of matrix.

    \[ V_1 = f(I_1,I_2) \]

    \[ V_2 = f(I_1, I_2) \]

(1)   \begin{equation*} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} \end{equation*}

Where, Z11, Z12, Z21, and Z22 are the impedance parameter.

This equation shows the matrix form of the impedance parameter. Now, separate both equations;

(2)   \begin{equation*} \begin{align} V_1 &= Z_{11} I_1 + Z_{12} I_2 \\ V_2 &= Z_{21} I_1 + Z_{22} I_2 \end{align} \end{equation*}

These equations are main equations of Z parameter and used to calculate other components.

Calculation of Z Parameters

To calculate the Z parameter, perform two cases. In the first case, the port-1 is open-circuited. And hence, the current of port-1 that is I1 is zero. In the second case, the port-2 is open-circuited. And hence, the current of port-2 that is I2 is zero.

First Case (Port-1 is open):

In this condition, the port-1 is open-circuited and the below figure shows this condition. Because of the open circuit, the current I1=0.

Impedance Parameter (Port-1 is open)
Impedance Parameter (Port-1 is open)

Now put the value of I1=0 in the eq-(2)

    \[ Z_{12} = \left. \frac{V_1}{I_2} \right\vert_{I_1 = 0} \]

    \[ Z_{22} = \left. \frac{V_2}{I_2} \right\vert_{I_1 = 0} \]

Second Case (Port-2 is open):

In this condition, the port-2 is open-circuited and the below figure shows this condition. because of the open-circuit, the current I2=0.

Impedance Parameter (Port-2 is open)
Impedance Parameter (Port-2 is open)

Now put the value of I2=0 in the eq-(2)

    \[ Z_{11} = \left. \frac{V_1}{I_1} \right\vert_{I_2 = 0} \]

    \[ Z_{21} = \left. \frac{V_2}{I_1} \right\vert_{I_2 = 0} \]

As we can see, the value of the Z parameter is a ratio of voltage and current. Hence, this parameter is also known as the Impedance parameter. Hence, the unit of all Z parameter is ohm.

Reciprocity Conditions for Z Parameter

A network is said to be reciprocal if the port-2 voltage is due to applied current at port-1 is the same as the port-1 voltage when the applied current is the same as port-2.

Generally, the network that consists of passive elements (resistor, inductor, and capacitor) only, that network are reciprocal networks and the network that consists of the active element (transistor or generator), that networks are not a reciprocal network.

Consider the below figure for the first condition,

Reciprocity Condition-1 for Z Parameter
Reciprocity Condition-1 for Z Parameter

From above figure,

    \[V_1=V_s \quad V_2=0 \quad I_2=-I_2' \]

Now, put these values in eq-(2),

Hence,

    \[V_s = Z_{11}I_1 + Z_{12}(-I_2') \]

    \[0 = Z_{21}I_1 + Z_{22}(-I_2') \]

Now,

    \[Z_{21}I_1 = Z_{22}(I_2') \]

    \[I_1 = \frac{Z_{22}}{Z_{21}} (I_2') \]

    \[V_s = Z_{11} \left[ \frac{Z_{22}}{Z_{21}} (I_2') \right] + Z_{12}(-I_2') \]

    \[V_s = \left[ \frac{Z_{11} Z_{22} - Z_{12}Z_{21}}{Z_{21}} \right] (I_2') \]

    \[\frac{V_s}{I_2'} = \left[ \frac{Z_{11} Z_{22} - Z_{12}Z_{21}}{Z_{21}} \right] \]

Consider the below figure for second condition,

Reciprocity Condition-2 for Z Parameter
Reciprocity Condition-2 for Z Parameter

    \[ V_2 = V_s \quad V_1 = 0 \quad I_1' = -I_1 \]

From equation-2;

    \[0 = Z_{11}(-I_1') + Z_{12}I_2 \]

    \[V_s = Z_{21}(-I_1') + Z_{22}I_2 \]

    \[Z_{11}I_1' = Z_{12}I_2 \]

    \[I_2 = \frac{Z_{11}}{Z_{12}} I_1' \]

    \[V_s = Z_{21}(-I_1') + Z_{22} \frac{Z_{11}}{Z_{12}} I_1' \]

    \[V_s = \left[ \frac{Z_{22}Z_{11}-Z_{12}Z_{21}}{Z_{12}} \right] I_1' \]

    \[\frac{V_s}{I_1'} = \left[ \frac{Z_{22}Z_{11}-Z_{12}Z_{21}}{Z_{12}} \right] \]

For condition of reciprocity,

    \[ \frac{V_s}{I_1'} = \frac{V_s}{I_2'} \]

    \[\frac{Z_{22}Z_{11}-Z_{12}Z_{21}}{Z_{12}} = \frac{Z_{11} Z_{22} - Z_{12}Z_{21}}{Z_{21}} \]

    \[ Z_{12} = Z_{21} \]

This is the condition for reciprocity for Z parameter.

Symmetry Conditions for Z Parameter

The electrical network is said to be symmetrical, if the input impedance is equal to the output impedance. It is not necessary that, this circuits are physically symmetrical.

Consider the below figure for the first Symmetry condition of Z parameter,

Symmetry Conditions-1 for Z Parameter
Symmetry Conditions-1 for Z Parameter

    \[ V_1 = V_s \quad I_2 = 0 \]

Put these values in the main equations of Z parameter.

    \[ V_s = Z_{11} I_1 \]

    \[ \frac{V_s}{I_1} = Z_{11} \]

Now, consider the below figure for the second Symmetry condition of Z parameter,

Symmetry Conditions-2 for Z Parameter
Symmetry Conditions-2 for Z Parameter

    \[ V_2 = V_s \quad I_1 = 0 \]

Put these values in the main equations of Z parameter.

    \[V_s = Z_{22} I_2 \]

    \[ \frac{V_s}{I_2} = Z_{22} \]

The impedance of both ports must be equal to fulfil the condition of symmetry. Hence,

    \[ \frac{V_s}{I_1} = \frac{V_s}{I_2} \]

Related Article:

Two-Port Network

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