# Verify Z-Parameters of Two-port Network

## Aim

To calculate and verify Z-parameters of two-port network.

## Apparatus

Breadboard, Batteries or DC regulated power supply, Resistors, Digital multimeter, Connecting wires, Alligator clips.

## Theory

Z parameters (also known as impedance parameters or open-circuit parameters) are properties used in electrical engineering to describe the electrical behaviour of linear networks.

These Z-parameters are used in Z-matrixes (impedance matrixes) to calculate the incoming and outgoing voltages and currents of a network.

Z parameter is a factor by which input voltage and current and output voltage and current of two port network is related with.

For any two-port network, input voltage V1and output voltage V2can be expressed in terms of input current I1and output current I2respectively.

Z parameter in terms of input voltage V1and output voltage V2and input current I1and output current I2is given as below.

[ V ] = [ Z ] [ I ]

Where [Z] is impedance matrix, [V] and [I] are voltage and current matrix. Therefore, in matrix form the input and output voltage and current can be represented as below.

Form the above matrix, we get,

V1= Z11I1+ Z12I2  …(1)

V2= Z21I1+ Z22I2 … (2)

The Z-parameters can be calculated by considering supply voltage at one port and the other port is open circuited.

There are four Z parameter for a two-port or four-terminal network. Their values are tabulated below.

 Z11 (V1/ I1) Condition: Output port of the two-port network is open i.e. I2= 0 Z21 (V2/ I1) Z12 (V1/ I2) Condition: Input port of the two-port network is open i.e. I1= 0 Z22 (V2/ I2)

### How to calculate Z-Parameters

Consider the following figure.

We know that,

V1= Z11I1+ Z12I2  …(1)

V2= Z21I1+ Z22I2 … (2)

Case1:Assume output port open i.e. I2=0, voltage across impedance Z3will be equal to V2.

V2= Z3I1

Z3= V2/ I1

But V2/ I1= Z21,

∴Z21= Z3 …(3)

Also, under the condition of output port open, applying Kirchhoff’s Loop Law in loop 1,

V1= I1Z1+ V2

Diving both side of above expression by I1, we get

(V1/ I1) = Z1+ (V2/ I1)

But (V1/ I1) = Z11and (V2/ I1) = Z21,

∴Z11= Z1+ Z21

= Z1+ Z3 [from (3)]

∴Z11= (Z1+ Z3)

Case2:Assume input port open i.e. I1=0, voltage across impedance Z3 will be equal to V1.

V1= Z3I2

Z3= V1/ I2

But V1/ I2= Z12,

∴Z12= Z3 …(4)

Also applying Kirchhoff’s Loop Law in loop 2,

V2= I2Z2+ V1

Diving both side of above expression by I2, we get

(V2/ I2) = Z2+ (V1/ I2)

But (V2/ I2) = Z22and (V1/ I2) = Z12,

∴ Z22= Z2+ Z12

= Z2+ Z3 [from (4)]

∴ Z22= (Z2+ Z3)

Hence,

Z11= (Z1+ Z3),

Z22= (Z2+ Z3),

Z12= Z3,

Z21= Z3.

### Significance of Different Z-Parameters

• Since Z11is the ratio of input voltage and current when the output port is open, therefore it is known as input driving point impedance. This can be understood as a transformer at no load. The input voltage is primary supply voltage Vsand the input current is excitation current Ie. Therefore, the input driving point impedance Z11for this will be (Vs/ Ie).
• Z22is the ratio of output voltage and current when input port is open, therefore it is called output driving point impedance of the network.
• Z12is the ratio of input voltage and output current when input port is open, therefore it is called reverse transfer impedance.
• Z21is the ratio of output voltage and input current when output port is open, therefore it is called forward transfer impedance.

## Circuit Diagram

[Recommended values: R1= 2KΩ, R2= 450Ω, R3= 1kΩ, V1= 10V and V2= 5V OR chose any resistor between 200Ω to 2KΩ and DC supply beween 5V to 12V]

## Procedure

1. Connect the circuit as shown in figure 3.
2. First open the output port and supply 10V to input port. Measure output voltage and input current.
3. Secondly, open input port and supply 5V to output port. Measure input voltage and output current using multi-meter.
4. Calculate the values of Z parameters using respective formulas (Shown in calculation section).
5. Switch ‘OFF’ the supply after taking the readings.

## Precautions

• Before circuit connection working condition of all the components must be checked.
• All the connection should be tight.
• Ammeter must be connected in series while voltmeter must be connected in parallel to the components (resistors).
• The electrical current should not flow the circuit for long time, otherwise its temperature will increase and the result will be affected.

## Observation table

 When output port is open circuited When input port is open circuited V1 V2 I1 V1 V2 I2

## Calculations

For Practical Values:

(a) When output is open circuited i.e. I2 = 0

Z11 = V1/I1 = ____Ω

Z21 =V2 /I1 = ____Ω

(b) When input is open circuited i.e. I1 = 0

Z12 = V1/I2 = ____Ω

Z22 = V2 /I2 = ____Ω.

For Theoretical Values:

Z11= (R1+ R3)= ____Ω

Z22= (R2+ R3)= ____Ω

Z12= R3= ____Ω

Z21= R3=____Ω

## Result

 Parameter→ Z11 Z12 Z21 Z22 Theoretical Practical

## Conclusion

Calculated and practical values of Z-parameters are found to be nearly equal hence ‘Z’ parameters are verified.