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