Table of Contents
ToggleAim
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 V_{1}and output voltage V_{2}can be expressed in terms of input current I_{1}and output current I_{2}respectively.
Z parameter in terms of input voltage V_{1}and output voltage V_{2}and input current I_{1}and output current I_{2}is 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,
V_{1}= Z_{11}I_{1}+ Z_{12}I_{2 } …(1)
V_{2}= Z_{21}I_{1}+ Z_{22}I_{2 } … (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.
Z_{11} | (V_{1}/ I_{1}) | Condition: Output port of the two-port network is open i.e. I_{2}= 0 |
Z_{21} | (V_{2}/ I_{1}) | |
Z_{12} | (V_{1}/ I_{2}) | Condition: Input port of the two-port network is open i.e. I_{1}= 0 |
Z_{22} | (V_{2}/ I_{2}) |
How to calculate Z-Parameters
Consider the following figure.
We know that,
V_{1}= Z_{11}I_{1}+ Z_{12}I_{2 } …(1)
V_{2}= Z_{21}I_{1}+ Z_{22}I_{2 } … (2)
Case1:Assume output port open i.e. I_{2}=0, voltage across impedance Z_{3}will be equal to V_{2}.
V_{2}= Z_{3}I_{1}
Z_{3}= V_{2}/ I_{1}
But V_{2}/ I_{1}= Z_{21},
∴Z_{21}= Z_{3} …(3)
Also, under the condition of output port open, applying Kirchhoff’s Loop Law in loop 1,
V_{1}= I_{1}Z_{1}+ V_{2}
Diving both side of above expression by I_{1}, we get
(V_{1}/ I_{1}) = Z_{1}+ (V_{2}/ I_{1})
But (V_{1}/ I_{1}) = Z_{11}and (V_{2}/ I_{1}) = Z_{21},
∴Z_{11}= Z_{1}+ Z_{21}
= Z_{1}+ Z_{3} [from (3)]
∴Z_{11}= (Z_{1}+ Z_{3})
Case2:Assume input port open i.e. I_{1}=0, voltage across impedance Z3 will be equal to V_{1}.
V_{1}= Z_{3}I_{2}
Z_{3}= V_{1}/ I_{2}
But V_{1}/ I_{2}= Z_{12},
∴Z_{12}= Z_{3} …(4)
Also applying Kirchhoff’s Loop Law in loop 2,
V_{2}= I_{2}Z_{2}+ V_{1}
Diving both side of above expression by I_{2}, we get
(V_{2}/ I_{2}) = Z_{2}+ (V_{1}/ I_{2})
But (V_{2}/ I_{2}) = Z_{22}and (V_{1}/ I_{2}) = Z_{12},
∴ Z_{22}= Z_{2}+ Z_{12}
= Z_{2}+ Z_{3} [from (4)]
∴ Z_{22}= (Z_{2}+ Z_{3})
Hence,
Z_{11}= (Z_{1}+ Z_{3}),
Z_{22}= (Z_{2}+ Z_{3}),
Z_{12}= Z_{3},
Z_{21}= Z_{3}.
Significance of Different Z-Parameters
- Since Z_{11}is 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 V_{s}and the input current is excitation current I_{e}. Therefore, the input driving point impedance Z_{11}for this will be (V_{s}/ I_{e}).
- Z_{22}is the ratio of output voltage and current when input port is open, therefore it is called output driving point impedance of the network.
- Z_{12}is the ratio of input voltage and output current when input port is open, therefore it is called reverse transfer impedance.
- Z_{21}is the ratio of output voltage and input current when output port is open, therefore it is called forward transfer impedance.
Circuit Diagram
[Recommended values: R_{1}= 2KΩ, R_{2}= 450Ω, R_{3}= 1kΩ, V_{1}= 10V and V_{2}= 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 | ||||
V_{1} | V_{2} | I_{1} | V_{1} | V_{2} | I_{2} |
Calculations
For Practical Values:
(a) When output is open circuited i.e. I_{2} = 0
Z_{11} = V_{1}/I_{1} = ____Ω
Z_{21} =V_{2} /I_{1 }= ____Ω
(b) When input is open circuited i.e. I_{1} = 0
Z_{12} = V_{1}/I_{2} = ____Ω
Z_{22} = V_{2} /I_{2 }= ____Ω.
For Theoretical Values:
Z_{11}= (R_{1}+ R_{3})= ____Ω
Z_{22}= (R_{2}+ R_{3})= ____Ω
Z_{12}= R_{3}= ____Ω
Z_{21}= R_{3}=____Ω
Result
Parameter→ | Z_{11} | Z_{12} | Z_{21} | Z_{22} |
Theoretical | ||||
Practical |
Conclusion
Calculated and practical values of Z-parameters are found to be nearly equal hence ‘Z’ parameters are verified.
Video Tutorial on Z-Parameters
Video tutorial "To Verify Z-parameters of two-port network"
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