Aim
To calculate and verify Zparameters of twoport 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 opencircuit parameters) are properties used in electrical engineering to describe the electrical behaviour of linear networks.
These Zparameters are used in Zmatrixes (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 twoport 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 Zparameters can be calculated by considering supply voltage at one port and the other port is open circuited.
There are four Z parameter for a twoport or fourterminal network. Their values are tabulated below.
Z_{11}  (V_{1} / I_{1})  Condition: Output port of the twoport 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 twoport network is open i.e. I_{1} = 0 
Z_{22}  (V_{2} / I_{2}) 
How to calculate ZParameters
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 ZParameters
 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 multimeter.
 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 Zparameters are found to be nearly equal hence ‘Z’ parameters are verified.
Video Tutorial on ZParameters
Video tutorial "To Verify Zparameters of twoport network"
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