Z parameter is a factor by which input voltage and current & 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. It is also known as open circuit impedance parameter.
Z parameter in terms of input voltage V_{1} and output voltage V_{2} & 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.
Need of Z Parameter:
Any electrical network can be represented by a black box as we may not know the internal detail of the network. In order to connect this box with other network, it must have to have terminals. If it has two pairs of terminals for external connection, it is known as two port network or four terminal network. One pair of terminal is called input port while the other port is called output port. Driving source is connected to input port. Figure below shows the diagrammatic representation of a two port network.
To analyse the behaviour of this black box, we need to known the relationship between the input and output quantities. Z parameter gives us this relationship.
Calculation of Z Parameter:
Let us consider a two port network as shown in figure below.
As per the definition,
V_{1} = Z_{11}I_{1} + Z_{12}I_{2} ……(1)
V_{2} = Z_{21}I_{1} + Z_{22}I_{2} …….(2)
Assuming the output of the two port network to be open, therefore
I_{2} = 0
Now putting I_{2} = 0 in (1), we get
V_{1} = Z_{11}I_{1}
Z_{11} = (V_{1} / I_{1})
Similarly putting I_{2} = 0 in (2), we get
V_{2} = Z_{21}I_{1}
Z_{21} = (V_{2} / I_{1})
Again assuming input port of the two port network to be open, the input current will be zero.
I_{1} = 0
Now putting I_{1} = 0 in (1), we get
V_{1} = Z_{12}I_{2}
Z_{12} = (V_{1} / I_{2})
Similarly putting I_{1} = 0 in (2), we get
V_{2} = Z_{22}I_{2}
Z_{22} = (V_{2} / I_{2})
Thus 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}) |
Significance of Different Z Parameter:
- 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.
Example:
Find the Z parameter for the network shown below.
Solution:
We know that,
V_{1} = Z_{11}I_{1} + Z_{12}I_{2}
V_{2} = Z_{21}I_{1} + Z_{22}I_{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}, therefore
Z_{21} = Z_{3} ……(3)
Also under the condition of output port open, applying Kirchoff’s Loop Law in loop 1,
V_{1} = I_{1}Z_{1} + V_{2}
Diving both side of above expression by I1, 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}, therefore
Z_{11} = Z_{1} + Z_{21}
= Z_{1} + Z_{3} [from (3)]
Hence, 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}, therefore
Z_{12} = Z_{3} ……(4)
Also applying Kirchoff’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}, therefore
Z_{22} = Z_{2} + Z_{12}
= Z_{2} + Z_{3} [from (4)]
Hence, 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} (Ans.)