Y Parameter of Two Port Network
Y parameter of two port network is a 2×2 admittance matrix. Since admittance is the ratio of circuit current and voltage, therefore this admittance matrix gives the relationship between the input and output current and voltage of the network. It is also known as short circuit admittance parameter.
The Y parameter for a two port network is defined as
[ I ] = [ Y ] [ V ]
where [ Y ] is the admittance matrix, [ I ] and [ V ] are the current and voltage matrix.
From the above matrix form representation of two port network, it is clear that there are four admittance parameters i.e. Y_{11}, Y_{12}, Y_{21} and Y_{22}. Each of them has a special meaning and significance which will be discussed latter in this post.
Calculation of Y Parameter:
Let us consider a two port network. Let V_{1}, I_{1}, V_{2} and I_{2} are the input voltage, input current, output voltage and output current respectively.
The relationship between the input and output quantities for the above network is obtained using (1) as below.
I_{1} = Y_{11}V_{1} + Y_{12}V_{2} ……(2)
I_{2} = Y_{21}V_{1} + Y_{22}V_{2} …….(3)
Assuming the output of the two port network to be short circuited, therefore
V_{2} = 0
Now putting V_{2} = 0 in (2), we get
I_{1} = Y_{11}V_{1}
Y_{11} = (I_{1} / V_{1})
Similarly putting V_{2} = 0 in (3), we get
I_{2} = Y_{21}V_{1}
Y_{21} = (I_{2} / V_{1})
Again assuming input port of the two port network to be short circuited, the input voltage will be zero.
V_{1} = 0
Now putting V_{1} = 0 in (2), we get
I_{1} = Y_{12}V_{2}
Y_{12} = (I_{1} / V_{2})
Similarly putting V_{1} = 0 in (3), we get
I_{2} = Y_{22}V_{2}
Y_{22} = (I_{2} / V_{2})
Thus there are four Y parameter for a two port or four terminal network. Their values are tabulated below.
Y_{11} | (I_{1} / V_{1}) | Condition: Output port of the two port network is short circuited i.e. V_{2} = 0 |
Y_{21} | (I_{2} / V_{1}) | |
Y_{12} | (I_{1} / V_{2}) | Condition: Input port of the two port network is short circuited i.e. V_{1} = 0 |
Y_{22} | (I_{2} / V_{2}) |
Significance of Different Y Parameter:
- Since Y_{11} is the ratio of input current and voltage when the output port is short circuited, therefore it is known as input driving point admittance.
- Y_{22} is the ratio of output current and voltage when input port is short circuited, therefore it is called output driving point admittance of the network.
- Y_{12} is the ratio of input current and output voltage when input port is short circuited, therefore it is called reverse transfer admittance.
- Y_{21} is the ratio of output current and input voltage when output port is short circuited, therefore it is called forward transfer admittance.
Equivalent Circuit Representation of Y Parameter:
The equivalent circuit of Y parameter for two port network can be represented using (2) and (3) as shown below.
In the above circuit, the current sources Y_{12}V_{2} and Y_{21}V_{1} are called Voltage Controlled Current Source.
Y Parameter Solved Problems:
To better understand the discussed concept of Y parameter, we will consider one example.
Example:
Find the Y parameter for the π circuit given below.
Solution:
Applying Kirchoff’s Current Law at node ‘a’,
I_{1} = I_{3} + I_{4}
= V_{1}Y_{A} + (V_{1} – V_{2})Y_{B}
= (Y_{A} + Y_{B})V_{1} + (-Y_{B})V_{2}
Thus, I_{1 }= (Y_{A} + Y_{B})V_{1} + (-Y_{B})V_{2}
But from (2),
I_{1} = Y_{11}V_{1} + Y_{12}V_{2}
Comparing the above two expressions, we get
Y_{11 }= (Y_{A} + Y_{B})
Y_{12} = -Y_{B}
Again, applying Kirchoff’s Current Law at node ‘b’
I_{2} = I_{5} – I_{4}
= Y_{C}V_{2} – (V_{1} – V_{2})Y_{B}
= (-Y_{B})V_{1} + (Y_{B} + Y_{C})V_{2}
Thus,
I_{2} = (-Y_{B})V_{1} + (Y_{B} + Y_{C})V_{2}
But from (3),
I_{2} = Y_{21}V_{1} + Y_{22}V_{2}
Comparing the above two expressions, we get
Y_{22 }= (Y_{A} + Y_{B})
Y_{21} = -Y_{B}
Hence, the four Y parameters are
Y_{11 }= (Y_{A} + Y_{B}), Y_{12} = -Y_{B}, Y_{22 }= (Y_{A} + Y_{B}), Y_{21} = -Y_{B }(Answer)