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Two-Port Parameter Conversions
In the previous chapter, we discussed about six types of two-port network parameters. Now, let us convert one set of two-port network parameters into other set of two port network parameters. This conversion is known as two port network parameters conversion or simply, two-port parameters conversion.
Sometimes, it is easy to find one set of parameters of a given electrical network easily. In those situations, we can convert these parameters into the required set of parameters instead of calculating these parameters directly with more difficulty.
Now, let us discuss about some of the two port parameter conversions.
Procedure of two port parameter conversions
Follow these steps, while converting one set of two port network parameters into the other set of two port network parameters.
Z parameters to Y parameters
Here, we have to represent Y parameters in terms of Z parameters. So, in this case Y parameters are the desired parameters and Z parameters are the given parameters.
Step 1 − We know that the following set of two equations, which represents a two port network in terms of Y parameters.
We can represent the above two equations in matrix form as
Equation 1
Step 2 − We know that the following set of two equations, which represents a two port network in terms of Z parameters.
Step 1 − Write the equations of a two port network in terms of desired parameters. Step 2 − Write the equations of a two port network in terms of given parameters. Step 3 − Re-arrange the equations of Step2 in such a way that they should be similar to the equations of Step1. Step 4 − By equating the similar equations of Step1 and Step3, we will get the desired parameters in terms of given parameters. We can represent these parameters in matrix form.
I 1 = Y 11 V 1 + Y 12 V 2
I 2 = Y 21 V 1 + Y 22 V 2
[^ I I^1 ] = [ ] [ ]
2
Y 11
Y 21
Y 12
Y 22
V 1
V 2
We can represent the above two equations in matrix form as
Step 3 − We can modify it as
Equation 2
Step 4 − By equating Equation 1 and Equation 2, we will get
Where,
So, just by doing the inverse of Z parameters matrix, we will get Y parameters matrix.
Z parameters to T parameters
Here, we have to represent T parameters in terms of Z parameters. So, in this case T parameters are the desired parameters and Z parameters are the given parameters.
Step 1 − We know that, the following set of two equations, which represents a two port network in terms of T parameters.
Step 2 − We know that the following set of two equations, which represents a two port network in terms of Z parameters.
Step 3 − We can modify the above equation as
V 1 = Z 11 I 1 + Z 12 I 2
V 2 = Z 21 I 1 + Z 22 I 2
[^ V V^1 ] = [ ] [ ]
2
Z 11
Z 21
Z 12
Z 22
I 1
I 2
[^ I I^1 ] = [ ]
2
[^ Z Z^11 ]
21
Z 12
Z 22
−1 V 1
V 2
[^ Y Y^11 ] =
21
Y 12
Y 22 [^ ]
Z 11
Z 21
Z 12
Z 22
−
⇒ [^ Y Y^11 ] =
21
Y 12
Y 22
[ −^ ZZ^22 ]
21
− Z 12
Z 11
Δ Z
Δ Z = Z 11 Z 22 − Z 12 Z 21
V 1 = A V 2 − BI 2
I 1 = C V 2 − DI 2
V 1 = Z 11 I 1 + Z 12 I 2
V 2 = Z 21 I 1 + Z 22 I 2
Equation 4
Step 4 − By equating Equation 3 and Equation 4, we will get
Where,
So, just by doing the inverse of Y parameters matrix, we will get the Z parameters matrix.
Y parameters to T parameters
Here, we have to represent T parameters in terms of Y parameters. So, in this case, T parameters are the desired parameters and Y parameters are the given parameters.
Step 1 − We know that, the following set of two equations, which represents a two port network in terms of T parameters.
Step 2 − We know that the following set of two equations of two port network regarding Y parameters.
Step 3 − We can modify the above equation as
Step 4 − The above equation is in the form of. Here,
[^ V V^1 ] = [ ]
2
[^ Y Y^11 ]
21
Y 12
Y 22
−1 I 1
I 2
[^ Z Z^11 ] =
21
Z 12
Z 22 [^ ]
Y 11
Y 21
Y 12
Y 22
−
⇒ [^ Z Z^11 ] =
21
Z 12
Z 22
[ −^ YY^22 ]
21
− Y 12
Y 11
Δ Y
Δ Y = Y 11 Y 22 − Y 12 Y 21
V 1 = A V 2 − BI 2
I 1 = C V 2 − DI 2
I 1 = Y 11 V 1 + Y 12 V 2
I 2 = Y 21 V 1 + Y 22 V 2
⇒ I 2 − Y 22 V 2 = Y 21 V 1
⇒ V 1 = ⟮ − YY^22 ⟯ − ⟮ ⟯
21
V 2 −1 Y
21
I 2
V 1 = A V 2 − BI 2
A = − YY^22
21 B = −1 Y 21
Step 5 − Substitute value of Step 3 in equation of Step 2.
Step 6 − The above equation is in the form of. Here,
Step 7 − Therefore, the T parameters matrix is
T parameters to h-parameters
Here, we have to represent h-parameters in terms of T parameters. So, in this case hparameters are the desired parameters and T parameters are the given parameters.
Step 1 − We know that, the following h-parameters of a two port network.
Step 2 − We know that the following set of two equations of two port network regarding T parameters.
Equation 5 Equation 6
Step 3 − Substitute in the above equations in order to find the two h-parameters, and.
V 1 I 1
I 1 = Y 11 {⟮ − YY^22 ⟯ − ⟮ ⟯ } +
21
V 2 Y^ −
21
I 2 Y 12 V 2
⇒ I 1 = ⟮ Y^12 Y^21 Y^ − 21 Y^11 Y^22 ⟯ V 2 − ⟮ − YY 2111 ⟯ I 2
I 1 = C V 2 − DI 2
C = Y^12 Y^21 Y − Y^11 Y^22
21 D = − YY^11 21
[^ A C^ B D ] =
− Y 22 Y 21 Y 12 Y 21 − Y 11 Y 22 Y 21
− Y 21 − Y 11 Y 21
h 11 =^ V I^1 , when = 0 1
V 2
h 12 =^ V V^1 , when = 0 2
I 1
h 21 =^ I I^2 , when = 0 1
V 2
h 22 = (^) V^ I^2 , when = 0 2
I 1
V 1 = A V 2 − BI 2
I 1 = C V 2 − DI 2
V 2 = 0 h 11 h 21 ⇒ V 1 = − BI 2 ⇒ I 1 = − DI 2
Step 1 − We know that, the following set of two equations of two port network regarding Z parameters.
Step 2 − We know that, the following set of two equations of two-port network regarding h-parameters.
Step 3 − We can modify the above equation as
The above equation is in the form of
Step 4 − Substitute V 2 value in first equation of step 2.
The above equation is in the form of. Here,
Step 5 − Therefore, the Z parameters matrix is
V 1 = Z 11 I 1 + Z 12 I 2
V 2 = Z 21 I 1 + Z 22 I 2
V 1 = h 11 I 1 + h 12 V 2 I 2 = h 21 I 1 + h 22 V 2
⇒ I 2 − h 21 I 1 = h 22 V 2
⇒ V 2 = I^2 − h 22 h^21 I^1
⇒ V 2 = ⟮ − hh^21 ⟯ + ⟮ ⟯ 22
I (^1) h^1 22
I 2
V 2 = Z 21 I 1 + Z 22 I 2. Here ,
Z 21 =^ − hh^21 22 Z 22 = h^1 22
V 1 = h 11 I 1 + h 21 {⟮ − hh^21 ⟯ + ⟮ ⟯ } 22
I (^1) h^1 22
I 2
⇒ V 1 = ⟮ h^11 h^22 h^ − h^12 h^21 ⟯ + ⟮ ⟯ 22
I 1^ h h^12 22
I 2
V 1 = Z 11 I 1 + Z 12 I 2
Z 11 = h^11 h^22 h^ − h^12 h^21 22 Z 12 =^ h h^12 22
[^ Z Z^11 ] =
21
Z 12
Z 22
h 11 h 22 − h 12 h 21 h 22 − h 21 h 22
h 12 h 22 h^1 22
In this way, we can convert one set of parameters into other set of parameters.
