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.

· 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.

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.

I1=Y11V1+Y12V2I1=Y11V1+Y12V2

I2=Y21V1+Y22V2I2=Y21V1+Y22V2

We can represent the above two equations in matrix form as

[I1I2]=[Y11Y21Y12Y22][V1V2][I1I2]=[Y11Y12Y21Y22][V1V2]Equation 1

Step 2 − We know that the following set of two equations, which represents a two port network in terms of Z parameters.

V1=Z11I1+Z12I2V1=Z11I1+Z12I2

V2=Z21I1+Z22I2V2=Z21I1+Z22I2s

We can represent the above two equations in matrix form as

[V1V2]=[Z11Z21Z12Z22][I1I2][V1V2]=[Z11Z12Z21Z22][I1I2]

Step 3 − We can modify it as

[I1I2]=[Z11Z21Z12Z22]−1[V1V2][I1I2]=[Z11Z12Z21Z22]−1[V1V2]Equation 2

Step 4 − By equating Equation 1 and Equation 2, we will get

[Y11Y21Y12Y22]=[Z11Z21Z12Z22]−1[Y11Y12Y21Y22]=[Z11Z12Z21Z22]−1

⇒[Y11Y21Y12Y22]=[Z22−Z21−Z12Z11]ΔZ⇒[Y11Y12Y21Y22]=[Z22−Z12−Z21Z11]ΔZ

Where,

ΔZ=Z11Z22−Z12Z21ΔZ=Z11Z22−Z12Z21

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.

V1=AV2−BI2V1=AV2−BI2

I1=CV2−DI2I1=CV2−DI2

Step 2 − We know that the following set of two equations, which represents a two port network in terms of Z parameters.

V1=Z11I1+Z12I2V1=Z11I1+Z12I2

V2=Z21I1+Z22I2V2=Z21I1+Z22I2

Step 3 − We can modify the above equation as

⇒V2−Z22I2=Z21I1⇒V2−Z22I2=Z21I1

⇒I1=⟮1Z21⟯V2−⟮Z22Z21⟯I2⇒I1=⟮1Z21⟯V2−⟮Z22Z21⟯I2

Step 4 − The above equation is in the form of I1=CV2−DI2I1=CV2−DI2. Here,

C=1Z21C=1Z21

D=Z22Z21D=Z22Z21

Step 5 − Substitute I1I1 value of Step 3 in V1V1 equation of Step 2.

V1=Z11{⟮1Z12⟯V2−⟮Z22Z21⟯I2}+Z12I2V1=Z11{⟮1Z12⟯V2−⟮Z22Z21⟯I2}+Z12I2

⇒V1=⟮Z11Z21⟯V2−⟮Z11Z22−Z12Z21Z21⟯I2⇒V1=⟮Z11Z21⟯V2−⟮Z11Z22−Z12Z21Z21⟯I2

Step 6 − The above equation is in the form of V1=AV2−BI2V1=AV2−BI2. Here,

A=Z11Z21A=Z11Z21

B=Z11Z22−Z12Z21Z21B=Z11Z22−Z12Z21Z21

Step 7 − Therefore, the T parameters matrix is

[ACBD]=⎡⎣Z11Z211Z21Z11Z22−Z12Z21Z21Z22Z21⎤⎦[ABCD]=[Z11Z21Z11Z22−Z12Z21Z211Z21Z22Z21]

Y parameters to Z parameters

Here, we have to represent Z parameters in terms of Y parameters. So, in this case Z parameters are the desired parameters and Y parameters are the given parameters.

Step 1 − We know that, the following matrix equation of two port network regarding Z parameters as

[V1V2]=[Z11Z21Z12Z22][I1I2][V1V2]=[Z11Z12Z21Z22][I1I2]Equation 3

Step 2 − We know that, the following matrix equation of two port network regarding Y parameters as

[I1I2]=[Y11Y21Y12Y22][V1V2][I1I2]=[Y11Y12Y21Y22][V1V2]

Step 3 − We can modify it as

[V1V2]=[Y11Y21Y12Y22]−1[I1I2][V1V2]=[Y11Y12Y21Y22]−1[I1I2]Equation 4

Step 4 − By equating Equation 3 and Equation 4, we will get

[Z11Z21Z12Z22]=[Y11Y21Y12Y22]−1[Z11Z12Z21Z22]=[Y11Y12Y21Y22]−1

⇒[Z11Z21Z12Z22]=[Y22−Y21−Y12Y11]ΔY⇒[Z11Z12Z21Z22]=[Y22−Y12−Y21Y11]ΔY

Where,

ΔY=Y11Y22−Y12Y21ΔY=Y11Y22−Y12Y21

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.

V1=AV2−BI2V1=AV2−BI2

I1=CV2−DI2I1=CV2−DI2

Step 2 − We know that the following set of two equations of two port network regarding Y parameters.

I1=Y11V1+Y12V2I1=Y11V1+Y12V2

I2=Y21V1+Y22V2I2=Y21V1+Y22V2

Step 3 − We can modify the above equation as

⇒I2−Y22V2=Y21V1⇒I2−Y22V2=Y21V1

⇒V1=⟮−Y22Y21⟯V2−⟮−1Y21⟯I2⇒V1=⟮−Y22Y21⟯V2−⟮−1Y21⟯I2

Step 4 − The above equation is in the form of V1=AV2−BI2V1=AV2−BI2. Here,

A=−Y22Y21A=−Y22Y21

B=−1Y21B=−1Y21

Step 5 − Substitute V1V1 value of Step 3 in I1I1 equation of Step 2.

I1=Y11{⟮−Y22Y21⟯V2−⟮−1Y21⟯I2}+Y12V2I1=Y11{⟮−Y22Y21⟯V2−⟮−1Y21⟯I2}+Y12V2

⇒I1=⟮Y12Y21−Y11Y22Y21⟯V2−⟮−Y11Y21⟯I2⇒I1=⟮Y12Y21−Y11Y22Y21⟯V2−⟮−Y11Y21⟯I2

Step 6 − The above equation is in the form of I1=CV2−DI2I1=CV2−DI2. Here,

C=Y12Y21−Y11Y22Y21C=Y12Y21−Y11Y22Y21

D=−Y11Y21D=−Y11Y21

Step 7 − Therefore, the T parameters matrix is

[ACBD]=⎡⎣−Y22Y21Y12Y21−Y11Y22Y21−1Y21−Y11Y21⎤⎦[ABCD]=[−Y22Y21−1Y21Y12Y21−Y11Y22Y21−Y11Y21]

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.

h11=V1I1,whenV2=0h11=V1I1,whenV2=0

h12=V1V2,whenI1=0h12=V1V2,whenI1=0

h21=I2I1,whenV2=0h21=I2I1,whenV2=0

h22=I2V2,whenI1=0h22=I2V2,whenI1=0

Step 2 − We know that the following set of two equations of two port network regarding T parameters.

V1=AV2−BI2V1=AV2−BI2Equation 5

I1=CV2−DI2I1=CV2−DI2Equation 6

Step 3 − Substitute V2=0V2=0 in the above equations in order to find the two h-parameters, h11h11 and h21h21.

⇒V1=−BI2⇒V1=−BI2

⇒I1=−DI2⇒I1=−DI2

Substitute, V1V1 and I1I1 values in h-parameter, h11h11.

h11=−BI2−DI2h11=−BI2−DI2

⇒h11=BD⇒h11=BD

Substitute I1I1 value in h-parameter h21h21.

h21=I2−DI2h21=I2−DI2

⇒h21=−1D⇒h21=−1D

Step 4 − Substitute I1=0I1=0 in the second equation of step 2 in order to find the h-parameter h22h22.

0=CV2−DI20=CV2−DI2

⇒CV2=DI2⇒CV2=DI2

⇒I2V2=CD⇒I2V2=CD

⇒h22=CD⇒h22=CD

Step 5 − Substitute I2=⟮CD⟯V2I2=⟮CD⟯V2 in the first equation of step 2 in order to find the h-parameter, h12h12.

V1=AV2−B⟮CD⟯V2V1=AV2−B⟮CD⟯V2

⇒V1=⟮AD−BCD⟯V2⇒V1=⟮AD−BCD⟯V2

⇒V1V2=AD−BCD⇒V1V2=AD−BCD

⇒h12=AD−BCD⇒h12=AD−BCD

Step 6 − Therefore, the h-parameters matrix is

[h11h21h12h22]=[BD−1DAD−BCDCD][h11h12h21h22]=[BDAD−BCD−1DCD]

h-parameters to Z parameters

Here, we have to represent Z parameters in terms of h-parameters. So, in this case Z parameters are the desired parameters and h-parameters are the given parameters.

Step 1 − We know that, the following set of two equations of two port network regarding Z parameters.

V1=Z11I1+Z12I2V1=Z11I1+Z12I2

V2=Z21I1+Z22I2V2=Z21I1+Z22I2

Step 2 − We know that, the following set of two equations of two-port network regarding h-parameters.

V1=h11I1+h12V2V1=h11I1+h12V2

I2=h21I1+h22V2I2=h21I1+h22V2

Step 3 − We can modify the above equation as

⇒I2−h21I1=h22V2⇒I2−h21I1=h22V2

⇒V2=I2−h21I1h22⇒V2=I2−h21I1h22

⇒V2=⟮−h21h22⟯I1+⟮1h22⟯I2⇒V2=⟮−h21h22⟯I1+⟮1h22⟯I2

The above equation is in the form of V2=Z21I1+Z22I2.Here,V2=Z21I1+Z22I2.Here,

Z21=−h21h22Z21=−h21h22

Z22=1h22Z22=1h22

Step 4 − Substitute V2 value in first equation of step 2.

V1=h11I1+h21{⟮−h21h22⟯I1+⟮1h22⟯I2}V1=h11I1+h21{⟮−h21h22⟯I1+⟮1h22⟯I2}

⇒V1=⟮h11h22−h12h21h22⟯I1+⟮h12h22⟯I2⇒V1=⟮h11h22−h12h21h22⟯I1+⟮h12h22⟯I2

The above equation is in the form of V1=Z11I1+Z12I2V1=Z11I1+Z12I2. Here,

Z11=h11h22−h12h21h22Z11=h11h22−h12h21h22

Z12=h12h22Z12=h12h22

Step 5 − Therefore, the Z parameters matrix is

[Z11Z21Z12Z22]=⎡⎣h11h22−h12h21h22−h21h22h12h221h22⎤⎦[Z11Z12Z21Z22]=[h11h22−h12h21h22h12h22−h21h221h22]

In this way, we can convert one set of parameters into other set of parameters.