Inter-relationship between the two-port Parameters

The following steps of operations can be used for the transformation one set of parameters in terms of other set of parameters:

Step 1. Write the standard equations for both sets.

Step 2. Solve or rearrange the second set of equations and writing them in terms of the independent variables of 1^st set of equations.

Step 3. Compare the equations obtained in step 2 with those of the first set to obtain the required parameters.

Z parameter in terms of Y parameters

Step-1: Writing Z and Y parameter equations

Z parameter equation	Y parameter equation
V1= Z11I1  + Z12I2 V2= Z21I1 + Z22I2	I1= Y11V1 + Y12V2 I2= Y21V1 + Y22V2

 

Step-2: Solve or rearrange the second set of equations and writing them in terms of the independent variables of 1^st set of equations.

I₁= Y₁₁V₁ + Y₁₂V₂         ——-1

I₂= Y₂₁V₁ + Y₂₂V₂          ——-2

Balancing equation (1) and (2) so as to omit V₂, so we multiply (1) by Y₂₂ and (2) by Y₁₂ and subtract

Y₂₂xI₁= Y₂₂xY₁₁V₁ + Y₂₂xY₁₂V₂        ———-(3)

Y₁₂xI₂= Y₁₂xY₂₁V₁ + Y₁₂xY₂₂V₂           ———-(4)

Subtracting (4) from (3) we get

Y₂₂xI₁  –  Y₁₂xI₂= Y₂₂xY₁₁V₁ + Y₂₂xY₁₂V₂   – ( Y₁₂xY₂₁V₁ + Y₁₂xY₂₂V₂ ) and rearranging we get:

Y₂₂xY₁₁V₁ – Y₁₂xY₂₁V₁ = Y₂₂xI₁  –  Y₁₂xI₂

V₁(Y₂₂xY₁₁  – Y₁₂xY₂₁)  = Y₂₂xI₁  –  Y₁₂xI₂        OR          ∆yV₁    = Y₂₂xI₁  –  Y₁₂xI₂

OR          V₁    = (Y₂₂ /∆y )I₁  –  (Y₁₂/∆y )I₂         ——–(5)

Now balancing the eqn (1) and (2) in terms of V₁ by multiplying (1) by Y₂₁ and (2) by Y₁₁ and subtracting

Y₂₁xI₁= Y₂₁xY₁₁V₁ + Y₂₁xY₁₂V₂        ———-(6)

Y₁₁xI₂= Y₁₁xY₂₁V₁ + Y₁₁xY₂₂V₂           ———-(7)

Y₂₁xI₁  –  Y₁₁xI₂= Y₂₁xY₁₁V₁ + Y₂₁xY₁₂V₂   –  Y₁₁xY₂₁V₁   –  Y₁₁xY₂₂V₂  and rearranging we get:

Y₂₁xY₁₂V₂ – Y₁₁xY₂₂V₂ = Y₂₁xI₁  –  Y₁₁xI₂

V₂(Y₂₁xY₁₂  – Y₁₁xY₂₂)  = Y₂₁xI₁  –  Y₁₁xI₂        OR          -∆yV₁    = Y₂₁xI₁  –  Y₁₁xI₂

OR          V₂    = -(Y₂₁ /∆y )I₁  +  (Y₁₁/∆y )I₂   ——–(8)

Step-3:

Rewriting eqn (5) and eqn(8)

V₁    = (Y₂₂ /∆y )I₁  –  (Y₁₂/∆y )I₂

V₂    = -(Y₂₁ /∆y )I₁  +  (Y₁₁/∆y )I₂

Comparing these equations with the general Z parameter equation we find that:

Z₁₁ = Y₂₂ /∆y             Z₁₂ = –  Y₁₂/∆y             Z₂₁ = -Y₂₁ /∆y          Z₂₂ = Y₁₁/∆y       

 

Z parameters in terms of Transmission parameter

Step-1

Z parameter equation	ABCD parameter Eqn
V1= Z11I1  + Z12I2 V2= Z21I1 + Z22I2	V1= AV2  –  BI2 I1= CV2   –  DI2

Step-2:

Solve or rearrange the second set of equations and writing them in terms of the independent variables of 1^st set of equations.

I₁= CV₂   –  DI₂

V₂   = (1/C)I₁    +  (D/C)I₂

Substituting this value in eqn(1) of ABCD parameter

V₁     =    A{(1/C)I₁    +  (D/C)I₂}  –  BI₂

V₁   =   (A/C) I₁        + {(AD – BC)/C} I₂

Step-3:

V₁   =   (A/C) I₁        + {(AD – BC)/C} I₂

V₂   = (1/C)I₁    +  (D/C)I₂

so, the Z parameter equation in terms of Transmission (ABCD) parameters:

Z₁₁  =  A/C             Z₁₂  =  {(AD – BC)/C           Z₂₁  =  1/C             Z₂₂  =   D/C

Z parameters in terms of hybrid parameter

 

Step-1

Z parameter equations	h parameter equations
V1= Z11I1  + Z12I2 V2= Z21I1 + Z22I2	V1= h11I1  + h12V2 I2   = h21I1 + h22V2

Step-2 : Solve or rearrange the second set of equations and writing them in terms of the independent variables of 1^st set of equations.

From eqn-2 of h parameter

I₂   = h₂₁I₁ + h₂₂V₂

V₂ = – (h₂₁/ h₂₂)I₁ + (1/ h₂₂)I₂ 

Substituting this in eqn(1) of h parameter

V₁= h₁₁I₁  + {h₁₂ (- (h₂₁/ h₂₂)I₁ + (1/ h₂₂) I₂}

V₁= {h₁₁ h₂₂ – h₁₂h₂₁) / h₂₂}I₁ + h₁₂/ h₂₂)I₂   =

 V₁=(∆h / h₂₂)I₁    +   (h₁₂/ h₂₂)I₂ 

Step-3: Rewriting

V₁ = (∆h / h₂₂)I₁    +   (h₁₂/ h₂₂)I₂ 

V₂ = – (h₂₁/ h₂₂)I₁ + (1/ h₂₂)I₂ 

so, we get the Z parameter equation in terms of h parameters as :

Z₁₁  =  ∆h / h₂₂     Z₁₂  =  h₁₂/ h₂₂     Z₂₁  =  – h₂₁/ h₂₂                    Z₂₂  =   1/ h₂₂

Summary of Z parameters in terms of other parameters

Y parameters in terms of other parameters

Transmission Parameters in terms of other Parameters

Hybrid parameters in terms of other parameters

 

Ex-1: Transformation of Parameters

Exercise:

1. If the Y parameter of a two-port network is given as:   Y₁₁ = 0.3 ohm; Y₁₂  = Y₂₁   = – 0.1 ohm and  Y₂₂ = 0.2 ohm. Determine the Z, transmission and hybrid parameters.