Inter-relationship between the two-port Parameters

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 1st 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 1st 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

Y22xI–  Y12xI2= Y22xY11V1 + Y22xY12V2   – ( Y12xY21V1 + Y12xY22V2 ) and rearranging we get:

Y22xY11V1 – Y12xY21V1 = Y22xI–  Y12xI2

V1(Y22xY11  – Y12xY21)  = Y22xI–  Y12xI2        OR          ∆yV1    = Y22xI–  Y12xI2

OR          V1    = (Y22 /∆y )I–  (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)

Y21xI–  Y11xI2= Y21xY11V1 + Y21xY12V2   –  Y11xY21V1   –  Y11xY22V2  and rearranging we get:

Y21xY12V2 – Y11xY22V2 = Y21xI–  Y11xI2

V2(Y21xY12  – Y11xY22)  = Y21xI–  Y11xI2        OR          -∆yV1    = Y21xI–  Y11xI2

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

Step-3:

Rewriting eqn (5) and eqn(8)

V1    = (Y22 /∆y )I–  (Y12/∆y )I2

V2    = -(Y21 /∆y )I+  (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 1st 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 1st set of equations.

From eqn-2 of h parameter

I2   = h21I1 + h22V2

V2 = – (h21/ h22)I1 + (1/ h22)I

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)I

Step-3: Rewriting

V1 = (∆h / h22)I1    +   (h12/ h22)I

V2 = – (h21/ h22)I1 + (1/ h22)I

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.
Updated: April 5, 2020 — 2:03 pm

Leave a Reply

Your email address will not be published. Required fields are marked *

care4you © 2020 Connect On Facebook DMCA.com Protection Status