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
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 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)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:
- 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.