Reciprocity Condition in two port Parameters
Reciprocity Theorem states that the two networks are said to be reciprocal of each other if the ratio of the source voltage t one port to response (current) in other branch remain unchanged upon interchanging the position of the excitation and the response
Figure-1: Network Reciprocity
From Network-1, the Response to Excitation ratio= I2’/ Vs
From Network-2, the Response to Excitation ratio= I1’/ Vs
therefore for the two networks to be reciprocal : I2’/ Vs = I1’/ Vs
Reciprocity in Z Parameter
The generalized Z parameters equations are:
V1 = Z11I1 + Z12I2
V2 = Z21I1 + Z22I2
| Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs | Case-2: Port-1 shorted, V1=0, I1=-I1’; and V2=Vs |
| So the equations become:
Vs = Z11I1 – Z12 I2’————-(1) 0 = Z21I1 – Z22 I2’ ————–(2) From (2) I1 = (Z22/ Z21) I2’ Putting it in eqn-1 for Vs Vs= Z11(Z22/ Z21) I2’ – Z12 I2’ Rearranging I2’/ Vs= Z21/{ Z11Z22 – Z12 Z21} ———-(5) |
So the equations become:
0= -Z11 I1’ + Z12 I2 —————–(3) Vs = -Z21 I1’ + Z22 I2—————-(4) From (3) I2 = (Z11/ Z12) I1’ Putting it in eqn-4 for Vs Vs= -Z21I1’ + Z22 (Z11/ Z12) I1’ Rearranging I1’/ Vs= Z12/{ Z11Z22 – Z12 Z21} ——–(6) |
Comparing equation (5) and (6) we find the for the LHS to be equal their RHS must also be equal.
Therefore the condition for reciprocity of Z parameters is: Z12 = Z21
Reciprocity of Y Parameters
The generalized Y parameter equations are:
I1= Y11V1 + Y12V2
I2= Y21V1 + Y22V2
| Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs | Case-2: Port-1 shorted, V1=0, I1=-I1’; and V2=Vs |
| Substituting these values, the equations become:
-I2’= Y21Vs Rearranging -I2’/ Vs = Y21———-(7) |
Substituting these values, the equations become:
-I1’= + Y12Vs Rearranging -I1’/ Vs = Y12——–(8) |
Comparing equation (7) and (8) we get the condition for reciprocity of Y parameters as Y12 = Y21
Condition of Reciprocity of Transmission Parameters:
The generalized equation of the transmission parameters are:
V1= AV2 + BI2
I1= CV2 + DI2
| Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs | Case-2: Port-1 shorted, V1=0, I1=-I1’; and V2=Vs |
| Substituting these conditions, the equations for V1 become:
Vs= 0 + B(-I2’) So the equations become: Vs= -BI2’ Rearranging -I2’/ Vs= 1/B ———-(9)
|
Substituting these, the equations become:
0 = AVs + BI2 —————-(10) -I1’= CVs + DI2 —————- (11) From eqn(10) I2= -(A/B)Vs Substituting in eqn(11) we get -I1’= CVs – D(A/B)Vs Rearranging: -I1’= {(AD – BC)/B}Vs ——–(12) |
Comparing eqn(9) and eqn(12) for the two networks to be reciprocal their ratio of response to excitation must be equal, therefore eqn (9) must be equal to eqn (12)
Therefore the reciprocal condition for transmission parameters is:
(AD – BC)/B = 1/B
AD – BC = 1
Writing the above equation in matrix form; the reciprocal condition of transmission parameters are:
Condition of Reciprocity in hybrid parameters
The generalized equation of hybrid (h) parameters are:
V1= h11I1 + h12V2
I2= h21I1 + h22V2
| Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs | Case-2: Port-1 shorted, V1=0, I1=-I1’; and V2=Vs |
| So the equations become:
Vs= h11I1 ………………… (13) -I2’= h21I1 or I1 =-I2’/ h21 Substituting value of I1 in eqn(13) we get: Vs= -(h11 / h21)I2’ I2’ / Vs = – h21 / h11 ———————- (14)
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So the equations become:
0= -h11I1’ + h12Vs Rearranging: h11I1’ = h12Vs I1’/ Vs = h12/ h11 ————–(15)
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For reciprocity, eqn 14 must be equal to eqn(15), so,
– h21 / h11 = h12/ h11
Therefore, the condition of reciprocity in h parameter is: h21 = – h12
Ex-1: Show that network is reciprocal or not
Ex-2: Reciprocity in Y Parameter
