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
From Network1, the Response to Excitation ratio= I_{2}’/ V_{s}
From Network2, the Response to Excitation ratio= I_{1}’/ V_{s}
The generalized Z parameters equations are:
V_{1 }= Z_{11}I_{1} + Z_{12}I_{2}
V_{2 }= Z_{21}I_{1} + Z_{22}I_{2}
Case1: Port2 shorted, V_{2}=0, I_{2}=I_{2}’; and V_{1}=V_{s} 
Case2: Port1 shorted, V_{1}=0, I_{1}=I_{1}’; and V_{2}=V_{s} 
So the equations become:
V_{s }= Z_{11}I_{1} – Z_{12} I_{2}’————(1) 0 = Z_{21}I_{1} – Z_{22} I_{2}’ ————–(2) From (2) I_{1} = (Z_{22}/ Z_{21}) I_{2}’ Putting it in eqn1 for Vs V_{s}= Z_{11}(Z_{22}/ Z_{21}) I_{2}’ – Z_{12} I_{2}’ Rearranging I_{2}’/ V_{s}= Z_{21}/{ Z_{11}Z_{22} – Z_{12} Z_{21}} ———(5) 
So the equations become:
0= Z_{11} I_{1}’ + Z_{12} I_{2 }—————–(3) V_{s} = Z_{21} I_{1}’ + Z_{22} I_{2}—————(4) From (3) I_{2} = (Z_{11}/ Z_{12}) I_{1}’ Putting it in eqn4 for Vs V_{s}= Z_{21}I_{1}’ + Z_{22} (Z_{11}/ Z_{12}) I_{1}’ Rearranging I_{1}’/ V_{s}= Z_{12}/{ Z_{11}Z_{22} – Z_{12} Z_{21}} ——–(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: Z_{12} = Z_{21}
The generalized Y parameter equations are:
I_{1}= Y_{11}V_{1} + Y_{12}V_{2}
I_{2}= Y_{21}V_{1} + Y_{22}V_{2}
Case1: Port2 shorted, V_{2}=0, I_{2}=I_{2}’; and V_{1}=V_{s} 
Case2: Port1 shorted, V_{1}=0, I_{1}=I_{1}’; and V_{2}=V_{s} 
Substituting these values, the equations become:
I_{2}’= Y_{21}V_{s} Rearranging I_{2}’/ V_{s} = Y_{21}———(7) 
Substituting these values, the equations become:
I_{1}’= + Y_{12}V_{s} Rearranging I_{1}’/ V_{s } = Y_{12}——–(8) 
Comparing equation (7) and (8) we get the condition for reciprocity of Y parameters as Y_{12 }= Y_{21}
Condition of Reciprocity of Transmission Parameters:
The generalized equation of the transmission parameters are:
V_{1}= AV_{2} + BI_{2}
I_{1}= CV_{2} + DI_{2}
Case1: Port2 shorted, V_{2}=0, I_{2}=I_{2}’; and V_{1}=V_{s} 
Case2: Port1 shorted, V_{1}=0, I_{1}=I_{1}’; and V_{2}=V_{s} 
Substituting these conditions, the equations for V_{1 }become:
V_{s}= 0 + B(I_{2}’) So the equations become: V_{s}= BI_{2}’ Rearranging I_{2}’/ V_{s}= 1/B ———(9)

Substituting these, the equations become:
0 = AV_{s} + BI_{2} —————(10) I_{1}’= CV_{s} + DI_{2 } ————— (11) From eqn(10) I_{2}= (A/B)V_{s} Substituting in eqn(11) we get I_{1}’= CV_{s} – D(A/B)V_{s} Rearranging: I_{1}’= {(AD – BC)/B}V_{s} ——–(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:
The generalized equation of hybrid (h) parameters are:
V_{1}= h_{11}I_{1} + h_{12}V_{2}
I_{2}= h_{21}I_{1} + h_{22}V_{2}
Case1: Port2 shorted, V_{2}=0, I_{2}=I_{2}’; and V_{1}=V_{s} 
Case2: Port1 shorted, V_{1}=0, I_{1}=I_{1}’; and V_{2}=V_{s} 
So the equations become:
V_{s}= h_{11}I_{1} ………………… (13) I_{2}’= h_{21}I_{1 } or I_{1 }=I_{2}’/ h_{21} Substituting value of I1 in eqn(13) we get: V_{s}= (h_{11} / h_{21})I_{2}’ I_{2}’ / V_{s} = – h_{21} / h_{11 }——————— (14)

So the equations become:
0= h_{11}I_{1}’ + h_{12}V_{s} Rearranging: h_{11}I_{1}’ = h_{12}V_{s} I_{1}’/ V_{s} = h_{12}/ h_{11} ————–(15)

For reciprocity, eqn 14 must be equal to eqn(15), so,
– h_{21} / h_{11 }= h_{12}/ h_{11}
Therefore, the condition of reciprocity in h parameter is: h_{21} = – h_{12}