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Reciprocal two-port and condition in T parameters

Short answer

A reciprocal two-port yields identical transfer ratios when excitation and response are swapped. From T equations, apply voltage at port1 with port2 shorted: I2 = −V/B. Apply same voltage at port2 with port1 shorted: I1 = V(BC−AD)/B. Reciprocity requires I2/V = I1/V, so AD − BC = 1.

What do you understand by reciprocal two-port network? Also derive the condition for the same in terms of T parameters.

A two-port network is reciprocal if the ratio of the response at one port to the excitation at the other port is the same when the positions of excitation and response are interchanged (the reciprocity theorem). Networks made only of R, L, C and mutual inductance (no dependent sources, no gyrators) are reciprocal.

Derivation for T (ABCD) parameters

The equations are V1 = A V2 − B I2 and I1 = C V2 − D I2.

Case 1. Voltage Va applied at port 1, port 2 short-circuited (V2 = 0):

Va = −B I2a    →   I2a = −Va/B

Case 2. Voltage Vb applied at port 2, port 1 short-circuited (V1 = 0):

0 = A Vb − B I2b   →   I2b = A Vb/B
I1b = C Vb − D I2b = C Vb − D A Vb/B = Vb (BC − AD)/B

For a reciprocal network the transfer admittances are equal: (current at port 2 due to voltage at port 1) = (current at port 1 due to the same voltage at port 2). That is I2a/Va = I1b/Vb:

−1/B = (BC − AD)/B   →   AD − BC = 1

Condition for reciprocity: AD − BC = 1 (i.e. determinant of the T matrix = 1).

(Reciprocity in other parameters: z12 = z21, y12 = y21. A reciprocal network is also symmetrical if A = D.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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