Transmission parameters in terms of Y-parameters

The transmission (ABCD) parameters are defined by V1 = A V2 − B I2 and I1 = C V2 − D I2. The Y-parameters are defined by I1 = y11 V1 + y12 V2 and I2 = y21 V1 + y22 V2. ("Transmission line parameters" here means the transmission parameters of the two-port.)

Derivation

Step 1. Solve the second Y equation for V1:

I2 = y21 V1 + y22 V2
V1 = −(y22/y21) V2 + (1/y21) I2

Comparing with V1 = A V2 − B I2:

A = −y22/y21          B = −1/y21

Step 2. Substitute V1 into the first Y equation:

I1 = y11 V1 + y12 V2
   = y11[−(y22/y21) V2 + (1/y21) I2] + y12 V2
   = [ y12 − y11 y22/y21 ] V2 + (y11/y21) I2
   = −[(y11 y22 − y12 y21)/y21] V2 + (y11/y21) I2

Comparing with I1 = C V2 − D I2, and writing Δy = y11 y22 − y12 y21:

C = −Δy/y21           D = −y11/y21

Result

A = −y22/y21      B = −1/y21
C = −Δy/y21       D = −y11/y21        with Δy = y11 y22 − y12 y21

Checks. (1) AD − BC = [y11y22 − Δy]/y21² = y12 y21/y21² = y12/y21, which equals 1 exactly when y12 = y21 (reciprocal). (2) For a series resistor R (y11 = y22 = 1/R, y12 = y21 = −1/R): A = 1, B = R, C = 0, D = 1, as expected.