Equivalent-π model of a long line

An equivalent-Π model of a long line is a lumped Π circuit (series $Z'$, shunt $Y'/2$ at each end) whose ABCD constants are exactly equal to those of the long line. It gives exact terminal results while being as simple as a nominal-Π circuit.

 IS          Z'                 IR
 o-->--+---[ Z' ]----+-->--o
       |             |
 VS  [Y'/2]       [Y'/2]      VR
       |             |
 o-----+-------------+-----o

ABCD of the Π circuit and of the long line

For any Π circuit:

$$A = D = 1 + \frac{Z'Y'}{2}, \quad B = Z', \quad C = Y'\left(1 + \frac{Z'Y'}{4}\right)$$

For the long line:

$$A = D = \cosh\gamma l, \quad B = Z_c\sinh\gamma l, \quad C = \frac{\sinh\gamma l}{Z_c}$$

Series arm $Z'$

Equating $B$:

$$Z' = Z_c\sinh\gamma l = \sqrt{\frac{z}{y}}\sinh\gamma l$$

Multiply and divide by $l\sqrt{zy}$ (note $Z = zl$, $\gamma l = l\sqrt{zy}$):

$$Z' = zl\,\frac{\sinh\gamma l}{l\sqrt{zy}} = Z\,\frac{\sinh\gamma l}{\gamma l}$$

Shunt arm $Y'/2$

Equating $A$:

$$\begin{aligned} 1 + \frac{Z'Y'}{2} &= \cosh\gamma l \\ \frac{Y'}{2} &= \frac{\cosh\gamma l - 1}{Z_c\sinh\gamma l} = \frac{1}{Z_c}\tanh\frac{\gamma l}{2} \end{aligned}$$

using $\cosh\gamma l - 1 = 2\sinh^2\frac{\gamma l}{2}$ and $\sinh\gamma l = 2\sinh\frac{\gamma l}{2}\cosh\frac{\gamma l}{2}$. Writing $\frac{1}{Z_c} = \sqrt{y/z}$ and multiplying and dividing by $\frac{\gamma l}{2}$:

$$\frac{Y'}{2} = \frac{Y}{2}\,\frac{\tanh(\gamma l/2)}{\gamma l/2}$$

(The $C$ equation is then automatically satisfied, because $AD - BC = 1$.)

Summary

The correction factors $\frac{\sinh\gamma l}{\gamma l}$ and $\frac{\tanh(\gamma l/2)}{\gamma l/2}$ tend to 1 as $\gamma l \to 0$, so for shorter lines the equivalent Π reduces to the nominal Π. For long lines, $Z'$ is slightly less than $Z$ and $Y'$ slightly more than $Y$.