ABCD of nominal-π and nominal-T compared

Both models lump the line's total series impedance $Z = R + jX$ and total shunt admittance $Y = j\omega C$. They differ only in where the shunt admittance is placed.

Nominal-Π model

Half the admittance ($Y/2$) is placed at each end.

 IS       IL    Z = R + jX        IR
 o-->--+-----[ R ]--[ jX ]-----+-->--o
       |                       |
 VS  [Y/2] Ic1           Ic2 [Y/2]  VR
       |                       |
 o-----+-----------------------+-----o

$$\begin{aligned} I_{C2} &= \tfrac{Y}{2}V_R, \quad I_L = I_R + \tfrac{Y}{2}V_R \\ V_S &= V_R + Z I_L = \left(1 + \tfrac{ZY}{2}\right)V_R + Z I_R \\ I_S &= I_L + \tfrac{Y}{2}V_S = Y\left(1 + \tfrac{ZY}{4}\right)V_R + \left(1 + \tfrac{ZY}{2}\right)I_R \end{aligned}$$ $$A = D = 1 + \frac{ZY}{2}, \quad B = Z, \quad C = Y\left(1 + \frac{ZY}{4}\right)$$

Nominal-T model

The full admittance $Y$ is placed at the middle and $Z/2$ in each arm.

 IS     Z/2            Z/2     IR
 o-->--[R/2 jX/2]--+--[R/2 jX/2]-->--o
                   |
 VS              [ Y ] Ic   (VC)     VR
                   |
 o-----------------+-----------------o

$$\begin{aligned} V_C &= V_R + \tfrac{Z}{2}I_R, \quad I_S = I_R + Y V_C = Y V_R + \left(1 + \tfrac{ZY}{2}\right)I_R \\ V_S &= V_C + \tfrac{Z}{2}I_S = \left(1 + \tfrac{ZY}{2}\right)V_R + Z\left(1 + \tfrac{ZY}{4}\right)I_R \end{aligned}$$ $$A = D = 1 + \frac{ZY}{2}, \quad B = Z\left(1 + \frac{ZY}{4}\right), \quad C = Y$$

Comparison

What can be said:

• $A$ and $D$ are identical in both, and both networks are symmetrical and reciprocal ($AD - BC = 1$).
• They are duals: the factor $(1 + ZY/4)$ multiplies $C$ in the Π model and $B$ in the T model.
• Since $ZY$ is small for medium lines, $(1 + ZY/4) \approx 1$, so both give nearly the same results; neither is exact. The exact (long-line) values of $B = Z(1 + ZY/6 + \cdots)$ and $C = Y(1 + ZY/6 + \cdots)$ lie between those of the two models.
• The Π model is preferred in load-flow studies because it adds no extra node.