Line model classification and short-line ABCD

Classification of transmission line models

Based on length (for 50 Hz lines):

ABCD parameters of a short line

The shunt capacitance is neglected, so the same current flows through the whole line.

 IS = IR      R        jX        IR
 o---->----[ R ]----[ jX ]---->----o
 VS                                VR
 o---------------------------------o

$$\begin{aligned} I_S &= I_R \\ V_S &= V_R + I_R(R + jX) = V_R + Z I_R \end{aligned}$$

Comparing with $V_S = AV_R + BI_R$ and $I_S = CV_R + DI_R$:

$$A = 1, \quad B = Z = R + jX\ \Omega, \quad C = 0, \quad D = 1$$

Check: $AD - BC = 1 - 0 = 1$.

Phasor diagrams

Take $V_R$ as reference. The drop $I_R R$ is parallel to $I_R$ and $I_R X$ leads $I_R$ by 90°. $\delta$ is the angle between $V_S$ and $V_R$.

Lagging p.f. load ($I_R$ lags $V_R$ by $\phi_R$):

                               + tip of VS
                    VS     .   |
                       .       | IR.X (90 deg
                   .           |  ahead of IR)
              .  d             |
   O------------------------> VR
    \ phiR                     \ IR.R (along IR)
     \                          +
      v IR

Here $V_S > V_R$ and $V_S \approx V_R + I_R R\cos\phi_R + I_R X\sin\phi_R$, so regulation is positive and large.

Leading p.f. load ($I_R$ leads $V_R$ by $\phi_R$):

             ^ IR
            /       IR.X   + tip of VS
           /     (90 deg     \
          /    ahead of IR)   \
         /         VS  .       + IR.R
        / phiR    .           /  (along IR)
       /     . d             /
   O------------------------> VR

Here $V_S \approx V_R + I_R R\cos\phi_R - I_R X\sin\phi_R$; the $I_R X$ drop tilts backwards, so $V_S$ can be less than $V_R$ (negative regulation).