Power angle characteristics of cylindrical rotor machine

The power angle characteristic is the curve of electrical power ($P$) against the power (load) angle $\delta$, the angle between the excitation emf $E$ and the terminal voltage $V$.

Derivation (armature resistance neglected)

For a cylindrical-rotor generator with $\bar E = E\angle\delta$, $\bar V = V\angle 0$:

$$\begin{aligned} \bar I &= \frac{E\angle\delta - V\angle 0}{jX_s} \\ \bar S &= 3\bar V\bar I^{*} \\ P &= \frac{3EV}{X_s}\sin\delta \\ Q &= \frac{3V}{X_s}\left(E\cos\delta - V\right) \end{aligned}$$

(per-phase values; for a motor, $\delta$ is negative and power flows into the machine).

The curve

  P
  |  generator       Pmax
  |  region       .-"""""-.
  |            .'           '.
  |          .'               '.
  |        .'                   '.
 -+------+------------+-----------+-----> delta
 -180   .0           90          180 deg
     .'
  .'   motor region (delta < 0)

Features

1. Power varies sinusoidally with $\delta$.
2. Maximum power occurs at $\delta = 90^\circ$: $$P_{max} = \frac{3EV}{X_s}$$ This is the steady-state stability limit. Beyond 90°, more $\delta$ gives less power, the machine cannot hold synchronism and falls out of step (pole slipping).
3. Positive $\delta$ (E leads V): generator action; negative $\delta$: motor action.
4. Normal operation is at $\delta$ = 20°–35°, leaving a margin for sudden load changes.
5. $P_{max}$ increases with excitation $E$ and with lower $X_s$; strong excitation improves stability.
6. Synchronising power coefficient $P_{syn} = \dfrac{dP}{d\delta} = \dfrac{3EV}{X_s}\cos\delta$ is the slope of the curve; it is largest at $\delta = 0$ and zero at $90^\circ$.