Power of salient pole machine, power angle curve

Per phase, with resistance neglected, a generator has terminal voltage $V$, excitation emf $E$ and load angle $\delta$ (the angle between $E$ and $V$). The two-reaction model gives $\vec E = \vec V + j\vec I_d X_d + j\vec I_q X_q$.

Derivation

Components of current:

$$I_q = \frac{V\sin\delta}{X_q}, \qquad I_d = \frac{E - V\cos\delta}{X_d}$$

The output power per phase is $V$ multiplied by the component of $I$ in phase with $V$. $I_q$ is in phase with $E$, so it makes angle $\delta$ with $V$, and $I_d$ (at $90^\circ$ to $I_q$) also has a component $I_d\sin\delta$ along $V$:

$$P_{ph} = V I\cos\phi = V I_q\cos\delta + V I_d\sin\delta$$ $$P_{ph} = \frac{V^2\sin\delta\cos\delta}{X_q} + \frac{V\sin\delta\,(E - V\cos\delta)}{X_d}$$ $$P_{ph} = \frac{VE}{X_d}\sin\delta + V^2\sin\delta\cos\delta\left(\frac{1}{X_q} - \frac{1}{X_d}\right)$$

Using $\sin\delta\cos\delta = \tfrac12\sin2\delta$:

$$\boxed{P_{ph} = \frac{VE}{X_d}\sin\delta + \frac{V^2}{2}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin 2\delta}$$

The three-phase power is $3P_{ph}$ (with phase values). For a cylindrical rotor, $X_d = X_q$ and the second term vanishes.

• First term: excitation power (fundamental, $\sin\delta$).
• Second term: reluctance power ($\sin2\delta$). It exists even if $E=0$.

Power-angle characteristic

 P
 |        total
 |     _.-'-._
 |   .'  excit.  `-.
 | .' ____---___     `-.
 |/.-'   reluct. `-._    `-.
 +--+----+----+----+----+----> delta
 0       45   ~75-80  90   180
 (max power below 90 deg)

• The total curve is the sum of a $\sin\delta$ curve and a $\sin2\delta$ curve.
• Maximum power occurs at $\delta < 90^\circ$ (usually 60 to 80°) for a salient-pole machine, and at $90^\circ$ for a cylindrical machine.
• Stable operation is on the rising part ($0 < \delta < \delta_{max}$), where $dP/d\delta > 0$. Beyond it the machine loses synchronism.