MODULE 7 — Synchronous Machines
7.1 EMF Equation & Winding Factors
Induced EMF
$$ E_{ph} = 4.44 f \Phi T_{ph} K_p K_d $$
Pitch Factor ($K_p$)
$$ K_p = \cos\left(\frac{\alpha}{2}\right) $$
Distribution Factor ($K_d$)
$$ K_d = \frac{\sin(m\gamma/2)}{m\sin(\gamma/2)} $$
$$K_p = \cos(15^\circ) = 0.966$$
7.2 Power Angle Characteristic
Cylindrical Rotor
$$ P = \frac{EV}{X_s} \sin\delta $$
Salient Pole Rotor
$$ P = \frac{EV}{X_d}\sin\delta +
\frac{V^2}{2}\left(\frac{1}{X_q} -
\frac{1}{X_d}\right)\sin2\delta $$
7.3 V-Curves & Compounding
$$ (\text{Under-excited}) \xrightarrow{\text{Normal}}
(\text{Over-excited}) $$
Leading PF
(Generator)
(Generator)
Unity PF
Lagging PF
(Generator)
(Generator)
Note: For Motors, Over-excited implies Leading PF (Synchronous Condenser).
7.4 Synchronization (Dark Lamp Method)
Conditions for Paralleling
- 1. Equal Voltage ($V_1 = V_2$)
- 2. Equal Frequency ($f_1 = f_2$)
- 3. Same Phase Sequence
Bus
$f_{gen} \approx f_{bus}$