Synchronous Generators and Motors

Synchronous machines run at exactly synchronous speed. Almost all the world's electricity comes from synchronous generators (alternators); synchronous motors are used where constant speed or power factor control is needed.

EMF equation

E_ph = 4.44 f N_ph Φ K_w, with winding factor K_w = K_p K_d:

• Pitch factor K_p = cos(α/2) (α = short-pitch angle).
• Distribution factor K_d = sin(mβ/2)/(m sin(β/2)) (m slots per pole per phase, β = slot angle).
• Short pitching reduces harmonics: to eliminate the nth harmonic, short-pitch by 180°/n.

Equivalent circuit and regulation

Per phase: E = V + I_a(R_a + jX_s), where X_s (synchronous reactance) = leakage reactance + armature reaction reactance.

• Voltage regulation = (E − V)/V. It is positive for lagging and unity pf, and can be negative for leading pf.
• Methods: EMF (synchronous impedance) method (pessimistic), MMF method (optimistic), ZPF (Potier) method (most accurate).
• From open- and short-circuit tests: Z_s = (open-circuit voltage)/(short-circuit current) at the same field current. Short-circuit ratio (SCR) = 1/X_s (per unit).

Power–angle characteristic

For a cylindrical rotor (R_a neglected):

P = (3 E V/X_s) sin δ (δ = load or power angle)

• Maximum power at δ = 90° (steady-state stability limit).
• Salient pole machines add a reluctance power term (V²/2)(1/X_q − 1/X_d) sin 2δ, so their maximum is at δ < 90°.

Parallel operation (synchronising)

Before connecting to the bus: same voltage, frequency, phase sequence and phase angle (checked with synchroscope or lamps). Once connected:

• Changing the prime-mover input changes the real power (and δ).
• Changing the field excitation changes the reactive power (and pf).

Synchronous motor

• Not self-starting (uses damper windings to start as an induction motor, or a pony motor).
• V-curves: armature current against field current at constant load; minimum current at unity pf.
• Over-excited: leading pf, supplies reactive power (a synchronous condenser for pf correction). Under-excited: lagging pf.

Worked example

A 3-phase alternator: 50 Hz, 100 turns per phase, flux 0.05 Wb, K_w = 0.95.

• E_ph = 4.44 × 50 × 100 × 0.05 × 0.95 = 1054.5 V.

Practice