Synchronous machines: basic principle and applications

Why it matters

Almost all the electricity you use comes from synchronous generators (alternators) in power stations, and large synchronous motors drive compressors, mills and pumps at exactly constant speed while correcting plant power factor. Understanding excitation, synchronous reactance and the power-angle relation explains how a generator's voltage is regulated, how much power it can deliver before losing synchronism, and how a synchronous condenser supplies reactive power.

Key ideas

Construction

• Stator (armature): a three-phase distributed winding in a laminated core, like an induction motor. In large machines the armature is stationary because it is easier to insulate high voltage and take out large currents without sliding contacts.
• Rotor (field): carries a DC field winding fed through slip rings or a brushless exciter (or uses permanent magnets in small machines).
  • Salient-pole rotor: many projecting poles, large diameter, short length; low speeds (hydro turbines, diesel sets); non-uniform air gap.
  • Cylindrical (non-salient) rotor: smooth, long, small diameter; 2 or 4 poles at 3000/1500 rpm at 50 Hz (steam and gas turbines); uniform air gap.
• Damper (amortisseur) winding: short-circuited bars in the pole faces, like a cage. Damps oscillations (hunting) and provides starting torque for motors.

Synchronous speed. The rotor turns at exactly Ns = 120·f/P. A generator's frequency is set by its speed; a motor's speed is fixed by the supply frequency at every load (until it pulls out).

Generated EMF. Eph = 4.44·f·Φ·Tph·kw, where Tph is turns per phase and kw = kp·kd is the winding factor (pitch factor × distribution factor, slightly less than 1 for short-pitched, distributed windings).

Synchronous reactance and phasor relation. On load, armature reaction plus leakage are modelled by a synchronous reactance Xs, with armature resistance Ra usually small. Per phase:

• Generator: E = V + I·(Ra + j·Xs).
• Motor: V = E + I·(Ra + j·Xs). For a lagging load the generator needs E much larger than V; for a leading load E can be less than V. Voltage regulation = (E − V)/V can therefore be large and even negative, which is why automatic voltage regulators adjust excitation.

Power–angle relation. With Ra neglected, power transferred per machine is P = 3·V·E·sin δ / Xs, where δ (load angle) is the angle between E and V. Maximum (steady-state) power is at δ = 90°. Beyond that the machine loses synchronism (pulls out of step) — the equivalent of overloading a synchronous motor or overdriving a generator.

Synchronous motor

• Not self-starting: the stator field rotates at Ns while the heavy rotor at rest experiences a torque that reverses every half cycle, averaging zero. It is started as an induction motor on its damper winding (field shorted through a resistor), by a pony motor or by a variable-frequency drive, then DC excitation is applied near Ns and the rotor pulls into step.
• Excitation controls power factor. At constant load, under-excitation makes the motor draw lagging current, normal excitation gives unity power factor (minimum armature current), and over-excitation makes it draw leading current. Plotting armature current against field current gives the V-curves; an over-excited motor run without load is a synchronous condenser for power-factor correction and voltage support.
• For a generator on an infinite bus the same physics reads the other way: an over-excited generator delivers lagging reactive power (VArs) to the system.

Parallel operation of alternators. Before closing the breaker: equal voltage, equal frequency, same phase sequence and voltages in phase (checked with a synchroscope or lamps).

Formulas

• Ns = 120·f / P — synchronous speed (rpm).
• Eph = 4.44·f·Φ·Tph·kw — rms EMF per phase (V); Φ flux per pole (Wb), Tph series turns per phase, kw winding factor.
• E = V + I·(Ra + j·Xs) — generator phasor equation per phase (motor: V = E + I·Z_s).
• Regulation = (E − V) / V × 100 % — generator, per phase.
• P = 3·V·E·sin δ / Xs — three-phase power (W), Ra neglected; V and E per-phase rms (V), Xs (Ω).
• Pmax = 3·V·E / Xs — steady-state stability limit (δ = 90°).

Worked examples

Example 1 (standard): EMF equation. Given: 3-phase, 6-pole, star-connected alternator running at 1000 rpm; 90 slots with 4 conductors per slot; flux per pole 0.05 Wb; winding factor 0.95.

1. Frequency: f = P·N/120 = 6 × 1000/120 = 50 Hz.
2. Conductors per phase = 90 × 4/3 = 120; turns per phase Tph = 120/2 = 60.
3. Eph = 4.44·f·Φ·Tph·kw = 4.44 × 50 × 0.05 × 60 × 0.95 = 632.7 V.
4. Line EMF (star) = √3 × 632.7 = 1096 V.

Example 2 (GATE level): excitation EMF, regulation and load angle. Given: 500 kVA, 3.3 kV, star-connected alternator with Xs = 4 Ω per phase, Ra negligible, delivering full load at 0.8 pf lagging.

1. Per-phase voltage V = 3300/√3 = 1905.3 V; current I = 500000/(√3 × 3300) = 87.48 A.
2. I = 87.48∠−36.87° A; j·Xs·I = 4 × 87.48∠53.13° = 349.9∠53.13° = 210.0 + j279.9 V.
3. E = 1905.3 + 210.0 + j279.9 = 2115.3 + j279.9 V; |E| = 2133.6 V per phase (3695 V line); δ = 7.54°.
4. Regulation = (2133.6 − 1905.3)/1905.3 × 100 = 12.0 %.
5. Check power: 3 × 1905.3 × 2133.6 × sin 7.54°/4 = 400 kW = 500 kVA × 0.8. Maximum power at this excitation = 3 × 1905.3 × 2133.6/4 = 3.05 MW.

Common mistakes

• Adding I·Xs arithmetically to V instead of as a phasor — it greatly overstates E.
• Using line voltage in per-phase phasor equations for a star-connected machine.
• Saying an over-excited synchronous motor has a lagging power factor; it draws leading current.
• Thinking a synchronous motor slows down under load; its speed is fixed until it pulls out of step.
• Confusing the rotor type with application: salient poles for low-speed hydro, cylindrical rotors for high-speed turbo-alternators.
• Forgetting the winding factor and the "conductors = 2 × turns" step in the EMF equation.

For GATE IN

The Instrumentation paper treats synchronous machines at the level of principle: Ns and frequency–pole–speed relations, the EMF equation, excitation EMF and regulation from a phasor diagram, power–angle relation and maximum power, why the motor is not self-starting, and V-curves / synchronous condenser. Practise phasor calculations with complex numbers on a calculator.

Quick check

1. At what speed must a 12-pole alternator run to give 50 Hz?
2. An over-excited synchronous motor draws current at what power factor?
3. At what load angle is the power of a cylindrical-rotor machine maximum?
4. What is the function of a damper winding?

Answers: 1. 500 rpm. 2. Leading. 3. δ = 90°. 4. To damp hunting oscillations and to provide starting torque as an induction motor.