Synchronous machines

Synchronous machines run at a speed locked to the supply frequency. As alternators they generate almost all of the world's electricity — every generator at Kali Gandaki A, Middle Marsyangdi or Upper Tamakoshi is a synchronous machine. As synchronous motors they drive constant-speed loads and correct power factor. This topic covers construction, windings and excitation, emf equation, speed–frequency relation, regulation, losses, synchronising and operation on an infinite bus, and synchronous motor basics.

Key concepts

• Construction: stationary armature (stator) and rotating field — easier to insulate high voltage on the stator, only two slip rings for low-voltage DC field, lighter rotor.
  • Salient-pole rotor: projecting poles, large diameter, short axial length; many poles, low speed — hydro and diesel sets. Damper windings in pole faces.
  • Cylindrical (non-salient) rotor: smooth, small diameter, long axial length; 2 or 4 poles, high speed (3000/1500 rpm) — steam/gas turbine turbo-alternators.
• Armature winding: three-phase, distributed in slots, usually double-layer, short-pitched. Pitch factor $k_p = \cos(\alpha/2)$ (α = chording angle) and distribution factor $k_d = \dfrac{\sin(m\beta/2)}{m\sin(\beta/2)}$ (m slots/pole/phase, β slot angle) reduce emf slightly but suppress harmonics.
• Field winding: on rotor, DC fed via slip rings or brushless exciter.
• Excitation systems: DC exciter (shaft-mounted DC generator), AC exciter with rectifiers, static excitation (thyristor fed from generator terminals), brushless (AC exciter + rotating diodes). See the "Excitation systems" topic.
• Speed and frequency: $f = PN/120$ — for 50 Hz, 2-pole runs at 3000 rpm, 4-pole 1500 rpm, a 24-pole hydro unit at 250 rpm.
• Voltage regulation: $\%VR = (E_0 - V)/V\times100$. Positive for lagging/unity pf, can be negative for leading pf. Determined by synchronous impedance (EMF) method (pessimistic), MMF method (optimistic), ZPF/Potier method (most accurate).
• Frequency regulation: frequency is governed by prime-mover speed; the governor's droop characteristic sets how load is shared between parallel units.
• Losses: stator copper, rotor (field) copper, core loss, friction and windage, excitation loss, stray loss. Large alternators reach 98–99% efficiency.
• Synchronising (parallel operation): incoming machine must match bus in (1) voltage magnitude, (2) frequency, (3) phase sequence, (4) phase angle. Checked with synchroscope, dark/bright lamp methods, or auto-synchroniser. Close breaker when synchroscope pointer is at 12 o'clock rotating slowly (fast direction = incoming slightly faster).
• Infinite bus: constant V and f regardless of load. On it:
  • Changing prime-mover input (steam/water) changes active power output (and load angle δ).
  • Changing excitation changes reactive power only: over-excited → supplies lagging VAR (acts like capacitor); under-excited → absorbs VAR.
• Power–angle: $P = \dfrac{EV}{X_s}\sin\delta$; maximum at δ = 90° (steady-state stability limit, cylindrical rotor).
• Synchronous motor: rotating stator field "locks" the DC-excited rotor poles; runs only at synchronous speed and is not self-starting (average torque zero at standstill).
  • Starting: damper (amortisseur) winding as induction motor, then apply DC field near synchronous speed; pony motor; variable-frequency drive.
  • V-curves: armature current vs field current for constant load — minimum at unity pf; inverted V-curves show pf.
  • Applications: constant speed drives (compressors, mills), pf correction — an over-excited motor on no load is a synchronous condenser.
  • Hunting: oscillation about synchronous speed with sudden load changes; damped by damper windings.

High-yield points

• $N_s = 120f/P$: 2-pole 3000 rpm, 4-pole 1500 rpm at 50 Hz.
• $E_{ph} = 4.44k_pk_df\Phi T_{ph}$ with $k_p = \cos(\alpha/2)$.
• Synchronising conditions: equal voltage, frequency, phase sequence and phase angle.
• On an infinite bus, excitation controls reactive power; prime-mover input controls active power.
• $P = (EV/X_s)\sin\delta$, maximum at δ = 90° (cylindrical rotor).
• Synchronous motor is not self-starting; over-excited it draws leading current (synchronous condenser).

Formulas

$E_{ph} = 4.44\,k_p k_d\, f\, \Phi\, T_{ph}$ $T_{ph}$ = series turns per phase, Φ = flux per pole (Wb), f = frequency (Hz).

$\vec E_0 = \vec V + \vec I(R_a + jX_s),\qquad N_s = \frac{120f}{P}$

Worked example

A 3-phase, 50 Hz, 20-pole alternator has 180 slots, 8 conductors/slot (single-layer, star), flux 25 mWb, full-pitch, $k_d = 0.96$. Find speed and line emf.

1. $N_s = 120\times50/20 = 300$ rpm.
2. Conductors/phase = 180×8/3 = 480 → $T_{ph} = 240$.
3. $E_{ph} = 4.44\times1\times0.96\times50\times0.025\times240 = 1278.7$ V.
4. $E_L = \sqrt3\times1278.7 \approx 2215$ V.

Exam tips

• Hydro = salient pole, many poles, low speed; thermal = cylindrical rotor, 2/4 poles.
• On an infinite bus: excitation → reactive power; mechanical input → active power.
• Synchronous motor is not self-starting; damper winding helps starting and stops hunting.
• Over-excited synchronous machine supplies lagging VAR (leading pf as a motor).
• EMF method is pessimistic, MMF method optimistic, Potier (ZPF) most accurate.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. A star-connected alternator has a per-phase terminal voltage of 2000 V and synchronous reactance $10\,\Omega$ per phase (resistance negligible). When it supplies 100 A per phase at unity power factor, its voltage regulation is about:

• A. 50 %
• B. 5.0 %
• C. 20.0 %
• D. 11.8 %

Answer: D. $E = \sqrt{V^2 + (IX_s)^2} = \sqrt{2000^2 + 1000^2} \approx 2236$ V; regulation $= (2236-2000)/2000 \approx 11.8\%$.

2. An over-excited synchronous motor run at no load solely to improve the power factor of a system is called a:

• A. Induction generator
• B. Phase advancer
• C. Synchronous condenser
• D. Synchronous reactor

Answer: C. When over-excited a synchronous motor draws leading current; operated unloaded for reactive power supply it is called a synchronous condenser (compensator).

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