Synchronous Machines

Introduction and Context

Synchronous machines are alternating-current machines whose mechanical speed is locked to the rotating magnetic field. They are used as turbogenerators, hydrogenerators, large industrial motors, synchronous condensers, and voltage-support devices. In the laboratory, they provide a unified platform for studying electromagnetic energy conversion, excitation control, voltage regulation, power-angle behavior, reactive-power exchange, synchronization, and parallel operation.

A synchronous machine may operate as:

• an alternator, converting mechanical power into three-phase electrical power;
• a synchronous motor, converting electrical power into mechanical power; or
• a synchronous condenser, operating without a mechanical load to generate or absorb reactive power.

The same machine can perform all three functions, although its construction and control settings may differ according to the application.

Learning Objectives

After completing this chapter, the student should be able to:

1. Explain the construction and operating principle of synchronous machines.
2. Determine synchronous speed, frequency, generated emf, and torque relationships.
3. Perform open-circuit, short-circuit, voltage-regulation, and load tests on an alternator.
4. Explain excitation systems and distinguish the effects of armature reaction and synchronous reactance.
5. Synchronize an alternator safely with an energized busbar.
6. Analyze synchronous-motor power factor, V-curves, reactive-power control, and power-angle characteristics.
7. Identify capability limits, stability limits, and practical causes of abnormal operation.
8. Operate machines in parallel while observing electrical safety, measurement accuracy, and applicable laboratory procedures.

1. Construction and Operating Principle

A synchronous machine has two principal magnetic systems: the stator, which normally carries a distributed three-phase winding, and the rotor, which carries a direct-current field winding or permanent magnets. The stator is usually stationary and the rotor rotates inside it.

The rotor is commonly one of two types:

• Salient-pole rotor: has projecting poles, a large diameter, and a short axial length. It is suitable for low- and medium-speed hydraulic machines.
• Cylindrical or nonsalient rotor: has a smooth forged-steel body with field conductors embedded in slots. It is mechanically strong and suitable for high-speed turbo-alternators.

When a three-phase stator winding is supplied by balanced currents, it produces a rotating magnetic field. Its synchronous speed is

where n_s is synchronous speed in revolutions per minute (r/min), f is electrical frequency in hertz, and P is the number of poles. Equivalently, angular synchronous speed is

in rad/s. These relationships assume a balanced system and the conventional definition of pole count. For example, a four-pole, 50-Hz machine operates at 1500 r/min.

In an alternator, a prime mover rotates the field. The rotating field cuts the stator conductors and induces three sinusoidal phase emfs displaced by 120 electrical degrees. In a motor, the stator field interacts with the rotor field and produces torque only when the rotor is brought close to synchronous speed and pulled into synchronism.

The rotor must rotate at exactly synchronous speed in steady operation. A small angular displacement between the rotor field axis and the resultant stator field axis is called the torque angle, load angle, or power angle, commonly denoted by δ.

2. Generated EMF and Machine Ratings

For a sinusoidally distributed winding, the rms generated phase emf is approximately

where E_phase is rms phase emf in volts, k_w is the winding factor, Φ is flux per pole in webers, and T is the number of series turns per phase. The winding factor includes distribution and pitch factors and is less than or equal to unity. This equation assumes sinusoidal flux variation and a sinusoidal fundamental component of the induced emf.

For a star-connected winding,

where V_L is line voltage and V_phase is phase voltage. For a delta-connected winding, V_L = V_phase. The line current relationships are reversed: in star, line current equals phase current; in delta, line current is √3 times phase current.

The apparent power rating of a balanced three-phase machine is

in volt-amperes. Real and reactive powers are

where φ is the phase-angle displacement between line voltage and line current. The sign convention for Q must be stated: positive reactive power may be defined as power absorbed by the machine, whereas generator studies sometimes use the opposite convention.

A machine’s nameplate ratings are simultaneous operating limits, not merely independent target values. Terminal voltage, stator current, field current, frequency, temperature rise, power factor, and speed must remain within manufacturer limits.

3. Alternator Tests

3.1 Preliminary laboratory procedure

Before energizing a machine, inspect earthing, guards, couplings, fuses, emergency-stop facilities, field supply polarity, instrument ranges, and the mechanical condition of the prime mover. Confirm that the alternator terminals are isolated until synchronization is intentionally performed. Never assume that an apparently open terminal is dead; verify with an approved meter.

Record the nameplate voltage, current, frequency, speed, field-current limit, connection, and rated power factor. Use a tachometer or speed sensor and calibrated electrical instruments. Begin with minimum field current and no load unless the laboratory procedure specifies otherwise.

3.2 Open-circuit characteristic

The open-circuit characteristic (OCC) is the relationship between generated terminal voltage and field current at rated speed with the armature open-circuited. The field current is increased in steps, and the terminal voltage is recorded after the readings stabilize. A descending set of readings may also be obtained to reveal hysteresis.

At low excitation, voltage rises almost linearly with field current. At higher excitation, the iron saturates and the curve bends. The OCC is useful for estimating excitation requirements, observing saturation, and comparing magnetization behavior. It must be taken at constant speed because generated emf is proportional to frequency as well as flux.

3.3 Short-circuit characteristic

The short-circuit characteristic (SCC) is the armature short-circuit current as a function of field current at rated speed. The test is performed cautiously with suitable current protection. In the unsaturated region, the SCC is often approximately linear because the magnetic circuit is weakly excited under short-circuit conditions.

A common approximate synchronous-impedance method uses the OCC voltage corresponding to a specified field current and the SCC current at the same field current:

The synchronous impedance is then resolved into resistance and reactance:

where R_a is armature resistance per phase. This method assumes balanced operation, sinusoidal quantities, constant speed, and a simplified representation of armature reaction by synchronous reactance. It usually gives a conservative voltage-regulation estimate because saturation is represented imperfectly.

3.4 Synchronous-impedance voltage regulation

For a generator using the generator sign convention, the approximate per-phase phasor equation is

where E_A is the internal generated emf per phase, V_φ is terminal phase voltage, I_A is armature current, R_a is armature resistance, and X_s is synchronous reactance. The resistance and reactance are in ohms per phase.

Voltage regulation is defined as

for a specified load current and power factor, with the field current adjusted so that rated-load terminal voltage is obtained. The definition assumes comparison between rated-load voltage and no-load voltage at the same speed and field excitation, unless a particular standard specifies another test condition.

The resistance can be measured by a dc test. For a star-connected winding, the measured line-to-line resistance is approximately twice the phase resistance, so R_a = R_LL/2, after correcting for temperature. Copper resistance increases with temperature; therefore, a cold resistance measurement can underestimate the operating value.

A more accurate voltage-regulation procedure is the ampere-turn method or Potier method, both of which attempt to separate leakage reactance from armature-reaction effects. The Potier method uses the OCC and a zero-power-factor characteristic. These methods are more involved and remain dependent on test quality and assumptions about magnetic saturation.

Worked Example 1: Synchronous impedance and regulation

A three-phase, star-connected alternator has a line-to-line open-circuit voltage of 400 V at a particular field current. At the same speed and field current, the short-circuit line current is 20 A. The armature resistance is 0.20 Ω per phase. Determine the approximate synchronous impedance and reactance per phase.

The open-circuit phase voltage is

E_oc,phase = 400/√3 = 230.9 V.

Because the star-connected short-circuit phase current equals line current,

Z_s = 230.9/20 = 11.55 Ω.

Thus,

X_s = √(11.55² − 0.20²) ≈ 11.55 Ω.

The resistance is small compared with the synchronous reactance. The result is an approximate unsaturated synchronous reactance, not a direct measurement of leakage reactance alone.

4. Armature Reaction, Excitation, and Voltage Control

Armature reaction is the effect of stator current mmf on the main magnetic field. Its effect depends on load power factor. At unity power factor it is approximately cross-magnetizing; at lagging power factor it tends to demagnetize; at leading power factor it tends to magnetize. The exact interpretation depends on machine convention, winding distribution, saturation, and the chosen phasor reference.

The excitation system supplies rotor field current and may include a dc exciter, brushless exciter, automatic voltage regulator (AVR), rotating rectifiers, and protective limiters. An AVR adjusts field current to maintain terminal voltage during changes in load. However, the AVR cannot overcome thermal, magnetic, or stability limits indefinitely.

Increasing excitation in a grid-connected alternator generally increases reactive-power export and tends to raise terminal voltage if the system is weak. In a grid-connected synchronous motor, overexcitation usually causes leading stator current and reactive-power generation; underexcitation causes lagging current and reactive-power absorption. The exact sign should be confirmed using the laboratory’s power convention.

5. Power-Angle Relationship and Stability

For a cylindrical-rotor machine connected to a strong bus through synchronous reactance, neglecting armature resistance, the real power is approximately

where P_e is three-phase electrical power, V_φ is bus phase voltage, E_A is internal emf per phase, and X_s is synchronous reactance per phase. This is the classical power-angle equation.

The maximum theoretical power in this simplified model occurs at δ = 90°:

Operation near this point is undesirable because the synchronizing torque becomes weak and transient stability margin is small. The stable operating region is normally well below the theoretical maximum.

For a salient-pole machine, the approximate power-angle equation includes reluctance power:

where X_d and X_q are direct- and quadrature-axis reactances. The first term is excitation torque; the second is reluctance torque. This equation assumes a simplified steady-state model, negligible resistance, and sinusoidal balanced conditions.

A sudden increase in mechanical input to a generator increases δ and real-power transfer. It does not, by itself, determine the reactive-power output. Reactive power is mainly controlled through excitation, while real power is mainly controlled through mechanical input.

6. Synchronization and Parallel Operation

Connecting an incoming alternator to a live busbar is synchronization. Before closing the breaker, the incoming machine must have:

1. the same phase sequence;
2. nearly the same rms voltage magnitude;
3. the same frequency; and
4. zero or sufficiently small phase-angle difference at the instant of closing.

The phase-sequence condition is fundamental. Closing with reversed phase sequence produces severe current and torque transients. Phase sequence may be checked with a sequence indicator or by an approved lamp method. Voltage magnitude is adjusted with excitation, while frequency is adjusted through prime-mover speed. The phase angle is allowed to drift slowly so that the breaker can be closed near phase coincidence.

The three-lamp method can indicate synchronism. Depending on the connection, lamps may brighten and darken together or in a sequence. A synchroscope directly indicates phase-angle difference and whether the incoming machine is fast or slow. Automatic synchronizers measure voltage, frequency, phase sequence, and phase angle and issue a close command within configured limits.

After connection, the alternator must not be controlled as though it were isolated. Increasing prime-mover torque increases exported real power. Increasing field excitation changes reactive-power exchange and may alter terminal voltage. Load sharing between parallel generators requires suitable governor droop and excitation control. Poor control can cause circulating reactive currents, unequal kW sharing, oscillations, or loss of synchronism.

Parallel-operation checklist

• Verify breaker rating, earthing, interlocks, and protection.
• Confirm phase sequence and correct instrument connections.
• Match voltage and frequency within prescribed limits.
• Close only when phase displacement is acceptable.
• Observe active power, reactive power, current, field current, and power factor.
• Transfer load gradually; never make abrupt governor or excitation changes.
• Open the breaker only after unloading or following the approved procedure.

7. Synchronous-Motor Tests

A synchronous motor is not self-starting when connected directly to a fixed-frequency supply because the average starting torque is zero while the rotor is stationary. Starting methods include a damper winding, a variable-frequency drive, or an auxiliary motor. Once near synchronous speed, field excitation is applied and the rotor locks into synchronism.

For a motor, the phasor relation is commonly written

with the sign and angle convention chosen for motor operation. The input power is

and, neglecting losses, the developed mechanical power is approximately equal to input electrical power. In practice,

The losses must be defined consistently; field power may be supplied separately and may or may not be included in the stated input power.

V-curves

At constant terminal voltage and constant mechanical load, armature current plotted against field current has a V-shaped characteristic. At low excitation, the motor is underexcited and draws lagging current. As excitation increases, current decreases until near-unity power factor is reached. Further excitation causes leading current and increases its magnitude again. These are the V-curves.

The corresponding curves of power factor versus field current are often called inverted V-curves. V-curves demonstrate that excitation controls reactive power without being the primary control of mechanical load. They also show why excessive excitation can exceed the field-current or stator-current limit.

Reactive-power control

A synchronous motor operating overexcited can supply reactive power to an industrial bus and improve power factor. Used without a mechanical load, it is a synchronous condenser. Underexcitation absorbs reactive power. In either case, the machine must remain within its capability curve and thermal limits.

8. Capability Limits

A generator capability diagram identifies combinations of real power and reactive power that can be sustained safely. Principal limits include:

• stator-current limit: approximately a constant apparent-power boundary, since S = √3 V_L I_L;
• field-current limit: limits overexcited operation and rotor heating;
• underexcitation limit: prevents end-region heating, loss of stability, and operating difficulties;
• prime-mover limit: restricts maximum mechanical input and real power;
• stability limit: restricts power-angle operation, especially during disturbances;
• terminal-voltage and cooling limits: depend on insulation, temperature, and ventilation.

A capability diagram is not a license to operate continuously at every boundary. Measurement uncertainty, ambient temperature, transient conditions, and protection settings require an operating margin. The actual diagram must come from the manufacturer or system study rather than from an assumed generic circle.

9. Illustrative Case: Synchronizing a Laboratory Alternator

Background. A 400-V, 50-Hz, three-phase alternator is driven by a variable-speed motor and is to be connected to a laboratory bus supplied by the utility. The alternator has a rated speed of 1500 r/min and a four-pole rotor.

Problem. During a previous exercise, the incoming machine was connected with a frequency difference that was too large. The breaker produced a current transient and the shaft experienced a noticeable torque shock.

Analysis. The machine’s synchronous speed is

n_s = 120(50)/4 = 1500 r/min.

The operator first confirms that the phase sequence matches the bus. The prime mover is adjusted so that the incoming frequency is only slightly above the bus frequency, allowing the phase angle to move slowly. Field current is adjusted until the incoming voltage is approximately 400 V line-to-line. A synchroscope is then used to identify phase coincidence.

Approach. The breaker-close permissive is set to require correct phase sequence, a small voltage difference, a small frequency difference, and an acceptable phase-angle window. The operator closes near the selected synchroscope position and immediately observes active power, reactive power, current, and field current.

Outcome. The alternator closes without a severe transient. The governor is then adjusted to increase exported real power, while excitation is adjusted to control reactive power. The machine remains below its stator-current and field-current limits.

Learning points. Synchronization is both an electrical and mechanical event. Matching only voltage magnitude is insufficient. Frequency, phase sequence, and phase angle must all be considered. After closing, mechanical input primarily controls real power and excitation primarily controls reactive power.

10. Measurement, Errors, and Safety

Laboratory results may differ from theoretical predictions because of winding resistance, saturation, harmonics, temperature variation, friction, windage, instrument loading, speed variation, and inaccurate phase-angle measurements. The synchronous-impedance method is especially sensitive to the chosen OCC point and to saturation assumptions.

Use true-rms meters where waveforms are distorted, and verify whether instruments display phase or line quantities. Apply the correct connection relationships. Record field current and speed with every voltage or current reading. Allow temperatures to stabilize when comparing tests.

Electrical hazards include lethal voltage, stored energy, rotating shafts, unexpected backfeed, and overspeed. The machine must be guarded, grounded, fused, and operated from a clearly defined switching procedure. Never change a current-transformer secondary connection while energized and never bypass an interlock without authorization. Protective relays and emergency stops must be tested according to institutional rules.

Ethically, engineers must not conceal unstable readings, falsify synchronization results, or operate beyond nameplate limits merely to obtain a desired graph. Professional practice requires traceable data, uncertainty awareness, appropriate calibration, and prompt reporting of abnormal noise, heating, vibration, or smell.

11. Contemporary Developments

Modern synchronous machines increasingly use digital AVRs, brushless excitation, permanent-magnet rotors, online thermal monitoring, and model-based protection. Grid-forming converters can reproduce some voltage-source functions of synchronous machines, while conventional synchronous condensers remain valuable for inertia, short-circuit strength, and voltage support. Digital twins and machine-learning tools can assist condition monitoring by identifying abnormal vibration, partial discharge, or temperature patterns.

These technologies do not remove the need for engineering judgment. Training data may be incomplete, sensor failures may produce misleading predictions, and automated control must remain subject to protection systems, cybersecurity requirements, and human oversight.

Best-Practice Test Sequence

1. Read the nameplate and laboratory risk assessment.
2. Inspect mechanical coupling, guards, grounding, and wiring.
3. Select safe meter ranges and verify zero readings.
4. Start the prime mover with field current at the prescribed minimum.
5. Stabilize speed before recording OCC or SCC data.
6. Do not exceed rated current, field current, speed, or temperature.
7. For synchronization, check sequence, voltage, frequency, and phase angle.
8. Record both electrical and mechanical variables.
9. Reduce load and field safely before shutdown.
10. Plot results with units, scales, operating conditions, and uncertainty.

Chapter Summary

Synchronous machines convert energy through interaction between a rotating rotor field and a three-phase stator field. Their speed is fixed by frequency and pole number. Alternator testing commonly includes the open-circuit and short-circuit characteristics, resistance measurement, and voltage-regulation determination. The synchronous-impedance method is useful but approximate because it represents armature reaction and leakage effects by a single reactance.

Excitation controls the rotor magnetic field and strongly influences reactive power and terminal voltage. Mechanical input primarily controls real-power transfer. The classical power-angle equation explains steady-state transfer and stability limits, while salient-pole machines also develop reluctance power. Synchronous motors require a starting method, operate at controllable power factor, and exhibit V-curves as excitation changes.

Safe parallel operation requires matching phase sequence, voltage, frequency, and phase angle. After synchronization, governor and excitation controls must be coordinated to share real and reactive power. Stator heating, rotor heating, underexcitation, prime-mover capability, voltage, and stability define the practical capability region. Accurate data, calibrated instruments, sound protection, and ethical reporting are essential to meaningful laboratory work.

Key Terms

Synchronous machine: AC machine whose rotor runs at synchronous speed.

Alternator: Synchronous machine operating as a generator.

Synchronous motor: AC motor whose rotor locks to the rotating stator field.

Synchronous speed: Speed determined by frequency and pole number.

Salient-pole rotor: Projecting-pole rotor used mainly at lower speeds.

Cylindrical rotor: Smooth rotor used mainly in high-speed machines.

Excitation: DC or equivalent magnetic-field supply to the rotor.

Open-circuit characteristic: Terminal voltage versus field current at no load.

Short-circuit characteristic: Armature short-circuit current versus field current.

Armature reaction: Effect of stator mmf on the main magnetic field.

Synchronous reactance: Approximate reactance representing leakage and armature-reaction effects.

Voltage regulation: Change from rated-load voltage to corresponding no-load voltage.

Power angle: Electrical angle between internal emf and terminal-voltage reference.

Synchronizing: Connecting an incoming alternator to an energized bus under matched conditions.

V-curve: Armature-current versus field-current characteristic of a synchronous motor.

Synchronous condenser: Unloaded synchronous machine used for reactive-power control.

Capability curve: Permissible region of real and reactive power operation.

Underexcitation: Field condition producing lagging motor current or reduced generator excitation.

Overexcitation: Field condition producing leading motor current or increased generator reactive export.

Automatic voltage regulator: Controller that adjusts excitation to regulate voltage.

Review Questions

Short Answer Questions

1. What determines the synchronous speed of an AC machine?
2. Why does an alternator’s open-circuit characteristic become nonlinear at high field current?
3. State the four principal synchronization conditions.
4. What is the purpose of a damper winding in a synchronous motor?
5. Why does increasing prime-mover torque primarily change real power after synchronization?

Long Answer Questions

1. Explain the construction, operation, and applications of salient-pole and cylindrical-rotor synchronous machines.
2. Describe the open-circuit and short-circuit tests and explain how the synchronous-impedance method estimates voltage regulation.
3. Derive and discuss the assumptions behind the cylindrical-rotor power-angle equation.
4. Explain synchronous-motor V-curves and the relationship between excitation, power factor, and reactive power.
5. Describe a complete and safe procedure for synchronizing an alternator with a live busbar.

Analytical / Application Questions

1. A four-pole alternator is connected to a 50-Hz system. Determine its synchronous speed and explain the consequences of operating it steadily at a different speed.
2. A synchronous motor’s field current is increased at constant mechanical load. Predict the sequence of changes in armature current, power factor, and reactive-power exchange, and identify the relevant thermal limits.
3. Two alternators operate in parallel. Explain how an operator can increase the real-power output of one machine without intentionally changing its reactive-power output, and identify practical reasons why some reactive-power change may still occur.

Multiple Choice Questions with Answers

Question 1

A four-pole synchronous machine connected to a 50-Hz supply has a synchronous speed of:

A. 750 r/min

B. 1000 r/min

C. 1500 r/min

D. 3000 r/min

Correct Answer: C. 1500 r/min

Question 2

In an open-circuit test of an alternator, the quantity normally varied is:

A. Field current

B. Number of poles

C. Stator resistance

D. Mechanical load torque only

Correct Answer: A. Field current

Question 3

For a star-connected three-phase winding, the line voltage is:

A. Equal to zero

B. √3 times the phase voltage

C. One-third of the phase voltage

D. π times the phase voltage

Correct Answer: B. √3 times the phase voltage

Question 4

Which condition is essential before paralleling an incoming alternator with a live busbar?

A. The incoming machine must have zero field current

B. The phase sequence must be identical

C. The prime mover must be stopped

D. The terminal voltage must be zero

Correct Answer: B. The phase sequence must be identical

Question 5

In the classical cylindrical-rotor power-angle equation, maximum theoretical power occurs at approximately:

A. δ = 0°

B. δ = 30°

C. δ = 90°

D. δ = 180°

Correct Answer: C. δ = 90°

Question 6

At constant motor load, an underexcited synchronous motor generally draws:

A. Lagging current

B. Leading current

C. Zero current

D. Only harmonic current

Correct Answer: A. Lagging current

Question 7

The minimum armature current on a synchronous-motor V-curve generally occurs near:

A. Rated overspeed

B. Unity power factor

C. Zero terminal voltage

D. Short circuit

Correct Answer: B. Unity power factor

Question 8

After a generator is synchronized to a stiff bus, increasing mechanical input primarily increases:

A. Real-power export

B. Number of poles

C. Frequency of the stiff bus

D. Armature resistance

Correct Answer: A. Real-power export

Question 9

The stator-current capability limit is closely associated with a limit on:

A. Rotor position only

B. Apparent power and stator heating

C. Phase sequence only

D. Mechanical bearing color

Correct Answer: B. Apparent power and stator heating

Question 10

A synchronous condenser is a synchronous machine operated primarily to:

A. Provide mechanical starting torque to an induction motor

B. Control reactive power without a useful mechanical load

C. Increase the number of supply poles

D. Measure winding resistance

Correct Answer: B. Control reactive power without a useful mechanical load