Synchronous Machines

Introduction and Context

Synchronous machines are alternating-current machines whose mechanical speed is locked to the frequency of the electrical system. They are used as generators in almost all large power stations and as motors where constant speed, high efficiency, or controllable power factor is required. In modern mechatronic systems, synchronous-machine principles also underpin permanent-magnet motors, high-performance servo drives, wind-turbine generators, and grid-connected converter systems.

A synchronous machine may operate as:

• a synchronous generator, converting mechanical power into electrical power;
• a synchronous motor, converting electrical power into mechanical power; or
• a synchronous condenser, exchanging reactive power with the grid without delivering significant mechanical output.

The central analytical ideas are excitation, phasor relationships, power-angle characteristics, voltage regulation, synchronization, transient stability, and the effects of rotor saliency.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain the construction and operating principle of synchronous machines.
2. Describe excitation systems and their influence on voltage and reactive power.
3. Construct and interpret generator and motor phasor diagrams.
4. Apply synchronous impedance and power-angle equations.
5. Calculate approximate voltage regulation and power-factor effects.
6. State the conditions and procedure for synchronizing a generator.
7. Explain steady-state and transient stability in qualitative and quantitative terms.
8. Distinguish cylindrical-rotor and salient-pole behavior.
9. Evaluate practical operating decisions involving excitation, loading, and grid connection.

Construction and Operating Principle

The stator normally contains a three-phase distributed armature winding. The rotor carries either a direct-current field winding or permanent magnets. The three-phase stator currents produce a rotating magnetic field with synchronous speed

where n_s is synchronous speed in revolutions per minute, f is electrical frequency in hertz, and P is the number of poles. In angular form,

where ω_s is mechanical synchronous angular speed in radians per second. These formulas assume a balanced polyphase system and a conventional pole count.

The rotor field rotates at the same mechanical speed as the stator field in steady operation. If the rotor speed differs from synchronous speed for a sustained period, the machine cannot remain synchronously locked, although transient oscillations can occur after a disturbance.

Rotor Types

The rotor field may be supplied through slip rings and brushes, brushless exciters, or static excitation equipment. Permanent-magnet rotors eliminate field winding losses but reduce direct control of excitation and may create demagnetization or fault-management challenges.

Excitation Systems

Excitation establishes the rotor magnetic field. In a conventional machine, a controlled direct current I_f produces the field flux. Increasing excitation generally increases the internal generated emf E and changes the reactive-power exchange with the network.

A practical excitation system commonly contains:

1. an exciter supplying rotor field current;
2. a voltage regulator measuring terminal voltage;
3. a stabilizing or limiting function;
4. protective elements such as overexcitation, underexcitation, and field-current limiters.

An automatic voltage regulator (AVR) adjusts excitation to maintain terminal voltage under changing load. Its action is not instantaneous: the field winding has inductance, the exciter has limits, and the control loop may produce oscillations if poorly tuned.

For a grid-connected generator, excitation primarily controls reactive power and terminal voltage, while the mechanical prime mover controls active power. For a grid-connected motor, excitation controls the motor power factor and reactive-power exchange. This separation is approximate because voltage, active power, reactive power, and network impedance are coupled.

Underexcitation and Overexcitation

An overexcited synchronous motor draws a leading current and can supply reactive power to the grid. An underexcited motor draws lagging current. A synchronous generator generally supplies reactive power when overexcited and absorbs reactive power when underexcited, subject to the sign convention and network conditions.

Excitation must remain within a capability curve. Excessive field current causes rotor heating; excessive stator current causes armature heating; and severe underexcitation can cause loss of synchronism or end-region heating. Therefore, excitation control is both an operating and a protection problem.

Per-Phase Phasor Model

Analysis is usually performed per phase using rms phasors. For a balanced three-phase machine connected in star, the phase voltage is related to the line voltage by V_ph = V_L/√3, and the three-phase apparent power is

where S is in volt-amperes, V_L is line-to-line rms voltage, and I_L is line current. For a balanced star-connected machine, I_L = I_ph.

The synchronous impedance is approximated by

where R_a is per-phase armature resistance in ohms and X_s is synchronous reactance in ohms. The reactance includes leakage reactance and the effect of armature reaction under the adopted linear model.

Generator Equation

Using the convention that current leaves the generator terminals,

where E is internal generated emf per phase, V is terminal voltage per phase, and I_a is armature current per phase. This equation assumes a balanced steady-state system, sinusoidal quantities, a fixed frequency, and the synchronous-impedance approximation.

Motor Equation

Using the convention that current enters the motor,

or equivalently E = V - I_a(R_a+jX_s). The sign difference from the generator equation arises from the direction of power flow and current reference.

If resistance is neglected, the generator relationship becomes E ≈ V+jX_sI_a, while the motor relationship becomes E ≈ V-jX_sI_a. Resistance should not be ignored when copper loss, voltage regulation, or low-power-factor operation is important.

Phasor Interpretation

For a generator supplying a lagging current, I_aR_a is in phase with current and jX_sI_a leads current by 90°. The internal emf is the vector sum of terminal voltage and the impedance drop. For a motor, the internal emf is obtained by subtracting the impedance drop from terminal voltage.

The load angle, or power angle, δ, is the angular displacement between the internal emf and terminal voltage. In generator operation, E normally leads V; in motor operation, it normally lags V. The load angle is not the same as the phase angle between terminal voltage and current.

Synchronous Generator Voltage Regulation

Voltage regulation indicates the change in terminal voltage between no-load and full-load conditions at constant speed and excitation. A common definition is

where E₀ is the no-load voltage at the specified excitation and V_FL is the full-load terminal voltage. The definition must state whether excitation is held constant and whether the comparison is made at a specified power factor.

Using the synchronous-impedance method, calculate E from the generator phasor equation and then use the regulation expression. The method is convenient but often gives a conservative result because synchronous reactance includes armature-reaction effects and assumes a linear magnetic circuit.

For a generator with resistance neglected:

• at unity power factor, the reactive drop is perpendicular to current;
• at lagging power factor, the internal emf is usually greater than terminal voltage;
• at leading power factor, regulation may be small or negative.

Negative regulation does not mean that the machine violates energy conservation. It means that terminal voltage at load can exceed the selected no-load reference under the specified excitation and power-factor condition.

Worked Example 1: Generator Regulation

A three-phase, star-connected generator delivers 10 MVA at 11 kV and 0.8 lagging power factor. Its synchronous impedance per phase is Z_s = 0.20 + j2.0 Ω. Determine the approximate internal emf per phase and voltage regulation using the synchronous-impedance method.

The line current is

The phase voltage is

The current angle is φ = cos^-1(0.8) = 36.87°, so I_a = 524.9∠−36.87° A. The impedance drop is approximately

Therefore,

and

The approximate regulation is

The result is an engineering estimate based on the stated impedance and assumptions; saturation and temperature effects may change the measured value.

Power-Angle Relations

For a cylindrical-rotor machine with negligible armature resistance connected to a stiff grid, the three-phase active power transfer is

Here V and E are per-phase rms magnitudes, X_s is synchronous reactance per phase, and δ is the internal load angle. This equation assumes a balanced sinusoidal system, constant frequency, negligible resistance, and a non-salient rotor.

The theoretical maximum power occurs at δ = 90°:

Operation near this limit is undesirable because synchronizing torque becomes weak and disturbances may cause loss of synchronism. Normal operation uses a substantially smaller load angle.

Mechanical torque is related to electromagnetic power by

where T_e is electromagnetic torque in newton-metres and ω_s is mechanical synchronous angular speed in radians per second.

For a motor, the same transfer equation applies when P_e is interpreted as electrical power converted to mechanical output plus losses. The motor load angle increases as shaft load increases. If the required angle exceeds the stable operating range, the motor may fall out of synchronism.

Salient-Pole Machines

A salient-pole rotor has different magnetic reluctance along the direct axis and quadrature axis. Consequently, it has two reactances: X_d along the field-pole axis and X_q perpendicular to it. The steady-state power-angle equation, neglecting resistance, is

An equivalent second term is (3V²/2)(1/X_q − 1/X_d)sin2δ. The first term is excitation torque; the second is reluctance power. The equation applies to a balanced, sinusoidal, steady-state system with constant voltage and negligible resistance.

For most salient-pole machines, X_d > X_q, so the reluctance-power term is positive for moderate positive load angles. A salient-pole synchronous motor can therefore produce torque even with reduced excitation, although field excitation is normally retained for control and stability.

The salient-pole model is important in hydro-generators, low-speed machines, and high-performance motors. It also explains why a single-reactance model can produce inaccurate power-angle and fault calculations.

Synchronous Motor Operation and Power-Factor Control

At constant terminal voltage and mechanical load, changing excitation changes the armature current. An underexcited motor draws lagging current; at an intermediate excitation it operates near unity power factor; and when overexcited it draws leading current.

The characteristic relation between armature current and field current at constant mechanical load is called the V-curve. The minimum armature current generally occurs near unity power factor. The corresponding inverted-V curve relates power factor to excitation.

A synchronous motor can therefore be used for power-factor correction. Compared with capacitor banks, it offers continuously adjustable reactive-power control and can support voltage regulation. Its disadvantages include higher capital cost, starting complexity, maintenance, and losses.

The power-factor angle satisfies

where P is three-phase active input power, V_L is line voltage, I_L is line current, and φ is the phase angle between voltage and current.

Synchronization of a Generator

A generator must be synchronized before it is connected in parallel with an energized bus. The essential conditions are:

1. equal rms voltage magnitude;
2. equal frequency;
3. identical phase sequence;
4. zero phase-angle difference between corresponding phase voltages at the instant of closing.

Failure to satisfy these conditions can produce severe transient currents, torque shocks, shaft stress, breaker damage, and system disturbances.

Synchronization may use lamps, a synchroscope, or an automatic synchronizer. A typical procedure is to bring the prime mover close to synchronous speed, establish the correct phase sequence, adjust excitation to match voltage, and close the breaker when the phase difference is acceptably small. After closing, the prime-mover torque is increased to raise active power, while excitation is adjusted to control reactive power and terminal voltage.

The generator must not be treated as a voltage source that can be connected arbitrarily to a live bus. The grid fixes frequency and largely fixes terminal voltage; the prime mover and excitation system then determine the machine's power exchange within operating limits.

Stability Basics

Synchronous stability is the ability of the machine or interconnected system to remain in synchronism after a disturbance. Stability is often discussed in three forms:

• steady-state or small-disturbance stability: ability to remain synchronized after small gradual changes;
• transient stability: ability to remain synchronized after large sudden disturbances such as faults or line switching;
• dynamic stability: response including control systems, damping, and oscillations over a longer time scale.

The synchronizing-power coefficient is the slope of the power-angle curve:

For a cylindrical machine,

with power per electrical radian when δ is expressed in radians. Positive K_s indicates restoring synchronizing power for small perturbations under the adopted convention. As δ approaches 90°, the coefficient decreases toward zero.

During a disturbance, the rotor accelerates when mechanical input exceeds electrical output and decelerates when electrical output exceeds mechanical input. The equal-area principle provides a graphical transient-stability test for a simplified single-machine infinite-bus system: synchronism is retained when the accelerating area is no greater than the available decelerating area.

Practical stability improvement measures include faster fault clearing, stronger transmission connections, high-speed excitation systems, power-system stabilizers, damping windings, controlled power-system devices, and appropriate operating limits.

Losses, Limits, and Capability

Important losses include stator copper loss, rotor copper loss, core loss, mechanical windage and friction, and stray-load loss. Approximate three-phase stator copper loss is

for a balanced machine using per-phase resistance. Efficiency is

where input and output are measured consistently as electrical or mechanical power according to the operating mode.

The capability curve of a grid-connected generator is bounded by:

• stator-current heating;
• rotor-field heating;
• prime-mover power limit;
• underexcitation limit;
• stability limit;
• end-region heating and network constraints.

A machine should not be operated merely at its nameplate MVA. The permissible combination of active and reactive power depends on voltage, coolant conditions, excitation, frequency, and duration.

Applied Case: Grid Connection of a Hydro-Generator

Background

This illustrative case concerns a 50-MVA, 11-kV, 50-Hz salient-pole hydro-generator being connected to a regional grid. The machine has a large number of poles and is driven by a water turbine. Its AVR regulates terminal voltage, while the turbine governor controls mechanical input.

Problem

During commissioning, the operator observes that the generator reaches rated speed but the synchronizing breaker produces a large current transient. After a successful connection, increasing turbine gate opening raises active power, but the terminal power factor becomes strongly lagging.

Analysis

The first event indicates inadequate synchronization. Reaching rated speed alone is insufficient. Phase sequence, voltage magnitude, frequency, and phase angle must all be checked. A phase-sequence error can cause a severe voltage opposition at the breaker terminals.

The post-connection behavior is consistent with insufficient excitation. Increasing turbine input raises the mechanical power and therefore the load angle. If excitation is not adjusted, the generator may absorb reactive power, producing a lagging power factor and increased stator current.

Because the rotor is salient, the cylindrical-rotor power equation may underestimate or misrepresent the reluctance-power contribution. The direct- and quadrature-axis parameters should be used for detailed studies.

Approach and Solution

The commissioning team verifies phase sequence, calibrates the voltage transformers, and uses an automatic synchronizer with a defined slip-frequency and phase-angle limit. The AVR is set to match bus voltage before closing. After closure, turbine input is increased gradually to establish the desired megawatt output. Excitation is then raised until reactive-power flow and power factor are within the capability curve.

Protection settings are checked for overexcitation, underexcitation, loss of field, reverse power, overcurrent, abnormal frequency, and pole slipping. A disturbance-recording system is enabled to observe rotor-angle oscillations.

Outcome

The generator closes without damaging transients, supplies the scheduled active power, and operates at the planned reactive-power set point. The case demonstrates that synchronization, mechanical control, excitation control, machine modeling, and protection must be treated as one integrated engineering problem.

Learning Points

1. Speed matching is necessary but not sufficient for synchronization.
2. Mechanical input primarily controls active power after grid connection.
3. Excitation strongly influences reactive power and power factor.
4. Saliency matters in low-speed hydro-generators.
5. Protection and capability limits are essential to safe operation.

Practical Measurement and Best Practices

Laboratory evaluation commonly includes open-circuit and short-circuit tests. The open-circuit characteristic relates generated voltage to field current at rated speed. The short-circuit characteristic relates armature current to field current with terminals shorted. Their combination can estimate synchronous impedance, although saturation and armature reaction require careful interpretation.

Best practices include:

• use per-phase quantities consistently;
• declare generator or motor current conventions;
• distinguish line and phase values;
• include resistance when regulation or losses matter;
• use X_d and X_q for salient-pole analysis;
• operate within the capability curve;
• coordinate AVR, governor, limiter, and protection settings;
• verify synchronization through independent measurements;
• assess transient behavior rather than relying only on steady-state calculations.

Digital twins, high-speed phasor measurement units, model-predictive control, and data-driven condition monitoring are increasingly used in machine operation. Artificial intelligence can assist anomaly detection and maintenance scheduling, but its outputs must be validated against physical models and protected measurement systems. Incorrect training data or unexplained alarms must not override statutory protection and operator responsibility.

Challenges and Limitations

The classical synchronous-impedance model is valuable but idealized. It assumes sinusoidal steady state, balanced operation, constant parameters, and often negligible saturation. Real machines exhibit magnetic saturation, damper-bar effects, harmonics, temperature-dependent resistance, rotor-field transients, negative-sequence heating, and nonlinear control interactions.

The power-angle equation does not by itself predict complete transient behavior. A stability study may require the swing equation, exciter and governor models, network reduction, fault impedance, and numerical time-domain simulation. Similarly, a nameplate power factor does not define every permissible operating point.

Chapter Summary

Synchronous machines operate at a speed fixed by electrical frequency and pole number. Their stator creates a rotating magnetic field, while rotor excitation establishes the internal emf. Generator and motor phasor equations differ according to current and power-flow conventions. Synchronous impedance provides a useful steady-state model, and voltage regulation can be estimated from the internal emf relative to terminal voltage.

For a cylindrical rotor, active power transfer follows a sine power-angle relation. Salient-pole machines require direct- and quadrature-axis reactances and can develop additional reluctance power. Excitation controls internal emf, reactive power, and power factor, while mechanical input primarily controls active power in grid-connected operation. Synchronization requires matching voltage, frequency, phase sequence, and phase angle. Stability depends on the slope and limits of the power-angle curve, disturbance size, fault-clearing time, and control-system performance. Safe professional practice combines analytical models, capability limits, protection, measurement, and controlled commissioning.

Key Terms

Synchronous machine: AC machine whose steady-state speed is locked to system frequency.

Synchronous speed: Mechanical speed determined by frequency and pole number.

Excitation: Production and control of rotor magnetic field.

Internal generated emf: The voltage induced in the machine's armature behind synchronous impedance.

Synchronous impedance: Approximate per-phase impedance combining armature resistance and synchronous reactance.

Synchronous reactance: Reactance used to represent leakage and armature-reaction effects.

Load angle: Angle between internal emf and terminal voltage.

Power-angle curve: Relationship between electromagnetic power and load angle.

Salient pole: Projecting rotor pole producing a nonuniform air gap.

Direct-axis reactance: Reactance along the rotor field axis.

Quadrature-axis reactance: Reactance perpendicular to the rotor field axis.

Voltage regulation: Relative change between specified no-load and full-load voltage.

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

Synchronizing: Connecting a generator to a live bus under matched electrical conditions.

Capability curve: Operating boundary defined by thermal, excitation, stability, and prime-mover limits.

V-curve: Relation between motor armature current and field current at constant load.

Transient stability: Ability to remain in synchronism after a large disturbance.

Synchronizing power coefficient: Derivative of electrical power with respect to load angle.

Synchronous condenser: Synchronous motor operating primarily to exchange reactive power.

Review Questions

Short Answer Questions

1. What determines the synchronous speed of a machine?
2. Why does excitation affect reactive power?
3. State the generator per-phase phasor equation.
4. What four conditions must be satisfied before synchronizing a generator?
5. Why are X_d and X_q used for salient-pole machines?

Long Answer Questions

1. Explain the construction, operating principle, and applications of cylindrical-rotor and salient-pole synchronous machines.
2. Derive and interpret the cylindrical-rotor power-angle equation under the assumptions of negligible resistance and a stiff grid.
3. Explain how excitation changes the power factor and reactive-power exchange of a synchronous motor.
4. Describe voltage regulation using the synchronous-impedance method and discuss its limitations.
5. Explain steady-state and transient stability and describe practical methods for improving stability.

Analytical / Application Questions

1. A three-phase, 6.6-kV generator delivers 5 MVA at 0.8 lagging power factor. Explain the steps required to calculate its internal emf when its per-phase synchronous impedance is known.
2. A grid-connected synchronous motor operates at constant shaft load while its field current is increased. Predict the changes in armature current, power factor, and reactive-power flow, and justify them using phasor reasoning.
3. A hydro-generator experiences a severe current transient when its breaker closes. Develop a diagnostic sequence covering synchronization measurements, excitation, phase sequence, protection, and data recording.

Multiple Choice Questions with Answers

Question 1

A 10-pole synchronous machine connected to a 50-Hz system has what synchronous speed?

A. 300 rpm

B. 600 rpm

C. 1,000 rpm

D. 1,500 rpm

Correct Answer: B. 600 rpm

Question 2

For a balanced three-phase star-connected machine, the phase voltage is:

A. √3 times the line voltage

B. Equal to the line voltage squared

C. The line voltage divided by √3

D. The line voltage divided by 3

Correct Answer: C. The line voltage divided by √3

Question 3

Using the generator convention, the approximate per-phase phasor equation is:

A. E = V + I_a(R_a + jX_s)

B. E = V − I_a(R_a + jX_s)

C. V = E + I_a(R_a − jX_s)

D. E = I_a/(R_a + jX_s)

Correct Answer: A. E = V + I_a(R_a + jX_s)

Question 4

Neglecting resistance, the cylindrical-rotor power-angle relation is proportional to:

A. cos δ

B. tan δ

C. sin δ

D. 1/sin δ

Correct Answer: C. sin δ

Question 5

An overexcited synchronous motor generally operates with:

A. Leading armature current

B. Zero armature current

C. Lagging armature current

D. Negative frequency

Correct Answer: A. Leading armature current

Question 6

Which condition is NOT a required synchronization condition?

A. Equal frequency

B. Identical phase sequence

C. Equal mechanical shaft torque before closing

D. Matching phase angle at closing

Correct Answer: C. Equal mechanical shaft torque before closing

Question 7

For a salient-pole machine, reluctance power arises primarily because:

A. The rotor has no magnetic field

B. Direct- and quadrature-axis reactances differ

C. The stator resistance is zero

D. The supply frequency is variable

Correct Answer: B. Direct- and quadrature-axis reactances differ

Question 8

The synchronizing-power coefficient is defined as:

A. P_e/ω_s

B. dP_e/dδ

C. V_L/I_L

D. I_a²R_a

Correct Answer: B. dP_e/dδ

Question 9

A synchronous condenser is primarily used to:

A. Provide mechanical speed variation

B. Exchange reactive power with the grid

C. Convert AC to DC without controls

D. Reduce the number of rotor poles

Correct Answer: B. Exchange reactive power with the grid

Question 10

For a three-phase machine rated 10 MVA and 11 kV line-to-line, the approximate line current is:

A. 262 A

B. 381 A

C. 525 A

D. 909 A

Correct Answer: C. 525 A