6. Synchronous Machines

Introduction and context

Synchronous machines occupy a central position in modern electric-power systems. Large turbo-generators and hydro-generators convert mechanical energy into electrical energy, while synchronous motors provide constant-speed mechanical drive, reactive-power support, and high-efficiency operation. Unlike induction machines, whose rotor speed differs slightly from synchronous speed during motoring, a synchronous machine operates in steady state with its rotor magnetic field locked to the rotating magnetic field produced by the stator.

The machine therefore links mechanical torque, electromagnetic power, excitation, terminal voltage, reactive power, and system stability in a particularly direct manner. This makes synchronous-machine theory essential for the analysis of generators connected to large grids, industrial motors, synchronous condensers, excitation control, transient stability, and renewable-energy interfaces.

For a balanced, sinusoidal, three-phase system, the synchronous speed is

\[

n_s=\frac{120f}{P}\quad \text{r/min}

\]

where \(f\) is electrical frequency in hertz and \(P\) is the number of poles. The corresponding angular speed is \(\omega_s=2\pi f/(P/2)\). In a generator, the prime mover drives the rotor at this speed; in a motor, the stator field pulls the rotor into synchronism.

Learning objectives

After studying this chapter, the reader should be able to:

1. Explain the construction, operating principle, and principal classifications of synchronous machines.

2. Distinguish salient-pole and cylindrical-rotor machines and relate their construction to applications.

3. Interpret excitation systems, armature reaction, synchronous impedance, and voltage regulation.

4. Develop and use per-phase phasor equations for generators and motors.

5. Explain the power-angle relationship and its connection with synchronizing torque and stability.

6. Describe the d–q model and its significance in steady-state and dynamic analysis.

7. Analyze capability curves, reactive-power control, and grid-connected operation.

8. Explain synchronization procedures, operating risks, and professional protection requirements.

9. Apply these ideas to practical generator and motor operating decisions.

Magnetic structure and operating principle

A synchronous machine has a three-phase armature winding, normally placed in slots on the stator, and a field system, normally mounted on the rotor. The rotor field may be produced by direct-current excitation supplied through slip rings and brushes or by a brushless exciter. Permanent-magnet rotors are also used in selected applications, especially high-performance drives and renewable-energy systems.

When balanced three-phase currents flow in the stator winding, they produce a rotating magnetic field whose speed is synchronous speed. A direct-current rotor field produces a fixed magnetic axis on the rotor. In steady operation, the rotor field axis maintains a constant angular displacement from the stator resultant field. This displacement is called the torque angle or load angle, usually denoted by \(\delta\). Mechanical power transfer changes \(\delta\), whereas excitation primarily changes the reactive-power balance and internal generated voltage.

The same machine can operate as a generator or motor. In generator operation, mechanical torque supplied by the prime mover leads the rotor field relative to the resultant stator field, and electrical power flows from the machine to the network. In motor operation, electromagnetic torque causes the rotor field to lag the stator field, and real power flows from the network to the machine. The electromagnetic energy-conversion mechanism is otherwise the same.

Rotor constructions

Salient-pole rotors have projecting poles with concentrated field coils. They possess different magnetic reluctances along the direct axis and quadrature axis and are mechanically suited to low and medium speeds. Hydro-generators commonly use salient-pole rotors because hydraulic turbines rotate relatively slowly and require many poles.

Cylindrical, or nonsalient, rotors have a smooth forged-steel body with field conductors embedded in slots. They have a nearly uniform air gap and withstand high centrifugal stresses. Two- and four-pole turbo-generators driven by steam or gas turbines generally use cylindrical rotors and operate at high speed.

The distinction is not merely structural. Saliency creates reluctance torque and causes the direct-axis and quadrature-axis synchronous reactances, \(X_d\) and \(X_q\), to differ. In a cylindrical machine, the approximation \(X_d\approx X_q\) is often acceptable for steady-state calculations.

Excitation systems and voltage control

The field current determines the magnitude of the rotor magnetomotive force. In a generator, changing field current changes the internal generated emf and therefore influences terminal voltage and reactive-power exchange. An excitation system includes the exciter, sensing and control circuits, power electronics where applicable, and protective limiters.

A simplified automatic voltage regulator (AVR) measures terminal voltage, compares it with a reference, and adjusts field voltage. A fast AVR improves voltage regulation and supports transient stability by increasing excitation during severe disturbances. However, excessive or prolonged field current can overheat the rotor, and rapid control action can interact with system oscillations. For this reason, excitation systems include overexcitation and underexcitation limiters, volts-per-hertz protection, field-current limits, and stabilizing signals.

In an overexcited synchronous motor, the internal emf is relatively large and the motor can supply leading reactive power to the network. In an underexcited condition, it absorbs lagging reactive power. This controllability is why a lightly loaded synchronous motor may be operated as a synchronous condenser for power-factor correction and voltage support.

Armature reaction and the phasor model

Armature reaction is the effect of stator current mmf on the main rotor-produced magnetic field. Its nature depends on the phase relationship between stator current and the internal emf. A component of armature reaction may demagnetize the main field, another may magnetize it, and a quadrature component may distort the air-gap flux. The exact interpretation is most reliable through phasor diagrams and d–q-axis analysis.

For a cylindrical-rotor synchronous generator, neglecting resistance initially, the per-phase relation is commonly written as

\[

\underline{E}=\underline{V}+jX_s\underline{I}

\]

where \(\underline{E}\) is the internal generated emf, \(\underline{V}\) is terminal voltage, \(\underline{I}\) is armature current, and \(X_s\) is synchronous reactance. Including armature resistance gives

\[

\underline{E}=\underline{V}+R_a\underline{I}+jX_s\underline{I}.

\]

For a motor, using the passive sign convention, the corresponding relation is

\[

\underline{V}=\underline{E}+R_a\underline{I}+jX_s\underline{I}.

\]

The sign conventions must be stated explicitly. Many apparent disagreements in machine calculations arise from applying the generator equation to a motor or from treating current direction inconsistently.

The synchronous impedance is \(Z_s=R_a+jX_s\). It represents an equivalent steady-state model, not a literal physical impedance concentrated at one location. It combines the effect of winding resistance and the fundamental component of the armature-reaction reactance. Saturation makes \(X_s\) dependent on operating condition, so linear calculations are approximations.

Voltage regulation

The voltage regulation of an alternator is often defined as

\[

\%VR=\frac{|E_0|-|V|}{|V|}\times100

\]

where \(E_0\) is the no-load excitation emf corresponding to the field current required to maintain the specified terminal voltage at the given load and power factor. The synchronous-impedance method estimates \(E_0\) from the open-circuit and short-circuit characteristics. It is convenient but tends to overestimate regulation because it treats the armature-reaction effect as a reactance and does not fully represent saturation.

The zero-power-factor method and Potier method separate leakage reactance from armature reaction more carefully. The Potier method uses the air-gap line and a zero-power-factor characteristic to estimate leakage reactance and the field ampere-turns required for armature reaction. These methods remain valuable for understanding machine behavior even when digital electromagnetic models are used in design.

Salient-pole analysis and the d–q model

The two-reaction theory resolves stator quantities into components along two perpendicular rotor axes. The direct axis, or d-axis, is aligned with the rotor field axis; the quadrature axis, or q-axis, leads it by 90 electrical degrees. In a salient-pole machine, the d-axis and q-axis have different permeances and therefore different reactances.

For a salient-pole generator, neglecting resistance, the steady-state power-angle equation is approximately

\[

P_e=\frac{3EV}{X_d}\sin\delta+\frac{3V^2}{2}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin2\delta.

\]

The first term is excitation, or electromagnetic-field, power. The second is reluctance power caused by rotor saliency. For a cylindrical rotor, \(X_d=X_q=X_s\), and the second term vanishes:

\[

P_e=\frac{3EV}{X_s}\sin\delta.

\]

These expressions assume a stiff infinite bus, negligible resistance, balanced operation, and a steady-state model. They are therefore analytical approximations, but they clearly show why the load angle increases when mechanical input increases and why a machine has a finite synchronizing limit.

The d–q model is more general than the simple phasor model. It represents stator currents, rotor flux linkages, field winding dynamics, damper windings, and transient and subtransient reactances in rotating coordinates. In transient studies, the internal emf behind transient reactance \(X_d'\), and in very fast disturbances behind subtransient reactance \(X_d''\), is used. Since \(X_d''<X_d'<X_d\) in typical machines, the initial fault current can be much larger than the sustained fault current.

Power-angle behavior and stability

The power-angle curve relates electrical output to rotor angle. For the cylindrical-rotor approximation, maximum theoretical power occurs near \(\delta=90^\circ\). Operation near this point is undesirable in a practical grid because small disturbances produce large angle changes and the synchronizing margin is small. Normal operating angles are substantially lower.

Synchronizing power is the incremental change in electrical power for a small change in angle:

\[

P_s=\frac{dP_e}{d\delta}.

\]

A positive value indicates restoring synchronizing action around the operating point. In a real interconnected system, stability also depends on inertia, damping, network strength, excitation controls, turbine-governor response, protection clearing time, and interactions among many machines.

Small-disturbance, or small-signal, stability concerns oscillations around an equilibrium point. Large-disturbance or transient stability concerns whether synchronism is retained after faults, line switching, generator trips, or major changes in mechanical input. The equal-area criterion provides an instructive first analysis for a single machine connected to an infinite bus. During a fault, electrical output may fall while mechanical input remains nearly constant, causing the rotor to accelerate. After fault clearing, the post-fault network must provide sufficient decelerating power to return the rotor to a stable trajectory.

Damping windings, power-system stabilizers, coordinated excitation control, fast fault clearing, and appropriate network design all contribute to stability. A high AVR gain alone is not a complete stability strategy; it may improve synchronizing torque while reducing damping unless supplementary control is properly designed.

Synchronization with the grid

Before a generator is connected to a live bus, its phase sequence, frequency, voltage magnitude, and phase angle must match the bus within prescribed limits. Connecting with an incorrect phase sequence can produce severe currents and torque. A frequency mismatch causes the phase angle to move continuously, while a voltage-magnitude mismatch produces reactive current and electromagnetic stress.

Synchronization may be performed using lamps, synchroscopes, synchronizing relays, or automatic synchronizers. The operator or controller adjusts prime-mover speed to reduce slip frequency and adjusts excitation to match voltage. The breaker is commanded to close at the appropriate phase coincidence, with allowance for breaker closing time. Once connected, the prime mover controls real-power transfer through torque input, while excitation controls reactive-power exchange and terminal voltage within system constraints.

A generator must not be treated as an isolated voltage source after connection to a strong grid. The grid largely fixes frequency and voltage angle; the machine’s mechanical input determines its real-power contribution, and excitation determines its reactive behavior. This separation is approximate because all controls and network quantities are coupled dynamically.

Capability curves and grid-connected operation

The capability curve of a synchronous generator defines permissible combinations of real power \(P\) and reactive power \(Q\), usually at specified terminal voltage, speed, and cooling conditions. The stator-current limit forms an approximately circular boundary because apparent power \(S=\sqrt{P^2+Q^2}\) is limited by armature heating. The rotor-field heating limit restricts overexcited operation, corresponding generally to reactive-power delivery. The underexcitation limit restricts operation on the opposite side because excessive flux patterns can cause end-region heating, loss of synchronizing margin, or instability.

Prime-mover capability may impose an additional real-power boundary. At low power factor, a machine may reach a field or stator-current limit before its mechanical rating is reached. Capability curves therefore provide an operational map rather than merely a design rating.

For a grid-connected generator, active-power commands are typically implemented through the turbine governor or prime-mover controller, while reactive-power or voltage commands are implemented through the AVR. Operators must consider system voltage, line loading, thermal limits, short-circuit strength, reserve requirements, and stability constraints. In industrial facilities, synchronous motors can improve overall power factor but may introduce starting complexity, torsional interactions, and excitation-maintenance requirements.

Starting and control of synchronous motors

A synchronous motor is not inherently self-starting when connected directly to a fixed-frequency supply because the average starting torque of a stationary rotor under a rapidly rotating field is approximately zero. Starting methods include damper windings, reduced-frequency drives, variable-frequency converters, pony motors, and mechanical run-up systems. With damper windings, the motor initially accelerates as an induction motor; the field is then energized near synchronous speed, allowing the rotor to lock into synchronism.

Variable-frequency drives provide superior starting and speed control. The controller establishes a rotating stator field whose frequency and voltage are increased according to a suitable flux-control strategy. Permanent-magnet synchronous motors eliminate rotor copper loss but require careful control, thermal design, and attention to demagnetization and fault behavior.

Applied case: hydro-generator voltage support after a transmission-line outage

Background

Consider an illustrative 200-MVA salient-pole hydro-generator connected through a transformer and two parallel transmission lines to a strong regional grid. The generator normally exports 120 MW at near-unity power factor. Its AVR maintains terminal voltage, and the plant participates in automatic voltage control.

Situation and problem

One transmission line trips because of a protection operation. The remaining line has a lower transfer capability and a higher effective reactance. If the generator continues to export the same real power without coordinated control, its load angle increases and the terminal voltage tends to decline. The operator must preserve synchronism while avoiding stator-current and rotor-field limits.

Analysis

The outage changes the network transfer reactance and therefore reduces the coefficient in the power-angle relationship. For the same mechanical input, the post-outage equilibrium requires a larger \(\delta\). The AVR can increase field current, raising internal emf and supporting voltage, but this action may move the generator toward its overexcitation limit. If the machine is already heavily loaded, the capability curve may prevent unlimited reactive-power production.

Approach and solution

The control system first increases excitation within the field-current limiter to support voltage. The governor does not immediately increase mechanical input; doing so could increase the rotor angle and reduce transient margin. The plant controller may request a modest reduction in active-power output if the post-contingency loading approaches stability or thermal limits. Network operators check the remaining line, transformer, and generator capability boundaries. If voltage remains inadequate, controlled load shedding or redispatch elsewhere may be required.

Outcome

With adequate reactive reserve and a timely control response, the generator remains synchronized, terminal voltage is restored within the permitted band, and the remaining line operates below its emergency rating. The generator’s operating point moves to a lower real-power and higher reactive-power region of its capability curve until the network is reinforced or the tripped line is returned.

Learning points

The case illustrates that excitation primarily supports voltage and reactive power, whereas mechanical input controls real-power transfer. It also shows that a generator can be voltage-secure yet stability-limited, and that capability curves, protection, controls, and network topology must be assessed together.

Challenges, limitations, and professional safeguards

The classical synchronous-machine model is valuable because it exposes physical relationships, but it omits saturation details, damper-winding transients, harmonics, thermal dynamics, shaft torsion, and control-system interactions. Parameters obtained from tests may vary with temperature, frequency, saturation, and operating point. Simulation results should therefore be validated against commissioning tests, disturbance records, and manufacturer data.

Common technical mistakes include confusing mechanical and electrical degrees, using line voltage where phase voltage is required, ignoring the generator–motor sign convention, treating synchronous reactance as constant under all conditions, and assuming that a large excitation increase can indefinitely support voltage. Another mistake is closing a breaker based only on voltage magnitude while neglecting phase sequence and phase angle.

Professional operation requires coordination among excitation limits, turbine-governor controls, protection relays, synchronizing equipment, and system-operator instructions. Protection functions may include loss of field, out-of-step, reverse power, over/under-voltage, over/under-frequency, negative-sequence current, stator and rotor overcurrent, differential protection, and volts-per-hertz protection. Settings must balance security against dependability and must be verified through studies and field testing.

Safety and governance are equally important. Switching procedures should be authorized, interlocked, and documented. Maintenance personnel must account for stored magnetic energy, rotating parts, induced voltages, and automatic remote commands. Digital excitation and protection systems require access control, change management, event logging, time synchronization, and cybersecurity practices. Machine-learning tools may assist anomaly detection using vibration, temperature, partial-discharge, or electrical-signature data, but their alarms should support—not replace—engineering judgment and validated protection functions.

Current developments and best practices

Modern synchronous machines increasingly use digital AVRs, static excitation systems, high-speed power-system stabilizers, and online condition monitoring. Phasor measurement units and wide-area measurement systems provide synchronized voltage and angle data for stability assessment. Reduced-order models are used for control design, while detailed finite-element and electromagnetic-transient models support design verification and fault analysis.

Decarbonized power systems create new operating requirements. Conventional synchronous generators continue to provide inertia, short-circuit strength, voltage support, and system-forming behavior, while inverter-based resources increasingly provide grid-support functions through advanced controls. Hybrid plants may combine synchronous condensers, batteries, and power-electronic converters to supply selected grid services. The engineering challenge is to coordinate these resources without exceeding thermal, stability, or protection limits.

Best practice is to begin with a clearly defined sign convention and base quantities, select a model appropriate to the study time scale, verify parameters, examine both steady-state and disturbance behavior, and interpret every calculated operating point against capability and protection limits. A mathematically feasible point is not necessarily an operationally permissible point.

Chapter Summary

Synchronous machines convert energy through magnetic fields that remain locked to synchronous speed in steady operation. Their principal construction types are salient-pole and cylindrical rotors, with saliency producing different d- and q-axis reactances. Excitation controls rotor field strength, terminal voltage, and reactive-power exchange, while prime-mover torque primarily controls real-power transfer after grid connection.

The phasor model uses synchronous impedance to estimate voltage regulation and steady-state behavior. The d–q model provides a more accurate framework for salient-pole effects, transient reactances, damper windings, and dynamic studies. The power-angle equation explains real-power transfer, synchronizing torque, and the finite stability margin. Capability curves combine stator heating, rotor-field heating, prime-mover limits, and underexcitation restrictions into an operating map.

Reliable operation requires correct synchronization, coordinated excitation and governor controls, validated protection, and awareness of thermal and stability constraints. Digital AVRs, phasor measurements, condition monitoring, and advanced grid-support controls are extending the role of synchronous machines in contemporary power systems.

Key Terms

Synchronous speed: The speed of the rotating magnetic field, determined by frequency and pole number.

Synchronous machine: An AC machine whose rotor runs at synchronous speed in steady operation.

Salient-pole rotor: A projecting-pole rotor with different direct- and quadrature-axis magnetic reluctances.

Cylindrical rotor: A smooth rotor with embedded field conductors, suited to high-speed operation.

Excitation system: The equipment that supplies and controls rotor field current.

Automatic voltage regulator: A controller that adjusts excitation to regulate terminal voltage or reactive output.

Armature reaction: The effect of stator-current magnetomotive force on the main magnetic field.

Synchronous reactance: An equivalent reactance representing leakage and fundamental armature-reaction effects.

Voltage regulation: The change from rated-load terminal voltage to corresponding no-load voltage at specified excitation conditions.

Load angle: The electrical angle between relevant rotor and stator magnetic-field axes.

Power-angle equation: A relationship between electrical power transfer and load angle.

Direct axis: The rotor axis aligned with the field magnetic axis.

Quadrature axis: The rotor axis electrically perpendicular to the direct axis.

Transient reactance: The effective reactance governing machine response after a disturbance over an intermediate time period.

Subtransient reactance: The very-short-time effective reactance immediately after a disturbance.

Capability curve: A map of permissible real- and reactive-power operating combinations.

Synchronizing torque: Restoring electromagnetic torque produced by a small displacement from synchronism.

Synchronous condenser: A synchronous motor operated without significant mechanical load to control reactive power.

Infinite bus: An idealized grid with constant voltage magnitude, frequency, and phase reference.

Out-of-step condition: Loss of stable angular synchronism between a machine and the power system.

Review Questions

Short Answer Questions

1. Why does a synchronous machine operate at a speed fixed by frequency and pole number?

2. State two constructional differences between salient-pole and cylindrical-rotor machines.

3. What functions are performed by an automatic voltage regulator?

4. Define load angle and explain its significance in generator operation.

5. Why is a directly connected synchronous motor not inherently self-starting?

Long Answer Questions

1. Derive and interpret the steady-state power-angle equation for a cylindrical-rotor generator connected to an infinite bus.

2. Explain armature reaction and develop the synchronous-impedance model for an alternator.

3. Describe the d–q-axis model and explain why it is important in transient and salient-pole-machine analysis.

4. Explain the construction and interpretation of a synchronous-generator capability curve.

5. Discuss synchronization procedures, excitation control, governor action, and protection requirements for grid-connected generators.

Analytical / Application Questions

1. A generator connected to a stiff grid experiences a transmission-line outage. Explain how the outage affects its power-angle characteristic, rotor angle, excitation requirement, and stability margin.

2. A synchronous motor is operating at constant mechanical load. Predict the qualitative changes in stator current, power factor, and reactive-power flow when its excitation is increased from underexcited to overexcited operation.

3. A generator calculation gives an operating point within its apparent-power rating but beyond the rotor-field limit. Should the point be accepted? Analyze the decision using the capability curve and explain the relevant operational risks.

22. Multiple Choice Questions with Answers

Question 1

A synchronous generator connected to a strong grid is required to increase its real-power output while terminal voltage is maintained. Which control action is primarily required?

A. Increase prime-mover mechanical input

B. Increase only the stator resistance

C. Reduce the number of rotor poles

D. Open the field circuit

Correct Answer: A. Increase prime-mover mechanical input

Question 2

For a cylindrical-rotor synchronous generator neglecting resistance, which expression represents the steady-state power-angle relationship?

A. P = 3EV/Xs cosδ

B. P = 3EV/Xs sinδ

C. P = 3V²/Xs tanδ

D. P = 3E²/Xs sinδ

Correct Answer: B. P = 3EV/Xs sinδ

Question 3

What is the principal reason for using a cylindrical rotor in a turbo-generator?

A. It provides a large number of projecting poles

B. It is suited to high-speed operation and centrifugal stress

C. It eliminates the need for excitation

D. It produces maximum reluctance torque at low speed

Correct Answer: B. It is suited to high-speed operation and centrifugal stress

Question 4

When a synchronous motor is overexcited, it generally operates as a machine that

A. absorbs leading reactive power

B. supplies leading reactive power

C. absorbs real power from the prime mover

D. cannot exchange reactive power

Correct Answer: B. supplies leading reactive power

Question 5

In a salient-pole machine, the reluctance-power term exists primarily because

A. the stator resistance is zero

B. the direct- and quadrature-axis reactances differ

C. the rotor has no field winding

D. the terminal voltage is fixed

Correct Answer: B. the direct- and quadrature-axis reactances differ

Question 6

Before closing a generator breaker onto a live bus, which condition is unacceptable even if voltage magnitudes match?

A. Correct phase sequence and small phase-angle difference

B. Matched frequency and voltage magnitude

C. Incorrect phase sequence

D. A small controlled slip frequency

Correct Answer: C. Incorrect phase sequence

Question 7

A generator operates within its apparent-power rating but beyond its rotor-field heating boundary. The appropriate conclusion is that

A. the operating point is always safe because apparent power is within rating

B. the operating point may overheat the rotor and is not permissible

C. the stator current must be zero

D. the machine has automatically become a motor

Correct Answer: B. the operating point may overheat the rotor and is not permissible

Question 8

Immediately after a severe fault, the machine current is governed more closely by which reactance?

A. Synchronous reactance Xd

B. Transient reactance Xd' only after steady state

C. Subtransient reactance Xd''

D. Leakage resistance only

Correct Answer: C. Subtransient reactance Xd''

Question 9

If mechanical input to a grid-connected generator increases while excitation and grid voltage remain approximately constant, the immediate steady-state tendency is for

A. load angle and real-power output to increase

B. load angle to become exactly zero

C. reactive power to become independent of excitation

D. frequency to increase without limit

Correct Answer: A. load angle and real-power output to increase

Question 10

Why can a high-gain AVR not be regarded as a complete solution to stability problems?

A. An AVR has no influence on voltage

B. It may improve synchronizing action but can interact adversely with damping and limits

C. It controls only mechanical torque

D. It prevents all transient currents

Correct Answer: B. It may improve synchronizing action but can interact adversely with damping and limits