9. Dynamic, Transient and Harmonic Performance

Introduction and context

Steady-state analysis is indispensable in electrical machines, but it does not describe the most demanding events encountered in service. A motor accelerates, decelerates, starts under load, and responds to disturbances in voltage or torque. A generator experiences short circuits, sudden load rejection, synchronization disturbances, and unbalanced faults. In each case, currents, flux linkages, torque, speed, and mechanical stress vary with time. In addition, practical windings and magnetic structures produce spatial and temporal harmonics that may cause parasitic torque, losses, acoustic noise, vibration, and electromagnetic interference.

Dynamic performance concerns the response of a machine when its operating point changes. Transient performance refers particularly to rapid changes, often involving electromagnetic states that have not yet reached their new steady values. Harmonic performance concerns the effects of nonsinusoidal magnetomotive force, flux, current, or voltage distributions. These subjects are closely related: a harmonic component can excite a mechanical resonance, while a transient can temporarily drive magnetic parts into saturation and alter the harmonic spectrum.

For postgraduate electrical engineers, the important task is not merely to calculate a transient waveform. It is to select an appropriate model, identify the dominant time scales, interpret electromagnetic and mechanical interactions, and judge whether a result is physically credible and operationally acceptable.

Learning objectives

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

1. distinguish electromagnetic, electromechanical, and mechanical transients;

2. formulate and interpret dynamic machine equations in phase and rotating reference frames;

3. explain short-circuit, starting, sudden-loading, and load-rejection phenomena;

4. analyse electromechanical oscillations using the swing equation and torque–speed relationships;

5. assess the effects of unbalanced supply, saturation, and negative-sequence quantities;

6. identify the origins and consequences of space and time harmonics;

7. evaluate parasitic torques, losses, vibration, acoustic noise, and electromagnetic compatibility;

8. select practical mitigation, measurement, protection, and modelling strategies.

9.1 Time scales and dynamic machine models

A machine contains several energy-storage mechanisms. Magnetic energy is stored in the stator and rotor fields, electrical energy in leakage inductances, and mechanical energy in the rotating masses. Their associated time constants are different. Subtransient currents may change within milliseconds, transient currents over several cycles, and rotor speed over tens of milliseconds to many seconds. A useful model must retain the states relevant to the disturbance while avoiding unnecessary detail.

For a synchronous machine represented in a rotating d–q reference frame, the stator voltage equations may be written in compact form as

\[

\mathbf v_{dq}=R_s\mathbf i_{dq}+\frac{d\boldsymbol\lambda_{dq}}{dt}+\omega_k\mathbf J\boldsymbol\lambda_{dq},

\]

where \(R_s\) is stator resistance, \(\boldsymbol\lambda\) is the flux-linkage vector, \(\omega_k\) is the reference-frame speed, and \(\mathbf J\) represents the cross-coupling operator. The rotor circuits have corresponding equations, with rotor speed entering through slip or relative angular velocity. Torque is obtained from magnetic co-energy; in a general form,

\[

T_e=\frac{\partial W'_m}{\partial \theta_m},

\]

where \(W'_m\) is magnetic co-energy and \(\theta_m\) is rotor position. For many machines this relationship becomes a bilinear function of d–q currents and flux linkages.

The mechanical equation is

\[

J\frac{d\omega_m}{dt}=T_e-T_L-B\omega_m,

\]

where \(J\) is total inertia, \(T_L\) is load torque, and \(B\) represents viscous friction. This equation explains why an electrical disturbance can become a mechanical problem. A transient electromagnetic torque changes acceleration, shaft torsion, bearing loads, and eventually speed.

Reference-frame choice is a modelling decision. Phase-variable models preserve asymmetry and harmonics but may be computationally expensive. The stationary \(\alpha\beta\) frame is convenient for converter-fed machines and spatial interpretation. The synchronous d–q frame transforms balanced fundamental quantities into approximately constant values, simplifying control and stability analysis. For unbalanced or harmonic operation, positive-, negative-, and zero-sequence variables, multiple rotating frames, or finite-element models may be necessary.

9.2 Electromagnetic transients

Starting and sudden loading

At starting, an induction motor has zero mechanical speed and nearly maximum slip. The rotor frequency is initially close to the supply frequency, and the motor may draw five to eight times rated current, depending on design and supply impedance. The starting current produces voltage depression in the network and thermal stress in the stator and rotor. The electromagnetic torque must exceed load torque, friction, and accelerating torque for successful acceleration.

The acceleration equation can be expressed as

\[

\frac{d\omega_m}{dt}=\frac{T_e(\omega_m)-T_L(\omega_m)}{J}.

\]

A high-inertia fan, pump, or compressor may require a long acceleration interval even when the final operating current is normal. A soft starter, autotransformer starter, or variable-frequency drive reduces inrush or controls the acceleration profile, but the resulting lower voltage can also reduce torque. Since induction-motor torque approximately varies with the square of applied voltage in the unsaturated region, a poor starting strategy may cause stalling rather than protection.

Sudden loading causes a temporary speed reduction and an increase in slip in an induction motor. In a synchronous motor, the rotor angle changes so that electromagnetic torque rises. If the demanded torque exceeds the pull-out torque, synchronism is lost. The resulting oscillation may be more severe than the final steady-state torque requirement suggests.

Short-circuit transients in synchronous machines

Immediately after a terminal short circuit, the flux linkages of the stator and rotor cannot change instantaneously. The induced current is therefore constrained by conservation of flux, producing a large initial current. The conventional sequence is described by subtransient, transient, and synchronous reactances:

\[

X''_d < X'_d < X_d.

\]

The initial symmetrical short-circuit current is approximately proportional to \(1/X''_d\), the later transient current to \(1/X'_d\), and the sustained current to \(1/X_d\), subject to resistance, network impedance, and saturation. The first current cycles may also include a decaying dc component caused by the point on wave at which the fault occurs. This component creates asymmetry and may produce a high peak current even when the rms symmetrical current is within an expected value.

Short-circuit analysis determines interrupting duties, thermal withstand, electrodynamic forces, generator capability, and protection settings. A model that ignores damper windings, field winding dynamics, or network impedance may substantially misrepresent the first cycles. Conversely, a detailed finite-element model is not justified for every protection study. Model complexity should follow the decision being made.

Load rejection and field transients

When a generator is suddenly disconnected from its load, electromagnetic torque falls rapidly while mechanical input does not. The rotor accelerates, the power angle changes, and terminal voltage may rise, particularly if excitation responds slowly. Turbine control, excitation control, overvoltage protection, and load-rejection capability must therefore be considered together. In motors, sudden loss of supply can generate a residual voltage and, in some systems, dangerous re-energization transients.

9.3 Electromechanical oscillations and stability

The central model for rotor-angle motion is the swing equation. In power-angle form for a synchronous machine,

\[

\frac{2H}{\omega_s}\frac{d^2\delta}{dt^2}=P_m-P_e-D\frac{d\delta}{dt},

\]

where \(H\) is the inertia constant, \(\delta\) is rotor angle relative to a synchronously rotating reference, \(P_m\) is mechanical input, \(P_e\) is electrical output, and \(D\) represents damping. The equation reveals three outcomes: a new equilibrium, a decaying oscillation, or instability.

For a simplified cylindrical-rotor generator connected through reactance to a stiff system,

\[

P_e=P_{\max}\sin\delta.

\]

A disturbance changes \(\delta\), and the synchronizing torque coefficient near an operating point is proportional to \(P_{\max}\cos\delta_0\). As the initial angle approaches 90 degrees, synchronizing stiffness decreases. Damping is supplied by damper windings, winding and core losses, controls, and the network. Poorly tuned excitation systems or fast power-electronic controls can reduce damping in particular frequency ranges.

Small-signal stability examines linearized deviations about an operating point. Eigenvalues of the state matrix indicate oscillation frequency and damping ratio. Large-disturbance stability requires time-domain simulation because the system may pass through saturation, limiters, current limits, and nonlinear torque-angle relationships. Shaft systems with several masses introduce torsional modes. A periodic electromagnetic torque near a torsional natural frequency can cause fatigue even when average torque is acceptable.

A practical investigation should compare electrical forcing frequencies with mechanical natural frequencies, calculate modal participation, and verify the result using measurements such as shaft strain, accelerometers, torque transducers, or high-resolution current analysis.

9.4 Unbalanced supply and asymmetrical operation

Unbalanced voltages produce positive-, negative-, and zero-sequence components. In a three-phase machine, positive-sequence voltage creates the desired forward rotating field. Negative-sequence voltage creates a field rotating backward relative to the rotor. For an induction motor, the negative-sequence field sees a slip close to two at normal speed, inducing a current of high frequency in the rotor and producing braking torque and additional heating.

Even a modest voltage unbalance can produce a substantially larger current unbalance because of the machine’s sequence impedances. The negative-sequence current causes rotor heating, while zero-sequence current may flow through the neutral or frame depending on winding connection and grounding. In synchronous machines, negative-sequence currents induce double-frequency currents in the rotor surface and damper structures, creating severe localized heating during faults or prolonged unbalance.

Assessment should use symmetrical components rather than phase magnitudes alone. Engineers must distinguish voltage unbalance definitions, because the ratio of negative- to positive-sequence voltage is more physically meaningful for rotating machines than a simple maximum deviation from average voltage. Protection may include negative-sequence relays, thermal models, phase-loss detection, and derating curves.

9.5 Saturation and its dynamic consequences

Magnetic saturation is not merely a steady-state nonlinearity. During a transient, flux may be displaced by dc offsets, sudden field changes, or asymmetrical faults. The effective inductance then changes with operating point. Saturation reduces incremental inductance, increases magnetizing current, distorts flux, modifies leakage coupling, and changes the predicted transient torque.

The air-gap flux is often limited by the iron’s magnetization curve. In a saturated region, a small voltage change can produce a relatively large current change. Saturation also generates harmonics, especially odd components under symmetrical excitation, and may cause local heating at tooth tips, pole shoes, retaining rings, or rotor surfaces. In synchronous machines, cross-magnetization means that d-axis and q-axis effects are not always independent.

Dynamic simulations may represent saturation through nonlinear inductance functions, lookup tables, or finite-element field solutions. The data should be obtained from no-load, short-circuit, or identification tests and should include the relevant transient path. Using a steady-state saturation curve indiscriminately in a fast transient model is a common source of error.

9.6 Origins and classification of harmonics

Harmonics arise from both spatial and temporal sources. A spatial harmonic is a variation of magnetomotive force or permeance around the air gap with a space order different from the fundamental. It may result from distributed winding layout, slotting, fractional-slot windings, salient poles, or magnetic saturation. A time harmonic is a frequency component different from the fundamental in voltage, current, or flux. It may be introduced by rectifiers, inverters, switching devices, nonsinusoidal excitation, or saturation.

The winding distribution factor and pitch factor determine the ability of a winding to produce or respond to a particular space harmonic. Slotting creates permeance variation and produces slot harmonics. In induction machines, harmonic fields rotate at speeds determined by their space order and sequence. A harmonic whose synchronous speed is close to rotor speed can produce a large torque response; other harmonics may produce crawling, braking torque, or torque ripple.

In a converter-fed machine, carrier-frequency sidebands and low-order voltage harmonics generate corresponding current and torque components. Pulse-width modulation reduces selected low-frequency harmonics but may increase high-frequency common-mode voltage and bearing-current risk. The spectrum must therefore be interpreted together with machine impedance, cable characteristics, switching frequency, and mechanical resonances.

9.7 Parasitic torque, losses, noise, and vibration

The useful electromagnetic torque is the average component that performs mechanical work. Parasitic torque includes ripple, cogging torque, pulsating torque caused by harmonic interaction, and asynchronous torque components. Cogging torque is produced by the tendency of rotor salient features to align with stator teeth, even without current in some permanent-magnet machines. Torque ripple can produce speed modulation, position error, acoustic noise, and torsional fatigue.

Harmonics increase losses through additional copper loss, rotor-bar or magnet eddy-current loss, core loss, stray-load loss, and bearing loss. High-frequency flux variations may cause localized loss that is not captured by a simple fundamental-frequency equivalent circuit. Permanent magnets are especially vulnerable to eddy-current heating and partial demagnetization under high-frequency fields or severe faults.

Vibration may be electromagnetic, mechanical, aerodynamic, or a combination. Electromagnetic radial forces arise from the Maxwell stress at the air gap. Force waves with spatial order \(m\) and frequency \(f\) can excite stator modes when their frequency and spatial distribution match structural modes. Twice-line-frequency forces, slot-passing forces, rotor eccentricity, and negative-sequence fields are common excitation sources. A vibration peak should not automatically be attributed to imbalance; order tracking and phase information are needed.

Mitigation may involve optimized winding distribution, skewing, rotor-bar design, lower-ripple converter modulation, improved balancing, structural stiffening, damping, filtering, shielding, grounding, and controlled switching transitions. Each measure involves trade-offs. Skewing can reduce cogging and slot harmonics but may lower average torque and increase leakage.

9.8 Electromagnetic compatibility and measurement

Electromagnetic compatibility requires that a machine and its converter operate without unacceptable interference to themselves or other equipment. Conducted emissions travel through supply, dc-link, grounding, and bearing-current paths. Radiated emissions arise from high-frequency current loops and cables. Common-mode voltage from fast inverter switching can cause shaft voltage, circulating bearing currents, insulation stress, and interference in sensors or communication systems.

A sound EMC design begins with source identification, coupling-path analysis, and victim assessment. Practical measures include short commutation loops, appropriate dc-link layout, screened motor cables, 360-degree shield termination where appropriate, common-mode chokes, dv/dt or sine-wave filters, insulated bearings, shaft grounding, and coordinated protective-earth connections. Filters must be selected with regard to resonance, leakage current, thermal rating, and drive stability.

Measurement quality is central to transient diagnosis. Voltage probes must have adequate common-mode rating and bandwidth; current transformers may saturate during faults; Rogowski coils are useful for large transient currents but require integration; accelerometers and tachometers require correct mounting and sampling. Anti-aliasing, synchronized sampling, time-frequency analysis, order tracking, and window selection influence the conclusions. A spectrum without a reliable speed reference can confuse electrical frequency, slip frequency, and mechanical orders.

9.9 Analytical and computational approaches

A practical hierarchy of methods is useful:

• equivalent-circuit and sequence models for preliminary studies and protection calculations;
• state-space d–q models for control design and electromechanical transients;
• phase-variable models for asymmetry, faults, and selected harmonics;
• harmonic-balance or frequency-domain methods for periodic nonsinusoidal operation;
• finite-element time-stepping for saturation, local force, loss, and detailed harmonic prediction;
• coupled electromagnetic–mechanical simulation for vibration and torsional response.

Model validation should proceed from steady-state quantities to transient waveforms and finally to mechanical observables. Parameters such as inertia, damping, saturation curves, damper-circuit constants, and stray-load loss are often uncertain. Sensitivity analysis is therefore more informative than presenting a single apparently precise result. Digital data acquisition and machine-learning methods can assist condition monitoring, but an alarm should remain linked to a physically interpretable indicator. A classifier trained only on balanced, healthy operation may fail under a changed load or a different converter control law.

Illustrative case study: vibration in a converter-fed induction motor

Background

A 250-kW induction motor drives a high-inertia process through a variable-frequency drive. The motor meets its average torque and speed requirements, but operators report a pronounced tonal vibration near a particular operating speed. Bearing temperature is gradually increasing, although stator current remains within its rated value.

Situation and analysis

Initial inspection finds no significant shaft misalignment or mechanical looseness. A synchronized measurement of phase currents, dc-link voltage, shaft speed, and housing acceleration is performed. Order tracking shows a vibration peak that increases when the speed passes a narrow band. Current spectrum analysis reveals a pair of sidebands around the switching-related component. A finite-element electromagnetic model predicts a radial force wave whose frequency intersects a stator structural mode at the reported speed. The converter’s common-mode voltage is also found to produce measurable shaft voltage and intermittent bearing currents.

The result illustrates why a fundamental-frequency motor model is insufficient. Average torque is satisfactory, but high-frequency electromagnetic forces and bearing currents create the observed consequences. A purely mechanical balancing correction would not remove the excitation.

Approach and solution

The drive is retuned to reduce the relevant voltage harmonic at the critical speed, and a dv/dt filter is installed after verifying its effect on cable resonance and drive stability. The motor cable shield and protective-earth arrangement are corrected, and an insulated bearing is used at the non-drive end. The control system is configured to pass rapidly through the critical speed where process conditions permit. Vibration, bearing current, and winding temperature are then monitored during a controlled operating survey.

Outcome and learning points

The tonal vibration and bearing-current events are reduced without changing the rated mechanical output. The case demonstrates four principles: average values can conceal harmful spectral components; electromagnetic and mechanical models must be coupled near resonance; EMC treatment must consider both common-mode and differential-mode paths; and diagnosis is strongest when current, voltage, speed, vibration, and temperature are time-correlated.

9.10 Engineering risks, limitations, and best practice

The most frequent errors are using steady-state parameters for fast transients, ignoring saturation, assuming perfect balance, treating rms current as a complete heating indicator, and interpreting every vibration peak as a mechanical defect. Other errors include neglecting the decaying dc component in fault studies, using an inappropriate reference frame, failing to include converter dead time, and applying a filter without checking resonance or leakage current.

Professional analysis should state assumptions, initial conditions, parameter sources, numerical step size, and validity range. Results should be checked against energy conservation, limiting cases, measured waveforms, and expected symmetry. Protection studies must account for worst-case fault location and clearing time. Designs must comply with applicable machine, drive, insulation, EMC, safety, and grid-interconnection standards; the exact standard depends on the installation and jurisdiction.

Ethically, engineers must not suppress an alarm merely because a machine continues to operate. A transient that is infrequent may still impose cumulative fatigue or insulation damage. Maintenance recommendations should distinguish evidence from inference, protect personnel during energized measurements, and preserve traceable records of settings and test conditions. Current developments include physics-informed digital twins, wide-bandgap converter drives, online demagnetization and bearing-current diagnosis, advanced model-predictive control, and integrated electromagnetic–thermal–mechanical optimization. These technologies improve capability but do not eliminate the need for validated models and engineering judgement.

Chapter Summary

Dynamic machine behaviour results from the interaction of magnetic, electrical, and mechanical energy storage. d–q, phase-variable, sequence, finite-element, and coupled electromechanical models serve different purposes. Starting, sudden loading, short circuits, load rejection, and unbalance produce transient currents, torque, flux, and speed changes that determine thermal, mechanical, and protection requirements. The swing equation provides a foundation for rotor-angle stability, while damping and torsional modes determine oscillatory performance. Saturation changes both steady and transient inductances and introduces additional harmonics.

Space and time harmonics arise from windings, slotting, saliency, saturation, and power-electronic converters. Their effects include torque ripple, crawling, additional loss, rotor and magnet heating, vibration, acoustic noise, and EMC problems. Reliable diagnosis combines appropriate models with synchronized measurements of electrical and mechanical variables. Best practice requires explicit assumptions, validation, spectral interpretation, attention to resonances, and compliance with relevant technical and safety requirements.

Key Terms

Dynamic performance: Machine response when electrical or mechanical operating conditions change.

Electromagnetic transient: Short-duration variation of currents, flux linkages, or electromagnetic torque.

Subtransient reactance: Effective reactance governing the earliest current response of a synchronous machine after a disturbance.

Transient reactance: Reactance governing the intermediate response before the synchronous steady state.

Swing equation: Differential equation relating accelerating power to rotor-angle motion.

Synchronizing torque: Restoring torque that tends to return a synchronous rotor toward its equilibrium angle.

Damping torque: Torque opposing speed or angle oscillations.

Sequence component: Positive-, negative-, or zero-sequence representation of an unbalanced three-phase quantity.

Voltage unbalance: Inequality among phase voltages, preferably assessed using sequence components.

Saturation: Nonlinear reduction of incremental magnetic permeability at high flux density.

Space harmonic: Spatial variation of magnetic quantities with a space order other than the fundamental.

Time harmonic: Frequency component other than the fundamental in a periodic electrical quantity.

Cogging torque: Position-dependent torque caused by magnetic alignment between rotor and stator teeth.

Torque ripple: Periodic variation superimposed on average electromagnetic torque.

Parasitic torque: Unwanted torque components that do not contribute usefully to the intended mechanical output.

Torsional resonance: Large shaft twist response when excitation frequency approaches a torsional natural frequency.

Common-mode voltage: Voltage shared by conductors relative to a reference such as protective earth.

Electromagnetic compatibility: Ability to operate without causing or suffering unacceptable electromagnetic interference.

Order tracking: Analysis that relates vibration or electrical components to shaft rotational order.

Digital twin: A computational representation linked to measured behaviour of a physical machine.

Review Questions

Short Answer Questions

1. Why are subtransient currents greater than sustained short-circuit currents in a synchronous machine?

2. State the mechanical equation of a rotating machine and explain each term.

3. How does negative-sequence voltage affect an induction motor?

4. Distinguish a space harmonic from a time harmonic.

5. Why can saturation alter a transient simulation?

Long Answer Questions

1. Explain the starting and sudden-loading transients of an induction motor, including the influence of inertia and load torque.

2. Derive and interpret the swing equation for a synchronous machine connected to a power system.

3. Discuss the origins, effects, and mitigation of harmonics in converter-fed machines.

4. Explain how electromagnetic forces can produce vibration and torsional stress.

5. Compare equivalent-circuit, d–q, phase-variable, and finite-element approaches to dynamic machine analysis.

Analytical / Application Questions

1. A synchronous generator experiences a three-phase terminal fault. Describe the current sequence from the first cycle to the sustained condition and identify the parameters required for protection design.

2. An induction motor shows normal rms current but excessive vibration at one speed when operated from a PWM inverter. Propose a measurement and diagnosis plan that separates mechanical resonance, electromagnetic force harmonics, and bearing-current effects.

3. A motor operates on a supply with persistent voltage unbalance. Explain how sequence components, heating, protection settings, and permissible derating should be considered before approving continued operation.

22. Multiple Choice Questions with Answers

Question 1

Which reactance primarily determines the earliest symmetrical short-circuit current of a synchronous machine?

A. Synchronous reactance Xd

B. Transient reactance X′d

C. Subtransient reactance X″d

D. Leakage resistance only

Correct Answer: C. Subtransient reactance X″d

Question 2

When a synchronous motor is suddenly loaded, its electromagnetic torque initially increases mainly because the rotor

A. changes its resistance

B. changes its power angle

C. loses all excitation

D. changes its number of poles

Correct Answer: B. changes its power angle

Question 3

At approximately normal speed, a negative-sequence field in an induction motor produces rotor currents whose relative slip is close to

A. zero

B. one-half

C. one

D. two

Correct Answer: D. two

Question 4

In the swing equation, a positive value of mechanical input minus electrical output initially causes

A. rotor acceleration

B. instantaneous zero torque

C. magnetic saturation only

D. negative-sequence current only

Correct Answer: A. rotor acceleration

Question 5

Which modelling approach is generally most suitable for representing detailed local saturation and electromagnetic force waves?

A. Simple steady-state equivalent circuit

B. Finite-element time-stepping analysis

C. Per-unit voltage calculation alone

D. Mechanical free-body diagram alone

Correct Answer: B. Finite-element time-stepping analysis

Question 6

A voltage unbalance can cause disproportionately large current unbalance because

A. sequence impedances and machine response differ by sequence

B. the rotor has no inertia

C. frequency becomes zero

D. all phase impedances are always identical

Correct Answer: A. sequence impedances and machine response differ by sequence

Question 7

Which condition most directly identifies a possible torsional resonance?

A. A constant dc current with no speed variation

B. A periodic torque component near a shaft natural frequency

C. A reduction in stator resistance only

D. A balanced supply at standstill only

Correct Answer: B. A periodic torque component near a shaft natural frequency

Question 8

A high-frequency common-mode voltage from a PWM drive is especially associated with

A. shaft voltage and bearing currents

B. elimination of all rotor losses

C. zero acoustic noise

D. removal of saturation

Correct Answer: A. shaft voltage and bearing currents

Question 9

Why can a machine with acceptable average torque still experience harmful vibration?

A. Average torque includes every mechanical resonance automatically

B. harmonic radial forces may excite a structural mode

C. vibration depends only on rated current

D. harmonics occur only during faults

Correct Answer: B. harmonic radial forces may excite a structural mode

Question 10

Which practice is most appropriate when validating a transient machine model?

A. Report only the most favourable simulation

B. Ignore initial conditions

C. Compare predictions with synchronized electrical and mechanical measurements

D. Use steady-state parameters regardless of disturbance duration

Correct Answer: C. Compare predictions with synchronized electrical and mechanical measurements