5. Induction Machines

Introduction and context

Induction machines are the dominant electromechanical energy-conversion devices in industrial power systems. Their importance arises from a robust rotor construction, relatively low cost, high reliability, and suitability for operation over a wide power range. In motor operation, electrical power supplied to the stator is converted into mechanical output without a direct electrical connection to the rotor. Energy transfer occurs by electromagnetic induction, which gives the machine its name.

The induction machine is also a useful analytical bridge between magnetic-circuit theory, rotating-field theory, transformer action, and power-electronic drive control. The same machine may operate as a motor, generator, or, under special conditions, a brake. This chapter develops the steady-state theory of three-phase and single-phase induction machines, derives their equivalent circuits and torque–speed characteristics, and examines starting, braking, speed control, harmonics, parameter estimation, and generator operation.

Learning objectives

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

1. Explain the production of a rotating magnetic field and define synchronous speed and slip.

2. Derive the approximate and exact equivalent circuits of a three-phase induction motor.

3. Interpret the flow of power, torque production, losses, and efficiency.

4. Analyze torque–speed characteristics and determine starting, maximum, and full-load performance.

5. Explain the operation of single-phase induction motors using double-revolving-field theory.

6. Compare starting, braking, and speed-control methods.

7. Estimate machine parameters from no-load, blocked-rotor, and resistance tests.

8. Evaluate harmonics, generator operation, modern drives, and practical design trade-offs.

The rotating field and the induction principle

Consider a balanced three-phase winding distributed in space by 120 electrical degrees and supplied by balanced currents. The three sinusoidal magnetomotive forces combine to produce a resultant field of nearly constant magnitude rotating at synchronous speed. For a supply frequency $f$ and a stator with $P$ poles,

$$n_s=\frac{120f}{P}\quad \text{r/min}, \qquad \omega_s=\frac{2\pi f}{p}\quad \text{rad/s},$$

where $p=P/2$ is the number of pole pairs. The direction of rotation is reversed by interchanging any two supply phases.

The rotor must rotate below synchronous speed in motoring operation. If rotor speed is $n_r$, slip is

$$s=\frac{n_s-n_r}{n_s}.$$

The rotor electrical frequency is therefore $f_2=sf_1$. At standstill, $s=1$ and the rotor frequency equals the supply frequency; near synchronous speed, slip and rotor frequency are small. Slip is not merely a speed error: it is the quantity that enables relative motion between the rotating field and rotor conductors, inducing rotor emf and current.

The electromagnetic torque is produced by the interaction of the stator rotating field and rotor mmf. In a well-designed motor, the rotor current has a phase relationship that allows a substantial tangential force. If the rotor reached synchronous speed, induced rotor emf would vanish, rotor current would disappear, and electromagnetic torque would fall to zero. Thus a loaded induction motor always operates with positive slip.

Construction and operating regimes

A three-phase induction machine consists of a laminated stator core carrying insulated distributed windings and a rotor separated from it by a small air gap. The squirrel-cage rotor uses conducting bars placed in slots and short-circuited by end rings. It is mechanically strong and requires no brushes. A wound rotor carries a three-phase insulated winding connected to slip rings, allowing external resistance or power-electronic equipment to be inserted during starting or control.

The machine operates as a motor for $0<s<1$, as a generator for $s<0$ when the rotor is driven above synchronous speed, and in plugging or other braking modes when the electromagnetic torque opposes rotation. In generator operation, the rotor supplies mechanical input and real electrical power is delivered to the stator terminals, but an induction generator requires magnetizing reactive power unless connected to a suitable converter or capacitor system.

Important terms include synchronous speed, slip, rotor frequency, air-gap power, rotor copper loss, mechanical power developed, breakdown torque, starting torque, magnetizing current, and power factor. Confusing air-gap power with mechanical output is a frequent source of error.

Per-phase equivalent circuit

The induction motor can be represented by a transformer-like equivalent circuit. Referred to the stator, the per-phase parameters are:

• $R_1$: stator resistance;
• $X_1$: stator leakage reactance;
• $R_2'$ and $X_2'$: rotor resistance and leakage reactance referred to the stator;
• $X_m$: magnetizing reactance; and
• $R_c$: core-loss resistance, when core loss is represented explicitly.

The exact circuit contains the shunt excitation branch and the series stator and rotor impedances. A commonly used approximate circuit places the excitation branch directly across the supply. The rotor branch is written as

$$\frac{R_2'}{s}+jX_2'.$$

This form is physically useful because

$$\frac{R_2'}{s}=R_2'+R_2'\frac{1-s}{s}.$$

The first term represents rotor copper loss, while the second represents the mechanical power-conversion component. If $I_2'$ is the referred rotor current, the air-gap power is

$$P_g=3I_2'^2\frac{R_2'}{s},$$

rotor copper loss is

$$P_{cu2}=3I_2'^2R_2'=sP_g,$$

and mechanical power developed is

$$P_m=(1-s)P_g.$$

Shaft output is lower than $P_m$ because of friction, windage, and additional mechanical losses. The electromagnetic torque is

$$T_e=\frac{P_g}{\omega_s}=\frac{P_m}{\omega_r},$$

where $\omega_r=(1-s)\omega_s$. These equations show why torque is associated with air-gap power divided by synchronous angular speed.

For an approximate Thevenin representation of the stator and excitation network, let $V_{th}$ and $Z_{th}=R_{th}+jX_{th}$ be the Thevenin voltage and impedance seen by the rotor branch. Then

$$T_e=\frac{3|V_{th}|^2(R_2'/s)}{\omega_s[(R_{th}+R_2'/s)^2+(X_{th}+X_2')^2]}.$$

This equation is central to performance analysis. At small slip, torque is approximately proportional to slip. At large slip, torque is approximately inversely proportional to slip. The maximum torque condition is approximately

$$\frac{R_2'}{s_m}=\sqrt{R_{th}^2+(X_{th}+X_2')^2}.$$

Hence increasing rotor resistance increases the slip at maximum torque but, ideally, does not significantly change the maximum torque. It does, however, increase starting torque and starting losses. The approximate maximum torque varies with the square of applied voltage, so voltage reduction severely affects overload capability.

Torque–speed behavior and performance interpretation

At standstill, $s=1$, and the motor develops starting torque while drawing a large current. As speed rises, torque generally increases to a breakdown or pull-out value and then decreases toward zero at synchronous speed. The stable operating region for ordinary motoring is the portion where a small increase in load causes speed to fall and torque to rise.

The torque–speed curve must be interpreted together with the mechanical load characteristic. A fan or pump generally has a torque demand approximately proportional to speed squared, whereas a conveyor may require nearly constant torque. The intersection of motor and load curves determines the operating point. Starting is satisfactory only if motor starting torque exceeds load torque throughout acceleration.

Efficiency is governed by stator copper loss, core loss, rotor copper loss, mechanical loss, and stray-load loss. At low slip, rotor copper loss is a small fraction of air-gap power. Power factor, however, may remain modest because the machine requires magnetizing current. High-efficiency designs use improved magnetic materials, optimized slot geometry, reduced resistance, and carefully controlled cooling.

Parameter estimation and testing

The no-load test is performed near rated voltage and frequency with the shaft unloaded. Since slip is small, rotor copper loss is small. The measured input power includes core, friction, windage, and stator copper losses. The test provides estimates of the magnetizing branch and rotational losses, although separation of core and mechanical loss requires additional assumptions or tests.

In the blocked-rotor test, the rotor is prevented from turning and a reduced voltage is applied until rated current flows. Since $s=1$, the rotor branch has its actual referred resistance and reactance. The test yields the combined series resistance and leakage reactance. Temperature correction is important because winding resistance changes significantly with temperature. The test must be conducted rapidly because the motor cannot dissipate the heat associated with rated current effectively while stationary.

A dc resistance test estimates stator resistance. For a star-connected winding, the line-to-line resistance is twice the phase resistance; for a delta-connected winding, the relationship differs and must be derived from the parallel paths. Parameter identification may also use frequency-response tests, online estimation, or least-squares fitting of measured voltage, current, speed, and torque.

Starting methods and speed control

Direct-on-line starting is simple and inexpensive but causes high inrush current and voltage disturbance. It is appropriate when the supply is stiff and the driven load permits the resulting torque. Star–delta starting reduces phase voltage and starting current, but starting torque is reduced approximately in proportion to the square of voltage. An autotransformer starter provides selectable reduced voltage with improved utilization of supply capacity.

Rotor-resistance starting is available with wound-rotor machines. External resistance increases starting torque and limits current, but the energy dissipated in the resistance reduces efficiency. Modern systems increasingly use soft starters or variable-frequency drives (VFDs). A VFD controls both frequency and voltage; under constant volts-per-hertz operation, air-gap flux is approximately maintained below base speed. Above base speed, voltage reaches its limit and the machine operates in a field-weakening region with approximately constant-power capability.

Common speed-control methods are:

• supply-frequency control using a VFD;
• pole changing, which provides discrete speeds;
• stator-voltage control, useful mainly for fan-type loads;
• rotor-resistance control for wound-rotor motors; and
• slip-power recovery, in which rotor slip energy is electronically returned to the supply or converted to useful power.

For high-performance drives, field-oriented control and direct-torque control regulate flux and torque through measured or estimated state variables. These methods require current sensors, position or speed estimation, and a motor model. Sensorless control is attractive but becomes difficult at very low speed because the induced-voltage information becomes weak.

Braking and four-quadrant operation

Plugging is obtained by reversing the phase sequence while the rotor is still moving. The rotating field reverses, producing torque opposite to rotor motion. It gives rapid braking but produces high current and substantial rotor heating; supply disconnection at zero speed is essential to prevent reversal unless reverse motoring is required.

Dynamic braking can be achieved by disconnecting the ac supply and applying dc to the stator. The stationary field induces rotor currents as the rotor moves, producing braking torque. Braking energy is dissipated in the rotor and machine losses. Regenerative braking occurs when the drive commands a speed below the actual rotor speed or when the load drives the motor above synchronous speed. Mechanical energy is then returned through the converter to the dc link or grid.

A four-quadrant drive permits forward motoring, forward braking or regeneration, reverse motoring, and reverse braking. Selection must consider thermal limits, dc-link energy management, braking resistor rating, and the mechanical strength of the driven system.

Single-phase induction motors

A single-phase stator winding supplied from a sinusoidal source produces a pulsating rather than a naturally rotating field. Double-revolving-field theory resolves this pulsating field into two oppositely rotating fields, each of half the original amplitude. At standstill, the forward and backward fields produce equal and opposite torques, so a single-phase induction motor is not self-starting.

Once the rotor is given an initial direction, the forward field has a smaller relative speed and larger effective torque than the backward field, allowing continued acceleration. Practical motors create starting torque by using an auxiliary winding displaced in space and phase from the main winding. Split-phase motors use resistance to create phase displacement; capacitor-start motors use a large starting capacitor; permanent-split-capacitor motors use a smaller capacitor continuously; and shaded-pole motors use a short-circuited shading coil to produce a weak sweeping field.

The auxiliary winding or starting switch must be selected according to starting torque, current, cost, noise, efficiency, and duty cycle. Capacitor-start motors suit compressors and pumps requiring high starting torque, whereas shaded-pole motors are appropriate only for small, lightly loaded applications.

Harmonics, design limits, and power quality

Distributed windings reduce space-harmonic content, but slotting, winding distribution, saturation, inverter switching, and asymmetry introduce harmonics. Harmonic fields may rotate forward or backward depending on their sequence. A backward-rotating harmonic increases relative rotor frequency, causing additional rotor loss, torque pulsation, vibration, acoustic noise, and heating.

Inverter-fed motors experience additional concerns: common-mode voltage, bearing currents, insulation stress from fast voltage edges, and electromagnetic interference. Mitigation includes suitable cable design, output filters, shaft grounding, insulated bearings where justified, correct switching strategies, and motor insulation systems rated for converter duty. Harmonic currents in the supply can also affect transformers, capacitors, and protection equipment.

Design and operation are constrained by thermal limits, insulation life, bearing condition, mechanical resonance, voltage imbalance, and overload duration. Negative-sequence voltage produces a field rotating opposite to the rotor and can cause disproportionate rotor heating. Protection should therefore include overload, phase loss, under- and overvoltage, earth fault, locked rotor, and thermal monitoring as appropriate.

Induction generators and contemporary applications

When a prime mover drives the rotor above synchronous speed, slip becomes negative and the electromagnetic torque opposes the prime mover. Real power flows from the mechanical system to the electrical system. A grid-connected induction generator obtains magnetizing reactive power from the grid. In isolated operation, capacitors may supply excitation, but voltage and frequency regulation become sensitive to speed, capacitance, and load.

Wound-rotor induction generators with slip-power converters historically supported variable-speed wind-energy systems. Doubly fed induction generators remain important where a converter rated for only a fraction of machine power can control rotor currents and permit variable-speed operation. Full-scale converter systems provide broader grid support and decouple machine frequency from grid frequency, but require more converter capacity.

Digital monitoring increasingly uses current signature analysis, vibration analysis, thermal data, and model-based observers to identify broken rotor bars, eccentricity, bearing defects, and insulation degradation. Machine-learning tools can assist diagnosis, but their predictions require representative data, physical validation, cybersecurity controls, and careful treatment of false alarms. Automated control must remain subordinate to protection coordination and safe maintenance procedures.

Illustrative case study: selecting a drive for a pump station

Background. A municipal pumping station uses a 90-kW, 400-V, 50-Hz induction motor to drive a centrifugal pump. The existing direct-on-line starter produces unacceptable voltage dips and water-hammer transients during starting.

Problem. The pump requires low torque at low speed but torque rises approximately with the square of speed. Operators also need controlled acceleration, energy savings during reduced flow, and reliable stopping during power disturbances.

Analysis. A star–delta starter would reduce starting current but would also reduce starting torque and provide no continuous speed control. A soft starter would improve acceleration and reduce hydraulic shock, but it would not offer efficient speed reduction during periods of low demand. A VFD can maintain approximately constant flux below base speed, control acceleration, and reduce pump power substantially at reduced speed, subject to system head and efficiency.

Approach. The selected VFD is sized for motor current rather than only nameplate kilowatts. The design includes input protection, harmonic assessment, a braking strategy appropriate to the pump inertia, shielded motor cable, grounding provisions, bypass capability, and control interlocks for dry running and low suction pressure. The acceleration ramp is chosen to satisfy both motor thermal limits and hydraulic constraints. Commissioning includes verification of phase sequence, motor parameters, minimum speed, overload response, and emergency-stop behavior.

Outcome. The drive eliminates the severe starting disturbance and allows the pump to follow demand rather than operate continuously at full speed with throttling. The principal engineering lesson is that starter selection must consider the load torque law, process requirements, power quality, thermal duty, and safety—not merely starting current.

Professional practice, limitations, and best practices

Equivalent-circuit calculations are steady-state approximations. They do not fully predict saturation, transient current, deep-bar effects, skin effect, temperature variation, spatial harmonics, mechanical vibration, or converter interactions. The use of constant parameters can produce misleading predictions during locked-rotor operation or rapid transients.

Good engineering practice includes obtaining reliable nameplate and test data, distinguishing line and phase quantities, maintaining consistent referred-side parameters, checking units, and validating calculations against measured current, speed, and temperature. For VFD applications, the motor, converter, cable, protection, cooling, and load should be treated as one system.

Safety and professional responsibility require isolation and lockout procedures, discharge verification for dc links and capacitors, guarded rotating parts, correct earthing, and compliance with applicable electrical and machinery standards. Regenerative systems require a defined path for returned energy. Maintenance decisions should be based on condition and risk, not on an isolated diagnostic indicator.

Current developments

Modern induction-machine research and practice emphasize high-efficiency materials, low-loss manufacturing, integrated sensors, model-predictive and predictive torque control, wide-bandgap converter switches, and digital twins for condition monitoring. Additive manufacturing and advanced cooling can improve power density, while data-driven maintenance seeks earlier detection of degradation. These developments do not eliminate the need for electromagnetic understanding; rather, they increase the value of accurate models, trustworthy measurements, and engineering judgment.

Chapter Summary

A three-phase induction machine develops torque because a rotating stator field induces rotor currents at a frequency proportional to slip. Synchronous speed depends on supply frequency and pole number, while slip determines rotor frequency, rotor copper loss, and mechanical power conversion. The transformer-like equivalent circuit permits calculation of current, power factor, torque, efficiency, and torque–speed behavior. Maximum torque occurs at a slip related to rotor resistance and leakage impedance; increasing rotor resistance shifts this condition toward higher slip.

No-load, blocked-rotor, and resistance tests provide practical parameter estimates. Starting and braking methods involve trade-offs among current, torque, efficiency, heating, and controllability. VFDs provide the most versatile modern speed control, while field-oriented and direct-torque methods support high-performance operation. Single-phase motors require an auxiliary starting mechanism because their pulsating field creates zero net starting torque. Harmonics, negative sequence, converter effects, and thermal limits must be considered in professional applications. With negative slip, the induction machine operates as a generator, requiring appropriate reactive-power and converter arrangements.

Key Terms

Synchronous speed: Speed of the rotating stator field, determined by frequency and pole number.

Slip: Difference between synchronous speed and rotor speed divided by synchronous speed.

Rotor frequency: Electrical frequency induced in the rotor, equal to slip times supply frequency.

Air-gap power: Power transferred electromagnetically from stator to rotor across the air gap.

Rotor copper loss: Resistive loss in the rotor conductors, equal to slip times air-gap power.

Breakdown torque: Maximum steady electromagnetic torque available from the motor.

Magnetizing reactance: Equivalent reactance representing the current required to establish air-gap flux.

Thevenin equivalent: Reduced source representation used to simplify torque and current calculations.

Squirrel-cage rotor: Rotor formed by conducting bars short-circuited by end rings.

Wound rotor: Rotor carrying an insulated polyphase winding connected through slip rings.

Plugging: Braking by reversing the rotating-field direction while the rotor is moving.

Regenerative braking: Operation in which mechanical energy is converted and returned electrically.

Double-revolving-field theory: Representation of a pulsating single-phase field as two counter-rotating fields.

Slip-power recovery: Conversion of rotor slip energy into useful electrical or mechanical output.

Field weakening: Operation above base speed with limited voltage and reduced air-gap flux.

Negative-sequence field: A field rotating opposite to the normal sequence and producing additional rotor heating.

Blocked-rotor test: Test at zero rotor speed used to estimate series resistance and leakage reactance.

Induction generator: Induction machine driven above synchronous speed to deliver electrical power.

Review Questions

Short Answer Questions

1. Why must a motoring induction machine operate with positive slip?

2. State the relationship between rotor copper loss and air-gap power.

3. What is the function of the magnetizing branch in the equivalent circuit?

4. Why is a single-phase induction motor not self-starting?

5. What information is obtained from a blocked-rotor test?

Long Answer Questions

1. Derive the power-flow relations of an induction motor from air-gap power to shaft output.

2. Explain the torque–speed characteristic using the Thevenin equivalent circuit.

3. Compare direct-on-line, star–delta, autotransformer, and VFD starting.

4. Explain double-revolving-field theory and the operation of capacitor-start motors.

5. Discuss induction-generator operation, excitation requirements, and modern variable-speed applications.

Analytical / Application Questions

1. A four-pole, 50-Hz motor runs at 1,440 r/min. Calculate synchronous speed, slip, and rotor frequency, and interpret the results.

2. A pump motor experiences unacceptable starting voltage dips but operates for long periods at reduced flow. Select and justify a starting and speed-control strategy.

3. A wound-rotor motor must provide high starting torque with limited current. Explain how rotor resistance changes the torque–speed curve and identify the principal energy penalty.

22. Multiple Choice Questions with Answers

Question 1

A six-pole induction motor supplied at 50 Hz has a synchronous speed of

A. 500 r/min

B. 750 r/min

C. 1000 r/min

D. 1500 r/min

Correct Answer: C. 1000 r/min

Question 2

If an induction motor operates at 4% slip from a 50 Hz supply, its rotor frequency is

A. 0.2 Hz

B. 2 Hz

C. 12.5 Hz

D. 46 Hz

Correct Answer: B. 2 Hz

Question 3

For an induction motor, rotor copper loss is equal to

A. (1-s) times air-gap power

B. s times air-gap power

C. air-gap power divided by s

D. mechanical output divided by s

Correct Answer: B. s times air-gap power

Question 4

Increasing rotor resistance in a wound-rotor motor primarily

A. eliminates magnetizing current

B. moves maximum torque to lower slip

C. moves maximum torque to higher slip

D. reduces synchronous speed

Correct Answer: C. moves maximum torque to higher slip

Question 5

The approximate maximum torque of an induction motor varies with applied voltage approximately as

A. V

B. V squared

C. 1/V

D. independently of V

Correct Answer: B. V squared

Question 6

A single-phase induction motor is not self-starting because at standstill

A. its stator has no resistance

B. the forward and backward field torques cancel

C. rotor frequency is zero

D. the auxiliary winding always leads by 180 degrees

Correct Answer: B. the forward and backward field torques cancel

Question 7

Which test is conducted with the rotor stationary and reduced applied voltage?

A. No-load test

B. Open-circuit test

C. Blocked-rotor test

D. Overspeed test

Correct Answer: C. Blocked-rotor test

Question 8

A centrifugal pump requiring reduced energy consumption at variable flow is most appropriately controlled using

A. a fixed resistor in the stator circuit

B. a variable-frequency drive

C. permanent plugging

D. a dc excitation winding

Correct Answer: B. a variable-frequency drive

Question 9

An induction machine delivers real electrical power to the grid when

A. its rotor runs below zero speed

B. its rotor runs exactly at synchronous speed

C. its rotor is driven above synchronous speed

D. its stator is disconnected from all excitation

Correct Answer: C. its rotor is driven above synchronous speed

Question 10

A negative-sequence voltage is particularly undesirable because it

A. eliminates rotor heating

B. produces a counter-rotating field and additional rotor losses

C. increases synchronous speed without changing frequency

D. prevents all harmonic currents

Correct Answer: B. produces a counter-rotating field and additional rotor losses