Electrical Machines for Drive Applications

Introduction and Context

An electrical drive converts electrical energy into controlled mechanical motion. Its central electromechanical converter is the motor, but practical performance also depends on the power converter, controller, sensors, mechanical load, and thermal environment. Selecting a motor therefore requires more than comparing rated power. The engineer must examine torque–speed capability, starting current, braking, overload capacity, efficiency, controllability, duty cycle, maintenance, cost, and safety.

This chapter develops the operating principles and steady-state models of the principal motors used in industrial drives: separately excited and series DC motors, three-phase induction motors, synchronous motors, permanent-magnet and brushless DC motors, switched-reluctance motors, and stepper motors. The emphasis is on physical interpretation and drive application rather than on construction alone.

Learning Objectives

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

1. Explain how electromagnetic torque is produced in major motor families.
2. Use appropriate equivalent circuits and steady-state equations.
3. Interpret torque–speed characteristics and identify operating regions.
4. Calculate losses, efficiency, slip, power factor, and basic performance quantities.
5. Select a motor technology for a specified drive application.
6. Recognize practical limitations, control requirements, and emerging trends.

1. Common Foundations of Motor Analysis

1.1 Electromagnetic power conversion

In every rotating motor, electrical power is converted into mechanical power through electromagnetic interaction. The developed mechanical power is

where P_mech,dev is developed mechanical power in watts, T_e is electromagnetic torque in newton-metres, and ω_m is mechanical angular speed in radians per second. If speed is given in revolutions per minute,

where n is speed in revolutions per minute.

The shaft output is lower than developed power because of rotational losses:

where P_rot includes bearing friction, windage, and core-related rotational losses. Motor efficiency is

provided input and output powers are measured over the same time interval and expressed in the same units.

For a balanced three-phase motor, electrical input power is

where V_L and I_L are line voltage and line current, and cosφ is the input power factor. This expression assumes sinusoidal, balanced three-phase quantities. With significant converter harmonics, true power must be obtained from the average of instantaneous voltage multiplied by instantaneous current.

1.2 Torque, speed, and load matching

A drive reaches steady state when accelerating torque is zero:

where T_L is load torque and T_loss represents friction and other opposing torques. More generally,

where J is the combined rotational inertia in kilogram-metres squared. The sign of the right-hand side determines acceleration or deceleration.

Loads are commonly classified as constant torque, constant power, or fan/pump loads. A constant-torque load has approximately fixed T_L over its speed range. A constant-power load satisfies T_L ∝ 1/ω_m. A centrifugal fan or pump often follows approximately T_L ∝ ω_m² and P_L ∝ ω_m³, although the actual exponent depends on the system.

1.3 Losses and thermal constraints

Motor losses include:

• copper losses, P_cu = I²R, in windings;
• core losses due to hysteresis and eddy currents;
• mechanical losses from friction and windage;
• stray-load losses caused by leakage flux and harmonics;
• inverter and switching losses in electronically supplied drives.

The temperature rise is governed by the balance between generated losses and heat removal. A motor may tolerate short overloads even when continuous operation at the same torque would cause excessive temperature. Duty class, enclosure, ambient temperature, cooling method, and installation altitude must therefore be included in selection.

2. DC Motors

2.1 Operating principle and equations

A current-carrying armature conductor in a magnetic field experiences force. In a conventional DC motor, a commutator reverses conductor current at the appropriate rotor position so that developed torque remains unidirectional.

For a separately excited or shunt motor with approximately constant flux, the steady-state armature equations are

where V_a is armature voltage, E_a is back emf, I_a is armature current, R_a is armature resistance, Φ is flux per pole, and k_e and k_t are machine constants. In SI units, consistent motor definitions give numerically equal torque and emf constants, so k_t = k_e when units are N·m/A and V·(rad/s)⁻¹ respectively.

Combining these equations gives the speed relation

Thus, voltage control provides speed control below base speed. With constant flux, torque is proportional to armature current and the torque–speed characteristic is nearly linear with a negative slope. Armature reaction, brush voltage drop, magnetic saturation, and commutation effects make the actual characteristic less ideal.

Above base speed, field weakening reduces Φ. The motor can then operate approximately at constant power because torque decreases as speed increases. The safe field-weakening range is limited by commutation, mechanical speed, insulation, and residual flux requirements.

2.2 DC motor types

In a series motor, field winding current is approximately armature current. Before saturation, Φ ∝ I_a, so T_e ∝ I_a². This gives very high starting torque. A series motor must not be operated without a mechanical load because its speed can rise dangerously.

A compound motor combines series and shunt excitation. It provides higher starting torque than a shunt motor while retaining better speed regulation than a pure series motor.

2.3 Advantages and limitations

DC drives offer simple, rapid torque control and a wide speed range. However, brushes and commutators require maintenance, produce electrical noise, and limit operation in explosive or contaminated environments. These disadvantages have led many modern installations to replace DC motors with inverter-fed AC or electronically commutated motors.

3. Three-Phase Induction Motors

3.1 Rotating magnetic field and slip

A balanced three-phase stator supply produces a rotating magnetic field. Synchronous speed is

where f₁ is supply frequency in hertz and P is the number of poles. The rotor must run below synchronous speed in motoring operation so that relative motion induces rotor current. Slip is

where n_m is rotor speed, and rotor electrical frequency is f₂ = sf₁. At standstill, s = 1; near rated operation, slip is commonly a few percent for efficient motors.

3.2 Equivalent circuit

The per-phase induction-motor equivalent circuit resembles a transformer with a rotating secondary. Referred to the stator, the rotor resistance term appears as R₂'/s. The air-gap power is

The rotor copper loss is

and the mechanical power developed is

These relationships assume a balanced sinusoidal steady state and use rotor quantities referred to the stator. They show that slip represents the fraction of air-gap power dissipated in rotor resistance.

3.3 Torque–speed behavior

A commonly used approximate torque expression is

where V₁ is the per-phase stator voltage, R₁ and X₁ are stator resistance and leakage reactance, R₂' and X₂' are referred rotor parameters, and ω_s is synchronous angular speed. The approximation neglects or simplifies the magnetizing branch.

At low slip, torque is approximately proportional to slip. At high slip, torque decreases approximately with increasing slip. Maximum torque occurs approximately when

for the approximate circuit. Increasing rotor resistance in a wound-rotor motor increases the slip at maximum torque but does not greatly change the maximum torque. In a squirrel-cage motor, rotor-bar shape determines effective resistance and reactance at different frequencies; deep-bar and double-cage designs exploit this for improved starting.

3.4 Induction-motor drive control

A variable-frequency drive changes frequency and voltage. Constant V/f operation approximately maintains air-gap flux below base speed:

Therefore, voltage is reduced approximately in proportion to frequency. Below base speed the motor can produce near-constant torque; above base speed voltage is limited and flux weakens, giving approximately constant power. Vector control and direct-torque control provide improved dynamic torque response by controlling flux and torque-producing current components.

4. Synchronous Motors

4.1 Principle and steady-state model

A synchronous motor rotor field locks to the stator rotating field. In steady state, rotor speed is exactly synchronous speed:

A cylindrical-rotor synchronous motor has the approximate per-phase phasor equation

where V is terminal phase voltage, E_f is excitation emf, I_a is armature current, R_a is armature resistance, and X_s is synchronous reactance. Neglecting resistance, electromagnetic power is approximately

where δ is the torque angle. The corresponding torque is T_e = P_e/ω_s. Stable operation requires operation on the stable portion of the power-angle curve; excessive load can cause loss of synchronism.

4.2 Excitation and applications

Changing field excitation changes power factor. An underexcited motor draws lagging reactive current, while an overexcited motor can supply leading reactive power and improve plant power factor. Synchronous motors are attractive for constant-speed, high-power drives, compressors, mills, and power-factor correction, but starting and synchronism require suitable methods such as damper windings, an auxiliary drive, or a variable-frequency converter.

5. Permanent-Magnet and Brushless DC Motors

5.1 Permanent-magnet synchronous motors

A permanent-magnet synchronous motor (PMSM) uses rotor magnets instead of field windings. It has high efficiency and power density because rotor copper loss is eliminated. A simplified surface-magnet torque relation is

where p is the number of pole pairs, λ_m is permanent-magnet flux linkage, and i_q is the torque-producing current component in rotor-oriented coordinates. This equation assumes sinusoidal flux distribution, negligible reluctance torque, and balanced steady-state currents.

An interior PMSM also develops reluctance torque because direct-axis and quadrature-axis inductances differ:

where L_d and L_q are d- and q-axis inductances and i_d is the direct-axis current. Maximum-torque-per-ampere control exploits both magnet and reluctance torque. At high speed, negative direct-axis current weakens the effective air-gap flux, but excessive current can demagnetize magnets or exceed inverter voltage limits.

5.2 Brushless DC motors

A BLDC motor is electronically commutated and commonly has trapezoidal back emf. Six-step commutation energizes two phases at a time, with the unenergized phase often used for sensorless back-emf detection. The ideal rectangular-current, trapezoidal-emf combination produces nearly constant torque, although torque ripple occurs during commutation and because of nonideal waveforms.

PMSM and BLDC terminology is partly waveform-based: PMSM drives generally use sinusoidal currents and field-oriented control, whereas BLDC drives commonly use trapezoidal commutation. Both use permanent magnets and require an electronic inverter.

6. Switched-Reluctance Motors

6.1 Operating principle

A switched-reluctance motor (SRM) has salient poles on both stator and rotor, with concentrated stator windings and usually no rotor windings or magnets. When a phase is energized, the rotor tends to move toward the position of minimum magnetic reluctance.

For a linear magnetic approximation, co-energy is

and torque is

where L(θ) is phase inductance as a function of rotor position θ, and i is phase current. Positive torque is produced when inductance increases with rotor angle. Saturation makes the actual inductance current-dependent, so accurate prediction uses nonlinear flux-linkage data or finite-element models.

6.2 Characteristics and control

The rotor is mechanically robust, operates at high temperature, and contains no rotor copper or permanent magnets. The converter independently switches phases according to rotor position. Main disadvantages are torque ripple, acoustic noise, electromagnetic interference, and the need for specialized control. Current profiling, optimized turn-on and turn-off angles, multiphase operation, and advanced torque-sharing functions reduce ripple.

7. Stepper Motors

A stepper motor converts commanded electrical pulses into discrete angular movement. A step angle is approximately

for a simplified variable-reluctance arrangement, where N_r is rotor tooth number and m is the number of independently excited phases. Actual hybrid and permanent-magnet stepper designs require their specific pole and tooth relationships.

The commanded step rate determines speed. A stepper can hold position without a position sensor when the load remains within its pull-in and pull-out torque limits. Its torque decreases with speed because winding inductance prevents current from reaching its commanded value and because back emf increases. Resonance, missed steps, and vibration are important practical limitations. Microstepping improves smoothness and acoustic behavior but does not always increase absolute positioning accuracy under load.

Stepper motors are suitable for printers, laboratory mechanisms, valve actuators, and low-to-moderate-power positioning systems. Servo motors are preferable where high acceleration, disturbance rejection, or guaranteed position verification is required.

8. Comparative Performance and Selection

Selection should begin with the load profile rather than with motor popularity. Important questions include: Is the load constant torque or variable torque? Is continuous or intermittent duty required? What starting torque and acceleration are necessary? Is regenerative braking required? What accuracy and disturbance rejection are needed? Are noise, maintenance, hazardous location, rare-earth material use, or fault tolerance important?

9. Worked Example: Induction-Motor Speed, Power, and Efficiency

A three-phase, four-pole induction motor is supplied at 50 Hz. At a particular operating point its speed is 1,440 r/min, air-gap power is 20 kW, and rotational plus stray-load losses total 0.80 kW. Stator losses are 0.60 kW. Determine slip, rotor copper loss, developed mechanical power, shaft output, input power, and efficiency.

Synchronous speed is

Slip is

Rotor copper loss is

Developed mechanical power is

Shaft output is

Input power is the air-gap power plus stator losses:

Therefore,

The example illustrates that slip is not itself a separate loss; it determines the fraction of air-gap power dissipated as rotor copper loss. The remaining fraction is converted into mechanical power before rotational losses are subtracted.

10. Applied Case: Selecting a Motor for a Conveyor

Background

Consider an illustrative packaging conveyor requiring 7.5 kW at the shaft. It operates continuously at approximately 1,450 r/min, has frequent starts with a loaded belt, and must provide controlled stopping. The plant has a three-phase 400 V, 50 Hz supply and prefers low maintenance.

Problem

A DC motor offers excellent torque control but requires brush maintenance. A stepper motor is inappropriate because the power and continuous-duty requirements are high. An SRM could meet the duty but would require a specialized converter and acoustic mitigation. A PMSM would provide excellent efficiency and compactness, but its cost and control complexity are higher.

Analysis and approach

A squirrel-cage induction motor with a variable-frequency drive is a strong baseline choice. The VFD can provide low-speed starting torque, controlled acceleration, dynamic braking, and energy savings during reduced-load operation. A vector-controlled VFD is preferred if starting torque and speed regulation are demanding. The motor should be selected with an appropriate service factor only after checking thermal duty; oversizing alone can reduce efficiency and power factor at light load.

The engineer should verify the conveyor torque–speed curve, acceleration time, reflected inertia, braking resistor requirements, overload rating of the VFD, enclosure, ambient temperature, and protection against belt jamming. If the conveyor operates for very long periods at low speed, independent motor cooling may be necessary because shaft-mounted fan cooling decreases with speed.

Outcome

The induction motor–VFD combination provides a practical balance of cost, ruggedness, controllability, and maintainability. A PMSM could be selected instead where energy efficiency, compact size, or very high duty-cycle efficiency justifies the additional investment.

Learning points

1. Motor choice must be based on the complete duty cycle.
2. The converter is part of the drive, not an accessory.
3. Thermal performance at low speed can determine reliability.
4. Efficiency, maintenance, control quality, and lifecycle cost should be considered together.

11. Challenges, Safety, and Contemporary Development

Power electronic switching creates common-mode voltages, bearing currents, insulation stress, and electromagnetic interference. Correct cable selection, grounding, shielding, filtering, shaft grounding, and inverter-rated insulation reduce these risks. Overspeed protection, emergency stopping, guarding, lockout/tagout, and thermal protection are essential safety practices.

Regulatory efficiency standards increasingly influence motor selection. Engineers must consider full-load and part-load efficiency, harmonic distortion, standby losses, and lifecycle energy consumption rather than nameplate efficiency alone. Permanent-magnet machines reduce rotor losses but raise questions concerning rare-earth supply, recycling, and end-of-life recovery. Sensorless control, model-predictive control, digital twins, condition monitoring, and machine-learning-based fault diagnosis are developing rapidly. These tools are useful only when supported by reliable measurements, validated models, cybersecurity controls, and engineering judgment.

Best practice is to document assumptions, distinguish measured from estimated quantities, validate thermal and mechanical limits, and test the drive under representative load conditions. No motor technology is universally superior; the correct choice is the one whose electromagnetic, thermal, mechanical, control, economic, and environmental characteristics match the application.

Chapter Summary

Electrical motors convert electrical power into controlled mechanical torque. Their steady-state performance is governed by electromagnetic torque, speed, losses, thermal limits, and the interaction between motor and load. DC motors provide straightforward torque and speed control but require commutators and brushes. Induction motors use slip to induce rotor currents and remain the dominant general-purpose industrial motor because they are rugged and economical. Synchronous motors run at synchronous speed and permit power-factor control. PMSM and BLDC motors offer high efficiency and power density through permanent-magnet excitation. SRMs use position-dependent inductance and have robust rotors but require specialized converters and mitigation of torque ripple and noise. Stepper motors provide discrete positioning but have limited high-speed torque and may lose synchronism.

Equivalent circuits and equations allow engineers to calculate slip, air-gap power, copper losses, developed power, torque, speed, and efficiency. Variable-frequency drives extend the controllability of AC motors through constant-flux and field-weakening regions. Effective motor selection requires analysis of load profile, starting and braking, duty cycle, thermal conditions, maintenance, safety, efficiency, lifecycle cost, and environmental considerations.

Key Terms

Electromagnetic torque: Torque produced by interaction between magnetic fields and currents.

Back emf: Voltage induced in a motor winding that opposes the applied voltage.

Slip: Normalized difference between synchronous speed and rotor speed in an induction motor.

Synchronous speed: Speed of the rotating stator field.

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

Rotor copper loss: Resistive loss in induction-motor rotor conductors.

Torque angle: Electrical angle between excitation emf and terminal voltage in a synchronous motor.

Field weakening: Reduction of effective air-gap flux to permit operation above base speed.

Permanent-magnet synchronous motor: Synchronous motor whose rotor field is produced by permanent magnets.

Brushless DC motor: Electronically commutated permanent-magnet motor commonly using trapezoidal emf.

Switched-reluctance motor: Motor whose torque results from rotor movement toward increasing phase inductance.

Co-energy: Energy function used to calculate torque in magnetically nonlinear systems.

Stepper motor: Motor that moves in commanded angular increments.

Constant-torque region: Speed range in which a drive can approximately maintain rated torque.

Constant-power region: Speed range in which torque decreases approximately inversely with speed.

Power factor: Ratio of real power to apparent power in an AC system.

Service factor: Permitted rating margin under specified operating conditions.

Regenerative braking: Operation in which mechanical energy is converted back to electrical energy.

Vector control: Control method that independently regulates flux-producing and torque-producing current components.

Duty cycle: Time-dependent pattern of load, speed, acceleration, and rest periods.

Review Questions

Short Answer Questions

1. Why must an induction-motor rotor run below synchronous speed during motoring?
2. State the relationship between air-gap power, rotor copper loss, and developed mechanical power.
3. Why does field weakening permit operation above base speed?
4. Identify two advantages and two limitations of permanent-magnet motors.
5. What is the principal operating principle of a switched-reluctance motor?

Long Answer Questions

1. Explain the equivalent circuit and torque–speed characteristic of a three-phase induction motor.
2. Compare separately excited DC, induction, synchronous, and PMSM drives for industrial applications.
3. Explain how a variable-frequency drive provides constant torque below base speed and constant power above base speed.
4. Derive the basic co-energy expression for torque in an SRM and discuss the effect of magnetic saturation.
5. Discuss the construction, control, advantages, and limitations of stepper motors.

Analytical / Application Questions

1. A four-pole, 50 Hz induction motor operates at 1,470 r/min. Calculate synchronous speed, slip, and rotor frequency.
2. A conveyor requires constant torque from zero to base speed and constant power from base speed to twice base speed. Explain how flux and torque should be controlled in both regions.
3. Select an appropriate motor and control strategy for a high-cycle robotic axis requiring high efficiency, rapid acceleration, position feedback, and low maintenance. Justify the choice against at least two alternatives.

Multiple Choice Questions with Answers

Question 1

For a four-pole induction motor supplied at 50 Hz, the synchronous speed is:

A. 750 r/min

B. 1,000 r/min

C. 1,500 r/min

D. 3,000 r/min

Correct Answer: C. 1,500 r/min

Question 2

In an induction motor, rotor copper loss is equal to:

A. (1 − s)P_ag

B. sP_ag

C. P_ag/s

D. P_ag + s

Correct Answer: B. sP_ag

Question 3

Below base speed, constant V/f control is primarily used to maintain approximately constant:

A. Rotor resistance

B. Air-gap flux

C. Mechanical friction

D. Slip frequency at zero

Correct Answer: B. Air-gap flux

Question 4

A synchronous motor in steady state runs at:

A. A speed proportional to load torque

B. Less than synchronous speed by rated slip

C. Exactly synchronous speed

D. Zero speed unless rotor resistance is added

Correct Answer: C. Exactly synchronous speed

Question 5

Which motor commonly uses electronic commutation and trapezoidal back emf?

A. Brushless DC motor

B. Wound-rotor induction motor

C. Universal motor

D. Reluctance-start induction motor

Correct Answer: A. Brushless DC motor

Question 6

For an ideal surface-magnet PMSM, electromagnetic torque is primarily proportional to:

A. Permanent-magnet flux linkage multiplied by q-axis current

B. Stator resistance multiplied by speed

C. Slip multiplied by rotor resistance

D. Mechanical friction multiplied by voltage

Correct Answer: A. Permanent-magnet flux linkage multiplied by q-axis current

Question 7

An SRM produces positive torque, in the linear approximation, when phase inductance:

A. Decreases with rotor angle

B. Is independent of rotor angle

C. Increases with rotor angle

D. Is zero

Correct Answer: C. Increases with rotor angle

Question 8

A major limitation of an open-loop stepper drive is the possibility of:

A. Operating exactly at synchronous speed only

B. Missed steps under excessive load or acceleration

C. Rotor copper loss at zero current

D. Power-factor correction failure in all conditions

Correct Answer: B. Missed steps under excessive load or acceleration

Question 9

A motor develops 19.2 kW mechanically and has 0.8 kW rotational losses. Its shaft output is:

A. 18.4 kW

B. 19.2 kW

C. 20.0 kW

D. 24.0 kW

Correct Answer: A. 18.4 kW

Question 10

Which motor type requires brushes and a mechanical commutator in its conventional form?

A. Squirrel-cage induction motor

B. Permanent-magnet synchronous motor

C. Separately excited DC motor

D. Switched-reluctance motor

Correct Answer: C. Separately excited DC motor