DC, Permanent-Magnet and Reluctance Machines

Introduction and learning objectives

Electrical machines convert energy between electrical and mechanical forms through magnetic fields. Although induction and synchronous machines dominate many high-power applications, DC, permanent-magnet (PM), switched-reluctance (SR), and synchronous-reluctance (SynRM) machines remain important because they offer distinctive combinations of controllability, efficiency, torque density, robustness, and cost.

This chapter develops design-oriented understanding of these machine families. On completing it, a student should be able to:

• explain the electromagnetic principles and construction of DC, PM, SR, and SynRM machines;
• apply basic output, emf, torque, and magnetic-circuit equations;
• select winding, commutation, magnet, rotor, and cooling arrangements;
• assess cogging torque, torque ripple, demagnetization, and rare-earth-material trade-offs;
• compare machine technologies for industrial and transportation applications; and
• interpret a practical machine-selection and preliminary-design problem.

1. Common design foundations

Machine design is an iterative compromise among electromagnetic loading, thermal loading, mechanical stress, manufacturability, noise, cost, reliability, and control complexity. A useful preliminary design begins with the required speed, torque, duty cycle, voltage, current, enclosure, cooling method, allowable temperature rise, and service environment.

The mechanical output is

where P_mech is mechanical power in watts, T is torque in newton-metres, and ω_m is mechanical angular speed in radians per second. If speed is specified in revolutions per minute, ω_m = 2πn/60, where n is speed in r/min.

The electromagnetic torque may also be related to the air-gap power by

subject to the sign convention that positive torque produces mechanical output. Losses include copper loss, core loss, mechanical loss, stray-load loss, semiconductor loss, and—in PM machines—magnet and sleeve losses caused by space and time harmonics.

Specific electric loading and magnetic loading are useful scaling ideas. The former represents ampere-conductors per metre of air-gap circumference; the latter represents average air-gap flux density. Increasing either tends to increase torque density, but also increases copper loss, saturation, force, temperature, or acoustic excitation.

Magnetic circuits

For a simple magnetic path,

where ℱ is magnetomotive force in ampere-turns, N is turns, I is current, Φ is flux in webers, and ℛ is reluctance in ampere-turns per weber. For a uniform section,

where l is magnetic-path length, A is cross-sectional area, and μ is permeability. This linear relation is approximate: iron saturation makes permeability current-dependent, and leakage and fringing make the actual flux differ from a simple series-circuit estimate.

The air gap usually dominates reluctance because air has much lower permeability than unsaturated iron. Consequently, small changes in gap length, pole shape, slot opening, or eccentricity can materially affect flux, torque ripple, and noise.

2. Conventional DC machines

A conventional DC machine has a stationary field system, a slotted armature, a commutator, and brushes. The armature conductors experience force in the magnetic field, while the commutator reverses conductor current at the appropriate rotor position so that torque remains unidirectional. The same machine operates as a motor or generator.

2.1 EMF and torque

For a lap- or wave-wound machine with Z armature conductors, P poles, A parallel paths, flux per pole Φ, and speed n in r/min, the generated emf is

where N is speed in r/min. The formula assumes uniformly distributed flux and conductors, and neglects brush voltage effects. For a motor, the back emf is approximately E_a = V_a - I_aR_a - V_br, where V_a is applied armature voltage, I_a is armature current, R_a is armature-circuit resistance, and V_br is total brush-contact drop.

Electromagnetic torque is

with torque in newton-metres when flux is in webers and current in amperes. Thus, with constant field flux, DC-machine torque is approximately proportional to armature current.

The motor balance is

and, neglecting rotational losses, electromagnetic power satisfies E_aI_a = T_eω_m. These equations explain the traditional advantages of separately excited and shunt DC motors: simple torque control and a wide speed range. Their disadvantages are brush wear, commutator maintenance, sparking, radio-frequency interference, and limitations at high speed or in explosive environments.

2.2 Armature reaction and commutation

Armature current creates a magnetomotive force that distorts and weakens the main field. This armature reaction shifts the magnetic neutral axis and may drive pole shoes into saturation. Compensating windings embedded in pole faces can counteract armature reaction in large machines; interpoles, connected in series with the armature, improve commutation by producing a commutating emf.

During commutation, a coil is temporarily short-circuited by a brush while its current reverses. A high inductance, poor brush position, excessive current, or inadequate interpole action produces sparking. Design measures include suitable brush material, equalizer connections, interpoles, compensating windings, appropriate mica undercutting, and controlled current density.

2.3 DC-machine design considerations

The armature must provide sufficient slot area for conductors while retaining tooth strength and acceptable flux density. Increasing conductor current raises torque but increases copper loss P_Cu = I²R. A practical design therefore checks current density, thermal paths, commutator peripheral speed, brush current density, shaft stress, bearing life, and fault behavior.

Modern electronically commutated PM machines often replace the mechanical commutator while retaining the useful DC-machine relationship between torque and current. This is one reason the DC machine remains a valuable conceptual reference.

3. Permanent-magnet machines

PM machines use magnets to establish the excitation field. They may be brushed DC machines, brushless DC (BLDC) machines, sinusoidal permanent-magnet synchronous machines (PMSMs), or axial-flux variants. The rotor magnets eliminate field copper loss and can produce high efficiency and power density.

3.1 BLDC and PMSM distinctions

A BLDC machine generally uses trapezoidal back emf and quasi-rectangular phase currents, often with six-step electronic commutation. A PMSM is commonly designed for sinusoidal back emf and controlled with sinusoidal currents or vector control. The distinction is based on waveform and control objectives rather than on the mere presence of magnets.

For a sinusoidal PMSM in the simplified non-salient model, electromagnetic torque is approximately

where p is the number of pole pairs, λ_m is rotor magnet flux linkage in weber-turns, and i_q is the q-axis current in amperes. The equation assumes sinusoidal windings, balanced three-phase currents, negligible reluctance torque, and a consistent peak-value convention. If RMS current or a different Park-transform convention is used, numerical coefficients change; the convention must therefore be stated.

For an interior PMSM with saliency, a commonly used model is

where L_d and L_q are d- and q-axis inductances, and i_d is d-axis current. The first term is magnet torque and the second is reluctance torque. In many interior machines L_q > L_d, so suitable negative d-axis current can support maximum torque per ampere and weaken the air-gap flux at high speed.

3.2 Magnet selection and sizing

Common magnet materials include ferrite, bonded rare-earth magnets, samarium-cobalt, and neodymium-iron-boron (NdFeB). Important properties are remanent flux density B_r, intrinsic coercivity H_ci, maximum energy product, reversible temperature coefficient, corrosion resistance, and mechanical strength.

A first-order magnet operating-point estimate uses a demagnetization curve and the magnetic load line. The magnet must provide the required air-gap flux while operating away from the knee of its recoil or demagnetization characteristic. The magnet thickness is not selected from flux density alone: leakage, permeance coefficient, pole coverage, armature reaction, temperature, and manufacturing tolerances must be included.

Demagnetization risk is highest when temperature is high and opposing d-axis or short-circuit current is large. The design must verify:

• irreversible loss at maximum temperature;
• transient current and inverter short-circuit conditions;
• local demagnetization near magnet corners and bridges;
• magnet retention under centrifugal stress; and
• corrosion protection and adhesive durability.

NdFeB offers very high energy density but is relatively sensitive to temperature and corrosion and depends on supply chains for rare-earth elements. Samarium-cobalt is more temperature-resistant and corrosion-resistant but generally more expensive and brittle. Ferrite is inexpensive and resource-abundant, but its lower energy density requires larger magnets and often a larger machine.

3.3 Cogging torque, ripple, and mitigation

Cogging torque is the position-dependent torque caused by interaction between rotor magnets and stator slot permeance variation. Electromagnetic torque ripple additionally arises from nonsinusoidal back emf, current distortion, inverter dead time, saturation, commutation, and control delay.

Mitigation methods include fractional-slot windings, skewing magnets or slots, optimized pole-arc-to-pole-pitch ratio, dummy slots, shaped magnet edges, distributed windings, current-harmonic compensation, accurate rotor-position estimation, and selective reduction of slot permeance variation. Each method may reduce winding factor, increase manufacturing complexity, or add copper and iron loss. The design target is not necessarily zero ripple, but acceptable torque quality over the operating range.

4. Switched-reluctance machines

An SR machine has salient stator and rotor poles, concentrated stator windings, and usually no rotor windings or magnets. Torque is produced because the rotor tends to move toward a position of increasing inductance. A phase is energized before alignment and turned off before the aligned position to produce positive torque over the desired interval.

For a phase with flux linkage λ and current i, co-energy is

where θ is rotor position. The electromagnetic torque at constant current is

For a linear magnetic model with λ = L(θ)i, this becomes

Positive torque occurs where inductance increases with rotor angle. The linear expression is a useful explanatory approximation, but real machines saturate; accurate design uses nonlinear flux-linkage-versus-current-and-position data, finite-element analysis, or experimentally identified maps.

4.1 SR design and control

Important variables are pole arcs, aligned and unaligned inductance, phase number, stroke angle, turn-on angle, turn-off angle, current reference, and DC-link voltage. Increasing phase overlap can reduce torque ripple but may increase converter size and negative-torque risk. A higher pole count can improve torque smoothness and reduce torque pulsation frequency, but it narrows the available conduction interval and complicates manufacturing.

The converter commonly uses one asymmetric half-bridge per phase, allowing independent excitation and demagnetization. This converter is robust and fault-tolerant in some configurations, but it requires more switches than a conventional three-phase inverter. Acoustic noise is a major issue because radial force changes excite stator vibration. Skew, current profiling, unequal pole arcs, structural stiffening, and vibration-aware control are possible remedies.

5. Synchronous-reluctance machines

A SynRM rotor has no rotor magnets and no rotor winding. Its flux barriers create different d- and q-axis reluctances. When the rotor aligns its low-reluctance d-axis with the stator field, reluctance torque is produced. In the dq model,

for sinusoidal balanced currents and the same peak-value convention used earlier. High saliency ratio L_d/L_q improves torque per ampere and power factor, although saturation reduces the effective inductances and can alter the optimum current angle.

Rotor barriers must provide magnetic saliency while preserving mechanical integrity. Thin bridges improve saliency but experience high stress and may saturate strongly. Thick bridges improve strength but allow unwanted flux leakage. High-speed designs require detailed stress analysis, burst-speed testing, balancing, and consideration of fatigue.

A PM-assisted SynRM inserts modest magnets into the flux barriers. This increases torque density and power factor while using fewer magnets than a fully PM-excited rotor. It also introduces magnet demagnetization and retention requirements, so the claimed material savings must be evaluated over the complete design and life cycle.

6. Comparative design selection

The correct choice depends on duty cycle rather than peak efficiency alone. A PM motor may be best where compactness and efficiency justify magnet cost. An SR machine may be attractive in harsh temperatures, high-speed operation, or applications requiring magnet avoidance. A SynRM may suit industrial drives where inverter control is available and moderate torque density is acceptable. A conventional DC motor remains useful for legacy systems, laboratory equipment, and applications where simple analog speed control is valuable.

7. Worked preliminary example: PM motor torque and power

An illustrative three-phase PMSM has two pole pairs, magnet flux linkage λ_m = 0.080 Wb, and q-axis current i_q = 30 A. Assume a non-salient rotor, sinusoidal currents, and the peak-current torque convention.

Using

we obtain

At 3,000 r/min,

Therefore,

The ideal electromagnetic output is approximately 2.26 kW. Actual shaft output is lower because of copper, core, mechanical, inverter, and stray losses. If the current were specified as RMS rather than peak, the torque calculation would require the appropriate transformation coefficient; this illustrates why current convention must always be documented.

8. Applied case: magnet-free traction auxiliary drive

Background

This illustrative case concerns a 30 kW electric-vehicle auxiliary compressor drive expected to operate from a variable-speed inverter. The vehicle manufacturer wants to reduce dependence on rare-earth magnets while maintaining high efficiency over a broad speed range.

Problem

A surface-PM motor offers high torque density, but the procurement team is concerned about NdFeB price volatility, high-temperature demagnetization, and recycling. A conventional induction motor avoids magnets but has rotor losses. An SR machine avoids rotor losses and magnets but may create unacceptable acoustic noise. A SynRM is considered as a compromise.

Analysis and approach

The design team first specifies continuous and peak torque, speed range, overload duration, coolant temperature, acoustic limit, available inverter voltage, and allowable mass. It then compares four candidates using the same duty-cycle simulation rather than comparing isolated peak values.

For the SynRM, rotor barriers are optimized to increase L_d/L_q while retaining burst-speed strength. Finite-element analysis evaluates saturation, torque ripple, unbalanced magnetic forces, and inverter current. A prototype is tested for efficiency maps, thermal rise, noise, vibration, and fault response. Control software uses rotor-position feedback and maximum-torque-per-ampere operation at moderate speed, followed by flux weakening at high speed.

Outcome

The magnet-free SynRM meets the continuous power requirement but is slightly larger than the PM candidate. Its life-cycle cost is favorable because magnet procurement risk is removed, although the inverter and acoustic-treatment costs are higher. The selected design is acceptable because the compressor has a relatively small number of operating hours at peak torque and the enclosure permits additional sound attenuation.

Learning points

1. Material selection must be linked to duty cycle, not only rated efficiency.
2. Rotor mechanical design is as important as electromagnetic saliency.
3. Torque ripple and noise must be measured at the system level.
4. A magnet-free design can shift cost from materials to control, inverter, and manufacturing.
5. Environmental claims should include manufacturing energy, recyclability, and supply-chain impacts.

9. Best practices, limitations, and contemporary developments

A sound design workflow begins with requirements, establishes analytical scaling estimates, develops a magnetic and thermal model, checks mechanical integrity, and then uses finite-element and control simulations. Prototypes should be tested under the real duty cycle, including overload, hot restart, voltage transients, inverter faults, and sensor failures.

Finite-element results are only as reliable as material data, boundary conditions, mesh quality, and manufacturing assumptions. Magnet properties vary with temperature and batch; lamination loss data may not represent high-frequency PWM excitation; and idealized tolerances can hide eccentricity, burrs, skew errors, or assembly variation.

Contemporary development directions include ferrite-assisted and rare-earth-lean PM machines, additive-manufactured cooling channels, hairpin and formed windings, model-predictive and sensorless control, digital-twin-based condition monitoring, acoustic optimization, and integrated motor-inverter units. Artificial intelligence can assist surrogate modelling, fault classification, and design-space exploration, but it does not replace validated electromagnetic, thermal, mechanical, and safety analysis. Training data, extrapolation error, explainability, and verification remain important engineering responsibilities.

Designers must also address electrical safety, insulation coordination, rotating-part containment, electromagnetic compatibility, thermal runaway prevention in associated power electronics, and responsible sourcing and recycling of magnets and copper. Standards and certification requirements vary by application and jurisdiction; the applicable machinery, drive, EMC, vehicle, and safety standards must be identified before final design release.

Chapter Summary

DC machines establish the fundamental relationship between flux, armature current, torque, and back emf, but brushes and commutators impose maintenance and speed limitations. PM machines provide high efficiency and torque density, while requiring careful magnet sizing, thermal protection, mechanical retention, and demagnetization analysis. Cogging torque and torque ripple result from slotting, harmonics, saturation, commutation, and control effects and must be reduced through coordinated electromagnetic and control design.

SR machines generate torque from the tendency of the rotor to move toward increasing inductance. They are magnet-free and thermally robust at the rotor, but their converters, acoustic noise, and torque ripple require attention. SynRMs use rotor saliency and reluctance torque; their central design challenge is balancing magnetic anisotropy against rotor strength and manufacturability. Machine selection should consider the complete duty cycle, supply-chain exposure, control and inverter requirements, thermal behavior, noise, reliability, life-cycle cost, and environmental impact.

Key Terms

Armature reaction: Distortion and weakening of the main field caused by armature current.

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

Brushless DC machine: PM machine commonly using trapezoidal back emf and electronic commutation.

Cogging torque: Position-dependent torque caused by rotor magnets interacting with stator slot permeance.

Commutation: Reversal or electronic reassignment of phase current to maintain torque direction.

Demagnetization: Irreversible reduction of a permanent magnet’s magnetization.

Flux linkage: Magnetic flux linked by a winding, including the effect of turns.

Inductance saliency: Difference between d-axis and q-axis inductance.

Interpoles: Auxiliary poles used to improve commutation in DC machines.

Magnetic loading: Air-gap magnetic flux density used as a machine-sizing measure.

Magnetomotive force: Magnetic driving force equal to turns multiplied by current.

PMSM: Permanent-magnet synchronous machine.

Reluctance: Opposition offered by a magnetic circuit to flux.

SR machine: Switched-reluctance machine with salient stator and rotor poles.

SynRM: Synchronous-reluctance machine using rotor saliency without rotor excitation.

Torque ripple: Variation of electromagnetic torque with time or rotor position.

Unaligned position: SR rotor position having relatively low phase inductance.

Vector control: Control method that regulates d-axis and q-axis current components.

Review Questions

Short Answer Questions

1. Why does the commutator allow a DC motor to produce unidirectional torque?
2. Define cogging torque and identify two methods of reducing it.
3. What is the physical meaning of d-axis and q-axis inductance in a salient machine?
4. Why are high temperature and opposing armature reaction important in PM-machine design?
5. State one major advantage and one major limitation of an SR machine.

Long Answer Questions

1. Derive the basic emf and torque relationships of a conventional DC machine and state their assumptions.
2. Explain the construction, operation, control, and principal design challenges of a PMSM.
3. Describe how co-energy is used to derive torque in a switched-reluctance machine.
4. Compare PM, SR, and SynRM technologies for a variable-speed industrial drive.
5. Discuss how rotor mechanical strength, acoustic noise, and electromagnetic performance interact in reluctance-machine design.

Analytical / Application Questions

1. A four-pole, lap-wound DC machine has 480 conductors, flux per pole of 0.025 Wb, and armature current of 20 A. Determine its electromagnetic torque using the DC-machine torque equation, and state the assumptions.
2. A PMSM uses three pole pairs, magnet flux linkage of 0.06 Wb, and q-axis current of 25 A under the peak-current convention. Calculate the ideal non-salient electromagnetic torque.
3. For an SR phase with current held constant, explain how the sign of dL/dθ determines motoring or generating torque and describe how turn-on and turn-off angles affect torque ripple.

Multiple Choice Questions with Answers

Question 1

Which component in a conventional DC machine reverses armature-coil current during rotation?

A. Slip ring

B. Commutator

C. Damper winding

D. Rotor cage

Correct Answer: B. Commutator

Question 2

For a DC machine, electromagnetic torque is proportional to which product?

A. Flux per pole multiplied by armature current

B. Voltage multiplied by speed only

C. Resistance multiplied by flux leakage

D. Frequency multiplied by armature resistance

Correct Answer: A. Flux per pole multiplied by armature current

Question 3

Cogging torque in a PM machine is primarily caused by interaction between rotor magnets and:

A. Brush contact resistance

B. Stator slot permeance variation

C. Rotor copper loss

D. Bearing lubrication

Correct Answer: B. Stator slot permeance variation

Question 4

In the simplified non-salient PMSM model, torque is proportional to:

A. λ_m i_q

B. λ_m/i_q

C. i_d/i_q

D. R_sω_m

Correct Answer: A. λ_m i_q

Question 5

Which condition increases the risk of irreversible PM demagnetization?

A. Low temperature and assisting d-axis current

B. High temperature and opposing magnetic field

C. Zero armature current only

D. Reduced mechanical speed only

Correct Answer: B. High temperature and opposing magnetic field

Question 6

For a linear SR model, positive motoring torque is produced when:

A. dL/dθ is positive during positive current

B. dL/dθ is zero at all positions

C. Current is zero and inductance decreases

D. Resistance is negative

Correct Answer: A. dL/dθ is positive during positive current

Question 7

What is the principal source of torque in a conventional SynRM?

A. Rotor copper resistance

B. Difference between d-axis and q-axis reluctances

C. Mechanical commutator action

D. Permanent-magnet hysteresis only

Correct Answer: B. Difference between d-axis and q-axis reluctances

Question 8

A two-pole-pair PMSM has λ_m = 0.08 Wb and i_q = 30 A using the stated peak-current convention. Its ideal non-salient torque is:

A. 2.40 N·m

B. 4.80 N·m

C. 7.20 N·m

D. 14.40 N·m

Correct Answer: C. 7.20 N·m

Question 9

Which machine family normally has no rotor winding and no permanent magnet?

A. Surface-PM machine

B. Wound-field DC machine

C. Switched-reluctance machine

D. Permanent-magnet synchronous machine

Correct Answer: C. Switched-reluctance machine

Question 10

A major trade-off of reducing rare-earth magnet use through a SynRM design is that the machine may require:

A. A mechanical commutator

B. Higher saliency, specialized control, and careful rotor-strength design

C. A field winding on every rotor pole

D. No inverter at all

Correct Answer: B. Higher saliency, specialized control, and careful rotor-strength design