Transformers and Electrical Machines

Introduction

Transformers and rotating electrical machines convert electrical energy into forms required by power systems, industry, transport and domestic equipment. A transformer transfers AC power between circuits through a magnetic field without continuous mechanical motion. Motors convert electrical energy into mechanical energy, while generators perform the reverse conversion.

These devices share several physical principles: magnetic flux, electromagnetic induction, force on current-carrying conductors, magnetic-field energy and losses. Their practical performance is limited by winding resistance, leakage flux, core losses, mechanical losses, heating and insulation capability.

Learning Objectives

After studying this chapter, students should be able to:

• explain electromagnetic and electromechanical energy conversion;
• distinguish magnetic-field energy from magnetic co-energy;
• model ideal and practical transformers;
• calculate transformer voltage regulation, losses and efficiency;
• interpret open-circuit and short-circuit tests;
• explain the construction and characteristics of DC machines;
• calculate DC-machine emf, torque and speed;
• explain induction-motor slip, torque, starting and speed control;
• describe synchronous generators and motors; and
• select suitable machines for common engineering applications.

1. Electromechanical Energy Conversion

1.1 Energy balance

An electromechanical converter uses a magnetic or electric field as an intermediate energy-storage mechanism. Magnetic-field devices dominate conventional machines because useful flux densities can be produced efficiently in ferromagnetic cores.

For a lossless, singly excited magnetic system, an incremental electrical input is

where i is current in amperes and λ = NΦ is flux linkage in weber-turns. Here N is the number of turns and Φ is magnetic flux in webers.

The general energy balance is

where dW_f is the change in stored magnetic-field energy, dW_m is mechanical output and dW_loss represents electrical, magnetic and mechanical losses. For rotational motion, dW_m = T_e\,dθ, where T_e is electromagnetic torque and θ is angular displacement.

1.2 Field energy and co-energy

For a conservative, hysteresis-free magnetic system at fixed mechanical position x, magnetic-field energy is

Magnetic co-energy is

They satisfy

The force obtained from co-energy is evaluated at constant current:

Equivalently, field energy gives

For rotational systems, replace force and displacement with torque and angle:

These relations assume that the magnetic state is uniquely determined; substantial hysteresis requires loss to be treated explicitly.

For a linear magnetic system, λ = L(x)i, so

and, at constant current,

Thus, a movable member tends to move toward a position of higher inductance when current is held constant.

1.3 Fundamental conversion mechanisms

Electrical machines principally employ two related effects:

1. Generator action: motion of a conductor through a magnetic field induces emf according to Faraday’s law.
2. Motor action: a current-carrying conductor in a magnetic field experiences force. For a straight conductor perpendicular to the field, F = BIl, where B is flux density, I is current and l is active conductor length.

In rotating machines, distributed forces produce electromagnetic torque.

2. Transformers

2.1 Construction and operating principle

A transformer contains at least two windings linked by a common alternating magnetic flux. Laminated electrical-steel cores reduce eddy-current loss. Common constructions are core-type and shell-type. Power transformers may additionally contain oil, radiators, bushings, tap changers, protection relays and conservators.

Applying a sinusoidal voltage to the primary winding establishes alternating core flux. Faraday’s law gives the rms induced emf of a sinusoidally excited winding:

where E is rms induced emf in volts, f is frequency in hertz, N is turns and Φ_m is maximum core flux in webers. The equation assumes sinusoidal flux.

For an ideal transformer,

and, in magnitude,

Here a is the primary-to-secondary turns ratio. An ideal transformer has no winding loss, leakage flux, core loss or magnetizing-current requirement, so input and output apparent powers are equal.

2.2 Practical equivalent circuit

A practical transformer is represented by:

• R₁ and R₂: winding resistances;
• X₁ and X₂: leakage reactances;
• R_c: equivalent resistance representing core loss; and
• X_m: magnetizing reactance.

Secondary quantities referred to the primary are

With all relevant quantities referred to the primary, the phasor relations are

where I₀ is the exciting current. These equations use the passive load-current convention and consistent primary-referred quantities.

For many regulation calculations, the exciting current is small enough to neglect. The approximate primary-referred series impedance is

Figure: Approximate transformer series equivalent circuit referred to the primary; the exciting branch is omitted for load calculations.

2.3 Voltage regulation

Transformer voltage regulation measures the change in secondary terminal voltage from no-load to full-load while primary voltage and frequency remain constant:

When the series voltage drop is small, approximate regulation at secondary current I₂ is

All quantities must be referred to the secondary. Use the plus sign for lagging power factor and the minus sign for leading power factor. A sufficiently leading load can produce negative regulation.

Worked Example 1: Transformer regulation

A 12 kVA, 240 V secondary winding supplies rated current at 0.8 power factor lagging. The secondary-referred equivalent parameters are R_eq,2=0.040\;Ω and X_eq,2=0.065\;Ω. Estimate the regulation.

Rated current is

For \cos φ=0.8, \sin φ=0.6. Therefore,

The estimated full-load regulation is 1.48%.

2.4 Transformer losses and efficiency

The principal losses are:

• Core loss: hysteresis and eddy-current losses, approximately constant at fixed voltage and frequency.
• Copper loss: I²R loss in the windings; proportional to current squared at constant winding resistance.
• Stray-load loss: eddy currents caused by leakage fields in structural and winding components.
• Dielectric loss: usually small, but significant in high-voltage insulation.

Efficiency is

If P_cu,r is rated copper loss and x is the fraction of rated load, then approximately P_cu=x²P_cu,r. For a specified power factor,

where S_r is rated apparent power. Neglecting stray-load variation, maximum efficiency occurs when

This condition follows by differentiating efficiency with respect to load while voltage, frequency and power factor are fixed.

Worked Example 2: Efficiency and maximum-efficiency load

A 50 kVA transformer has a core loss of 400 W and full-load copper loss of 900 W. Find its efficiency at 75% load and 0.8 power factor, and the load fraction for maximum efficiency.

At x=0.75,

For maximum efficiency,

Maximum efficiency occurs at approximately 66.7% of rated load.

Distribution transformers are also evaluated by all-day efficiency, defined as energy output divided by energy input over 24 hours. This metric is important because their load varies while core loss continues whenever they are energized.

2.5 Open-circuit and short-circuit tests

In the open-circuit test, rated voltage is usually applied to the low-voltage winding while the other winding is open. The wattmeter reading approximates core loss because current and copper loss are small. Measurements determine the shunt branch parameters.

In the short-circuit test, one winding is shorted and a reduced voltage is applied to circulate rated current. Flux and core loss are small, so input power approximates rated copper loss. The test determines equivalent series resistance and leakage reactance.

These tests avoid direct full-power loading, but instruments, temperature corrections and waveform quality affect accuracy.

2.6 Three-phase transformers and applications

Three-phase transformation may use one three-phase unit or a bank of three single-phase units. Star and delta connections influence phase shift, neutral availability, insulation requirements and harmonic-current paths. Common vector groups include delta-star connections for distribution and star-delta connections for industrial transformation.

Transformer applications include transmission voltage conversion, isolation, measurement, impedance matching, rectifier supplies, renewable-energy interfaces and electronic power conversion. Autotransformers use a shared winding and offer material savings but do not provide galvanic isolation.

3. DC Machines

3.1 Construction

A conventional DC machine contains a stationary field system and rotating armature. Major components include the yoke, poles, field windings, armature core, armature winding, commutator, brushes, shaft and bearings. The commutator mechanically rectifies internally alternating armature emf, producing DC at generator terminals or unidirectional torque in motor operation.

DC machines may be separately excited, shunt, series or compound connected.

3.2 Generated and back emf

The average armature emf is

where E is emf in volts, P is the number of poles, Φ is flux per pole in webers, Z is the total number of armature conductors, n is speed in revolutions per minute and A is the number of parallel armature paths.

For a generator, terminal voltage is approximately

For a motor,

where I_a is armature current, R_a is armature resistance and V_b represents brush-contact drop.

3.3 Electromagnetic torque and speed

Converted electromagnetic power in a motor is P_conv=E_bI_a. Since P_conv=T_eω_m,

where ω_m is mechanical angular speed in rad/s and k_t is the torque constant in SI-compatible units.

From the emf equation,

Thus, speed is approximately proportional to back emf and inversely proportional to flux.

3.4 Characteristics

A shunt motor has approximately constant flux and therefore nearly constant speed. A series motor has flux that initially increases with armature current, producing high starting torque. It must not ordinarily be operated without mechanical load because weakening current can cause dangerously high speed. Compound motors combine shunt and series characteristics.

For generators, voltage regulation depends on armature resistance, armature reaction and field connection. Cumulative compounding can compensate for internal voltage drop.

3.5 Starting and speed control

At starting, E_b=0, so direct application of rated voltage can produce excessive armature current:

Because R_a is small, a starter inserts external resistance and removes it as speed and back emf rise. Modern drives instead use controlled power-electronic converters with current limiting.

DC-motor speed can be controlled by:

• varying armature voltage below base speed;
• weakening field flux above base speed; or
• adding armature resistance, an inefficient method used only in limited applications.

Armature-voltage control provides approximately constant-torque operation, while field weakening provides an approximately constant-power region subject to commutation and mechanical limits.

4. Three-Phase Induction Machines

4.1 Construction and rotating magnetic field

The stator contains a three-phase distributed winding. The rotor is either a squirrel-cage structure or a wound rotor with slip rings. Balanced three-phase currents produce a rotating field at synchronous speed

where n_s is in rpm, f is supply frequency in hertz and P is the number of poles.

Rotor speed n_r differs from synchronous speed. Slip is

For ordinary motoring, 0<s≤1. At standstill, s=1; near synchronous speed, s≈0. Rotor-current frequency is

4.2 Equivalent circuit and power flow

The per-phase induction-machine equivalent circuit resembles that of a transformer. Referred to the stator, the effective rotor resistance is R'₂/s. For motoring slips 0<s≤1, this quantity is minimum at starting, where s=1 and R'₂/s=R'₂. It increases as the rotor approaches synchronous speed.

The resistance term can be separated as

The first term represents rotor copper loss, while the second represents converted mechanical power in the equivalent circuit.

If P_g is air-gap power,

Mechanical output equals converted power minus mechanical losses.

Using a Thévenin equivalent of the stator network, torque is

where V_th, R_th and X_th describe the stator Thévenin equivalent, X'₂ is referred rotor leakage reactance and ω_s is synchronous mechanical angular speed in rad/s.

Maximum torque occurs when

This condition applies to the stated constant-frequency equivalent circuit. Increasing rotor resistance shifts maximum torque to higher slip but does not substantially change its magnitude when other parameters remain fixed.

Worked Example 3: Induction-motor slip and rotor frequency

A 50 Hz, four-pole induction motor runs at 1440 rpm. Determine synchronous speed, slip and rotor frequency.

4.3 Starting

A squirrel-cage motor can draw several times rated current when connected directly to the supply. Starting methods include:

• direct-on-line starting for smaller motors and strong supplies;
• star-delta starting, where appropriate for delta-rated motors;
• autotransformer starting;
• soft starters using controlled voltage; and
• variable-frequency drives (VFDs), which control both voltage and frequency.

Wound-rotor motors permit external rotor resistance, increasing starting torque and limiting current. The resistance is removed as acceleration proceeds.

4.4 Speed control

Induction-motor speed can be controlled by changing supply frequency, pole number, stator voltage or rotor resistance. VFD control is dominant because synchronous speed varies directly with frequency. Maintaining an approximately constant V/f ratio below base speed preserves air-gap flux. Above base frequency, voltage is limited and flux weakens, producing an approximately constant-power operating region.

Sensorless and vector-control algorithms estimate flux, torque and speed from electrical measurements. Such estimates depend on accurate machine parameters, especially at low speed and changing temperature.

5. Synchronous Machines

5.1 Construction and operating principle

A synchronous machine normally has a three-phase stator winding and a rotor magnetic field produced by DC excitation or permanent magnets. Rotors may be cylindrical, as in high-speed turbine generators, or salient-pole, as in low-speed hydroelectric generators.

The electrical frequency and mechanical speed satisfy

In steady state, the rotor runs exactly at synchronous speed. A synchronous generator converts mechanical input to AC electrical power. A synchronous motor locks its rotor field to the rotating stator field.

5.2 Power-angle relation

For a cylindrical-rotor, three-phase synchronous machine with negligible armature resistance, balanced operation and constant terminal voltage, transferred power is

where V and E are per-phase rms terminal and internal emf magnitudes, X_s is synchronous reactance per phase and δ is the torque or power angle. The associated electromagnetic torque is T_e=P/ω_s.

This sinusoidal relation is not a general formula for salient-pole machines; reluctance torque adds another term. Stable steady operation normally requires a substantial margin below the theoretical pull-out angle.

5.3 Excitation and power factor

Changing rotor excitation alters reactive-power exchange and stator power factor. An overexcited synchronous motor can supply reactive power and assist power-factor correction, while an underexcited motor absorbs reactive power. A synchronous condenser is an unloaded synchronous motor operated mainly for voltage and reactive-power support.

5.4 Starting and control

A conventional synchronous motor is not inherently self-starting from a fixed-frequency supply because average starting torque is zero. Starting methods include damper windings, a pony motor or a variable-frequency drive. Permanent-magnet synchronous motors and wound-field machines supplied by electronic drives can start under controlled frequency and current.

Grid-connected synchronous speed is fixed by frequency and pole count. Variable-speed operation requires variable-frequency power electronics.

6. Machine Selection and Applications

Selection must consider duty cycle, starting torque, speed range, supply quality, efficiency class, enclosure, cooling, acoustic noise, maintenance capability and lifecycle cost—not only purchase price.

7. Illustrative Case: Converting a Pumping Station to Variable-Speed Operation

Background

An illustrative municipal pumping station uses a 55 kW, four-pole induction motor supplied at 50 Hz. Flow is controlled by throttling while the motor runs near full speed. Measurements show prolonged operation at approximately 80% of rated flow.

Problem

Throttling dissipates pressure across the valve, wasting energy. Direct starting also causes voltage dips and mechanical stress. Engineers must determine whether a VFD and a high-efficiency replacement motor are justified.

Analysis

For a centrifugal pump, the affinity laws approximately give

where Q is flow, H is developed head, P is shaft power and n is speed. These laws apply most accurately to geometrically unchanged pumps operating near comparable efficiency regions; static head and system losses can materially alter savings.

If 80% flow can be obtained at 80% speed, the idealized shaft-power fraction is

Thus, a 55 kW shaft requirement at full speed would ideally fall to about 55(0.512)=28.2\;kW. Actual input power must account for pump, motor and drive efficiencies and the system head curve.

Approach

The engineering team should:

1. log flow, head and electrical demand across the operating cycle;
2. construct the actual system and pump curves;
3. check motor insulation suitability for pulse-width-modulated voltage;
4. provide output filtering or insulated bearings where cable length and common-mode currents require them;
5. verify harmonic compliance, protection coordination and emergency bypass needs; and
6. calculate lifecycle savings using measured duty data.

Outcome

A VFD enables soft starting, process-based speed control and lower energy use. However, the project should be approved only after confirming required minimum flow, cooling at low speed, critical mechanical speeds and harmonic limits.

Learning Points

Variable-speed control can produce large savings in variable-torque loads, but the cubic power relationship must not be applied blindly. System-level measurements and operational constraints determine the realizable benefit.

8. Safety, Standards and Contemporary Practice

Transformers and machines involve hazardous voltage, stored magnetic energy, high fault current, rotating components and hot surfaces. Safe practice includes isolation, lockout/tagout, earthing, discharge verification, guarding, thermal monitoring and arc-flash assessment. Transformer oil must be managed for fire risk and environmental contamination.

Standards from IEC, IEEE, NEMA and national authorities define ratings, tests, insulation systems, efficiency classes, temperature rise and protection. Efficiency regulation increasingly encourages premium-efficiency motors and low-loss transformers.

Contemporary developments include amorphous-metal transformer cores, ester insulating fluids, permanent-magnet motors, silicon-carbide converters, digital twins and condition monitoring. Machine-learning systems can identify anomalous vibration, current signatures and thermal patterns, but they should support rather than replace verified protection systems and engineering judgment. Cybersecurity is essential where networked drives and monitoring relays affect critical processes.

9. Engineering Limitations and Best Practices

Equivalent circuits are approximations. Saturation, harmonics, skin effect, temperature-dependent resistance, unbalanced supply and mechanical dynamics can make parameters operating-point dependent. Nameplate ratings specify permissible operation but do not alone guarantee suitability for a particular duty cycle.

Good professional practice includes:

• referring impedances and currents consistently to one transformer side;
• stating whether quantities are phase or line values;
• correcting winding resistance for operating temperature;
• checking starting voltage drop and acceleration time;
• allowing thermal and stability margins;
• considering harmonics and bearing currents with electronic drives;
• using measured duty profiles for lifecycle analysis; and
• validating analytical models against manufacturer or test data.

Chapter Summary

Electromechanical conversion uses magnetic fields to couple electrical and mechanical systems. Field energy is obtained from \int i\,dλ, while co-energy is obtained from \int λ\,di; force from co-energy is evaluated at constant current.

Transformers operate through mutual induction. Practical models include winding resistance, leakage reactance, magnetizing current and core loss. Voltage regulation depends on equivalent impedance and load power factor, while efficiency reflects core, copper, stray and dielectric losses. Open-circuit and short-circuit tests determine model parameters economically.

DC machines use commutation to produce DC terminal behavior and unidirectional torque. Their emf is proportional to flux and speed, while torque is proportional to flux and armature current. Induction motors operate below synchronous speed in the motoring region; slip controls rotor frequency, rotor loss and torque. Synchronous machines operate at synchronous speed and are important for generation, constant-speed drives and reactive-power control.

Machine selection requires technical, economic, safety and environmental evaluation. Power-electronic drives improve controllability and efficiency but introduce harmonics, insulation stress, bearing-current and cybersecurity considerations.

Key Terms

Air-gap power: Electromagnetic power transferred from an induction-machine stator to its rotor.

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

Co-energy: Complementary magnetic energy function calculated as \int λ\,di at fixed position.

Commutator: Mechanical switching device used in conventional DC machines.

Core loss: Magnetic loss caused mainly by hysteresis and eddy currents.

Efficiency: Ratio of useful output power or energy to input power or energy.

Field energy: Magnetic energy calculated as \int i\,dλ at fixed position.

Flux linkage: Product of winding turns and linked magnetic flux, λ=NΦ.

Leakage reactance: Reactance associated with flux that does not link all coupled windings.

Power angle: Angular displacement between relevant synchronous-machine internal and terminal voltage phasors.

Rotor frequency: Induction-motor rotor-current frequency, equal to sf.

Slip: Fractional difference between synchronous and rotor speeds.

Synchronous speed: Speed of a rotating magnetic field, 120f/P.

Transformer regulation: Change in secondary terminal voltage between no-load and load under stated conditions.

Turns ratio: Ratio of primary to secondary winding turns.

Variable-frequency drive: Power-electronic system that controls motor voltage and frequency.

Vector group: Designation describing transformer winding connections and phase displacement.

Review Questions

Short Answer Questions

1. Distinguish magnetic-field energy from magnetic co-energy.
2. Why is transformer core loss approximately constant at fixed voltage and frequency?
3. Why does a DC motor require current limitation during starting?
4. Define induction-motor slip and rotor frequency.
5. Why is a conventional synchronous motor not inherently self-starting?

Long Answer Questions

1. Derive the force relation for a linear singly excited magnetic system using co-energy.
2. Explain the practical transformer equivalent circuit and the purpose of each parameter.
3. Discuss the construction, emf, torque, starting and speed control of DC motors.
4. Explain the induction-motor power flow and derive the relationships among air-gap power, rotor copper loss and converted power.
5. Compare induction and synchronous motors in terms of construction, characteristics, starting, control and applications.

Analytical / Application Questions

1. A 25 kVA transformer has rated copper loss of 600 W and core loss of 300 W. Calculate its efficiency at 80% load and 0.9 power factor, and determine the load fraction for maximum efficiency.
2. A six-pole, 60 Hz induction motor runs at 1164 rpm. Determine synchronous speed, slip and rotor frequency, and explain the effect of increasing load on slip.
3. Evaluate the technical factors that should be considered before applying a VFD to an existing centrifugal-pump motor.

Multiple Choice Questions with Answers

Question 1

For a conservative singly excited magnetic system, which expression correctly defines magnetic co-energy at fixed mechanical position?

A. W'_f=\int₀ⁱ λ(i',x)\,di'

B. W'_f=\int₀^λ i(λ',x)\,dλ'

C. W'_f=i/λ

D. W'_f=λ/i

Correct Answer: A. W'_f=\int₀ⁱ λ(i',x)\,di'

Question 2

Force derived from magnetic co-energy is found by differentiating co-energy with respect to displacement while holding which quantity constant?

A. Flux linkage

B. Voltage

C. Current

D. Resistance

Correct Answer: C. Current

Question 3

A 12 kVA, 240 V transformer has secondary-referred R_eq=0.040\;Ω and X_eq=0.065\;Ω. At rated load and 0.8 power factor lagging, its approximate voltage regulation is:

A. 0.74%

B. 1.48%

C. 7.40%

D. 14.8%

Correct Answer: B. 1.48%

Question 4

A transformer has a core loss of 400 W and a rated copper loss of 900 W. Approximately what fraction of rated load gives maximum efficiency?

A. 44.4%

B. 50.0%

C. 66.7%

D. 75.0%

Correct Answer: C. 66.7%

Question 5

Which DC-machine emf equation uses n as speed in revolutions per minute?

A. E=PΦZn/(60A)

B. E=60PΦZ/(An)

C. E=PΦZ/(60An)

D. E=60An/(PΦZ)

Correct Answer: A. E=PΦZn/(60A)

Question 6

Why must the starting current of a conventional DC motor be limited?

A. Its flux is always zero at starting

B. Its back emf is zero and armature resistance is small

C. Its field frequency equals the supply frequency

D. Its commutator becomes an open circuit

Correct Answer: B. Its back emf is zero and armature resistance is small

Question 7

A 50 Hz, four-pole induction motor runs at 1440 rpm. What is its slip?

A. 2%

B. 4%

C. 6%

D. 8%

Correct Answer: B. 4%

Question 8

For induction-motor operation over 0<s≤1, the term R'₂/s is minimum when:

A. s=0

B. s=0.5

C. s=1

D. the rotor reaches synchronous speed

Correct Answer: C. s=1

Question 9

Which method normally provides the most efficient wide-range speed control of a squirrel-cage induction motor?

A. Adding fixed stator resistance

B. Using a variable-frequency drive

C. Increasing mechanical load

D. Operating permanently in star connection

Correct Answer: B. Using a variable-frequency drive

Question 10

An overexcited synchronous motor connected to an AC system typically:

A. supplies reactive power to the system

B. runs below synchronous speed

C. requires slip to produce steady torque

D. operates only at unity power factor

Correct Answer: A. supplies reactive power to the system