Transformers and Electrical Energy Conversion

Introduction

A transformer is a static electrical device that transfers alternating-current (AC) energy from one circuit to another through electromagnetic induction. It normally changes voltage and current levels without changing frequency. Transformers are fundamental to electric power systems, industrial automation, battery chargers, welding equipment, instrumentation, and safety isolation.

In a mechatronic system, a transformer may perform several functions simultaneously: it can reduce a mains voltage for control electronics, provide galvanic isolation, match an electrical load to a source, or supply a three-phase motor drive. Correct selection requires more than matching the nominal voltage. Rating, frequency, insulation, regulation, losses, cooling, short-circuit behavior, environmental conditions, and applicable safety standards must also be considered.

Learning Objectives

By the end of this chapter, you should be able to:

1. Explain transformer construction and electromagnetic operation.
2. Relate turns ratio to voltage, current, and impedance transformation.
3. Develop and interpret the practical equivalent circuit.
4. Calculate losses, efficiency, voltage regulation, and basic ratings.
5. Describe open-circuit, short-circuit, polarity, and insulation tests.
6. Select a transformer for isolation, power-supply, and industrial applications.
7. Identify safety, reliability, sustainability, and compliance requirements.

1. Operating Principle

When an alternating current flows in a primary winding, it establishes a time-varying magnetic flux in a ferromagnetic core. The changing flux links the secondary winding and induces an electromotive force (emf). Faraday’s law gives

where e is induced emf in volts, N is the number of turns, and Φ is magnetic flux in webers. The negative sign expresses Lenz’s law: the induced emf opposes the change producing it.

For sinusoidal flux, the rms induced emf is

where E is rms emf in volts, f is frequency in hertz, N is turns, and Φ_max is maximum flux in webers. This expression assumes sinusoidal flux and negligible waveform distortion. It shows why excessive voltage or insufficient frequency can drive the core into saturation.

For an ideal transformer,

where V₁ and V₂ are primary and secondary rms voltages, N₁ and N₂ are winding turns, and a is the turns ratio. With no losses, input power equals output power, so

and therefore

A step-down transformer has a > 1; a step-up transformer has a < 1 when a is defined as N₁/N₂. The frequency remains essentially unchanged.

2. Construction and Materials

A transformer consists principally of a magnetic core, insulated windings, insulation systems, terminals, and a mechanical enclosure. Small units may use bobbins and laminated steel cores. Distribution and power transformers may use tanks, radiators, bushings, tap changers, conservators, and protective relays.

Magnetic core

Power-frequency transformers commonly use thin, insulated laminations of grain-oriented electrical steel. Laminations reduce eddy-current loss by increasing the electrical resistance of circulating current paths. The core may be constructed as a core-type or shell-type arrangement. The core is clamped mechanically but insulated from unintended circulating currents.

Ferrite cores, which have high electrical resistivity, are common in high-frequency transformers used in switch-mode power supplies. Core material must be selected for frequency, flux density, temperature, and power level.

Windings and insulation

Copper or aluminium conductors are wound with enamel, paper, polymer film, or other insulation. The high-voltage winding requires greater insulation clearance to ground and between layers. Low-voltage windings generally carry higher current and therefore require larger conductor cross-sectional area.

Insulation must withstand normal operating voltage, transient overvoltage, thermal stress, mechanical vibration, moisture, and partial discharge. Transformer oil in larger units provides insulation and cooling; dry-type transformers use air or solid insulation.

Cooling

Losses produce heat. Common cooling methods include natural air, forced air, natural oil circulation, and forced oil circulation. The permissible temperature rise depends on insulation class, ambient temperature, loading duration, and cooling method. Overheating accelerates insulation ageing and reduces service life.

Three-phase transformers

Three-phase transformers may be built as one three-phase unit or as three single-phase units. Windings can be connected in star (Y) or delta (Δ). The connection affects line voltage, phase voltage, phase displacement, neutral availability, and harmonic behavior. Vector-group designation is essential when transformers are operated in parallel.

3. Practical Transformer Equivalent Circuit

An ideal transformer does not represent real effects. A practical model includes:

• primary winding resistance R₁ and leakage reactance X₁;
• secondary winding resistance R₂ and leakage reactance X₂;
• core-loss resistance R_c, representing hysteresis and eddy-current losses;
• magnetizing reactance X_m, representing the current required to establish flux.

Leakage flux links one winding more strongly than the other and appears electrically as leakage reactance. The excitation branch, R_c parallel to X_m, is normally placed across the primary in the approximate model.

To refer secondary quantities to the primary side, use

where Z₂ is a secondary impedance and Z₂′ is its equivalent primary-referred impedance. Thus

The approximate equivalent series parameters are

These relations assume consistent rms phasor quantities and a linear, balanced model. They are especially useful for voltage-drop, short-circuit-current, efficiency, and regulation calculations.

4. Transformer Losses

Copper loss

Copper loss is caused by winding resistance:

or, using primary-referred quantities,

The approximation is valid when the current used is the primary-referred current. At high frequency, skin and proximity effects can increase effective AC resistance.

Core loss

Core loss consists mainly of hysteresis and eddy-current loss. A commonly used engineering representation is

where P_h and P_e are hysteresis and eddy-current losses, f is frequency, B_max is maximum flux density, and k_h, k_e, and n depend on core material and construction. These are empirical relationships, not universal exact laws.

Core loss is approximately constant at fixed voltage and frequency because flux is approximately proportional to V/f. It is represented by the current through R_c.

Additional losses

Stray-load loss results from leakage flux inducing currents in tank walls, clamps, and structural parts. Dielectric loss occurs in insulation, especially at high voltage or high frequency. Mechanical vibration can produce acoustic noise and fatigue. In real designs, all losses must be included in thermal calculations.

Total loss may be written as

5. Efficiency and Voltage Regulation

Transformer efficiency is

For a load of apparent power S and power factor cosφ,

where S is in volt-amperes and P_out is in watts. Efficiency depends on load magnitude and power factor. Core loss is approximately constant, while copper loss varies approximately with current squared. Maximum efficiency occurs approximately when

This condition identifies maximum efficiency, not necessarily the rated operating point.

Voltage regulation measures the change in secondary voltage between no-load and load, commonly with primary voltage held constant:

where V_nl is no-load secondary voltage and V_fl is full-load secondary voltage. A useful approximate expression for a lagging power factor is

Here all parameters must be referred to the same side, and φ is the load power-factor angle. For a leading power factor, the reactive term changes sign. The approximation neglects higher-order phasor effects and is most reliable for moderate voltage drops.

Worked Example 1: Efficiency

A transformer delivers 10 kVA at 0.8 lagging power factor. Its core loss is 120 W and copper loss at this load is 180 W. The output power is

Total loss is 120 + 180 = 300 W. Hence

The result is realistic for a properly loaded power transformer.

Worked Example 2: Turns and current ratio

A 2300/230 V transformer supplies 5 kVA. The turns ratio, assuming the 2300 V winding is primary, is

The rated secondary current is

The ideal primary current is

Actual currents are slightly higher because of losses.

6. Ratings and Practical Selection

A transformer is commonly rated in VA or kVA rather than watts because winding heating depends primarily on rms voltage and current, while the load power factor may vary. For a single-phase transformer,

For a balanced three-phase transformer,

where V_L and I_L are line voltage and line current.

Selection should consider:

1. Primary and secondary voltage: Include nominal voltage, tolerance, taps, and expected voltage drop.
2. Frequency: A 50 Hz transformer must not automatically be connected to 60 Hz or vice versa without checking flux and thermal design.
3. kVA rating: Include continuous load, inrush, duty cycle, motor starting, and future expansion.
4. Isolation: An autotransformer does not provide galvanic isolation because primary and secondary share a conductor. Use a two-winding transformer when isolation is required.
5. Regulation: Sensitive control systems may require low regulation or downstream voltage regulation.
6. Insulation and safety: Check working voltage, impulse withstand, creepage, clearance, earthing, and enclosure.
7. Inrush current: Energization can produce a transient current several times rated current, affecting fuses and circuit breakers.
8. Cooling and environment: Consider ambient temperature, altitude, dust, moisture, vibration, hazardous areas, and enclosure rating.
9. Harmonics and nonlinear loads: Rectifiers, variable-frequency drives, and switched-mode supplies can increase heating and neutral currents.
10. System compatibility: Confirm phase, vector group, impedance, short-circuit strength, tap position, and parallel-operation conditions.

A transformer should not be selected merely by adding connected watts. Loads must be converted to apparent power, starting requirements must be evaluated, and diversity assumptions must be documented.

7. Transformer Tests

Open-circuit test

The secondary is open and rated voltage is applied, usually to the low-voltage side for convenience. Measure voltage, current, and input power. Since the current is small, copper loss is small; the measured input power approximates core loss. The test provides estimates of R_c and X_m.

Short-circuit test

One winding is short-circuited and a reduced voltage is applied until rated current flows. The applied voltage is small, so core loss is negligible. Measured input power approximates full-load copper loss. The test yields equivalent resistance, reactance, and impedance.

If V_sc, I_sc, and P_sc are measured,

These equations use rms quantities and consistent side-referred values.

Polarity and phase tests

Polarity testing identifies additive or subtractive voltage relationships and prevents incorrect series or parallel connections. Three-phase units require phase-sequence and vector-group verification before paralleling.

Insulation and condition tests

Insulation resistance, winding-resistance, turns-ratio, dielectric withstand, power-factor or dissipation-factor, and dissolved-gas analysis tests may be used depending on transformer type. Test voltages and procedures must follow manufacturer instructions and applicable standards. High-voltage testing is hazardous and must be performed by competent personnel.

8. Applications and Case Study

Applications

• Isolation transformers: Separate a circuit from the supply for safety, noise reduction, or servicing.
• Control transformers: Supply contactors, relays, sensors, and PLC control circuits.
• Power supplies: Reduce mains voltage before rectification and filtering.
• Instrument transformers: Current transformers and voltage transformers scale quantities for meters and protection systems. A current-transformer secondary must not be left open while energized.
• Industrial systems: Provide distribution, impedance matching, harmonic mitigation, and voltage adaptation for motors and automation equipment.
• High-frequency converters: Ferrite transformers transfer energy in isolated DC–DC converters and switch-mode supplies.

Illustrative case: transformer selection for a robotic cell

Background. An automated robotic cell has a 400 V three-phase supply. Its control cabinet requires 230 V for single-phase controls, sensors, and auxiliary equipment. The connected control loads are 2.8 kW at an average power factor of 0.85, with a short-duration contactor inrush.

Problem. Select a suitable isolation transformer and identify issues beyond steady-state power.

Analysis. The continuous apparent power is

A nominal 3.3 kVA transformer would have little margin for inrush, temperature, future additions, and voltage regulation. A practical preliminary choice is a 5 kVA, 400/230 V, 50 Hz, three-phase-to-single-phase arrangement only if the system architecture and phase loading permit it; otherwise a three-phase transformer with an appropriate 230 V secondary is selected. The designer must verify secondary current, which at 230 V and 5 kVA is approximately 21.7 A, protective-device coordination, short-circuit impedance, enclosure, earthing, and control-circuit fusing.

Solution approach. Confirm that galvanic isolation is required; specify a two-winding transformer. Check that the transformer tolerates the contactor inrush without unacceptable voltage collapse. Select a thermal class and cooling method suitable for the cabinet ambient temperature. Provide primary and secondary overcurrent protection, bond exposed conductive parts, and separate protective earth from functional control wiring as required by the installation standard.

Outcome. The 5 kVA choice provides operating margin, but it is not automatically final. Measurement of actual inrush, manufacturer impedance data, fault-current calculations, and compliance review are required before procurement.

Learning points. Apparent power, inrush, isolation, thermal environment, protection, and standards all influence selection. A larger transformer is not always better: it may increase inrush, fault current, cost, and no-load loss.

9. Limitations, Safety, and Contemporary Practice

The simple equivalent circuit assumes linear magnetic behavior, sinusoidal operation, balanced conditions, constant parameters, and negligible temperature variation. Saturation, hysteresis, frequency-dependent resistance, stray capacitance, harmonics, and transient phenomena require more detailed models.

Electrical safety requires de-energization, lockout/tagout, verification of absence of voltage, suitable personal protective equipment, and controlled high-voltage testing. Transformers must be installed according to applicable electrical codes and product standards. Protection against overload, short circuit, fire, electric shock, and oil leakage is both an engineering and governance responsibility.

Modern developments include amorphous-metal cores with reduced no-load loss, high-frequency planar transformers, digital thermal monitoring, fiber-optic temperature sensing, and online condition monitoring. Data analytics and machine-learning methods can help identify abnormal temperature, dissolved-gas, vibration, or load patterns, but they do not replace verified sensors, engineering judgment, or protective relays. Sustainable selection considers lifetime energy loss, recyclability, material use, noise, maintenance, and end-of-life disposal.

Chapter Summary

A transformer transfers AC energy by mutual electromagnetic induction and normally preserves frequency while changing voltage and current. The turns ratio determines ideal voltage and current relationships, while the practical equivalent circuit accounts for winding resistance, leakage reactance, core loss, and magnetizing current. Copper loss varies with current squared; core loss is approximately constant at fixed voltage and frequency. Efficiency is output power divided by input power, and maximum efficiency occurs approximately when copper and core losses are equal. Voltage regulation describes load-related secondary-voltage change and depends on equivalent resistance, reactance, and power factor. Open-circuit and short-circuit tests provide practical circuit parameters. Professional selection must consider kVA, voltage, frequency, isolation, regulation, inrush, cooling, harmonics, protection, standards, and life-cycle performance.

Key Terms

Transformer: Static device that transfers AC energy by electromagnetic induction.

Mutual induction: Induction of voltage in one winding by changing flux produced by another.

Turns ratio: Ratio of primary turns to secondary turns.

Core: Magnetic path that concentrates transformer flux.

Lamination: Thin insulated magnetic sheet used to reduce eddy-current loss.

Leakage flux: Flux linking one winding but not the other.

Magnetizing reactance: Equivalent reactance associated with establishing core flux.

Core loss: Hysteresis and eddy-current loss in the magnetic core.

Copper loss: Resistive winding loss caused by current.

Voltage regulation: Percentage change between no-load and full-load voltage.

Efficiency: Ratio of output power to input power.

Apparent power: Product of rms voltage and current, measured in VA.

Isolation transformer: Two-winding transformer providing galvanic separation.

Autotransformer: Transformer with electrically connected portions of one winding.

Inrush current: Temporary high current at energization.

Open-circuit test: Test used mainly to determine core-loss and excitation parameters.

Short-circuit test: Test used mainly to determine equivalent impedance and copper loss.

Vector group: Designation of three-phase winding connection and phase displacement.

Impedance: Combined opposition of resistance and reactance to AC current.

Review Questions

Short Answer Questions

1. Why are transformer cores laminated?
2. What is the physical meaning of the turns ratio?
3. Distinguish copper loss from core loss.
4. Why are transformers rated in VA or kVA?
5. What information is obtained from a short-circuit test?

Long Answer Questions

1. Explain transformer construction, including core, windings, insulation, and cooling.
2. Develop the practical equivalent circuit and explain the purpose of each element.
3. Derive the ideal voltage and current relationships from conservation of power.
4. Discuss the causes of transformer losses and their effect on efficiency and temperature.
5. Explain the procedure, assumptions, and uses of open-circuit and short-circuit tests.

Analytical / Application Questions

1. A 15 kVA, 2400/240 V transformer supplies a load at 0.75 lagging power factor. Calculate the secondary current and output real power.
2. A transformer has 200 W core loss and 450 W copper loss at rated load. Determine its efficiency when delivering 20 kVA at 0.9 power factor.
3. Design a selection checklist for a transformer supplying a mechatronic control cabinet with nonlinear loads, contactor inrush, and a requirement for galvanic isolation.

Multiple Choice Questions with Answers

Question 1

A transformer fundamentally operates through:

A. Electromagnetic induction

B. Electrostatic attraction

C. Chemical reaction

D. Thermionic emission

Correct Answer: A. Electromagnetic induction

Question 2

A 2300/230 V transformer has a primary-to-secondary turns ratio of:

A. 0.01

B. 0.1

C. 10

D. 100

Correct Answer: C. 10

Question 3

Which loss is approximately constant when voltage and frequency remain fixed?

A. Copper loss

B. Core loss

C. Load-dependent stray loss only

D. Contact loss

Correct Answer: B. Core loss

Question 4

A transformer delivers 8 kW and has total losses of 0.5 kW. Its efficiency is approximately:

A. 88.9%

B. 93.8%

C. 94.1%

D. 106.3%

Correct Answer: C. 94.1%

Question 5

The main purpose of an open-circuit test is to estimate:

A. Core-loss and excitation parameters

B. Full-load copper loss only

C. Mechanical shaft power

D. Insulation thickness only

Correct Answer: A. Core-loss and excitation parameters

Question 6

If a secondary impedance is referred to the primary side, it is multiplied by:

A. a

B. 1/a

C. a²

D. √a

Correct Answer: C. a²

Question 7

For a balanced three-phase transformer, apparent power is calculated using:

A. VI/√3

B. √3 V_L I_L

C. V_L/I_L

D. 3V_L²I_L

Correct Answer: B. √3 V_L I_L

Question 8

Which transformer provides galvanic isolation?

A. Autotransformer

B. Two-winding transformer

C. Current-limiting resistor

D. Voltage divider

Correct Answer: B. Two-winding transformer

Question 9

Maximum transformer efficiency occurs approximately when:

A. Copper loss equals core loss

B. Voltage equals zero

C. Power factor equals zero

D. Copper loss is zero at full load

Correct Answer: A. Copper loss equals core loss

Question 10

A major hazard associated with leaving an energized current-transformer secondary open is:

A. Zero primary current

B. Excessive secondary voltage

C. Permanent reduction in frequency

D. Core temperature becoming exactly zero

Correct Answer: B. Excessive secondary voltage