Magnetically Coupled Circuits and Transformers

Introduction

Magnetically coupled circuits transfer energy or information through a shared magnetic field. They appear in power transformers, switch-mode converters, communication interfaces, induction sensors, wireless chargers and electrical machines. Their analysis extends ordinary inductance by accounting for mutual inductance, through which a changing current in one coil induces voltage in another.

Transformers are the most important application of magnetic coupling. They provide voltage and current conversion, impedance matching and galvanic isolation. Real transformers also exhibit winding resistance, leakage flux, core loss, finite magnetizing current, heating and insulation limitations. Correct engineering analysis must therefore distinguish between ideal models and practical equivalent circuits.

Throughout this chapter, the transformer turns ratio is defined consistently as

where N₁ is the primary turns count and N₂ is the secondary turns count. Thus, a>1 denotes a step-down transformer when the primary is supplied.

Learning Objectives

After studying this chapter, students should be able to:

• calculate mutual inductance and coupling coefficient;
• apply the dot convention to determine induced-voltage polarity;
• formulate time-domain and phasor equations for coupled coils;
• analyze ideal transformers using a consistent turns ratio;
• reflect impedances, voltages and currents between transformer sides;
• construct and use practical transformer equivalent circuits;
• estimate voltage regulation, losses and efficiency;
• evaluate transformer applications involving isolation and impedance matching; and
• identify safety, thermal and regulatory considerations.

1. Self-Inductance and Mutual Inductance

1.1 Self-inductance

For a linear inductor, flux linkage \lambda is proportional to current:

Here, L is inductance in henries (H), \lambda is flux linkage in weber-turns and i is current in amperes. Faraday’s law gives the passive-sign-convention voltage:

This relationship assumes constant inductance. In a nonlinear magnetic core, \lambda is not proportional to current, and the incremental slope d\lambda/di may vary with operating point.

1.2 Mutual inductance

Consider two coils positioned so that part of the flux produced by either coil links the other. If current i₁ produces flux linkage \lambda₂₁ in coil 2, the mutual inductance is

Similarly, M₁₂=\lambda₁₂/i₂. In a linear, passive and reciprocal magnetic medium,

The unit of mutual inductance is the henry. Reciprocity may not hold in active, nonlinear, time-varying or nonreciprocal magnetic systems.

1.3 Coupling coefficient

Not all flux produced by one winding necessarily links the other. The coupling coefficient is

where L₁ and L₂ are self-inductances. For passive coupled inductors,

Consequently,

The sign associated with mutual effects is normally represented by the dot convention rather than by assigning a negative numerical value to M. Tight coupling has k near unity; loose coupling has a much smaller value.

2. Dot Convention and Coupled-Coil Equations

The dot convention identifies the relative instantaneous polarity of mutually induced voltages. If current enters the dotted terminal of one coil, the mutually induced voltage makes the dotted terminal of the other coil positive relative to its undotted terminal.

Figure: Reference currents entering dotted terminals produce positive mutual terms for the indicated voltage references.

When both reference currents enter dotted terminals and both voltages are measured positive from dotted to undotted terminals, the equations are

If one reference current enters a dotted terminal while the other enters an undotted terminal, the mutual terms are negative. A compact representation is

where s=+1 for aiding reference orientations and s=-1 for opposing orientations.

For sinusoidal steady state at angular frequency \omega rad/s, differentiation becomes multiplication by j\omega:

2.1 Stored magnetic energy

For linear coupled inductors using aiding references, stored energy is

For opposing references, the final term is negative. Requiring nonnegative energy for all current combinations gives M²≤L₁L₂, which explains the physical bound on k.

Worked Example 1: Mutual inductance

Two coils have L₁=80\text{ mH}, L₂=45\text{ mH} and k=0.75. Find M. If di₁/dt=30\text{ A/s} while i₂ is constant, determine the magnitude of the induced voltage in coil 2.

Therefore,

Its polarity is determined by the dot convention.

3. The Ideal Transformer

An ideal transformer assumes:

• zero winding resistance;
• no leakage flux, so coupling is perfect;
• infinite core permeability and negligible magnetizing current;
• no hysteresis or eddy-current loss; and
• no insulation or frequency limitation.

These assumptions are approximations, but they provide the foundation for transformer analysis.

3.1 Voltage ratio

If the same time-varying core flux \phi links every turn, Faraday’s law gives

Hence, for consistent terminal polarities,

Thus,

The ratio applies to instantaneous voltages and RMS phasors under ideal operation.

3.2 Current and power ratios

An ideal transformer absorbs no real or reactive power. Using current magnitudes,

Combining this result with the voltage ratio gives

or

Signed phasor equations may include a minus sign depending on whether both current references are defined into dotted terminals. The magnitude relations above avoid confusing reference-direction signs with physical conversion ratios.

3.3 Impedance reflection

Let a load Z_L be connected to the secondary. The impedance observed at the primary is

where Z_L' is the load referred to the primary. Conversely, a primary impedance referred to the secondary is

Resistance and reactance are each multiplied by the same squared ratio. Impedance angle and power factor are unchanged by an ideal transformer.

Worked Example 2: Ideal step-down transformer

An ideal transformer has N₁=1200 turns and N₂=120 turns. A 240\text{ V RMS} source supplies the primary, and an 8\,Ω resistive load is connected to the secondary.

The turns ratio is

The secondary voltage and current are

The primary current magnitude is

The load referred to the primary is

Indeed, 240/800=0.30\text{ A}, confirming the result.

3.4 Impedance matching

Suppose a source delivers maximum power to a resistive load when the input resistance is R_S. If the actual load is R_L, an ideal transformer can provide matching when

Therefore,

For complex impedances, maximum average power requires conjugate matching. A transformer changes impedance magnitude by a² but does not independently change impedance angle, so additional reactive compensation may be needed.

4. Practical Transformer Equivalent Circuit

An actual transformer contains several nonideal elements.

When secondary quantities are referred to the primary,

The total series parameters referred to the primary are

In a commonly used approximate equivalent circuit, the excitation branch is placed directly across the supply and the series impedance feeds the referred load.

Figure: Approximate equivalent circuit referred to the primary. The separate R_c and jX_m branches are in parallel across the supply; they are not short-circuited.

The excitation current is approximately

The core-loss component I_c is in phase with voltage, while the ideal magnetizing component I_m lags voltage by 90°. Therefore, I₀ is not generally equal to their arithmetic scalar sum; phasor addition is required.

4.1 Parameter determination

A no-load test is performed with one winding open. Its measurements estimate core loss and excitation parameters because winding current and copper loss are relatively small.

A short-circuit test is performed with one winding shorted while a reduced voltage produces rated current. The reduced core flux makes core loss relatively small, allowing estimation of R_eq and X_eq. These interpretations are approximations and depend on rated-frequency testing and appropriate temperature correction.

5. Voltage Regulation, Losses and Efficiency

5.1 Voltage regulation

Transformer voltage regulation describes the change in secondary terminal voltage from no-load to load at a specified primary voltage:

For a lagging-power-factor load, an approximate expression using secondary-referred series parameters is

Here, \phi is the load power-factor angle, and the formula assumes a relatively small series voltage drop. For a leading load, the reactance term changes sign; regulation can become negative.

5.2 Losses

The principal transformer losses are:

• Copper loss: P_cu=I₁²R₁+I₂²R₂. It varies approximately with current squared.
• Core loss: hysteresis and eddy-current loss, commonly represented by V²/R_c at a specified voltage and frequency.
• Stray-load loss: eddy-current and leakage-field effects in structural and conductor components.
• Dielectric loss: energy dissipated in insulation, especially important at high voltage or high frequency.

The simplified core-loss model is not universal: hysteresis and eddy-current losses have different frequency and flux-density dependencies.

5.3 Efficiency

Efficiency is

For the simplified loss model,

At a fixed voltage and power factor, maximum efficiency occurs approximately when variable copper loss equals constant core loss:

This condition maximizes efficiency with respect to load current under the stated simplified assumptions; it is not a universal optimum when voltage, frequency, temperature or auxiliary losses vary.

Worked Example 3: Regulation and efficiency

A 5 kVA, 240/120 V transformer has primary-referred parameters R_eq=0.18\,Ω, X_eq=0.30\,Ω and R_c=3000\,Ω. It supplies 4 kW at 0.8 lagging power factor and approximately 120 V.

The turns ratio and apparent load power are

Thus,

Secondary-referred parameters are

With \cos\phi=0.8 and \sin\phi=0.6,

The primary-referred load current is 41.67/2=20.83\text{ A}. Therefore,

At approximately 240 V,

Neglecting other losses,

6. Isolation, Matching and Safety

6.1 Galvanic isolation

An isolation transformer transfers energy magnetically without an intentional conductive path between primary and secondary. It can interrupt ground loops, support medical or laboratory protection systems and establish a separately derived circuit.

Isolation does not make the secondary inherently harmless. Contact with both secondary conductors can still produce a dangerous current, and parasitic capacitance permits high-frequency common-mode current. Insulation breakdown, incorrect bonding or excessive transient voltage can defeat isolation.

6.2 Autotransformers

An autotransformer uses a common winding section shared by input and output. It is often smaller and more efficient than a two-winding transformer of similar rating, but it does not provide galvanic isolation. It must not replace an isolation transformer where separation is required for safety.

6.3 Protection and professional practice

Transformer installations may require:

• primary and secondary overcurrent protection;
• protective earthing and correctly designed neutral bonding;
• insulation coordination and adequate creepage and clearance;
• thermal protection and ventilation;
• inrush-current accommodation;
• protection against internal faults and external short circuits; and
• compliance with applicable electrical, product and fire-safety standards.

Relevant requirements depend on jurisdiction and application. Examples include IEC transformer standards, national wiring rules and product-specific medical, industrial or utility requirements. Engineers must use the current legally adopted standards rather than relying solely on generic design formulas.

7. Illustrative Application Case: Isolated Control Supply

Background

This fictional case concerns an industrial control panel requiring a 24 V isolated supply from a 230 V source. The load consumes 360 VA at 0.75 lagging power factor. The transformer has an estimated full-load efficiency of 94%. Its magnetizing branch consumes approximately 35 var, and leakage reactive demand is neglected for an initial system-level estimate.

Problem

The engineer must select the turns ratio, determine the input power factor, evaluate source current and decide whether reactive compensation is useful. Safety requires genuine galvanic isolation.

Analysis

The required turns ratio is

The load real and reactive powers are

The load impedance at 24 V is

Equivalently, this follows from R=P_LV₂²/|S|² and X=Q_LV₂²/|S|². Referred to the primary,

Using the stated efficiency, estimated input real power is

Without external compensation, estimated input reactive power is

Thus,

and the source current is approximately

A hypothetical 0.95 or higher input power factor cannot arise from this uncompensated system merely because a transformer is present. Suppose an explicitly installed capacitor supplies 200 var at the input. The source then supplies approximately 273.1-200=73.1\text{ var}, giving

The source current falls to approximately 296.4/230=1.29\text{ A}. Capacitor selection must also consider tolerance, harmonics, switching transients and resonance.

Approach and Outcome

A two-winding 230/24 V transformer is selected rather than an autotransformer because isolation is mandatory. Its VA rating must exceed the load requirement after allowance for temperature, enclosure conditions, duty cycle and future growth. Primary and secondary fuses, protective earthing, insulation clearances and thermal controls are included. Reactive compensation is optional and, if used, is clearly external to the transformer.

Learning Points

• Turns ratio controls voltage conversion, while impedance changes by the square of that ratio.
• A lagging load plus magnetizing current generally worsens source power factor.
• Real losses increase input real power; they do not eliminate reactive demand.
• High input power factor requires low reactive demand or explicit compensation.
• Isolation, grounding and overcurrent protection are separate design responsibilities.

8. Limitations and Contemporary Developments

Equivalent circuits are frequency- and operating-point-dependent. Core saturation makes magnetizing inductance nonlinear, winding resistance changes with temperature, skin and proximity effects increase AC resistance, and interwinding capacitance becomes significant at high frequency. Fast switching also produces common-mode currents and electromagnetic interference that low-frequency models do not fully predict.

Modern developments include planar magnetics, high-frequency ferrite transformers, resonant converters, solid-state transformers and improved nanocrystalline or amorphous cores. Digital monitoring systems estimate winding temperature, dissolved gases, partial discharge and insulation aging. Machine-learning tools may assist fault classification, but their outputs require validated sensors, representative data and engineering review; they do not replace protection systems or compliance testing.

Best practice is to begin with the simplest model that captures the required behavior, verify its assumptions, compare calculations with test or manufacturer data and add parasitic or nonlinear effects when frequency, transient response, safety or thermal design demands them.

Chapter Summary

Magnetically coupled circuits are governed by self-inductance, mutual inductance and the polarity information encoded by the dot convention. The coupling coefficient satisfies |k|≤1, and magnetic-energy constraints require M²≤L₁L₂.

With the consistent definition a=N₁/N₂, an ideal transformer obeys V₁/V₂=a, |I₁|/|I₂|=1/a and Z_L'=a²Z_L. Practical transformers add winding resistance, leakage reactance, core loss and magnetizing reactance. These parameters permit calculation of voltage regulation, copper loss, core loss and efficiency.

Transformers support voltage conversion, impedance matching and galvanic isolation, but isolation does not remove all shock or transient hazards. Sound engineering requires suitable protection, grounding, insulation coordination, thermal management and compliance with applicable standards. At high frequency or near saturation, more detailed models are necessary.

Key Terms

Mutual inductance: The property by which changing current in one coil induces voltage in another.

Self-inductance: The relationship between a coil’s current and its own flux linkage.

Coupling coefficient: The dimensionless measure k=M/√(L₁L₂) of magnetic coupling.

Dot convention: A notation indicating relative induced-voltage polarity in coupled windings.

Flux linkage: Magnetic flux linked by all winding turns, measured in weber-turns.

Ideal transformer: A lossless, perfectly coupled transformer model.

Turns ratio: The ratio a=N₁/N₂ used for voltage, current and impedance transformation.

Reflected impedance: An impedance represented on the opposite transformer side through the squared turns ratio.

Leakage flux: Flux produced by one winding that does not link the other winding.

Leakage reactance: Equivalent reactance associated with leakage flux.

Magnetizing current: Current required to establish core flux.

Core loss: Magnetic-core power loss caused primarily by hysteresis and eddy currents.

Copper loss: Resistive winding loss proportional approximately to current squared.

Voltage regulation: Change in output voltage between no-load and load at specified input conditions.

Efficiency: Ratio of transformer output real power to input real power.

Galvanic isolation: Separation with no intentional direct conductive path.

Autotransformer: A transformer with a winding section common to input and output.

Inrush current: Large transient current that can occur when a transformer is energized.

Saturation: A nonlinear core condition in which large increases in magnetizing force produce limited additional flux.

Review Questions

Short Answer Questions

1. Define mutual inductance and state its SI unit.
2. What does the dot convention indicate?
3. State the voltage, current and impedance relations for a=N₁/N₂.
4. Distinguish leakage reactance from magnetizing reactance.
5. Why does an isolation transformer not eliminate every shock hazard?

Long Answer Questions

1. Derive the time-domain and phasor equations for two linearly coupled inductors, including dot-polarity effects.
2. Derive the ideal-transformer impedance-reflection formula from voltage and current ratios.
3. Explain every element of the practical transformer equivalent circuit and its physical origin.
4. Discuss transformer losses, voltage regulation and the condition for maximum efficiency under a simplified loss model.
5. Compare two-winding transformers and autotransformers with respect to cost, efficiency, isolation and safety.

Analytical / Application Questions

1. Two coils have L₁=50\text{ mH}, L₂=200\text{ mH} and k=0.6. Calculate their mutual inductance and determine induced voltage for a specified current slope of 100 A/s.
2. A 480/120 V ideal transformer supplies a 12+j9\,Ω load. Determine the primary-referred impedance and primary current when the primary voltage is 480 V RMS.
3. Develop a preliminary equivalent-circuit and protection analysis for an isolated transformer supplying a nonlinear electronic load. Identify which ideal-model assumptions are inadequate.

Multiple Choice Questions with Answers

Question 1

Two coils have L₁=40\text{ mH}, L₂=90\text{ mH} and k=0.5. What is their mutual inductance?

A. 15 mH

B. 30 mH

C. 45 mH

D. 60 mH

Correct Answer: B. 30 mH

Question 2

Both reference currents enter the dotted terminals, and both voltages are referenced positive from dotted to undotted terminals. What is the sign of the mutual terms?

A. Positive in both equations

B. Negative in both equations

C. Positive only in the first equation

D. Zero in both equations

Correct Answer: A. Positive in both equations

Question 3

An ideal transformer has N₁/N₂=5. What impedance is observed at the primary when a 20\,Ω load is connected to the secondary?

A. 0.8 Ω

B. 4 Ω

C. 100 Ω

D. 500 Ω

Correct Answer: D. 500 Ω

Question 4

An ideal transformer has a=N₁/N₂=4 and a primary voltage of 240 V RMS. What is the secondary voltage?

A. 60 V RMS

B. 120 V RMS

C. 480 V RMS

D. 960 V RMS

Correct Answer: A. 60 V RMS

Question 5

Which condition must hold for passive, linear coupled inductors?

A. |M|>√(L₁L₂)

B. |M|≤√(L₁L₂)

C. M=L₁+L₂

D. k>1

Correct Answer: B. |M|≤√(L₁L₂)

Question 6

A transformer supplies 900 W while its total losses are 100 W. What is its efficiency?

A. 81%

B. 90%

C. 91%

D. 100%

Correct Answer: B. 90%

Question 7

Which transformer type provides no galvanic isolation because input and output share part of a winding?

A. Two-winding isolation transformer

B. Current transformer

C. Autotransformer

D. Pulse transformer with separate windings

Correct Answer: C. Autotransformer

Question 8

In the illustrative case, why could the source power factor increase to approximately 0.969?

A. The turns ratio eliminated all reactive power

B. An external capacitor supplied 200 var

C. Transformer losses became negative

D. The load power factor automatically became unity

Correct Answer: B. An external capacitor supplied 200 var

Question 9

Which equivalent-circuit element principally represents hysteresis and eddy-current power loss?

A. R_c

B. X_m

C. X_eq

D. The ideal turns ratio

Correct Answer: A. R_c

Question 10

A transformer supplies 20 A at 0.8 lagging power factor. Its secondary-referred parameters are R_eq,2=0.10\,Ω and X_eq,2=0.20\,Ω. If V₂=100\text{ V} and \sin\phi=0.6, what is the approximate voltage regulation?

A. 1.6%

B. 2.0%

C. 4.0%

D. 8.0%

Correct Answer: C. 4.0%