Transformer Losses and Efficiency — Enrichment

Enrichment — beyond the explicitly named H2 Physics 9749 transformer outcome. The syllabus requires the simple iron-core principle and ideal ratio. This branch preserves useful practical-transformer material from the teacher anchor; treat it as extension unless your teacher specifies otherwise.

Main topic: Transformers

Overview

Real transformers depart from the ideal model because their windings and cores dissipate energy and their magnetic coupling is imperfect. Efficiency compares useful real output power with real input power.

Core Ideas

• Winding resistance, eddy currents and hysteresis dissipate real power as heat.
• Flux leakage is an imperfect-coupling and voltage-regulation effect, not a direct heat-loss term.
• Each non-ideal effect has a distinct physical cause and reduction method.
• Efficiency is a power ratio, not a voltage ratio.

1. Why a real transformer is not ideal

An ideal transformer transfers all input power to the secondary:

$$P_{\mathrm{out}}=P_{\mathrm{in}}.$$

A real transformer has

$$P_{\mathrm{out}}<P_{\mathrm{in}}$$

because some input energy becomes thermal energy in the windings or core. A real transformer also has imperfect magnetic coupling: some primary flux does not link the secondary. Flux leakage affects the secondary voltage and voltage regulation, but it is not by itself a direct thermal-energy dissipation mechanism.

2. Four main departures from the ideal model

Figure: Read each card as a causal chain. Copper, eddy-current and hysteresis effects dissipate real power as heat. Flux leakage is shown separately as imperfect coupling, not as a fourth heat-loss mechanism. Lamination primarily addresses eddy currents; it is not a general cure for every non-ideal effect.

Copper loss in the windings

Real wire has resistance. The rms winding current produces heating at a rate

$$P_{\mathrm{copper}}=I_{\mathrm{rms}}^{2}R.$$

Thicker, low-resistivity copper wire reduces $R$ and therefore reduces this heating for a given current.

Eddy-current loss in the core

The changing core flux induces currents within the conducting core itself. These eddy currents dissipate energy as heat.

Thin laminations are electrically insulated from one another. They interrupt large circulating current paths, increasing their resistance and reducing eddy-current heating.

Hysteresis loss in the core

The core is magnetised first in one direction and then the other during each AC cycle. Some energy supplied during magnetisation is not recovered during demagnetisation and becomes thermal energy.

A soft magnetic material with a narrow hysteresis loop reduces the energy lost per cycle.

Flux leakage: imperfect coupling

In a real transformer, not all flux produced by the primary links every secondary turn. Leakage flux therefore does not contribute fully to the desired secondary emf. It is best treated as a coupling and voltage-regulation effect rather than as energy that is directly converted to heat.

Suitable core geometry and closely coupled windings reduce leakage.

3. Efficiency

Transformer efficiency is

$$\eta =\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}.$$

As a percentage,

$$\eta =\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times 100\%.$$

If both input and output are treated as unity-power-factor quantities in a simplified calculation, then

$$\eta =\frac{V_s{I}_s}{V_p{I}_p}\times 100\%.$$

The missing power is

$$P_{\mathrm{loss}}=P_{\mathrm{in}}-P_{\mathrm{out}}.$$

Efficiency below $100\%$ does not mean charge is lost. It means some input energy per unit time is not delivered as useful output power.

For a general AC circuit, apparent power $V_{\mathrm{rms}}{I}_{\mathrm{rms}}$ need not equal real power; a power-factor term may be required. A wattmeter directly measures real power.

4. Worked example

A wattmeter measures the real input power of a transformer as $120\text{W}$. Its secondary supplies $18\text{V rms}$ and $5.5\text{A rms}$ to a resistive load.

Input power is the wattmeter reading:

$$P_{\mathrm{in}}=120\text{W}.$$

Output power:

$$P_{\mathrm{out}}=18(5.5)=99\text{W}.$$

Efficiency:

$$\eta =\frac{99}{120}\times 100\%=82.5\%.$$

Total power loss:

$$P_{\mathrm{loss}}=120-99=21\text{W}.$$

This calculation finds the total real-power loss. It does not by itself tell us how the $21\text{W}$ is divided among copper, eddy-current and hysteresis heating. Flux leakage may affect the operating currents and voltage regulation, but it is not itself a direct heat term to add to this power balance.

5. Misconception checks

• “Laminating the core reduces hysteresis.” No. Lamination primarily reduces eddy currents; low-hysteresis material reduces hysteresis loss.
• “Thicker wire removes all transformer loss.” No. It reduces winding resistance but does not remove core loss or leakage.
• “Flux leakage is heating in the core.” No. It is imperfect magnetic coupling; not all primary flux links the secondary.
• “Efficiency is the voltage ratio.” No. Efficiency is an output-power to input-power ratio.

Exam Relevance

This material is enrichment under the explicit 9749 transformer wording. If it appears in a school question or supplied context, state the physical mechanism precisely, match each reduction method to the correct effect, and use real power—not apparent power without justification—in efficiency calculations.

Summary table

Departure from ideal model	Main effect	Reduction
winding resistance	$I^{2}R$ heating in coils	thick, low-resistance copper
eddy currents	heating in core	insulated laminations
hysteresis	heating from repeated magnetisation	low-hysteresis soft magnetic material
flux leakage	imperfect coupling and voltage regulation	suitable core and close winding arrangement

Links

• Transformers
• Electromagnetic Induction
• RMS and AC Power