---
title: "Magnetic hysteresis and energy per cycle"
subject: "Electrical Machines"
chapter: 1
key: "semester_4/electrical_machines:hysteresis-energy-loop-area"
source: https://www.bctengineer.com/q/semester_4/electrical_machines/hysteresis-energy-loop-area
site: BCT Engineering
---

# Magnetic hysteresis and energy per cycle

Electrical Machines · Chapter 1: Magnetic Circuits and Induction

- Asked in: 2065 Chaitra (old course), 4+4 marks
- Video: [The Hysteresis loop explained](https://www.youtube.com/watch?v=ZyVSMfSgOq0) — The Electric Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machines/hysteresis-energy-loop-area
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-1

## Question

Explain the magnetic hysteresis and show that energy spent per cycle per unit volume is equal to the area of the hysteresis loop.

## Short answer

Hysteresis is the lag of B behind H due to domain friction. In one cycle, energy supplied per unit volume is ∮H dB, which equals the area enclosed by the B-H loop. This area represents energy lost as heat per cycle per unit volume.

## Answer

**Magnetic hysteresis** is the lagging of the flux density $B$ behind the magnetising force $H$ in a ferromagnetic material. When $H$ is increased, reduced, reversed and brought back, the $B$–$H$ curve does not retrace itself but forms a closed loop.

```text
      B
      |     ____ saturation
      |   /
   Br |--/--.
      | /    \
 -----+-+-----+------ H
     -Hc     Hc
      |  loop area = energy lost
```

**Explanation:** the material is made of small magnetic domains. When $H$ rises they rotate and align, and $B$ increases up to saturation. When $H$ falls to zero, some domains stay aligned, so a residual flux density $B_r$ (retentivity) remains. A reverse field $-H_c$ (coercive force) is needed to bring $B$ to zero. Overcoming this internal friction between domains costs energy, which appears as heat.

#### Energy per cycle equals loop area

Take a core of cross-section $A$, mean length $l$ and $N$ turns carrying current $i$.

$$
\begin{aligned}
e &= N\frac{d\phi}{dt} = NA\frac{dB}{dt} \\
H &= \frac{Ni}{l} \Rightarrow i = \frac{Hl}{N}
\end{aligned}
$$

Energy supplied by the source in time $dt$ is

$$
\begin{aligned}
dW &= e\,i\,dt = NA\frac{dB}{dt}\cdot\frac{Hl}{N}\,dt \\
&= (Al)\,H\,dB
\end{aligned}
$$

Here $Al = V$ is the volume of the core. Energy per unit volume is therefore

$$
\frac{dW}{V} = H\,dB
$$

For one full cycle,

$$
w = \oint H\,dB = \text{area of the } B\text{–}H\text{ loop}
$$

In the rising part of the cycle, part of this energy is stored in the field and is returned when the field falls. The rest is not returned and is lost as heat, which is the area enclosed by the loop. So the energy lost per cycle per unit volume equals the loop area (in J/m$^3$). For frequency $f$ and volume $V$, $P_h = f V \times (\text{loop area})$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is magnetic hysteresis?
   A. B lags behind H, so the B-H curve forms a closed loop
   B. B always equals mu H
   C. H lags behind the current in a coil
   D. Loss of magnetism at high temperature
   Answer: A. Hysteresis is the lagging of flux density behind the magnetising force. (Key point: Definition of hysteresis)

2. Why does hysteresis produce heat?
   A. Domain walls are cooled by air
   B. Eddy currents flow in the domains
   C. Residual flux generates emf
   D. Work is done against internal friction when domains realign
   Answer: D. Reversing the domains against internal friction costs energy that becomes heat. (Key point: Origin of loss)

3. In the derivation, the energy supplied in time dt is dW = (Al) H dB. What is dW/V?
   A. B dH
   B. H dB
   C. H B
   D. H/B
   Answer: B. Dividing by volume V = Al gives H dB; integrating over a cycle gives the loop area. (Key point: Energy per unit volume)

4. Of the energy supplied in a cycle, which part is lost as heat?
   A. All of it
   B. The part stored in the field
   C. The part that is not returned when the field falls, equal to the loop area
   D. None, it is all returned
   Answer: C. Stored energy is returned as the field falls; the unreturned part, the loop area, is lost. (Key point: Return of energy)

## Related in Electrical Machines

- [Hysteresis loop area and energy loss](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ahysteresis-loop-area-energy-loss) — Electrical Machines I, chapter 1, asked 2×
- [Hysteresis loop energy, hard and soft magnetic materials](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ahysteresis-area-energy-hard-soft) — Electrical Machines I, chapter 1
- [Significance of hysteresis loop, volume dependence](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Ahysteresis-loop-significance) — Electrical Machines, chapter 1
- [Retentivity, coercivity and hysteresis energy](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Aretentivity-coercivity-hysteresis-energy) — Electrical Machines I, chapter 1

## Related in other subjects

- [Retentivity, coercivity, energy per cycle](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Aretentivity-coercivity-energy-loop) — Electrical Machine, chapter 1
- [Progressive wave energy per unit volume](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aprogressive-wave-energy-per-volume) — Engineering Physics, chapter 2
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- [Silver: states per unit volume vs atoms](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asilver-states-per-volume) — Electrical Engineering Material, chapter 2, asked 2×