---
title: "Hysteresis loop area and energy loss"
subject: "Electrical Machines I"
chapter: 1
key: "semester_4/electrical_machines_i:hysteresis-loop-area-energy-loss"
source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/hysteresis-loop-area-energy-loss
site: BCT Engineering
---

# Hysteresis loop area and energy loss

Electrical Machines I · Chapter 1: Magnetic Circuits and Induction

- Asked in: 2078 Chaitra, 8 marks; 2070 Magh, 8 marks (asked 2 times)
- Video: [Magnetic circuits 11 Hysteresis loss [Electric Machinery]](https://www.youtube.com/watch?v=uATmtZv_raM) — Ognjen Grozdanovic
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machines_i/hysteresis-loop-area-energy-loss
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-1

## Question

Explain the hysteresis loop of a magnetic material used in electrical machine. Prove that the area of the loop is proportional to the energy loss per cycle per unit volume.

## Short answer

The hysteresis loop is a B–H curve where B lags H, enclosing area representing energy lost as heat per cycle per unit volume. The energy supplied per unit volume in a small change is dw = H dB; integrating over a full cycle gives w = ∮ H dB, which equals the loop area, proving area is proportional to hysteresis energy loss.

## Answer

The **hysteresis loop** is the closed B–H curve traced by a ferromagnetic material when the magnetising force H is taken through one complete cycle (+Hmax → 0 → −Hmax → 0 → +Hmax). Because B lags behind H, the curve for decreasing H does not retrace the rising curve, and the enclosed area represents energy lost as heat in every cycle.

#### The hysteresis loop

```text
               B
               |      a (saturation)
           b   |  ____*
            *--|-'   /
           /   |    /
     c    /    |   /
  ---*---/-----+--/----*--- H
        /      |  /    f
       /   ____|_*
  d  *'        |  e
               |
```

- **o–a:** an unmagnetised sample is magnetised along the initial curve to saturation (point a).
- **a–b:** when H is reduced to zero, B does not fall to zero. The value **ob is the residual flux density (retentivity)**.
- **b–c:** a reverse H is needed to make B zero. **oc is the coercive force (coercivity)**.
- **c–d:** further reverse H gives reverse saturation at d.
- **d–e–f–a:** reducing H to zero gives the negative residual flux density oe, and a positive H (of) is needed to bring B to zero; the loop closes at a.

Reason: the magnetic domains line up with H but do not all turn back when H is removed. Turning them back needs extra energy, which appears as heat.

#### Proof: loop area ∝ energy loss per cycle per unit volume

Take a ring of mean length $l$, area $A$, wound with $N$ turns carrying current $i$.

$$
H = \frac{Ni}{l} \Rightarrow i = \frac{Hl}{N}
$$

If the flux density changes by $dB$ in time $dt$, the induced emf is

$$
e = N\frac{d\phi}{dt} = NA\frac{dB}{dt}
$$

The source must supply power $ei$ against this emf. Energy supplied in time $dt$:

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

Energy per unit volume (volume $= Al$): $dw = H\,dB$. Over one complete cycle:

$$
w = \oint H\,dB \quad \text{J/m}^3 \text{ per cycle}
$$

$\oint H\,dB$ is exactly the area enclosed by the B–H loop. Part of the energy given while B rises is returned while B falls; the difference, equal to the loop area, is lost as heat. Hence the hysteresis power loss is

$$
P_h = (\text{area of loop}) \times \text{volume} \times f \quad \text{watts}
$$

If the loop is drawn to scales of $x$ A/m per cm and $y$ T per cm, loop area (in J/m³) = measured area (cm²) × $xy$.

#### Practical result

Steinmetz found empirically that $P_h = \eta B_{max}^{1.6} f V$. Machines therefore use silicon steel, whose loop is narrow (small area), to keep hysteresis loss low.

## Check yourself

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

1. What does the area enclosed by the B-H loop represent?
   A. Energy stored permanently in the core
   B. Energy lost as heat per cycle per unit volume
   C. Total flux in the core
   D. Residual flux density
   Answer: B. The integral of H dB over one cycle equals the loop area, and it is the net energy converted to heat each cycle. (Key point: Meaning of the loop area)

2. When H is reduced from saturation to zero, B does not fall to zero. This remaining B is called:
   A. Coercive force
   B. Saturation flux density
   C. Residual flux density (retentivity)
   D. Permeability
   Answer: C. The value ob left at H = 0 is the residual flux density; the reverse H needed to make B zero is the coercive force. (Key point: Retentivity and coercivity)

3. In the proof, the energy supplied in time dt is e*i*dt. After substituting e = NA dB/dt and i = Hl/N, the energy per unit volume is:
   A. B dH
   B. H dB
   C. H B
   D. H^2 dB
   Answer: B. dW = (Al) H dB, so dividing by volume Al gives dw = H dB. (Key point: Energy per unit volume)

4. Which choice correctly gives the cause of the loss and a way to keep it low?
   A. Eddy currents in the core; use solid iron
   B. Domains do not all turn back when H is removed; use silicon steel with a narrow loop
   C. Resistance of the winding; use thicker wire
   D. Air gap leakage; remove the gap
   Answer: B. Extra work to turn the domains back appears as heat, and a narrow loop (silicon steel) means small area and small loss. (Key point: Why hysteresis loss occurs / reduction)

## Related in Electrical Machines I

- [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
- [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

- [Magnetic hysteresis and energy per cycle](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Ahysteresis-energy-loop-area) — Electrical Machines, chapter 1
- [Retentivity, coercivity, energy per cycle](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Aretentivity-coercivity-energy-loop) — Electrical Machine, chapter 1
- [Hysteresis and eddy current losses; loop area](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Ahysteresis-eddy-loss-area) — Electrical Machine, 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