---
title: "Retentivity, coercivity, energy per cycle"
subject: "Electrical Machine"
chapter: 1
key: "semester_4/electrical_machine:retentivity-coercivity-energy-loop"
source: https://www.bctengineer.com/q/semester_4/electrical_machine/retentivity-coercivity-energy-loop
site: BCT Engineering
---

# Retentivity, coercivity, energy per cycle

Electrical Machine · Chapter 1: Magnetic Circuits and Induction

- Asked in: 2069 Poush
- Video: [What is Hysteresis Loop? | Hysteresis Loss Explained in Hindi | B-H Curve, Retentivity, Coercivity](https://www.youtube.com/watch?v=VCoDHTIWUaA) — Electrical Engineering MCQ's
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machine/retentivity-coercivity-energy-loop
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-1

## Question

Define retentivity and coercivity of core using hysteresis loop of a magnetic material used in electrical machine. Prove that energy spent per cycle per unit volume is proportional to the area of loop.

## Short answer

Retentivity Br is residual flux at H=0; coercivity Hc is reverse field for zero flux. From e = N dΦ/dt and i=H l/N, energy dW = V H dB, so per cycle energy/volume = ∮ H dB = area of B-H loop.

## Answer

#### Retentivity and coercivity
```text
              B
              |      .------- +Bs
              |    /
          Br -+---/---.
              |  /    |
  ------------+-/-----+-------- H
            -Hc /     +Hc
     -Bs ------'      |
              |
```
- **Retentivity (Br):** the flux density left in a magnetic material when the magnetising force H is brought back to zero after saturating it. It is the B-axis intercept of the hysteresis loop.
- **Coercivity (Hc):** the reverse magnetising force (A/m) needed to bring the residual flux density to zero. It is the H-axis intercept of the loop.
- Core materials for machines should have low Br and low Hc, so that the loop is narrow. Permanent magnets need both high.

#### Energy spent per cycle per unit volume is proportional to loop area
Consider a core of length *l*, cross-section *a*, wound with *N* turns carrying a current *i* that varies slowly.

```text
Field strength     H = N i / l          ->  i = H l / N
Flux               φ = B a
Induced emf        e = N dφ/dt = N a dB/dt
```
The source must supply power *e·i* against this emf. In a time *dt* the energy taken from the source is

```text
dW = e i dt
   = (N a dB/dt)(H l / N) dt
   = a l · H dB
```
With *a·l = V*, the volume of the core, the energy per unit volume in this step is

dW/V = H dB

For one complete cycle of magnetisation the energy per unit volume is

**W = ∮ H dB** = area of the B–H loop (J/m³ per cycle)

The scales of the loop are H in A/m and B in tesla, so the area has the unit (A/m)(T) = J/m³. Hence the energy lost per cycle per unit volume is directly proportional to (in fact equal to) the area of the hysteresis loop. At frequency *f* the hysteresis loss is Pₕ = f × V × (loop area) watts.

## Check yourself

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

1. On the hysteresis loop, where are retentivity and coercivity read?
   A. Br on the H-axis and Hc on the B-axis
   B. Both on the B-axis
   C. Br on the B-axis and Hc on the H-axis
   D. Both at the saturation points
   Answer: C. Br is the B-axis intercept (H = 0); Hc is the H-axis intercept (B = 0). (Key point: Intercepts of loop)

2. What properties should a machine core material have?
   A. Low Br and low Hc for a narrow loop
   B. High Br and high Hc
   C. Low Br and high Hc
   D. High Br and low Hc
   Answer: A. Cores need a narrow loop to keep hysteresis loss low; permanent magnets need both high. (Key point: Core material)

3. In the proof, why does dW per unit volume equal H dB?
   A. Because i = e/R
   B. Because power equals B H
   C. Because H = B/mu0
   D. Because e = N a dB/dt and i = H l/N, so e i dt = a l H dB
   Answer: D. Substituting e = N a dB/dt and i = Hl/N gives dW = (a l) H dB, and a l is the volume. (Key point: Energy step)

4. What unit does the loop area have, and what does it mean?
   A. T/m, flux gradient
   B. J/m^3, the energy lost per cycle per unit volume
   C. A/m, field strength
   D. W/m^3, power per volume
   Answer: B. (A/m) x T = J/m^3, the energy per unit volume per cycle. (Key point: Unit check)

## Related in Electrical Machine

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- [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
- [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
- [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

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