---
title: "Hysteresis with AC excitation"
subject: "Electrical Machines I"
chapter: 1
key: "semester_4/electrical_machines_i:hysteresis-ac-excitation"
source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/hysteresis-ac-excitation
site: BCT Engineering
---

# Hysteresis with AC excitation

Electrical Machines I · Chapter 1: Magnetic Circuits and Induction

- Asked in: 2078 Baisakh, 8 marks
- Video: [Electrical Machines | Lecture - 1M | Magnetic Circuits | AC Excitation in Magnetic Circuits](https://www.youtube.com/watch?v=i3GRxU3xzOs) — Engineering Devotion
- 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-ac-excitation
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-1

## Question

Explain hysteresis phenomenon with ac excitation.

## Short answer

Hysteresis is the lag of flux density B behind magnetising force H in ferromagnetic material. Under AC excitation, H cycles between positive and negative maxima, tracing a closed hysteresis loop once per cycle. The loop area represents energy lost as heat per unit volume per cycle.

## Answer

**Hysteresis** is the lagging of flux density B behind the magnetising force H in a ferromagnetic material. Under ac excitation H goes through a full cycle every period, so the material traces a closed **hysteresis loop** each cycle and loses energy as heat.

#### Mechanism

- A ferromagnetic material is made of small **domains**, each fully magnetised. With no field they point randomly, so the net B is zero.
- As H rises, domains in line with H grow and others turn, so B rises steeply and then saturates when almost all are aligned.
- When H is reduced, many domains do not turn back; B stays higher than on the rising curve. Turning them back requires work, which is lost as heat. This is the cause of hysteresis.

#### Hysteresis loop with ac excitation

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

With an ac current $i = I_m \sin\omega t$, H varies sinusoidally between $+H_m$ and $-H_m$:

1. H rises 0 → $+H_m$: B rises to $+B_m$ (point a).
2. H falls to 0: B falls only to **residual flux density** $B_r$ (point b), called retentivity.
3. H reversed to $-H_c$: B becomes zero (point c); $H_c$ is the **coercive force**.
4. H to $-H_m$: B reaches $-B_m$ (point d).
5. H back through 0 ($-B_r$, point e) and $+H_c$ (point f) to $+H_m$: the loop closes.

The loop is traced **f times per second**. For each cycle the energy lost per unit volume is $\oint H\,dB$, the area of the loop.

#### Hysteresis loss

$$
P_h = \eta\, B_m^{1.6} f\, V \quad \text{W}
$$

where $\eta$ is the Steinmetz coefficient (J/m³), $B_m$ the peak flux density, $f$ the frequency and $V$ the core volume. The index 1.6 varies from 1.5 to 2.5 for different materials.

#### Effects in machines

- Heats the core and lowers efficiency of transformers, motors and generators.
- Makes the exciting current non-sinusoidal and slightly leading the flux (core-loss component).
- Reduced by using **soft magnetic materials** (silicon steel, CRGO steel) with a narrow loop, and by keeping $B_m$ moderate.

## Check yourself

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

1. Why is energy lost as heat in hysteresis?
   A. Domains are turned back only by doing extra work after H is reduced
   B. Eddy currents flow in the core
   C. Winding resistance
   D. Flux leakage
   Answer: A. Domains do not all return when H falls; turning them back requires work that appears as heat. (Key point: Cause of hysteresis)

2. On the loop, the value of B when H returns to zero is the ___, and the reverse H that makes B zero is the ___.
   A. coercive force; residual flux density
   B. residual flux density; coercive force
   C. saturation flux; permeability
   D. retentivity; retentivity
   Answer: B. B at H = 0 is Br (retentivity); Hc is the coercive force. (Key point: Loop points)

3. With a supply of frequency f, how often is the loop traced and what is the energy lost per cycle per unit volume?
   A. Once per second; the loop area
   B. f times per second; the loop area (integral of H dB)
   C. 2f times; zero
   D. f times; B_m
   Answer: B. One loop per cycle, so f loops per second, each costing the loop area in energy per unit volume. (Key point: Loop traced per second)

4. The hysteresis loss is given by:
   A. P_h = eta B_m^1.6 f V
   B. P_h = K B_m^2 f^2 t^2 V
   C. P_h = eta B_m f^2 V
   D. P_h = I^2 R
   Answer: A. Steinmetz: P_h = eta B_m^1.6 f V; the other form is eddy-current loss. (Key point: Steinmetz formula)

## Related in Electrical Machines I

- [Magnetization characteristic of iron core with AC excitation](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Amagnetization-characteristic-ac) — Electrical Machines I, chapter 1