--- title: "Retentivity, coercivity and hysteresis energy" subject: "Electrical Machines I" chapter: 1 key: "semester_4/electrical_machines_i:retentivity-coercivity-hysteresis-energy" source: https://www.bctengineer.com/q/semester_4/electrical_machines_i/retentivity-coercivity-hysteresis-energy site: BCT Engineering --- # Retentivity, coercivity and hysteresis energy Electrical Machines I · Chapter 1: Magnetic Circuits and Induction - Asked in: 2071 Bhadra, 2+6 marks - 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/retentivity-coercivity-hysteresis-energy - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines_i/chapter-1 ## Question What do you mean by retentivity and coercivity of a core? Prove that the energy spent in hysteresis loop is Wh = ∮H.dB. ## Short answer Retentivity is the residual flux density Bᵣ remaining when H returns to zero after saturation; coercivity is the reverse Hₙ required to reduce flux density to zero. The hysteresis energy Wh = ∮H dB is proved using e = N A dB/dt and i = Hl/N, giving energy per unit volume dW = Al H dB; integrating over a cycle yields the loop area. ## Answer #### Retentivity and coercivity - **Retentivity (residual flux density, $B_r$):** the flux density that remains in a magnetic material when the magnetising force H is reduced to zero after saturating it. It measures how much magnetism the material keeps. - **Coercivity (coercive force, $H_c$):** the reverse magnetising force needed to reduce the residual flux density to zero. It measures how hard it is to demagnetise the material. ```text B | __ +Bsat Br _|-' / / | / -Hc / | / ---*---/---+-/----- H / |/ ``` Permanent magnets need high $B_r$ and high $H_c$; transformer cores need low $H_c$. #### Proof that $W_h = \oint H\,dB$ Consider a ring of magnetic material with mean length $l$ (m), cross-sectional area $A$ (m²), uniformly wound with $N$ turns carrying current $i$. 1. By Ampere's circuital law the field intensity is $$ H l = N i \;\Rightarrow\; i = \frac{H l}{N} $$ 2. If the flux density changes by $dB$ in time $dt$, the flux changes by $A\,dB$, and the emf induced in the coil is $$ e = N\frac{d\phi}{dt} = N A \frac{dB}{dt} $$ 3. To keep the current flowing against this emf, the source supplies power $p = e\,i$. The energy supplied in time $dt$ is $$ \begin{aligned} dW &= e\, i\, dt \\ &= N A \frac{dB}{dt} \cdot \frac{Hl}{N}\, dt \\ &= A\,l\,H\,dB \end{aligned} $$ 4. $Al$ is the volume of the core, so the energy per unit volume is $dw = H\,dB$. 5. Over a complete cycle of magnetisation (B going from $+B_m$ to $-B_m$ and back): $$ W_h = \oint H\,dB \quad \text{J/m}^3 \text{ per cycle} $$ During the parts of the cycle where H and dB have the same sign energy is taken from the source; where they have opposite signs energy is returned. The integral round the closed loop is the net energy absorbed, which equals the **area of the hysteresis loop** and is dissipated as heat. Hence hysteresis power loss for a core of volume $V$ at frequency $f$: $$ P_h = f\,V \oint H\,dB \quad \text{W} $$ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is retentivity (Br)? A. The reverse H needed to bring B to zero B. The maximum flux density at saturation C. The flux density remaining when H is reduced to zero after saturating the material D. The ratio of B to H at the origin Answer: C. Retentivity, or residual flux density, measures how much magnetism the material keeps. (Key point: Retentivity) 2. What does coercivity (Hc) measure? A. How much flux remains at H = 0 B. How hard it is to demagnetise: the reverse H that reduces residual B to zero C. The energy lost per cycle D. The saturation value of H Answer: B. Hc is the reverse magnetising force needed to cancel the residual flux density. (Key point: Coercivity) 3. In the proof, what is the energy supplied to the core per unit volume for a change dB? A. dw = B dH B. dw = H^2 dB C. dw = H dB, because dW = e·i·dt = A l H dB D. dw = B dB / μ Answer: C. Substituting e = NA dB/dt and i = Hl/N gives dW = A l H dB, and Al is the core volume. (Key point: Energy per volume) 4. What does ∮H dB over a complete cycle represent? A. The energy returned to the source each cycle B. The residual flux density C. The energy stored permanently in the core D. The area of the hysteresis loop, energy dissipated as heat per unit volume per cycle Answer: D. Energy taken minus energy returned is the net absorbed, equal to the loop area, and it appears as heat. (Key point: Meaning of loop integral) 5. Which core properties suit a permanent magnet and a transformer? A. Permanent magnet: low Hc; transformer: high Hc B. Both need high Hc C. Permanent magnet: low Br; transformer: high Br and Hc D. Permanent magnet: high Br and high Hc; transformer: low Hc Answer: D. Magnets must resist demagnetisation; transformer cores need a narrow loop for low loss. 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