Magnetic circuits

A magnetic circuit is the closed path followed by magnetic flux, just as an electric circuit is the closed path of current. Every transformer, motor, generator, relay and contactor is designed around one. This topic covers flux linkage, inductance and stored magnetic energy, the properties of magnetic materials (permeability, hysteresis, saturation, soft versus hard materials), and the two effects that make machines work: induced emf (Faraday/Lenz) and force on a current-carrying conductor. The analogy between electric and magnetic circuits is a favourite source of NEC MCQs.

Key concepts

• Magnetomotive force (MMF): the "driving force" of flux, $F = NI$, in ampere-turns (AT).
• Flux (Φ): in weber (Wb). Flux density $B = \Phi/A$ in tesla (Wb/m²).
• Magnetic field intensity $H = NI/l$ in AT/m; $B = \mu H = \mu_0\mu_r H$.
• Reluctance: opposition to flux, $S = l/(\mu A)$, unit AT/Wb (or H⁻¹). Its reciprocal is permeance.
• Electric–magnetic analogy:

Electric	Magnetic
EMF (V)	MMF (AT)
Current I	Flux Φ
Resistance R	Reluctance S
Conductivity σ	Permeability μ
$I = V/R$	$\Phi = F/S$

Difference: flux does not "flow" and no energy is consumed to maintain a constant flux (only to establish it); there is no perfect magnetic insulator, so leakage flux always exists. An air gap has huge reluctance and dominates the MMF requirement; flux also bulges at the gap (fringing).

• Flux linkage: $\lambda = N\Phi$ (Wb-turns).
• Self-inductance: $L = \lambda/I = N\Phi/I = N^2/S$ (henry). Doubling turns quadruples L.
• Mutual inductance: $M = k\sqrt{L_1L_2}$, coupling coefficient $0\le k\le1$.
• Energy stored: $W = \tfrac12 LI^2$; energy density $w = B^2/(2\mu)$ J/m³ — most of the energy in a gapped core is stored in the air gap.
• Magnetic materials:
  • Diamagnetic ($\mu_r$ slightly < 1: copper, bismuth), paramagnetic ($\mu_r$ slightly > 1: aluminium, platinum), ferromagnetic ($\mu_r \gg 1$: iron, nickel, cobalt and their alloys).
  • Ferromagnetism disappears above the Curie temperature (about 770 °C for iron).
  • B–H curve: linear at low H, then knee, then saturation. μ is not constant.
  • Hysteresis loop: retentivity/remanence ($B_r$, flux left when H = 0) and coercivity ($H_c$, reverse H needed to bring B to zero). Loop area = hysteresis energy loss per cycle per unit volume.
  • Soft materials (silicon steel, CRGO, permalloy, ferrite): narrow loop, low coercivity — used in transformer and machine cores. Hard materials (alnico, ferrite magnets, NdFeB): wide loop, high $B_r$ and $H_c$ — permanent magnets.
• Induced emf — Faraday: $e = -N\,d\Phi/dt$; Lenz: the induced emf opposes the change causing it (the minus sign).
  • Statically induced (flux changes, conductor still): transformer; self- and mutually-induced emf.
  • Dynamically induced (conductor moves): $e = Blv\sin\theta$; direction by Fleming's right-hand rule (generator).
• Force on conductor: $F = BIl\sin\theta$; direction by Fleming's left-hand rule (motor). Force between magnetised surfaces: $F = B^2A/(2\mu_0)$ — basis of relays and lifting magnets.
• Core losses: hysteresis $P_h = \eta B_m^{1.6} f V$ (Steinmetz) and eddy current $P_e \propto B_m^2 f^2 t^2$; reduced by silicon steel and thin laminations respectively.

High-yield points

• Reluctance $S = l/(\mu A)$ (AT/Wb) is the magnetic analogue of resistance; MMF $= NI$ is the analogue of emf.
• $L = N^2/S$, so inductance is proportional to $N^2$.
• Stored energy $W = \tfrac12 LI^2$; energy density $B^2/(2\mu)$ — most energy of a gapped core sits in the air gap.
• Ferromagnetism is lost above the Curie temperature (about 770 °C for iron).
• Hysteresis loss $\propto B_m^{1.6}f$ (Steinmetz); eddy loss $\propto B_m^2f^2t^2$.
• Fleming's right-hand rule: generator ($e = Blv$); left-hand rule: motor ($F = BIl$).

Formulas

$\Phi = \frac{NI}{l/(\mu_0\mu_r A)}$ where N = turns, I = current (A), l = path length (m), A = area (m²), $\mu_0 = 4\pi\times10^{-7}$ H/m.

$L = \frac{N^2\mu_0\mu_r A}{l}\ \text{(H)},\qquad W = \tfrac12 LI^2\ \text{(J)}$

Worked example

An iron ring has mean length 0.5 m, cross-section 4 cm², $\mu_r = 1000$, and 500 turns. Find the inductance.

1. $S = \dfrac{0.5}{4\pi\times10^{-7}\times1000\times4\times10^{-4}} = \dfrac{0.5}{5.027\times10^{-7}} \approx 9.95\times10^{5}$ AT/Wb.
2. $L = N^2/S = 250\,000/9.95\times10^{5} \approx 0.251$ H.
3. At 2 A, stored energy $= 0.5\times0.251\times4 \approx 0.50$ J.

Exam tips

• Reluctance is analogous to resistance, permeance to conductance, MMF to EMF.
• $L \propto N^2$; inserting an air gap lowers L but makes it more linear.
• Soft = transformer core; hard = permanent magnet. Silicon reduces hysteresis, lamination reduces eddy current.
• Fleming's right hand = generator, left hand = motor.
• Energy density $B^2/2\mu$: for the same B, an air gap stores ~μr times more energy than iron.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. The area enclosed by the B–H (hysteresis) loop of a magnetic material represents:

• A. The energy lost per cycle per unit volume
• B. The retentivity of the material
• C. The permeability of the material
• D. The saturation flux density

Answer: A. The loop area $\oint H\,dB$ equals the energy dissipated as hysteresis loss per unit volume in each cycle of magnetization.

2. The direction of an induced emf, such that it always opposes the change of flux producing it, is given by:

• A. Biot–Savart law
• B. Coulomb's law
• C. Lenz's law
• D. Ampere's circuital law

Answer: C. Lenz's law states that the induced emf (and current) acts to oppose the change causing it; this is the minus sign in $e = -N\,d\Phi/dt$.

Practise AElE02 questions →