Magnetic circuits and electromagnetic fields

In short

A magnetic circuit is the closed path that flux follows through iron and air when a current-carrying coil drives it, analysed with an Ohm's-law analogy: MMF (ampere-turns) drives flux against reluctance. Electromagnetic field theory is the underlying physics - Coulomb, Gauss, Biot-Savart, Ampere and Faraday - from which inductance, capacitance, induced EMF and electromagnetic force all follow. Together they explain every transformer, motor, generator, relay, choke, solenoid and current sensor, and they decide core size, exciting current and iron loss. The topic carries high weight in university exams and in GATE EE (Electromagnetic Fields and Magnetic Circuits), and is standard interview ground for transformer and motor design, testing and protection roles at BHEL, Siemens, ABB, Crompton, Kirloskar Electric, Havells, Tata Power, NTPC and the state utilities.

Explained

Field quantities and the magnetisation (B-H) curve

• MMF (F): the driving "pressure", F = NI ampere-turns (AT). Magnetic field strength H = F/l = NI/l in A/m (depends only on current and geometry, not on the material).
• Flux density B in tesla (T = Wb/m²); flux Φ = B·A in weber. B = μH, where μ = μ₀μ_r, μ₀ = 4π × 10⁻⁷ H/m.
• Relative permeability μ_r: 1 for air/copper/aluminium, 2,000-8,000 for silicon steel at moderate B, up to 10⁵ for mumetal; it is *not* a constant - it falls sharply as the iron saturates.
• B-H curve regions: initial (low slope) → steep linear region (working region) → knee → saturation (slope approaches μ₀). Silicon steel saturates near 1.9-2.1 T, ferrites near 0.3-0.5 T.
• Machines and transformers are designed just below the knee (typically 1.4-1.7 T in CRGO cores) - above it exciting current grows out of proportion to flux.
• Magnetisation M and susceptibility χ_m: B = μ₀(H + M), μ_r = 1 + χ_m. Diamagnetic (χ_m slightly negative), paramagnetic (slightly positive), ferromagnetic (large, nonlinear), with domain alignment the mechanism; above the Curie temperature ferromagnetism disappears.

Magnetic circuits: reluctance and the electric analogy

• Hopkinson's law (magnetic Ohm's law): Φ = F/S, where reluctance S = l/(μA) in AT/Wb (H⁻¹). Permeance = 1/S.
• Series elements: S_total = S₁ + S₂ + … and the same flux passes through each; MMF drops add (Kirchhoff's MMF law, ΣNI = ΣHl around a loop).
• Parallel branches: permeances add, 1/S = 1/S₁ + 1/S₂; fluxes at a junction sum to zero (ΣΦ = 0, from ∇·B = 0).
• Solution steps (series circuit, flux specified): (1) find Φ from required B and A; (2) for each section compute B = Φ/A; (3) get H from B/μ or from the given B-H curve; (4) multiply by mean length to get the MMF drop Hl; (5) add drops and divide by N to get the current.
• If the *current* is specified instead and the iron is nonlinear, the problem is solved by trial (assume Φ, compute MMF, iterate) or graphically with the air-gap load line drawn on the B-H curve.
• Limits of the analogy: reluctance depends on flux (nonlinear), flux leaks everywhere whereas current does not, and sustaining flux needs no continuous energy input (no I²R-type loss in the "flow"), while current does.

Air gaps, leakage and fringing

• An air gap of a few millimetres usually dominates the circuit because μ_r of air is 1 - a 2 mm gap can take 80-90% of the total MMF, and most of the stored energy sits there.
• Consequences: a gap linearises the circuit, pushes saturation further away and reduces inductance - deliberately used in DC-filter chokes, swinging chokes and current transformers meant to resist saturation; unwanted in transformer joints, where it raises magnetising current.
• Fringing: flux bulges outward at a gap, increasing the effective area; corrected approximately as A_eff = (a + l_g)(b + l_g) for a rectangular face, which *reduces* gap reluctance.
• Leakage flux: the part of the flux that does not follow the intended path. Leakage factor = total flux in the core / useful flux in the gap, typically 1.1-1.25. Leakage causes transformer leakage reactance, motor stray loss and local eddy heating in tank walls and clamps.
• Magnetic refraction at a boundary: tanθ₁/tanθ₂ = μ₁/μ₂, so flux leaves iron almost perpendicular to the surface - the reason iron acts as a flux guide and as magnetic shielding.
• Stacking factor (0.90-0.97) accounts for insulation and burrs between laminations when converting gross to net iron area.

AC excitation: hysteresis, eddy currents and core loss

• Hysteresis loop: B lags H; the loop area is energy lost per cycle per unit volume (J/m³). Coercivity H_c and remanence B_r define the loop width.
• Steinmetz hysteresis loss: P_h = k_h·f·B_mⁿ·V, with the Steinmetz exponent n ≈ 1.6-2.0.
• Eddy-current loss: P_e = k_e·f²·B_m²·t²·V/ρ - reduced by thin laminations (0.23-0.50 mm), silicon alloying to raise resistivity, insulated and interleaved joints, and by ferrite or powdered cores at high frequency.
• Iron (core) loss = hysteresis + eddy; it depends on frequency and peak flux density, not on load, and is measured in W/kg by the Epstein frame test (IS 649 methods). Separation into the two components from tests at two frequencies at constant B_m is a standard exam problem.
• Soft vs hard materials (circuit view only): soft (CRGO/CRNGO silicon steel, ferrites, amorphous metal) - thin loop, low loss, for cores; hard (alnico, ferrite, NdFeB, SmCo) - fat loop, high B_r·H_c product, for permanent magnets, operated in the second quadrant on their demagnetisation curve with a defined load line and (BH)_max energy product.
• Indian practice and codes: IS 648 (cold-rolled non-oriented electrical steel), IS 3024 (grain-oriented electrical steel), IS 649 (magnetic testing), IS 1180 (distribution transformers up to 2500 kVA, with BEE star-labelled loss levels), IS 2026 (power transformers), IS 12615 (energy-efficient induction motors, IE classes). Core loss limits are enforced through BEE star labelling under the Energy Conservation Act 2001; India imports most CRGO, which is why amorphous-core distribution transformers appear in utility tenders.
• Transformer EMF equation follows directly: E_rms = 4.44 f N Φ_m (exactly 2π/√2 = 4.4429), and the exciting current has a magnetising (reactive) plus a core-loss (in-phase) component.

Electrostatic fields

• Coulomb's law: F = Q₁Q₂/(4πεr²); field intensity E = F/q in V/m; flux density D = εE in C/m² with ε = ε₀ε_r, ε₀ = 8.854 × 10⁻¹² F/m.
• Gauss's law: ∮D·dA = Q_enclosed, differential form ∇·D = ρ. Using symmetry: point charge E = Q/(4πεr²); infinite line E = λ/(2πεr); infinite sheet E = σ/(2ε); between parallel plates E = σ/ε = V/d.
• Potential V = −∫E·dl, E = −∇V; equipotential surfaces are perpendicular to field lines; work done around a closed path is zero (conservative field, ∇×E = 0 statically).
• Poisson ∇²V = −ρ/ε and Laplace ∇²V = 0 (charge-free region) are solved for field plotting and electrode design.
• Dielectrics: polarisation P, D = ε₀E + P; boundary conditions - tangential E continuous, normal D differs by surface charge. In a series (layered) insulation the *weaker* material with lower ε_r carries the higher stress - the cause of void breakdown and the reason for graded cable insulation.
• Capacitance C = Q/V: parallel plate εA/d, coaxial 2πε/ln(b/a) per metre, isolated sphere 4πεr. Energy = ½CV² = ½∫εE² dv, energy density ½εE² J/m³.

Magnetostatics, Faraday's law and inductance

• Biot-Savart law: dH = (I dl × âᵣ)/(4πr²) - gives H = I/(2πr) around a long straight wire, H = NI/(2r) at a circular loop centre, H = NI/l inside a long solenoid, H = NI/(2πr) in a toroid.
• Ampere's circuital law: ∮H·dl = I_enclosed, differential ∇×H = J (+ ∂D/∂t for time-varying fields) - the basis of all magnetic-circuit MMF balances.
• Lorentz force: F = q(E + v × B); on a conductor F = BIl sinθ (motor action); torque on a coil T = NBIA cosα.
• Faraday's law: e = −dλ/dt = −N dΦ/dt (Lenz's sign). Statically induced EMF (transformer action, flux changes in time) and dynamically induced EMF (e = Blv, motional). Eddy currents, induction heating and induction-disc relays all follow.
• Self inductance: L = NΦ/I = N²/S = μN²A/l; mutual inductance M = N₂Φ₂₁/I₁ = N₁N₂/S, with M₁₂ = M₂₁.
• Coupling coefficient k = M/√(L₁L₂), 0 ≤ k ≤ 1 (near 1 in transformers on a closed core, ~0.01 in air-cored coils). Series combination: L = L₁ + L₂ ± 2M (plus for aiding fluxes, by the dot convention).
• Stored energy W = ½LI² = ½∫B·H dv; density ½B²/μ (J/m³). Force from energy: F = −∂W/∂x at constant flux, or F = +∂W′/∂x (coenergy) at constant current - the attractive force on a pole face is F = B²A/(2μ₀).
• Maxwell's equations (time-varying): ∇·D = ρ, ∇·B = 0, ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t; the displacement-current term makes propagation possible, with plane waves of velocity 1/√(μ₀ε₀) = 3 × 10⁸ m/s, intrinsic impedance √(μ₀/ε₀) ≈ 377 Ω and power flow given by the Poynting vector S = E × H (W/m²).
• Skin effect: depth δ = √(2/ωμσ) = √(ρ/πfμ); AC resistance rises, which is why large conductors are stranded, transposed (Litz, Roebel bars) or made hollow. Current GATE EE lists the field syllabus up to self and mutual inductance; wave propagation, transmission lines and Poynting's theorem appear in many university papers and in GATE EC, so check your own syllabus.

Key points

• MMF - NI in ampere-turns, the driving quantity; flux is the response, reluctance the opposition (Φ = NI/S).
• H vs B - H is set by current and path length (A/m); B is the material's response (T), B = μ₀μ_rH.
• Reluctance - S = l/(μA); in series it adds, in parallel the permeances add.
• Air gap - dominates reluctance, stores most of the energy, linearises the circuit and raises exciting current.
• Leakage factor - core flux ÷ useful flux, usually 1.1-1.25; leakage gives transformer leakage reactance.
• Fringing - flux spreading at a gap; increases effective gap area and lowers gap reluctance.
• Saturation - beyond the knee μ_r collapses, so flux no longer rises proportionally with current.
• Hysteresis loss - ∝ f·B_mⁿ (n ≈ 1.6-2), the loop area per cycle; cut by better grain-oriented steel.
• Eddy-current loss - ∝ f²·B_m²·t², cut by thin, insulated laminations and higher-resistivity alloys.
• Core loss is constant with load - it depends on voltage (hence B_m) and frequency; copper loss varies with load.
• Faraday's law - e = −N dΦ/dt; static (transformer) and dynamic (e = Blv) induction.
• Transformer EMF - E = 4.44 f N Φ_m, which fixes the turns for a given core area.
• Inductance - L = N²/S; M = k√(L₁L₂); series aiding L = L₁+L₂+2M.
• Energy density - ½B²/μ magnetically, ½εE² electrically; force on a pole face F = B²A/(2μ₀).
• Gauss's law - ∮D·dA = Q_enc, the quickest route to E and C for symmetric geometries.
• Boundary rules - tangential E and normal B are continuous; flux leaves iron almost normal to the surface.
• Series dielectrics - the lower-ε_r layer suffers the higher electric stress, so voids and air gaps fail first.
• Skin depth - δ = √(ρ/πfμ): about 9 mm in copper at 50 Hz, tens of microns at MHz.

Formulas and examples

Example 1 - series circuit with an air gap. A cast-steel ring, mean length 0.40 m, cross-section 2 cm², has a 2 mm radial air gap and a 500-turn coil. Required gap flux density 0.8 T; μ_r of steel = 1,000; neglect leakage and fringing. Find the coil current.

• Φ = 0.8 × 2 × 10⁻⁴ = 1.6 × 10⁻⁴ Wb (same in iron and gap).
• Iron: H = 0.8/(4π×10⁻⁷ × 1000) = 636.6 A/m; iron length = 0.400 − 0.002 = 0.398 m → MMF = 636.6 × 0.398 = 253.4 AT.
• Gap: H = 0.8/(4π×10⁻⁷) = 636,620 A/m → MMF = 636,620 × 0.002 = 1,273.2 AT.
• Total MMF = 1,526.6 AT → I = 1,526.6/500 = 3.05 A. The 2 mm gap takes 1,273/1,527 = 83% of the MMF although it is 0.5% of the path length.

Example 2 - inductance and stored energy of the same circuit.

• S_iron = 0.398/(1.2566×10⁻³ × 2×10⁻⁴) = 1.584 × 10⁶ AT/Wb; S_gap = 0.002/(1.2566×10⁻⁶ × 2×10⁻⁴) = 7.958 × 10⁶ AT/Wb; S_total = 9.542 × 10⁶ AT/Wb.
• L = N²/S = 500²/9.542×10⁶ = 26.2 mH (check: L = NΦ/I = 500 × 1.6×10⁻⁴/3.053 = 26.2 mH).
• W = ½LI² = 0.5 × 0.0262 × 3.053² = 0.122 J.
• Energy in the gap alone = [B²/(2μ₀)] × volume = (0.64/2.513×10⁻⁶) × (2×10⁻⁴ × 0.002) = 254,650 × 4×10⁻⁷ = 0.102 J - again 83% of the total.

Example 3 - lifting magnet. A horseshoe electromagnet has two pole faces, each 25 cm², working at B = 1.0 T. Find the load it can hold.

• Per pole: F = B²A/(2μ₀) = (1.0² × 25×10⁻⁴)/(2 × 4π×10⁻⁷) = 25×10⁻⁴/2.513×10⁻⁶ = 994.7 N.
• Two poles: F = 1,989 N ≈ 1.99 kN → mass = 1,989/9.81 = 202.8 kg (no safety factor; scrapyard magnets are rated with one).

Example 4 - separating core loss. At constant B_m, total core loss is 60 W at 50 Hz and 25 W at 25 Hz. Take P_h = Af and P_e = Bf².

• 50A + 2,500B = 60 and 25A + 625B = 25 → from the second, A = 1 − 25B; substituting: 50 − 1,250B + 2,500B = 60 → B = 0.008, A = 0.8.
• At 50 Hz: hysteresis = 0.8 × 50 = 40 W, eddy = 0.008 × 2,500 = 20 W (sum 60 W ✓).
• At 25 Hz: 20 W + 5 W = 25 W ✓. Halving lamination thickness would cut the 20 W eddy component to about 5 W.

Example 5 - skin depth in copper. ρ = 1.72 × 10⁻⁸ Ω·m, μ = μ₀.

• At 50 Hz: δ = √(1.72×10⁻⁸ / (π × 50 × 4π×10⁻⁷)) = √(8.71×10⁻⁵) = 9.33 × 10⁻³ m = 9.3 mm - so a 400 mm² solid bar carries current unevenly; busbars are made flat or laminated.
• At 1 MHz: δ scales as 1/√f → 9.33 mm × √(50/10⁶) = 66 µm, hence Litz wire and ferrite cores in SMPS magnetics.

Where it is used

• Transformer design and testing: core area and turns from E = 4.44fNΦ_m, no-load test for iron loss, BEE star levels under IS 1180 - BHEL, Crompton Greaves, Kirloskar, Voltamp, and every state discom's transformer tender.
• Rotating machines: air-gap flux density, MMF per pole and magnetisation curves for induction and synchronous machines; open-circuit characteristic of a DC generator is a B-H curve in disguise (IS 12615 IE efficiency classes).
• Switchgear, relays and actuators: contactor and solenoid pull-in force from B²A/2μ₀, induction-disc and current-transformer design, MCB tripping coils - L&T, Siemens, Schneider, Havells.
• Power electronics magnetics: ferrite and amorphous cores for SMPS, EV on-board chargers and solar inverters; gapped inductors to avoid saturation, Litz wire to beat skin effect.
• Instrumentation and sensing: current transformers, Rogowski coils, Hall-effect and fluxgate sensors, metal detectors, induction cooktops, magnetic levitation and MRI gradient coils.
• High-voltage insulation and metrology: field plotting and stress grading in cables, bushings and GIS; capacitance of coaxial cables, cable-charging current and CVT design - CPRI Bengaluru and ERDA Vadodara test to these standards.

Interview questions

• Q: Why does a small air gap need a large share of the MMF? A: Reluctance is l/μA, and air has μ_r = 1 against 1,000-5,000 for iron, so even 1-2 mm of air has more reluctance than the whole iron path.
• Q: Where is the energy of a magnetic circuit stored? A: Mostly in the air gap - energy density is B²/2μ, and with B nearly the same in iron and gap, the low-μ gap holds far more energy per cubic metre.
• Q: Difference between hysteresis and eddy-current loss, and how is each reduced? A: Hysteresis is domain-realignment loss ∝ fB_mⁿ, reduced by grain-oriented or amorphous material; eddy loss is induced circulating current ∝ f²B_m²t², reduced by thin insulated laminations and higher resistivity.
• Q: Why does transformer core loss stay constant when load changes? A: It depends on peak flux density and frequency, which are fixed by the applied voltage; only copper loss follows load current.
• Q: Why is the magnetic-circuit/electric-circuit analogy only approximate? A: Permeability varies with flux (nonlinear), flux leaks into the surrounding air, and maintaining flux dissipates no power the way current through resistance does.
• Q: What is the coupling coefficient and typical values? A: k = M/√(L₁L₂); close to 1 for windings on a common closed iron core, well below 0.1 for air-cored or deliberately gapped coils.
• Q: Statically versus dynamically induced EMF? A: Statically induced - the conductor is still and the flux varies in time (transformer); dynamically induced - the flux is steady and the conductor moves, e = Blv (generator).
• Q: Why are inductors for DC filter chokes deliberately gapped? A: The gap adds linear reluctance so the core does not saturate under DC bias, keeping inductance nearly constant at the cost of a lower value.
• Q: In layered insulation, which layer sees the highest electric stress? A: The layer with the lowest permittivity, since normal D is continuous and E = D/ε - which is why air voids break down first.
• Q: State Maxwell's correction to Ampere's law and why it matters. A: The displacement-current term ∂D/∂t; it makes current continuous through a capacitor and allows electromagnetic waves to propagate.
• Q: Why do flux lines leave an iron surface almost perpendicular? A: From tanθ₁/tanθ₂ = μ₁/μ₂, with μ_iron >> μ_air the angle in air is nearly normal to the surface - the principle of magnetic shielding.
• Q: What does skin depth tell a designer at 50 Hz? A: About 9 mm in copper, so conductors thicker than roughly 2δ gain little useful area - hence stranded, transposed or hollow conductors and laminated busbars.

Common mistakes

• Treating μ_r as a constant and using the linear formula beyond the knee instead of reading the given B-H curve.
• Using the full mean length for the iron instead of subtracting the air-gap length, or forgetting the gap altogether.
• Confusing B with H, or quoting reluctance in ohms; MMF in "amperes" without the "turns" context.
• Mixing units: cm² with m², mm of gap with metres, ampere-turns per metre with per centimetre.
• Adding reluctances of parallel branches directly instead of adding permeances, and ignoring ΣΦ = 0 at the node.
• Applying P_e ∝ f² and P_h ∝ f without holding B_m constant - loss-separation problems are only valid at constant flux density (i.e. V/f constant).
• Taking lifting force as B²A/μ₀ for the whole magnet, or forgetting that a horseshoe magnet has two pole faces.
• Dropping the negative sign (Lenz) in Faraday's law and mis-setting the dot convention, giving L₁+L₂−2M where the fluxes actually aid.

Related topics

Network theorems and circuit analysis · AC circuits and three-phase systems · Electrical engineering materials · Transformers · DC machines and induction motors · Electrical measurements and instrumentation