Magnets 201
Advanced Magnetics Engineering

Magnetostatics: B, H, M, and the Self-Demagnetizing Field

Three vector fields describe a magnet, and design errors usually trace to conflating them:

• M (magnetization) — dipole moment per unit volume; the material's own contribution. In a hard magnet, J = µ₀M (polarization) is what the intrinsic curve plots.
• H (field strength) — the field from external sources plus the magnet's own demagnetizing field. Inside an isolated magnet, H opposes M.
• B (flux density) — the divergence-free total. Flux lines of B are always closed loops; lines of H begin and end on the poles.

The central fact of open-circuit operation: a magnet demagnetizes itself. Its own poles create an internal field H_d opposing M:

N is exactly defined only for ellipsoids (uniform internal field), but effective values guide intuition and connect directly to the permeance coefficient: Pc = (1−N)/N.

Analytical Field Solutions You Should Have on Hand

For uniformly magnetized parts, the charge-sheet model gives closed-form fields that match FEA within a few percent for common geometries — invaluable for sanity checks and first-pass sizing. All formulas below give the axial flux density on the magnetization axis, at distance z from the pole face, for a magnet of remanence Br.

Axially magnetized cylinder (radius R, length L)

Setting z = 0 gives the pole-face center field; letting z → 0 with L ≫ R recovers B → Br/2, the theoretical ceiling for a bare pole face — a useful reality check on gaussmeter readings.

Axially magnetized ring (outer R₂, inner R₁, length L)

Rectangular block (dimensions 2a × 2b, length L, magnetized along L)

Far field: the dipole approximation

Beyond ~3 characteristic lengths, any magnet is a point dipole of moment m = M·V = (Br/µ₀)·V:

Two consequences worth internalizing: (1) at sensing distances, only m = (Br/µ₀)·V matters — a cheap large N35 beats a small N52 of lower moment; (2) doubling reach requires ~8× the moment.

Magnetic Circuit Analysis: The Reluctance Model

When soft steel routes the flux, field superposition gives way to the lumped magnetic circuit — Ohm's law with flux as current. The magnet is a flux source with internal reluctance; air gaps dominate the external reluctance; well-designed steel is nearly a short circuit.

The workhorse result: gap flux density

For a magnet (length Lm, area Am) driving an air gap (length Lg, area Ag) through ideal steel, combining Ampère's law and flux conservation with the linear NdFeB characteristic gives:

Real circuits need two empirical corrections: the leakage factor k_L = Φm/Φg ≈ 1.1–3 (flux that bypasses the gap) and the reluctance/loss factor k_r ≈ 1.05–1.2 (MMF dropped in steel and joints). Compact, well-closed circuits sit near the low ends; open horseshoe-type circuits leak half their flux or more. Estimating k_L is exactly the job FEA does well (Chapter 12) — the lumped model with k_L from a quick 2D solve is a powerful hybrid workflow.

Design rules that fall out of the model

• Magnet length buys gap-driving MMF; magnet area buys flux. Match the magnet's B·H operating point to (BH)max — historically the "minimum magnet volume" condition: V_magnet ∝ Bg²·V_gap / (BH)max.
• Steel must not saturate: size cross-sections so B_steel stays under ~1.5 T (1008 steel knees near 1.6–1.8 T). A saturated yoke silently becomes an air gap.
• Series gaps add; parallel leakage steals. Every joint, plating layer, and clearance in the flux path is a series reluctance — spec joint flatness accordingly.

Demagnetization Analysis: The Calculation That Prevents Field Failures

This is the single most valuable calculation in permanent magnet engineering: verifying that the worst-case operating point stays above the intrinsic knee at the worst-case temperature. Here is the complete procedure, then a worked example.

The procedure

1. Compute Pc for the magnet in its circuit (Chapter 1 factors; add the mirror-image bonus for steel backing — a magnet on a steel plate behaves like a magnet of twice its length, roughly doubling effective Pc).
2. Translate properties to temperature T:
   Br(T) = Br(20°C) · [1 + α·(T−20)]   Hcj(T) = Hcj(20°C) · [1 + β·(T−20)]
   NdFeB typical: α ≈ −0.0011/°C, β ≈ −0.0055/°C
3. Locate the knee at T. For sintered NdFeB, a serviceable estimate of the knee field is H_k ≈ 0.8–0.9 × Hcj(T); use the manufacturer's curve family when available.
4. Compute the internal demagnetizing field at the operating point, adding any applied external field H_ext (stator reaction, adjacent magnets in repulsion, magnetizer stray fields in fixtures):
   H_total = M/(Pc·µ_rec + 1) + H_ext   [first term ≈ Br(T)/(µ₀(Pc+1)) for µ_rec≈1]
5. Check margin: require H_total < H_k(T) with margin — 10–20% for benign products, more where a single demag event is intolerable.

Worked example: Ø10 × 2 mm N42 disc, open air, 80 °C

Fixes, checked with the same arithmetic: steel backing (mirror-image, Pc → ≈0.44) drops H_d to ≈8.4 kOe — better, but still failing against the 6.8 kOe knee; a thin open disc is that unforgiving. Stepping to N42H raises Hcj(80) to ≈11.4 kOe (knee ≈9.7) — marginal bare, but steel backing + N42H gives 8.4 vs. 9.7 kOe, a real pass with ~15% margin. Or simply thicken the magnet: Ø10 × 5 mm N42 (Pc ≈ 0.55, H_d ≈ 7.8) with steel backing (Pc ≈ 1.1, H_d ≈ 5.8) passes on geometry alone. This one page of arithmetic is why "same grade, different shape" parts fail in the field while the datasheet says they shouldn't.

Computing Force and Torque

Magnetic force calculations come in three tiers of rigor. Use the cheapest one that answers the question.

Tier 1 — Maxwell stress on a pole face

Where flux crosses an interface at density B over area A, the tensile stress is:

For a magnet in contact with thick steel, take B as the gap/contact flux density (from Chapter 3's circuit model, including leakage) — not Br. This single formula, honestly fed, lands within ±20% of measured pull for well-behaved circuits, and it explains why the steel cup works: the cup raises contact B and adds a second working face.

Tier 2 — Virtual work / energy gradient

Differentiating gap energy vs. displacement gives force-vs-distance curves and handles shear: sliding a magnet across a plate changes overlap area, and dW/dx yields the restoring force. Shear breakaway, though, is frictional — F_slide = µ·F_normal with µ ≈ 0.2–0.3 for Ni-plate on steel — which is why rated (tensile) pull overstates sliding capacity ~4×.

Tier 3 — Magnet-to-magnet and FEA territory

Coaxial magnet pairs have exact but unwieldy series solutions; the dipole limit is the useful pocket formula:

Anything with lateral offsets, tilts, or nonlinear steel goes to FEA — but always bracket the answer with a Tier 1/2 estimate first. An FEA force that disagrees with B²A/2µ₀ by 5× is a meshing error, not a discovery.

Synchronous coupling torque

For a magnetic coupling with p pole pairs, torque follows a sine of the load angle δ (the angular lag between halves):

• More poles → stiffer coupling, lower slip angle, but faster field decay across the gap (field falls ~e^(−p·g/r)); large gaps (thick containment barriers) favor fewer poles.
• Metallic containment barriers see eddy drag ∝ speed; electrically resistive barriers (ceramic, PEEK, Hastelloy over stainless) cut that loss.
• Design slip as a feature: the coupling is a torque fuse — size T_max between worst-case process torque and the weakest downstream component.

Halbach Arrays: Theory and Practical Reality

Rotate the magnetization direction continuously along an array and the flux concentrates on one side while nearly cancelling on the other. Klaus Halbach's ideal results are strikingly clean:

That second result deserves a pause: a bore field greater than the material's remanence, with no steel and no external field — r_o/r_i = e ≈ 2.72 gives B_bore = Br. Nested rotating Halbach cylinders make variable-field sources; k > 2 arrangements make quadrupoles and sextupoles for beam optics.

Engineering the real thing

• Segmentation penalty: n discrete segments per wavelength capture sin(π/n)/(π/n) of the ideal — 4 segments ≈ 90%, 8 ≈ 97%. Four per wavelength is the usual value point.
• Assembly forces are violent. Segments repel exactly where you must place them; fixtures must capture each piece positively. Assemble-then-magnetize is often impossible (the pattern needs pre-magnetized segments), so plan tooling early.
• Internal demag: some segments sit in their neighbors' reverse field — run Chapter 4's analysis per segment; corner segments often need a higher coercivity class than the rest.
• Where they earn their cost: ironless motors/actuators (no cogging, no back-iron mass), maglev and eddy-current systems, self-shielded sensor assemblies, MRI-adjacent hardware, undulators/wigglers, and torque-dense couplings.

Magnets in PM Machines: The Hard Parts

Motors and generators are the most demanding magnet application: elevated temperature, deliberate opposing fields, high-frequency flux ripple, and centrifugal load — simultaneously. Four analyses dominate magnet specification in a machine.

1 — Demag under stator reaction

The armature's d-axis field adds directly to the magnet's self-demagnetizing field. The design case is a fault: short-circuit or maximum flux-weakening current at maximum rotor temperature.

This is why traction magnets are UH/EH class even though continuous rotor temperature might justify only SH: the spec is set by 300% current events, not steady state. Interior-PM (IPM) rotors partially shield magnets behind steel bridges; surface-PM rotors expose them fully.

2 — Eddy losses inside the magnets

Sintered NdFeB is a conductor (ρ ≈ 1.4–1.6 µΩ·m). Slot harmonics and PWM ripple induce eddy currents that heat the magnet — the nastiest kind of heat, generated where cooling is worst. For a magnet segment of width w (transverse to flux variation) below skin-effect frequencies:

Hence laminated/segmented magnets in high-speed machines: 3–6 axial (or circumferential) segments with insulating adhesive typically cut magnet loss by 70–90%. Bonded magnets, being ~100× more resistive, sidestep the problem at the cost of remanence.

3 — Skin depth sanity check

When segment size approaches δ, the simple w² scaling saturates — and so does the benefit of further segmentation. Match segmentation to the dominant loss harmonic, not to superstition.

4 — Mechanical retention

• Surface magnets need sleeves (carbon fiber, Inconel, Ti) above ~50–100 m/s tip speed; sleeve thickness trades directly against magnetic gap — a coupled electromagnetic-structural optimization.
• Adhesive selection is a temperature/lifetime spec (Chapter 9 logic); IPM bridges trade retention for leakage.
• True radial ring rotors eliminate segment-retention joints entirely for small machines — one hoop-stress calculation instead of n bond lines, and inherently balanced (Magnets 101, Ch. 7).

Advanced NdFeB Materials & Processing

Coercivity engineering is where NdFeB development lives now — getting high-temperature performance with less (or no) heavy rare earth. What follows is the practical map of the modern options an advanced buyer should know by name.

Grain boundary diffusion (GBD)

Instead of alloying Dy/Tb through the bulk (which dilutes Br everywhere), GBD coats finished magnets with Dy/Tb compounds and diffuses them along grain boundaries at ~900 °C — putting the coercivity-boosting element only in the thin shell of each grain where reverse domains nucleate.

• Result: +3–8 kOe Hcj for 60–80% less heavy rare earth, with Br loss of only ~0.1–0.3 kG vs. ~0.5–1 kG for bulk alloying.
• Limitation: diffusion depth ~3–6 mm — GBD works on thin magnets (exactly the motor-segment form factor) and loses effectiveness in thick blocks.
• Spec note: GBD grades often carry manufacturer-specific suffixes; ask for the guaranteed Hcj at temperature and treat the letter grade as marketing shorthand.

The rest of the toolbox

The physics scoreboard

That gap is the R&D frontier: coercivity is nucleation-limited at grain boundaries, which is why every technique above is ultimately grain-boundary engineering.

Thermal Design & Output-Drift Budgeting

Chapter 8 of Magnets 101 covered what heat does; here we quantify how to budget for it in precision systems.

Magnet material thermal properties

Flux drift budget for precision devices

Sensor and instrument designers need output stability, not just survival. Total field drift decomposes as:

• Reversible term: compensate in software (temperature sensor + α), or in hardware with Ni-Fe thermal-shunt alloys (e.g. 30% Ni "compensator" strips whose permeability falls with temperature, bypassing more flux when cold) — the classic trick in speedometers and magnetrons, still the right answer when electronics aren't available.
• Irreversible term: eliminate by thermal stabilization — bake parts 10–20 °C above worst service temperature for hours before calibration; loss taken in the factory doesn't drift in the field.
• Aging term: for properly stabilized NdFeB/SmCo, structural aging is <0.1%/decade-scale — usually below the mechanical term.
• Mechanical term: often dominant in the end: gap dimensions move with CTE; a 25 µm gap change in a 1 mm gap is 2.5% — budget the housing, not just the magnet.

Measurement & Metrology: Trust, but Verify Correctly

Most "bad magnet" disputes are really bad measurement plans. The right instrument depends on which question you're asking.

Helmholtz moment measurement — the 60-second math

Because m is volume-integrated, it's insensitive to chips, probe placement, and minor geometry — and it catches under-magnetization and wrong-grade substitution instantly. Specify it on drawings as "magnetic moment: X mWb·mm ± 3% per Helmholtz coil" (±3% is a typical commercial magnetic tolerance; tighter costs sorting).

Statistical acceptance

• Put moment (or fixture-defined flux) under SPC like any critical dimension: X̄-R charts, Cpk ≥ 1.33 for automotive PPAP programs.
• Watch distribution shape, not just means: a bimodal moment distribution signals mixed lots or a magnetizer fixture drifting mid-run.
• Angle error (magnetization axis vs. mechanical datum, typically ±3° standard, ±1° precision) needs a rotation stage or mapper — it silently destroys sensor calibrations if unspecified.

Tolerancing & Writing a Bulletproof Magnet Specification

A complete magnet drawing controls four domains — dimensional, magnetic, material, and environmental — and states how each is verified. Here's the checklist we wish every incoming print followed.

Finite Element Analysis: Getting Answers You Can Trust

Modern magnetostatic FEA (Ansys Maxwell, COMSOL, Simcenter MagNet, FEMM, or the excellent open-source FEMM/Elmer for 2D) is accurate to ~1–3% — when the inputs and mesh deserve it. The failure modes are consistent and avoidable.

Model setup rules

• 2D first. Axisymmetric or planar models solve in seconds and expose 90% of design behavior. Reserve 3D for end effects, skew, and true 3D geometry (a diametral ring in a planar problem is 3D; an axial ring is axisymmetric).
• Magnet definition: linear model with Br and µ_rec (≈1.05) is correct for NdFeB above the knee. For demag studies, use the manufacturer's temperature-dependent curve family including the knee, and a solver demag model — a linear magnet can't fail, so a linear demag study is self-deception.
• Steel BH data: use measured curves for the actual alloy and anneal state; the difference between annealed 1008 and cold-worked 1008 is real flux. Never extrapolate a BH table flat — give the solver the saturated µ→µ₀ tail.
• Mesh the gap. Force and torque accuracy lives in air-gap elements: 3+ element layers across every working gap; use force via virtual work or Maxwell stress on a contour in the gap center, not on the steel surface.
• Boundary distance: outer boundary ≥ 5–10× model size, or use infinite/ballooning boundaries; a close Dirichlet boundary is an invisible steel box squeezing your field.

Validation discipline

1. Reproduce a closed-form case from Chapter 2 within 2% before trusting the real model.
2. Cross-check FEA force against B²A/2µ₀ using the FEA's own gap B — factor-level disagreement means mesh or contour errors.
3. Mesh-refine until the answer moves <1% between refinements; report that number, not the pretty contour plot.
4. Close the loop with hardware: one Helmholtz moment + one gaussmeter map on a prototype calibrates the whole model chain.

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