Why couple EM fields and temperature in induction heating analysis?

Heating depends on resistivity and permeability, both strongly temperature-dependent. Steel's resistivity rises several-fold with temperature and it loses magnetism above the Curie point (about 770 °C). For the 100 mm bar, power rose from 0.70 to 1.66 MW/m² at constant field, then dropped abruptly to 0.14 MW/m² past Curie. Solving the EM field once without tracking temperature misses this.

What happens to heating above the Curie point?

Losing magnetism widens skin depth from 0.71 mm to 17 mm, so far less power enters the surface at the same field. At 1 kHz and constant field it fell to 0.14 MW/m², roughly equal to radiation loss from a surface above 1,000 °C, so after 2,000 s the surface reached only 1,057 °C. Supply output or frequency must be raised.

How is the surface-to-core temperature difference handled?

Heat enters from the surface, so the core lags when the surface reaches target. Holding 0.70 MW/m², the core was 1,018 °C — 182 K lower — when the surface hit 1,200 °C. A 37 s soak holding the surface at 1,200 °C brought the core to 1,150 °C. Pre-forging lines size their length including this soak.

Induction Heating Analysis

Category: Coupled Analysis – EM-Thermal | Revised 2026-10-01

Theory: heating that changes with temperature

Overview

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Isn't induction heating analysis just computing heat from the EM field and passing it to a thermal model?

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Doing it once goes badly wrong. Steel's resistivity rises several-fold as it heats, and above the Curie point (about 770 °C) it loses magnetism. Skin depth then widens abruptly and the power absorbed at the same coil current changes drastically. So coupled analysis re-solves the EM field as temperature advances. Let's see the difference in heating a 100 mm steel bar to 1,200 °C at 1 kHz.

EM side: skin depth and surface power

$$ \delta = \sqrt{\frac{2\rho(T)}{\omega\,\mu_0\,\mu_r(T)}},\qquad \frac{P}{A} \approx \frac{H_0^2\,\rho(T)}{\delta(T)}\,F\!\left(\frac{a}{\delta}\right) \propto H_0^2\sqrt{\rho\,\mu_r\,f} $$

$H_0$ is surface field (rms), $F$ a factor set by radius over skin depth. At constant surface field, power scales with the square root of resistivity times permeability: rising resistivity increases it, and permeability dropping from 100 to 1 at Curie cuts it sharply.

Thermal side: internal heating and surface radiation

$$ \rho c_p \frac{\partial T}{\partial t} = \frac{1}{r}\frac{\partial}{\partial r}\left(k r \frac{\partial T}{\partial r}\right) + \rho_e(T)\,|J(r;T)|^2,\qquad -k\frac{\partial T}{\partial r}\Big|_a = \varepsilon\sigma_{SB}(T_s^4 - T_\infty^4) $$

Heating follows the eddy-current distribution $J(r;T)$, and the surface loses heat by radiation. Above 1,000 °C radiation exceeds 0.1 MW/m² and can no longer be ignored against the heating power.

Coffee Break Trivia

How induction heating changed forging lines

Pre-forging steel was once heated in gas or oil furnaces, but induction heating spread in the 1950s and 60s. Furnaces need tens of minutes and grow thick oxide scale; induction takes minutes with little scale. In continuous lines where bars pass through coils one after another, the front section is often designed for the magnetic low-temperature phase and the rear for the non-magnetic high-temperature phase, with different frequencies or coils. The property change at the Curie point is the starting point of line design.

Worked example: 100 mm steel bar

The bar in a long coil was solved in 1D radially. Relative permeability is 100 below the Curie point (770 °C) and 1 above; resistivity rises linearly from 2×10⁻⁷ Ω·m at room temperature to 1.2×10⁻⁶ Ω·m at 800 °C. Conductivity 35 W/mK and volumetric heat capacity 5.0×10⁶ J/m³K are constant; surface emissivity 0.8. A simplified model evaluates EM properties at the surface temperature.

Change of skin depth

Frequency	Room temperature	Above Curie (800 °C)
1 kHz	0.71 mm	17.4 mm
5 kHz	0.32 mm	7.8 mm

Heating conditions and results

Condition	To Curie	Power change	Result
1 kHz, constant field (50 kA/m)	33 s	0.70 → 1.66 → 0.14 MW/m²	Surface stalls at 1,057 °C even after 2,000 s
1 kHz, constant power (0.70 MW/m²)	98 s	Held at 0.70 MW/m²	Surface 1,200 °C at 226 s, core 1,018 °C
5 kHz, constant field (50 kA/m)	7 s	1.57 → 3.75 → 0.35 MW/m²	Surface 1,200 °C at 372 s, core 1,133 °C

Under constant power, holding the surface at 1,200 °C after it arrives brought the core to 1,150 °C 37 s later (263 s from the start).

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Same field, yet power drops twelvefold the moment it passes Curie, and it stops heating beyond that.

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Just before Curie, resistivity is high and permeability still 100, so power climbs to 1.66 MW/m² — over twice the start. When permeability drops to 1, power, proportional to its square root, collapses. At 1 kHz it becomes 0.14 MW/m², about what a 1,050 °C surface radiates away; heat in balances heat out and temperature stops. Constant-power control raises coil current past Curie to hold power, reaching a 1,200 °C surface in 226 s. Raising frequency to 5 kHz still delivers 0.35 MW/m² above Curie, so heating continues. Real installations choose among these within supply voltage and current limits.

How to run the coupled analysis

Solve EM in the frequency domain and pass time-averaged heating to the thermal model.
Re-solve EM whenever temperature changes by a set amount (e.g. 20 to 50 K); refine near Curie.
Give resistivity, permeability (B-H), specific heat and conductivity as functions of temperature; steel's specific heat peaks near Curie.
Include surface radiation and convection; radiation dominates above 1,000 °C.
Reflect the supply control mode (constant current, power or temperature) in the model.

Coffee Break Trivia

“It heated fast in the simulation”

In one design, EM was solved once with room-temperature properties and that heating held constant in the thermal model, predicting a 1,200 °C surface in under four minutes. The prototype's surface stalled around 1,000 °C. Power had collapsed past the Curie point, which the analysis ignored. Switching to coupled analysis that re-solved EM with temperature reproduced the prototype, and revising supply capacity and frequency fixed it.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Solving EM only once	Heating overestimated	Re-solve with temperature
Ignoring Curie point	Wrong high-T power	Temperature-dependent permeability
Ignoring radiation	Final temperature overestimated	Include radiation
Judging by surface only	Core underheated	Evaluate soak time
Ignoring supply control	Mismatch with hardware	Model control mode

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I'd like to learn related topics.

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Try the skin depth tool and eddy current simulator. Related pages include induction heating (EM), induction hardening, eddy current heating and Joule heating.

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