The pan a better conductor will not heat

An induction hob is a flat coil of copper wire under a sheet of glass ceramic, driven by an inverter at twenty to a hundred thousand cycles a second. Nothing in it gets hot by design. The oscillating magnetic field passes up through the glass into whatever sits on it, and the field that makes the other does the rest: a changing magnetic flux through a conductor drives a circulating electric field, the electric field drives eddy currents, and the currents heat the metal they flow in. The pan is its own heating element.

The first surprise for anyone who has used one is which pans work. A cast-iron skillet or an enamelled steel pot heats as fast as on any gas flame. A good aluminium saucepan, or a copper one, does nothing at all, and the hob refuses it, beeping. Aluminium conducts electricity six times better than carbon steel and copper ten times better, and a naïve reading of the physics — the better the conductor, the larger the eddy currents, the more the heating — predicts the opposite of what happens. The resolution is that a good conductor carries large currents and dissipates almost nothing in them, and the quantity that sets the heating is not conductivity but the resistance the pan presents to the coil.

A current that stays near the surface #

The field from the coil does not penetrate the pan’s base uniformly. How far a field gets into metal derives the skin depth: an oscillating field entering a conductor diffuses rather than propagates, and its amplitude, and the amplitude of the currents it induces, fall off exponentially with depth over a distance $\delta = \sqrt{2\rho/\omega\mu}$, where $\rho$ is the resistivity, $\mu$ the permeability and $\omega$ the angular frequency.

At twenty-five kilohertz the skin depth in carbon steel is a tenth of a millimetre, because steel’s permeability, taken here as a hundred times that of free space, squeezes the field to the surface. In aluminium and copper, which are not magnetic, it is about half a millimetre. Every pan base is several skin depths thick, so in every case the current flows in a thin layer on the underside of the pan and the rest of the metal carries none. The heat is generated in that layer and conducts up into the food; a steel pan is heated in a film thinner than a sheet of paper.

The resistance a pan offers #

Once the current is confined to a skin, the pan behaves as if it were a sheet of thickness $\delta$ with resistivity $\rho$. A sheet’s resistance per square is its resistivity divided by its thickness, so the pan presents the coil with a surface resistance

$$R_s = \frac{\rho}{\delta} = \sqrt{\pi f \mu \rho}.$$

For a given current in the coil, and so a given field at the pan’s surface, the power dissipated per unit area is proportional to $R_s$. That is the quantity that decides what heats.

The formula holds the whole explanation. Surface resistance rises with resistivity, not with conductivity, and it rises with permeability. Carbon steel has six times aluminium’s resistivity and, in a hob’s field, something like a hundred times its permeability, and the product of the two under the square root gives it nineteen times aluminium’s surface resistance. Ferritic stainless steel, the magnetic kind used for the bases of induction-compatible cookware, has a still higher resistivity and does better still. Austenitic stainless steel, the common non-magnetic kind of most cutlery and many older pans, has a high resistivity but no permeability, and lands between: it heats, weakly, and many hobs reject it too.

The paradox dissolves once the circuit is seen whole. A coil carrying a current induces a voltage in the pan, and the pan’s response to that voltage is a current. If the pan were a perfect conductor, it would carry exactly the current needed to cancel the field inside it, and dissipate nothing, since no resistance means no loss: a superconducting pan would sit cold on a hob forever, expelling the field. A slightly resistive pan carries almost the same current and dissipates a little; a more resistive pan dissipates more. The same balance sets the drag on the magnet that falls slowly down a copper tube, and the drag that falls as the magnet speeds up is its high-frequency limit, where the tube’s currents, like a perfect conductor’s, cancel the field instead of dissipating it. In the pan the current is limited by the coil and the geometry, not by the pan, and so the dissipation, current squared times resistance, rises with resistance. That is the regime the skin depth puts every metal pan in.

Half the power, in the wrong place #

Low surface resistance would merely mean slow cooking if the coil were a perfect conductor. It is not. The coil’s copper has its own resistance, and the current that heats the pan heats the coil as well.

The pan and the coil form two resistances in series, the pan’s reflected through the magnetic coupling as in any transformer whose secondary is a circuit that fights its own change, and the power divides between them in proportion. Taking the coil’s loss to be equivalent to a pan surface of fifty microhms — a reasonable figure for a coil wound with litz wire, many fine insulated strands woven so that each carries current through the whole cross-section despite the skin effect in the copper itself — a steel pan takes ninety-five per cent of the power and the coil five. An aluminium pan takes half, and the coil the other half. To deliver the same heat to the aluminium as to the steel, the current would have to be four and a half times larger, and the coil’s own loss, going as the square of the current, nineteen times larger. The hob would cook its own coil. Its electronics detect the low load and decline.

The same series circuit explains the mechanical behaviour of light aluminium on a hob, which the ring that jumps because its current is late demonstrates with Elihu Thomson’s ring: a good conductor in an alternating field carries a current lagging the field by nearly a quarter-cycle, and the force between that current and the field is repulsive on average. A thin aluminium plate on an induction coil is pushed upward, and with enough current floats. The steel pan, whose current is limited by its resistance and lags less, is pulled down instead by its own magnetisation.

The thin sheet that heats like iron #

The skin depth explains a further trap. A pan base thicker than the skin depth presents the same surface resistance whatever its thickness, since the current never reaches the far side. A sheet thinner than the skin depth is different: the current fills the whole thickness, and the resistance per square is just the resistivity divided by the actual thickness, which for a thin enough sheet can be large.

Kitchen foil is about twenty micrometres thick, a twenty-fifth of aluminium’s skin depth at hob frequencies. The current is forced through the whole of that thin film, which presents twenty-six times the resistance of a thick aluminium pan and more than a thick steel one. Foil laid on an induction hob does not sit inertly; it heats rapidly, and with nothing to carry the heat away, it can melt or ignite whatever is on it. The behaviour is exactly the same physics as the pan’s, read at a different thickness: the metal matters, and so does its thickness compared with the skin depth.

Between the two limits, near one and a half skin depths, the curve dips about eight per cent below its thick value, because the field returning from the far surface of the sheet partly cancels the current near the face. The dip is a curiosity; the practical consequence is at the other end. Induction cookware made of aluminium or copper carries a plate of ferritic stainless steel bonded into its base, to give the coil something with a high surface resistance to work on while the aluminium or copper spreads the heat. The induction-compatible copper pan is a copper pan with a steel disc in its bottom.

How the hob knows what is on it #

A hob does not measure the pan’s material; it measures the pan’s resistance, and it does so with the same circuit that delivers the power. The coil is one half of a resonant circuit, a capacitor the other, and the inverter drives it near the circuit’s resonant frequency, where a modest voltage produces a large circulating current. How sharply the circuit resonates — its quality factor, the measure the frequency that gets an answer is about, the number of cycles a burst of oscillation takes to die away — is set by the resistance in the loop, and that resistance is the coil’s own plus the pan’s reflected into it.

With nothing on the hob, only the coil’s resistance damps the circuit, and a test pulse rings for dozens of cycles. With a steel pan in place the pan’s large surface resistance dominates and the ringing dies in a few cycles. With an aluminium pan it dies more slowly, in a number of cycles closer to the empty hob’s than to the steel pan’s. Counting the cycles of ringing after a short pulse, every second or so, is how the hob decides whether a suitable pan is present, how large it is and whether it has been lifted off. A fork dropped on the glass rings almost as long as an empty hob, and is ignored.

The resonance also explains the noises induction pans make. The eddy currents and the pan’s own magnetisation exert forces on the base that oscillate at twice the drive frequency, far above hearing, but two zones of a hob running at slightly different frequencies beat against each other, and a pan straddling them, or the hob’s own structure, can buzz at the difference — an audible tone from two inaudible ones. Layered pan bases whose layers are imperfectly bonded rattle under the same forces. The hum at twice the mains is the slower cousin of this sound: a force even in the field, made audible by a frequency that happens to fall in the range of the ear.

Raising the frequency, and its limit #

Since $R_s$ rises as the square root of the frequency, a hob could in principle heat aluminium as well as steel by running faster. The figure for surface resistance shows by how much: to raise aluminium’s surface resistance to steel’s at 25 kilohertz, the frequency would have to rise nearly four hundred times, to about nine megahertz.

That is in the range of radio transmitters, not kitchen appliances. At such frequencies the inverter’s switching losses grow, the coil’s own skin effect makes its resistance rise as fast as the pan’s, the glass and nearby objects begin to absorb energy, and the radiated interference becomes a regulatory problem. “All-metal” hobs that do heat aluminium exist; they run their coils at somewhat higher frequencies, with coils designed for lower loss, and accept a lower efficiency on non-magnetic pans. They are a compromise, not a solution, and the arithmetic of the square root is why: every factor of two in frequency buys only a factor of 1.4 in surface resistance.

The same arithmetic runs the other way in industrial induction heating. Melting furnaces for steel and copper run at a few hundred hertz to a few kilohertz, where the skin depth in molten metal is centimetres and the heat reaches deep into the charge; surface hardening of steel gears uses hundreds of kilohertz, where the skin depth is a fraction of a millimetre and only the surface is heated before the core has time to warm. Choosing the frequency is choosing the depth.

The steel that switches itself off #

Steel’s advantage on a hob is its permeability, and permeability is a property of ferromagnetic order — the order the magnetism classical physics forbids explains — which disappears at the Curie temperature, 770 °C for iron.

As steel warms its resistivity rises, roughly tenfold between room temperature and red heat, and its surface resistance rises with the square root of that. Approaching the Curie point the permeability collapses, the skin depth jumps tenfold, and the surface resistance falls by nearly an order of magnitude. A pan never gets near this, but the effect is used deliberately. Some soldering irons have a tip clad in a ferromagnetic alloy with a chosen Curie temperature, heated by a coil at high frequency. Below the Curie point the alloy presents a large resistance and absorbs power; at it, the resistance collapses and the power falls away. The tip holds its temperature at the Curie point with no sensor and no thermostat: the material’s magnetism is the thermostat. The same principle runs self-regulating heaters in pipelines and in medical hyperthermia with magnetic particles.

Why the glass stays cool, and when it does not #

The glass-ceramic plate between coil and pan is an electrical insulator, with a resistivity around $10^{12}$ times steel’s. Its skin depth at 25 kilohertz would be kilometres; nothing in it circulates; the field passes through it untouched. The hob’s surface is heated only by contact with the hot pan above, which is why it cools quickly when the pan is lifted, and why spilled food does not burn onto it as it does on a radiant ring. The same is true of the food itself: water is a poor conductor, and the hob heats a pan of soup through the pan, not the soup.

One permeability and one flat base #

The material properties are representative room-temperature values, and the permeability of steel is taken as a single number, a hundred, whereas in fact it depends on the field strength, on the alloy and on the history of the metal, and falls at high fields as the steel nears saturation; hob fields at the pan’s surface are a few milliteslas, where effective permeabilities of a hundred or so are typical. The pan base is treated as a flat sheet in a uniform tangential field, ignoring the geometry of the coil, the fringing at the pan’s edge and the field’s variation with radius, which in a real hob is weakest at the centre of the coil. The coil’s own loss is represented as a fixed equivalent resistance; real coils have losses that rise with frequency. The temperature dependence of steel’s resistivity is a smooth fit, and the collapse of permeability at the Curie point is drawn as a steep fall over seventy degrees rather than at the precise transition.

Still open: how efficient a hob can be #

Induction hobs convert about eighty-five per cent of the electrical energy from the mains into heat in the pan, against about seventy-five per cent for a radiant electric ring and forty per cent for gas; the remaining losses are in the inverter, the coil and the pan’s own radiation and convection from its sides. Pushing past ninety per cent requires lower coil losses — litz wire of finer strands, or coils of other conductors — and inverter switches with lower losses at high frequency, which wide-bandgap semiconductors such as silicon carbide and gallium nitride now provide. Whether an all-metal hob can heat aluminium at efficiencies close to steel’s, without a steel insert, depends on whether coil losses can be brought down as fast as aluminium’s surface resistance rises with frequency, and on the two square roots in the problem that is not obvious.

The habit worth carrying away is to ask what a load presents rather than what it conducts. An induction hob’s coil sees a pan as a surface resistance $\sqrt{\pi f\mu\rho}$, which at 25 kilohertz is nineteen times larger for steel than for aluminium because steel’s permeability and resistivity both raise it — so the better conductor heats less, shares half its power with the coil, and is refused, while a sheet of the same aluminium foil, thinner than its skin depth, heats harder than steel. In a field that changes fast enough, resistance is what turns current into heat, and a metal good at carrying current is good at keeping cool.