The field that turns with nothing turning

A coil of wire carrying a steady current makes a magnetic field along its axis, like a bar magnet. Feed it alternating current instead and the field alternates: it grows to a maximum pointing one way along the axis, shrinks to nothing, grows to a maximum pointing the other way, and shrinks again, fifty or sixty times a second. At no moment does it point anywhere except along the axis. A compass needle at the centre would shiver along that line and never be pulled round.

Now take three identical coils and set their axes at 120° to one another, like the spokes of a wheel with three spokes, all passing through the same centre. Feed the first with an alternating current, the second with the same current a third of a cycle later, the third with it two thirds of a cycle later. Each coil’s field still does nothing but pulse along its own axis. And at the centre, a compass needle is pulled round in a circle, steadily, once per cycle of the supply, by a field whose strength never changes.

Nothing in the arrangement rotates. There is no moving part, no switch, no commutator. The rotation exists only in the sum. That fact, found independently by Galileo Ferraris in Turin in 1885 and by Nikola Tesla, whose patents were filed in New York in 1887 and 1888, is the reason almost every electric motor in industry is an induction motor running off three-phase supply, and it is the reason the world’s electricity grids carry three phases rather than one or two. It is also, underneath the engineering, an exact statement about what adding waves does: a quantity that oscillates along a line is already the sum of two quantities rotating in opposite senses, and a rotating field is what is left when an arrangement makes one of the two cancel.

Three fields that never turn #

The addition is simple enough to carry out at six moments of one cycle and look at.

At the start of the cycle the first coil carries its full current, and the other two carry minus a half each, since the cosine of 120° and of 240° are both −½. The first coil’s field points along its axis at full strength. The other two point backwards along their own axes at half strength — and because those two axes are 120° either side of the first, their backward-pointing halves have components along the first axis that add to it, and components across it that cancel. The total points along the first coil’s axis with strength $1 + \tfrac14 + \tfrac14 = 1.5$.

A twelfth of a cycle later the first coil’s current has fallen to $\cos 30° = 0.87$, the second’s has risen to zero, the third’s has fallen to $-0.87$. The first field is shorter, the second has vanished, the third points backwards along its axis — and the sum is again 1.5 long, now pointing 30° further round. At every moment the arithmetic comes out the same way. The field has constant length and points at the angle $\omega t$, where $\omega$ is the supply’s angular frequency.

The algebra behind that is two trigonometric identities, and it is worth seeing once. Writing coil $k$’s axis as the unit vector at angle $\alpha_k = 2\pi k/3$ and its current as $\cos(\omega t - \alpha_k)$, the total field is

$$\mathbf{B}(t) = \sum_{k=0}^{2} \cos(\omega t - \alpha_k)\,\begin{pmatrix}\cos\alpha_k\\ \sin\alpha_k\end{pmatrix}.$$

Each product of cosines splits into a half-sum of two cosines: one of $\omega t$ alone, the same for every coil, and one of $\omega t - 2\alpha_k$, which differs from coil to coil by 240°. Summed over the three coils, the first kind adds up to $\tfrac32(\cos\omega t, \sin\omega t)$ and the second kind cancels, because three unit vectors at 0°, 240° and 480° point symmetrically round a circle and add to nothing. What is left is a vector of length 1.5 turning at the angular frequency of the supply.

That split — one part common to every coil, one part that differs and cancels — is the whole mechanism. Everything that follows is about reading what the two parts are.

Every pulsing field is two turning ones #

Take a single coil alone. Its field is $\cos\omega t$ along the $x$ axis. Now notice that

$$\cos\omega t\,\begin{pmatrix}1\\0\end{pmatrix} = \frac12\begin{pmatrix}\cos\omega t\\ \sin\omega t\end{pmatrix} + \frac12\begin{pmatrix}\cos\omega t\\ -\sin\omega t\end{pmatrix}.$$

The right-hand side is two vectors, each of half the peak length, one turning anticlockwise at $\omega$ and the other turning clockwise at the same rate. Their components along the axis are equal and add; their components across it are equal and opposite and cancel. The sum stays on the axis and pulses.

This is not a convenient fiction. It is exactly as real as the pulsating field itself, in the sense that anything responding to rotation — a rotor, a spinning nucleus, an electron with a preferred sense of circulation — responds to each of the two halves separately, and to each according to whether it turns with it or against it. The decomposition is the same one that writes light shaking along a line as two circularly polarised beams of opposite handedness added together, and it is the same mathematics: a real oscillation is half of $e^{i\omega t}$ plus half of $e^{-i\omega t}$, and in a plane those two exponentials are rotations in opposite senses.

With that in hand the three-coil result reads differently. Each of the three coils contributes a forward-turning half and a backward-turning half. The forward halves all start in step — each coil’s forward half is at the coil’s own axis when its current peaks, and the peaks come round in the same order as the axes — so all three forward halves point the same way at every instant and add to $3 \times \tfrac12 = 1.5$. The backward halves start at angles 0°, 240° and 480°, spread evenly round the circle, and add to nothing at every instant. The geometry of the coils and the timing of the currents have been matched so that one sense of rotation adds coherently and the other cancels. It is interference, in the plainest sense of the word, performed on two rotating vectors instead of two waves.

The path the tip traces #

The cleanest way to see the forward and backward parts together is to follow the tip of the total field over one cycle. A pure forward rotation traces a circle; equal forward and backward parts trace a line; anything in between traces an ellipse, with half-axes equal to the sum and the difference of the two parts.

The second panel is Ferraris’s arrangement: two coils at right angles, fed with currents a quarter cycle apart. It makes a perfectly circular rotating field with only two coils, because $\cos\omega t$ along one axis plus $\sin\omega t$ along the other is the rotation itself. Two-phase supply was used in the first installations, including the Niagara Falls power station in the 1890s. Three phases won because the three currents of a balanced three-phase supply add to zero at every instant, so three wires carry it rather than four, and because a three-phase generator delivers power at a constant rate rather than one that pulses at twice the supply frequency — the same cancellation, applied to power instead of field.

The fourth panel is what happens when the three currents are not equal, and it is the first of the things that go wrong.

What a weak coil adds #

Suppose two coils carry their full current and the third carries a fraction $k$ of it, with the timing unchanged. Then the forward halves no longer add to 1.5 and the backward halves no longer cancel. The forward halves still all point the same way, so they add to $(1 + 1 + k)/2$. The backward halves still point at 0°, 240° and 480°, but one of them is shorter than the others, so the three no longer close into a triangle, and what is left over is $|k - 1|/2$ turning the wrong way.

The result is exact and simple, and it says something that is not obvious from the snapshots. An unbalanced three-phase supply does not make a weaker or slower rotating field. It makes the full-speed forward field, scaled a little, plus a second field of the same speed turning the other way. With the third coil at 70 per cent, the backward part is 11 per cent of the forward one. With the third coil disconnected entirely, the backward part is 0.5 and the forward part 1.0: the backward part is half, and the field’s tip traces a flat ellipse.

That backward field is the one that matters for a motor, because of how an induction motor works. Its rotor is a cage of conducting bars, and a field sweeping past the bars induces currents in them — a changing flux drives a current that opposes the change — and those currents, sitting in the field, are pushed along with it. The rotor therefore chases the forward field and runs slightly slower than it, since a rotor that caught up exactly would see no change of flux and carry no current. The slip is a few per cent at full load.

The backward field sweeps past the same rotor from the other direction, at the supply speed plus the rotor’s own, nearly twice the supply frequency. To a rotor running at 97 per cent of synchronous speed, the backward field looks like the field a stationary rotor would see from a supply at 1.97 times the frequency, and a stationary rotor is the condition under which an induction motor draws its largest currents. So a small backward field induces large rotor currents, which heat the rotor and pull back on it. Motor standards derate machines sharply for supply imbalance for this reason: a few per cent of voltage imbalance produces a backward current several times as large in proportion, and a temperature rise that grows roughly as the square of it. The decomposition into forward and backward parts — generalised in 1918 by Charles Fortescue to any unbalanced set of three-phase quantities as the method of symmetrical components — is still how power engineers analyse every unbalanced fault on a grid.

Why one coil cannot start a motor it can keep running #

The single coil is the extreme case, and it has a surprising consequence. Its field is two equal fields turning opposite ways, so a stationary rotor in it is pulled forwards by one and backwards by the other with exactly equal torque. It does not move.

But once the rotor is turning, even slowly, the balance is broken. Turning forwards, it lags the forward field by less than it lags the backward one, and in an induction motor the torque from a field depends on how fast the field sweeps past the rotor. The two torques no longer match, the net torque is in the direction of motion, and the rotor accelerates until it is running at a little under synchronous speed — in the model, up to 99 per cent of it at no load, with the net torque largest near 84 per cent. Push it the other way first and it runs up the other way.

That is exactly how single-phase induction motors behave, and it is why every one of them — in a refrigerator compressor, a washing machine, a bench grinder — carries some means of making a little rotating field to get started. The commonest is a second winding at right angles to the first, fed through a capacitor that shifts its current by roughly a quarter of a cycle: Ferraris’s two-coil arrangement, imperfectly realised, for a second or two, before a switch disconnects it. The cheapest, the shaded-pole motor in small fans, puts a copper ring round part of each pole so that the flux through the ringed part lags the rest, which makes a weak, lopsided rotating field and a weak, reliable start.

The field round the gap is not a sine wave #

The coils so far have been ideal, each making a field that points along its axis at the centre. A real machine has a rotor filling the centre, and what matters is the field across the narrow air gap between rotor and stator, as a function of angle round it. A coil wound in a single pair of slots makes the crudest pattern possible: one polarity across half the circumference and the other across the other half, a square wave in angle.

Three of them at 120°, fed three-phase, give a staircase. It is not a sine wave, and its shape changes from one instant to the next: at the start of the cycle it has six steps at 2, 1, −1, −2, −1, 1; a twelfth of a cycle later it has only three levels, 1.73, 0 and −1.73. What moves round the gap rigidly, at exactly the supply rate, is its fundamental — the sine wave of the same period that best fits the staircase. The rest is higher harmonics, and the three-way sum sorts those with the same arithmetic that sorts forward from backward.

A spatial harmonic of order $h$ in each coil’s pattern varies as $\cos(h\theta)$ round the gap. Multiply by the coil’s current and split the product as before, and each coil contributes a part turning forwards at $\omega/h$ and a part turning backwards at $\omega/h$. Now the phases the three coils bring are $\alpha_k$ from the timing and $h\alpha_k$ from the geometry, and whether a part adds or cancels depends on whether the two combine to a multiple of 360° — which depends only on the remainder of $h$ after division by three.

Harmonics whose order is a multiple of three cancel in both directions: every coil’s contribution is in step with every other’s in time but spread round the circle in space, or the other way round, and neither combination adds. Harmonics one more than a multiple of six — 7, 13, 19 — turn forwards at a seventh, a thirteenth, a nineteenth of the field’s speed. Harmonics one less than a multiple of six — 5, 11, 17 — turn backwards. The fundamental is the case $h = 1$, forward at full speed.

These are not curiosities. A forward seventh harmonic turning at a seventh of synchronous speed can capture a motor’s rotor during starting and hold it there, a fault engineers call crawling. A backward fifth harmonic drags against the rotor at all speeds. The remedy is to shape the windings rather than the currents: distributing each coil over several slots, and spanning slightly less than half the circumference — five sixths of it is common — reduces the fifth and seventh harmonics to a few per cent without much weakening the fundamental, which is why the windings of a real stator look nothing like three simple coils.

The same arithmetic applies to harmonics in time rather than space. A current distorted by an electronic load carries harmonics of the supply frequency, and in a three-phase system the fifth-order current harmonics form a backward-rotating set, the seventh a forward one, and the triples are in step in all three phases at once and do not rotate at all — they add in the neutral wire instead of cancelling there. The three-way sum decides their fate in exactly the way it decides the fate of the spatial harmonics in the figure.

The half that was thrown away #

The split of an oscillating field into two counter-rotating halves has a second life, in a place that has nothing to do with motors, and it is where the argument reaches its most precise form.

A nucleus with spin in a strong steady magnetic field precesses about it at a fixed frequency, the Larmor frequency, always in the same sense. To tip it, a second, much weaker field is applied at right angles, oscillating at the Larmor frequency — from a single coil, so it pulses along one line. Split it, and it is two rotating fields of half the amplitude. One turns in the same sense as the precession and at the same rate, so in the frame of the precessing spin it is a steady field, and it tips the spin steadily over. The other turns the opposite way and, in the spin’s frame, whirls round at twice the Larmor frequency, averaging almost to nothing.

Discarding that second half is the rotating-wave approximation. It is how magnetic resonance is taught, how the pulses of every MRI scanner are designed, and — with the spin replaced by an atom and the coil by a light wave — how the interaction of atoms with laser light is solved in almost every textbook. It is also an approximation, and it has a measurable error. The counter-rotating half is not quite zero on average: it shifts the resonance slightly upward, by an amount proportional to the square of the oscillating field’s strength divided by the Larmor frequency. The shift was calculated by Felix Bloch and Arnold Siegert in 1940 and bears their names, and it is measured in precise resonance experiments exactly because the field that drives them is linear, not circular.

A motor designer with three coils cancels the backward half by geometry. A resonance experiment with one coil cannot, and relies instead on the backward half being too far from resonance to matter. Both are applications of the same decomposition, and both show why the decomposition deserves to be taken as physical: the backward half has consequences — a heated rotor, a shifted line — that can be measured.

It also connects to how phases decide the shape of a sum more broadly. There, the same set of waves added with equal phases or random ones gave a train of pulses or a steady glow. Here, three identical coils fed with currents in one order or the other give a field turning one way or the other; fed with equal phases, they give no field at the centre at all, since three equal vectors at 120° to one another add to nothing. Swapping any two supply leads reverses a three-phase motor — the order of the phases is the direction of rotation, and it is carried in the phases alone, the way a lattice’s position is carried in its beams’ phases while its shape is not.

What the decomposition does not say #

The splitting of a field into forward and backward parts assumes superposition holds for the thing being split. For the magnetic field in air it does exactly. In an iron-cored machine it holds only approximately, because iron saturates: when the forward field is strong enough to push the core towards saturation, the backward field’s effect depends on where the forward one has left the iron, and the two are no longer independent. Symmetrical components are a linear method applied to a system that is linear most of the time, which is why they work beautifully for faults and less well for machines near their limits.

The torque model in the figure is also a caricature: one rotor resistance and one leakage reactance, the simplest circuit that captures the shape of the curve. Real rotors have deep bars whose effective resistance changes with the frequency of the currents in them — a current at higher frequency crowds into the surface of a conductor — and designers exploit this to make a rotor that is resistive and high-torque when starting and efficient when running. None of that changes the conclusion at standstill, where the single coil’s two halves cancel by symmetry whatever the rotor is made of.

And the energy bookkeeping, as always, is quadratic. The forward and backward fields add linearly, but the power they deliver goes as their squares, so the cross terms matter and the torque on a rotor in an elliptical field pulses at twice the supply frequency — the hum of a transformer or a single-phase motor, and the reason such motors vibrate more than three-phase ones of the same power.

Still open: when the half that was thrown away cannot be thrown away. The rotating-wave approximation is excellent when the driving field is weak compared with the frequency it drives, and in ordinary atoms and nuclei it always is. In superconducting circuits and some semiconductor cavities, the coupling between a two-level system and a field mode can now be made a sizeable fraction of the mode’s frequency, and there the counter-rotating half is no longer a small shift. It creates and destroys excitations in pairs, lets the ground state contain photons, and changes the physics qualitatively. The single-mode version of the problem, the Rabi model, was solved exactly only in 2011; what happens with many modes, many emitters and losses in this regime, and whether the photons bound into the ground state can be released and detected, is being worked out now.