Illustrated Physics
physics, drawn
Electromagnetism

The coupling a resonance rescues

Two coils a hand's breadth apart share a few thousandths of their magnetic flux, and as a transformer they deliver almost nothing. Tune each with a capacitor to the same frequency and the same pair can carry most of the power across. Resonance does not strengthen the coupling, which is fixed by the geometry and falls as the cube of the distance. It changes how many times the energy goes round before it is lost. The efficiency then depends on one number, the coupling times the quality factor, and that one number explains a toothbrush on its stand, a phone on its pad, a car parked over a plate in the road and a light bulb lit two metres away.
16 min read 5 figures The same equation againMade into an instrument

Assumes: The coupling that is the same both ways · The circuit that fights its own change

The field that makes the other found that a changing current in one loop drives a current in another, by way of the flux the first threads through the second. The coupling that is the same both ways found that the mutual inductance M linking them is one number, the same whichever loop is the source. The circuit that fights its own change gave each loop its own self-inductance, the flux it threads through itself. Between them those three essays define the coupling coefficient,

k=ML1L2,k = \frac{M}{\sqrt{L_1 L_2}},

the fraction of one loop’s flux that the other shares, and a transformer is a pair of coils wound to make k as close to one as iron and careful winding allow.

This essay takes two coils for which k is small, because they are not wound together but sit some distance apart in air, and asks how power can be got from one to the other at all. The answer is resonance, and the interesting thing about it is what resonance does not do. It does not make the coupling any stronger. It changes the competition between the coupling and the losses, and it reduces the whole question of efficiency to a single dimensionless number.

How fast two coils lose touch

How quickly two coils stop sharing their field. The coupling coefficient k = M/L between two identical coaxial loops of radius a, made of wire a fiftieth of a as thick, against their separation in units of a, on logarithmic scales, from Maxwell's exact formula for the mutual inductance; dashed, the far-field limit, in which each loop sees the other as a small magnetic dipole and k falls as the cube of the distance. Close together most of one loop's flux threads the other: k = 0.77 at a twentieth of a radius apart. At one radius it is 0.099, at two radii 0.028, at five 0.0028. An inductive charger working at a few millimetres has k of a half or more; the same coils a hand's breadth apart share a few parts in a thousand of their flux, and a plain transformer made of them would deliver almost nothing.
Fig. 1 The coupling coefficient between two identical coaxial loops against their separation in loop radii, from the exact mutual inductance, with the dipole limit dashed. k = 0.77 at a twentieth of a radius, 0.099 at one radius, 0.028 at two and 0.0028 at five.

The mutual inductance of two coaxial circular loops was worked out exactly by Maxwell in terms of elliptic integrals, and it is computed exactly in the figure, divided by each loop’s self-inductance. Close together the loops share most of their flux. Once they are more than a radius or two apart, each sees the other as a small magnetic dipole, whose field falls as the cube of the distance, and k falls the same way: at five radii it is under three parts in a thousand.

That cube law is the reason inductive power transfer is a near-field technique. A coil carrying an alternating current does radiate, but a coil much smaller than the wavelength radiates very little — a loop of ten centimetres at a megahertz is three thousand times smaller than its wavelength, and its radiated power is smaller than its stored energy turned over each cycle by many orders of magnitude. Almost all of its field is the stored, non-radiating near field, which belongs to the coil and returns its energy each half-cycle. Power transfer through that field is not broadcasting. A second coil inside it borrows some of the stored energy, and only what it borrows can be delivered.

What resonance changes is easiest to see in the mechanical version, which the two pendulums that will not stop swapping drew. Two identical pendulums joined by a weak spring pass their energy back and forth completely, however weak the spring: the transfer takes longer as the coupling weakens, about one over k cycles, but it is complete. Two pendulums of different frequencies, by contrast, never transfer more than a fraction of their energy, because the small pushes from one fall out of step with the other’s swing and cancel.

Two resonant circuits tuned to the same frequency are those identical pendulums. The current in the first induces an electromotive force in the second each cycle, a fraction k of what would be needed to drive it fully; in step, those fractions add, and after about 1/k cycles the energy has crossed. A coil without a capacitor has no natural frequency of its own to accumulate in, and each cycle’s push is mostly undone by the next.

So with resonance the question becomes a race. The energy needs about 1/k cycles to cross; it survives about Q cycles before the coil’s resistance has turned it into heat, where Q is the quality factor that the width that is a lifetime identified with the number of cycles a resonator rings for. If Q is much larger than 1/k the energy crosses long before it is lost. If Q is much smaller, it is lost first. The ratio of the two is kQ, and that is the figure of merit.

The race can be watched in time rather than in frequency. Start with all the energy in the first circuit and none in the second, and the energy moves across and back at a slow beat frequency of about k times the resonant frequency — the beat of two modes split by k, the same arithmetic as two notes that beat. With no losses it would slosh back and forth for ever, completely, however small k was. With losses each circuit’s energy also decays, at a rate set by its Q. A load in the second circuit is a deliberate loss, and the design of a link is a choice of how fast to drain the second circuit: fast enough to catch the energy on its first arrival, before it sloshes back to be wasted in the first circuit’s resistance, and not so fast that the second circuit is damped too heavily to resonate at all.

One number

One number decides the efficiency. The best efficiency a link of two resonant circuits can reach, with its load chosen to maximise it, against the figure of merit U = k√(Q₁Q₂) on a logarithmic scale: the coupling coefficient times the geometric mean of the two coils' quality factors. The curve is η = U²/(1 + √(1 + U²))², and nothing else about the coils, the frequency or the distance enters: 0.25 per cent at U = 0.1, 17 per cent at U = 1, 52 per cent at U = 3, 82 per cent at U = 10, 94 per cent at U = 30. A weak coupling is rescued by a high Q: k = 0.003 with Q = 1,000 on each side gives U = 3 and 54 per cent, where the same coils used without tuning capacitors, as a plain transformer, deliver under one per cent, with the source driving mostly reactive current that does nothing.
Fig. 2 The best efficiency a pair of resonant circuits can reach, with the load chosen to maximise it, against U=kQ1Q2U = k\sqrt{Q_1Q_2}. The curve is U2/(1+1+U2)2U^2/(1 + \sqrt{1 + U^2})^2: 17 per cent at U = 1, 52 at U = 3, 82 at U = 10, 94 at U = 30.

Made exact, for two series-tuned circuits at their shared resonance with resistances R1R_1 and R2R_2 and a load RLR_L in the second, the efficiency is a product of two fractions. The load takes its share RL/(R2+RL)R_L/(R_2 + R_L) of the power reaching the second circuit, and the second circuit takes its share of the power the source supplies, competing with the first circuit’s own resistance. Both depend on the coupling and the resistances only through

U=kQ1Q2,U = k\sqrt{Q_1 Q_2},

and choosing the load to maximise the product gives

ηmax⁡=U2(1+1+U2)2.\eta_{\max} = \frac{U^2}{\left(1+\sqrt{1+U^2}\right)^2}.

Nothing else enters: not the frequency, not the sizes of the coils, not the distance except through k. A link with U = 1 can do no better than 17 per cent; with U = 10, 82 per cent; with U = 30, 94. The weak coupling of two coils several radii apart is rescued entirely by their quality factors, and a coupling of three parts in a thousand with Q of a thousand on each side reaches more than half.

The optimum load is itself informative. It is RL=R21+U2R_L = R_2\sqrt{1 + U^2}, larger than the coil’s own resistance by the same factor that appears in the efficiency. A load too small wastes power in the second coil’s resistance; a load too large draws too little current to take the energy out of the field before the first coil’s losses claim it. That is impedance matching of the kind the mismatch no network can remove treated, with the coupling standing where the transmission line stood.

The sender can hear the receiver

Because the coupling is the same both ways, whatever the second circuit does is felt by the first. Its current induces a voltage back in the first coil, and from the source’s side the receiver looks like an extra resistance, ω2M2/(R2+RL)\omega^2 M^2/(R_2 + R_L), appearing in series with the transmitting coil. That reflected resistance is how the power gets out: the source does work against it, and the work is what crosses.

It is also a channel for messages. When the receiver switches its load between two values, the reflected resistance changes and the current drawn from the transmitter changes with it, and the transmitter can read the pattern. Phone chargers use this load modulation for the receiver to report how much power it wants and to say when the battery is full; contactless cards use it to answer a reader without any transmitter of their own, signalling by changing how much of the reader’s field they absorb. A device with no battery and no radio talks back by being a slightly different load.

Range bought with quality

Range bought with quality factor. The best efficiency of a link between two identical resonant loops against their separation in loop radii, on a logarithmic scale, for quality factors of 30, 100, 300 and 1,000 on each side. Efficiency stays high while k·Q is well above one and collapses once the cubic fall of k overtakes the Q: the link passes 50 per cent out to 1.0 radii for Q = 30, 2.1 radii for Q = 100, 3.3 radii for Q = 300, 5.0 radii for Q = 1,000. Each tenfold gain in Q stretches the range by a little more than the cube root of ten, because of the d⁻³ law: quality is the only lever, and it is a weak one.
Fig. 3 The best efficiency of two identical resonant loops against their separation in radii, for Q = 30, 100, 300 and 1,000. The link passes 50 per cent out to 1.0, 2.1, 3.3 and 5.0 radii respectively.

Since U is a product, a weaker coupling can always be bought back with a higher quality factor. What the cube law dictates is the exchange rate. Each step outward multiplies k down by the cube of the distance ratio, so a tenfold gain in Q, which is a great deal of engineering — silver plating, careful insulation, coils wound to keep the current near the surface — extends the range at which a link reaches 50 per cent by little more than the cube root of ten, a factor of about two. Coils of thirty-centimetre radius with Q near a thousand reach about two metres.

That is what a group at MIT demonstrated in 2007, lighting a sixty-watt bulb across two metres with coils of that size resonating at about ten megahertz, at an efficiency near forty per cent. By the formula, forty per cent needs U of about 2.1, and with Q of 950 that means a coupling of about two parts in a thousand. The demonstration was widely described as a new kind of wireless power using “strongly coupled magnetic resonances”, and the coupling was, in the sense of k, extremely weak. What was strong was kQ.

The idea is older than the electronics. Nikola Tesla spent the 1890s lighting lamps without wires with resonant coils in his laboratories, and imagined a world powered by resonance on the scale of the whole Earth, which the cube law was never going to allow; his large-scale scheme, built at Wardenclyffe on Long Island, was abandoned unfinished. More modestly, two French engineers, Maurice Hutin and Maurice Le Blanc, patented in 1894 a scheme for powering electric railway vehicles by induction from a cable in the track, with tuned circuits. The physics of every one of those schemes was the formula above; what changed in the following century was the availability of cheap high-frequency switching, low-loss conductors and capacitors, and a market of small devices that wanted to be sealed.

When the coupling is too strong

Two resonators that split when they couple too well. The power reaching the load of the second of two identical resonant loops, each with a loaded quality factor of 100, against the drive frequency as a fraction of their shared resonance, for couplings k = 0.5/Q, 1/Q and 4/Q, scaled to the largest. Weakly coupled, the pair answers at the shared frequency with a single peak that grows as k². At k = 1/Q, critical coupling, the peak reaches its maximum and flattens. Coupled more strongly, the two resonators behave as one system with two modes, in phase and out of phase, at ω₀/√(1 ± k): 0.981 and 1.021 for k = 0.04, with a dip between. A charger that holds its frequency fixed sees its power fall as a phone is brought closer — one reason chargers track the frequency or adjust the load.
Fig. 4 Power reaching the load of two identical resonant loops with Q = 100, against drive frequency, for couplings of 0.5/Q, 1/Q and 4/Q. Past critical coupling the response splits into two peaks at ω0/1±k\omega_0/\sqrt{1 \pm k} — 0.981 and 1.021 for k = 0.04 — with a dip between.

The opposite limit has its own surprise. Bring the coils so close that k exceeds 1/Q and the pair stops behaving as two resonators passing energy to each other and starts behaving as one system with two modes, the currents in step and the currents opposed, at frequencies ω0/1±k\omega_0/\sqrt{1 \pm k}. The single peak of the weakly coupled pair splits into two, and at the original frequency there is a dip. A charger that holds its frequency fixed while a receiver is brought closer sees the power delivered rise, peak at critical coupling, and then fall.

This is the same splitting that two identical pendulums show when strongly coupled, and that two tails that swap everything found for two optical waveguides side by side: coupled modes repel in frequency by an amount proportional to the coupling. For a charger it is a nuisance to be managed. Chargers track the frequency to follow one of the peaks, or vary the load presented by the receiver, or limit how close the coils can come — which is part of why a phone has to be set down roughly in the right place on its pad.

The figure of merit of six inductive links. The figure of merit U = k·Q, for six inductive links with rough, typical values of the coupling coefficient and of the quality factor on each side, on a logarithmic scale, with the best efficiency it allows written beside each bar. Assumed values — toothbrush in its stand: k ≈ 0.6, Q ≈ 10, U ≈ 6.0, at most 72 per cent; phone on a charging pad: k ≈ 0.5, Q ≈ 50, U ≈ 25.0, at most 92 per cent; car on a ground pad: k ≈ 0.2, Q ≈ 300, U ≈ 60.0, at most 97 per cent; implant through skin: k ≈ 0.05, Q ≈ 80, U ≈ 4.0, at most 61 per cent; MIT, 2007: coils 2 m apart: k ≈ 0.0022, Q ≈ 950, U ≈ 2.1, at most 40 per cent; card at a reader: k ≈ 0.01, Q ≈ 30, U ≈ 0.30, at most 2 per cent. A close coupling makes a low quality factor enough, and a distant one needs a high one; the 2007 demonstration lit a 60-watt bulb two metres away with a coupling of about two parts in a thousand, by tuning coils with Q near a thousand. A contactless card works at a low efficiency and does not need more: it draws a few milliwatts at most, and the reader can afford to waste the rest.
Fig. 5 The figure of merit k·Q for six inductive links with typical values, and the best efficiency it allows: a toothbrush 72 per cent, a phone pad 92, a car over a ground pad 97, an implant 61, the 2007 demonstration at two metres 40, a contactless card 2.

The single number puts very different devices on one axis. An electric toothbrush sits in its stand with its coils a few millimetres apart and a coupling over a half, so its coils need little quality and are made cheaply, sealed in plastic against water. A phone on a charging pad, working at a little over a hundred kilohertz, couples about as well and uses better coils, which buys efficiency. A car parked over a pad in the road has a gap of fifteen or twenty centimetres, much larger coils and a coupling of a few tenths, and with quality factors of hundreds reaches the high nineties, which matters when the power is several kilowatts.

At the weak end, a medical implant powered through the skin, such as a heart pump, has to cross a centimetre or two of tissue to a coil small enough to bury, and its link is engineered to reach a moderate efficiency without heating the tissue. A contactless payment card pressed near a reader at 13.56 megahertz has a coupling of about a hundredth and a figure of merit well below one, and it does not matter: the chip needs milliwatts, and the reader can afford to waste the rest. In each case the design question is the same: what product of coupling and quality factor the application needs, and whether it is cheaper to buy it with geometry or with coils.

Buying coupling with iron

The other way to raise U is to raise k, and the close-range links do that with magnetic material rather than with quality. A phone’s receiving coil is laid on a thin sheet of ferrite, and so is the pad’s transmitting coil. The ferrite has a permeability of hundreds or thousands and almost no conductivity, so it gives the flux an easy path behind each coil, guiding the field that would otherwise spread backwards round to the front, where the other coil is. It raises each coil’s self-inductance and the mutual inductance more, so the coupling rises, and it shields what lies behind: the phone’s battery and circuit board, which as sheets of metal would otherwise be heated by eddy currents and would detune the coil.

Electric-vehicle pads go further, shaping the ferrite into bars that steer the flux across the gap between the road and the car, since raising the coupling from a tenth to a quarter is worth as much as raising both coils’ quality factors two and a half times. A transformer is the limit of the same idea, with the iron closing the magnetic circuit completely and k close to one, at which point resonance stops being necessary and a plain winding ratio does the work.

Why the body is not in the way

A charger’s field passes through whatever lies between the coils, and for the systems above that is usually air, a phone case or, for an implant, a person. Tissue is a poor conductor and almost entirely non-magnetic, so at frequencies up to a few megahertz a magnetic near field passes through it nearly undisturbed, which is why the electric toothbrush can be sealed and why implant links work. What tissue does respond to is the electric field that accompanies the magnetic one, and higher-frequency systems are designed so that the strong electric fields, between the turns of the coil and across its tuning capacitor, are kept inside the device. Exposure limits for such systems are written in terms of the induced current density and the heating, and the near field’s rapid fall with distance helps: a person standing a metre from a charging pad is in a field reduced by the cube of the distance ratio.

Metal objects are another matter. A coin or a key between a pad and a phone is a short-circuited loop of very low resistance, and the coupling, which is weak between two coils, can be strong from coil to coin; the eddy currents that slow a magnet falling down a copper pipe heat it. Charging standards include ways of detecting a foreign object, from measuring the power that goes missing to watching for the shift of resonant frequency the metal causes.

Where the circuit model stops

The figures treat each coil as an ideal inductor in series with a resistor and a capacitor, with a single fixed quality factor. Real coils have resistances that rise with frequency, as the current crowds into a thin skin of the wire and into the side of each turn facing its neighbours, and they have capacitance between turns that gives them a self-resonance of their own; the MIT coils used that self-resonance instead of an added capacitor. The coupling figure assumes thin circular loops on a common axis. Offset or tilted coils couple less, and the coupling can pass through zero for a coil turned edge-on, which is part of why a phone must be laid flat. The efficiency figure is the best a link can reach with its load matched, which real chargers approach only over a range of positions and loads.

The model is for the near field, coils much smaller than the wavelength and closer together than a wavelength. Beyond that the stored field is gone and only radiation remains, which spreads as the inverse square and is collected in proportion to an antenna’s area. Power beamed by microwaves to a receiving antenna obeys entirely different rules and has its own efficiencies.

Still open: how much further a near field can be made to reach

Beyond a few coil radii the cube law wins against any achievable quality factor, and attempts to stretch the near field have looked to resonators with higher Q, such as superconducting coils, which raise Q into the tens or hundreds of thousands at the cost of refrigeration, and to structures made of many small resonators — metamaterial slabs — that can concentrate or relay the near field from one coil to the next. Relays of resonant coils have carried power along a chain, each passing to its neighbour. How far such schemes can be pushed while keeping the efficiency, the safety of the surrounding fields and a tolerance for the receiver moving about, and whether they can compete with a cable over any useful distance, is the subject of continuing work rather than settled engineering.

The habit worth carrying away is to ask what a resonance actually changes. Tuning two coils does not strengthen their coupling, which is fixed by geometry and falls as d−3d^{-3}; it lets a fraction k of the energy accumulate over Q cycles, so the best efficiency depends on U=kQ1Q2U = k\sqrt{Q_1Q_2} alone — 17 per cent at U = 1, 94 per cent at U = 30. A weak link with a long memory beats a strong link with a short one, and when the link is too strong for the memory, it splits.

Part 7 of 7

This essay is one argument about Induction. The others:

The drag peaks and then gives way to lift. The two forces on a magnet moving parallel to a thin conducting sheet, as fractions of the repulsion a perfect mirror image would exert, against the magnet's speed divided by the sheet's characteristic speed w = 2/μ₀σd, on a logarithmic axis. Drag grows in proportion to the speed at first, peaks at 1.27 w with 0.300 of the full repulsion, and then falls as w/v. Lift starts as the square of the speed, equals drag at v = w, where both are 0.293, and approaches the full repulsion at high speed: 0.900 at 10 w. The same eddy currents cause both. What changes with speed is how much of the image they form has time to sink away before the magnet has passed. The drag that falls as the magnet speeds up Part 6 — Drop a magnet down a copper pipe and it creeps, because the currents it stirs up in the metal pull back on it in proportion to its speed. Push it faster and the proportion fails: the drag rises to a peak and then falls away, while a second force, pointing straight up, grows to replace it. A magnet moving fast enough over a sheet of aluminium is not braked but held up, with lift thirty times its drag at the speed of a fast train, and the reason is that at high speed the metal has no time to forget the magnet has passed.

Where this goes next

Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields. The coupling that is the same both ways A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand. The two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else. The two pendulums that will not stop swapping Coupled oscillators joined by a weak spring appear to hand energy back and forth. Nothing is handed anywhere: the system has only a pair of motions that keep their shape, at √(k/m) and √((k+2kc)/m), and the apparent traffic is the beat between them — 51 swings from one handover to the next at a coupling of one part in fifty. Extend the same arithmetic to N masses and it produces a dispersion relation with a hard ceiling, near 7 THz in copper. The same number read off a decay and off a linewidth. A lightly damped oscillator released and left alone, above, and the power spectrum of exactly those samples, below. The decay falls to 1/e of its starting amplitude after 8.0 cycles, which makes the quality factor π times that, or 25.0. The spectrum peaks at 1.0000 radians per second and falls to half its power 0.04001 radians per second wide, which makes the quality factor the peak divided by the width, or 25.0. The two disagree by 0.02 per cent, which is the resolution of the frequency grid rather than a difference in the physics. They cannot disagree by more, because they are the same statement: a resonance is narrow because its ringing is long, and the transform that turns one into the other is not an approximation but an identity. A measurement of either is a measurement of both — which is why a bell can be characterised by hitting it and listening, or by driving it and sweeping, and why the two instruments never argue. The width that is a lifetime Hit a bell and time how long it rings; drive it and measure how narrow its response is. The two numbers are the same number, and they cannot disagree — not because the physics conspires but because a decay and a linewidth are one function seen in two coordinate systems.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Coupling coefficientImpedance matchingMode splittingMutual inductanceNear fieldQuality factorResonanceWireless power