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9.4 — Wireless Charging and Resonant Induction
Put a phone on a pad and it charges. No connector, no plug, no wear, and it works through a case.
It is a transformer with an air gap, and everything difficult about it comes from that gap. Chapter 9.1's transformer had a magnetic core coupling its windings at over 99%. Here the coupling is 20 to 50%, and recovering usable efficiency from that is what resonance is for.
1. The basic transfer
A coil in the pad, driven with alternating current, produces an alternating magnetic field. A coil in the phone sits in that field, and Faraday's law induces a voltage:
V_2=-M\frac{dI_1}{dt}
where M is the mutual inductance — how much of one coil's flux links the other.
The coupling coefficient:
k=\frac{M}{\sqrt{L_1L_2}}, \qquad 0\le k\le1
| Arrangement | k |
|---|---|
| Transformer with a shared core | 0.99+ |
| Phone on a Qi pad | 0.3–0.6 |
| Coils 5 cm apart | 0.1–0.2 |
| Coils 20 cm apart | 0.01–0.05 |
Coupling falls roughly as 1/d^3 once the separation exceeds the coil radius, which is the near-field relationship of Chapter 8.4.
A poorly coupled transformer is a very bad transformer. Most of the primary's flux does not reach the secondary at all, so most of the primary current does no useful work and simply heats the coil.
That fraction has a name and a symbol you will meet in every datasheet: the coupling coefficient k, running from 0 (the coils know nothing of each other) to 1 (every line of flux from one passes through the other). A transformer with a shared iron core, from Chapter 9.1, achieves k above 0.99, which is why it is so efficient. Two coils in air, separated by a phone case, manage 0.5 to 0.7 at best — and that drops sharply the moment the phone slides off centre, which is exactly the annoyance of a charging pad that stops working when nudged.
The falloff is steep. Coupling drops roughly as the cube of separation, so doubling the gap cuts the linked flux by about eight. This is why "wireless" charging requires contact, and why the schemes promising room-scale power keep failing: the geometry, not the electronics, is what defeats them. The next section shows the one trick that partly rescues a weak coupling.
2. Why resonance rescues it
Tune both coils to the same resonant frequency with a series or parallel capacitor.
f_0=\frac{1}{2\pi\sqrt{LC}}
What that buys is exactly the current magnification of Chapter 1.6. At resonance the coil's reactance and the capacitor's cancel, so the source sees only the small remaining resistance, and a large current circulates in the tuned loop:
I_{circulating}=Q\times I_{source}
A large circulating current means a large magnetic field — for the same power drawn from the source. So the primary produces a much stronger field than it otherwise could, and the secondary, also resonant, extracts far more from that field than an untuned coil would.
The figure of merit that decides whether it works:
k\sqrt{Q_1Q_2}
Read it as the product of how well the coils are coupled and how good each resonator is. Poor coupling can be compensated by high Q, which is precisely the point.
| k\sqrt{Q_1Q_2} | Maximum efficiency |
|---|---|
| 1 | 17% |
| 3 | 38% |
| 10 | 67% |
| 30 | 85% |
| 100 | 95% |
Worked example. Qi charging at k=0.4 with Q=100 on both coils:
k\sqrt{Q_1Q_2}=0.4\times100=40 \;\Rightarrow\; \approx88\%
And loosely coupled at 5 cm, k=0.15 with Q=300:
0.15\times300=45 \;\Rightarrow\; \approx89\%
Nearly the same efficiency at three times the distance, purely from higher Q. That is the entire argument for resonant wireless power, and it is why the loosely coupled systems used for vehicles operate at high Q and low frequency where copper losses are manageable.
The theoretical maximum:
\eta_{max}=\frac{k^2Q_1Q_2}{\left(1+\sqrt{1+k^2Q_1Q_2}\right)^2}
3. The Qi standard
The dominant standard, from the Wireless Power Consortium, and it is worth walking through because every detail is a solution to a specific problem.
Frequency: 87 to 205 kHz, and the transmitter varies it to control power.
Coils: typically 10 to 20 turns of Litz wire — many fine strands individually insulated and woven — because at 150 kHz the skin depth in copper is
\delta=\sqrt{\frac{\rho}{\pi f\mu}}=\sqrt{\frac{1.68\times10^{-8}}{\pi\times150{,}000\times4\pi\times10^{-7}}}=0.17\ \text{mm}
Solid wire thicker than about 0.34 mm would have most of its cross-section carrying no current. Litz wire keeps every strand thinner than the skin depth, so all the copper works.
Ferrite shielding behind each coil, doing two jobs: concentrating the flux towards the other coil rather than letting it spread, and shielding the phone's electronics and battery from the field.
The protocol
1 — Analog ping. The transmitter pulses briefly every few hundred milliseconds and watches its own coil's impedance. A receiver's presence loads the coil detectably — the same load-sensing principle as NFC in Chapter 8.4.
2 — Digital ping. A longer pulse, enough to power up the receiver's electronics.
3 — Identification. The receiver reports its identity and maximum power.
4 — Negotiation. They agree on a power level.
5 — Power transfer, with continuous feedback.
Communication is by load modulation — the receiver switches a capacitor or resistor across its coil, and the transmitter detects the change in its own current. The same trick as an NFC card, at 2 kbit/s. No separate radio is involved, which keeps the receiver simple and cheap.
Power control is a closed loop: the receiver measures its rectified voltage, compares it with what it wants, and sends an error packet. The transmitter adjusts by shifting frequency away from resonance, changing duty cycle, or changing its input voltage. This is Chapter 6.1's feedback loop, running over a load-modulated link.
Foreign object detection
The safety function, and it is essential. A metal object between the coils absorbs energy through eddy currents and heats up — a coin can reach 100 °C in seconds.
Two detection methods, used together:
Power loss accounting. The transmitter knows how much it is sending; the receiver reports how much it is receiving. An unexplained discrepancy means something else is absorbing it.
P_{transmitted}-P_{received}\gt\text{threshold} \;\Rightarrow\; \text{stop}
Quality factor measurement. Before transferring power, the transmitter measures its coil's Q. A metal object lowers it measurably, so a Q below the expected value means an obstruction.
Neither is perfect. Power loss accounting must distinguish a foreign object from normal system losses, and the threshold is a compromise — too tight and legitimate charging is interrupted, too loose and a small object goes undetected. This is why standards work on foreign object detection has continued for a decade, and why higher-power systems need better methods than 5 W ones.
4. Power levels and why they are limited
| Standard | Power | Use |
|---|---|---|
| Qi BPP | 5 W | basic phones |
| Qi EPP | 15 W | modern phones |
| Qi2 | 15 W + magnets | aligned phones |
| Proprietary | 50–120 W | some manufacturers |
| AirFuel | up to 50 W | loosely coupled |
| SAE J2954 | 3.7–22 kW | electric vehicles |
Why phones stopped around 15 W in the standard, despite wired charging reaching 100 W:
Heat. At 75% efficiency, 15 W of delivered power means 5 W dissipated — much of it inside the phone, right next to the battery. And Chapter 9.3 established that heat is the primary cause of calendar ageing. A phone that wireless-charges warm every night ages its battery measurably faster.
Alignment. Efficiency depends strongly on coil alignment, and a phone dropped onto a pad at an angle may transfer at 50% or not at all.
This is what Qi2's magnets solve, and it is a genuinely good piece of engineering: rather than making the electronics tolerate misalignment, remove the misalignment. A ring of magnets aligns the coils within a millimetre every time, which lets the system be designed for the aligned case rather than for the worst case.
5. Electric vehicle charging
The same principle at a thousand times the power, and the differences are instructive.
Frequency: 85 kHz, standardised in SAE J2954. Lower than Qi because at kilowatt levels the switching losses at higher frequency become unmanageable.
Gap: 100 to 250 mm between the ground pad and the vehicle pad.
Coupling: k = 0.15 to 0.3 — much worse than a phone.
Compensated by very high Q, achieved with large Litz coils and low-loss capacitors, giving 85 to 93% efficiency at 11 kW.
Alignment is the real problem. A 30 cm lateral offset can halve the transfer, so vehicles use magnetic field sensing to guide the driver, and some use the field itself for the final positioning.
Foreign object detection is far more critical here. A steel object in an 11 kW field reaches red heat in seconds. Systems use dedicated sensing coil arrays across the pad surface, plus living object detection using radar or capacitive sensing — because a cat sleeping on a warm charging pad is a foreseeable event.
Why it has not displaced cables. A cable is 95% efficient, costs a hundred pounds, and works today. Wireless is 90% efficient, costs several thousand, and requires infrastructure in the ground.
Where it makes genuine sense is where the cable is the problem: buses that charge at every stop for thirty seconds, taxi ranks, and — the case with the strongest argument — dynamic charging embedded in a road, which would let a vehicle carry a much smaller battery. Several countries have test sections.
6. The other approaches
Capacitive coupling. Use an electric field between plates instead of a magnetic field between coils. Simpler and lighter, with no magnetic core, and it needs very small gaps or very high voltages. Used in some rotating and sliding applications where a coil would not fit.
Radiative — microwaves or lasers. Genuinely long range, and the efficiency is poor and the safety implications severe. A beam that can deliver useful power at ten metres is a beam you do not want to walk through.
Its one serious application is space-based solar power, where a satellite would collect sunlight continuously and beam it down at 2.45 GHz to a large rectifying antenna. The power density at the ground would be kept below sunlight levels by making the receiver enormous — kilometres across. The physics works; the economics have never come close.
Ultrasonic. Transmit acoustic energy and convert it with a piezoelectric receiver. Milliwatts at best, and mainly of interest for implanted medical devices where nothing else can pass through tissue.
Backscatter and energy harvesting. Not transferring power so much as scavenging it — from ambient radio, vibration, light, or thermal gradients. Microwatts, and enough for a sensor that sleeps almost permanently, which is exactly the Chapter 3.6 duty-cycling argument taken to its conclusion.
7. Is it safe
The exposure limits come from ICNIRP guidelines, and for the frequencies involved the concern is induced currents in tissue rather than heating.
At 100 kHz the public exposure limit is 27 µT.
A Qi pad produces 3 to 5 mT between the coils — far above the limit — and it falls as 1/r^3. At 10 cm from the pad it is under the limit, and at 30 cm it is a thousand times under it.
So the field is intense in the gap and negligible everywhere else, which is a direct consequence of near-field physics rather than a design choice.
Pacemakers. The main documented concern. Manufacturers advise keeping wireless chargers, and phones with charging magnets, at least 15 cm from an implanted device. The risk is that a strong static or low-frequency field can switch some pacemakers into a fixed-rate test mode.
Metal objects. Genuinely the main practical hazard, and it is why foreign object detection is mandatory rather than optional.
Radio frequency interference. A Qi charger's switching harmonics extend into the AM broadcast band, and poorly designed ones are audible on a nearby radio. This is a compliance requirement rather than a safety one, and it is why shielding and edge-rate control matter.
8. The honest assessment
Where wireless charging is genuinely better:
- Sealed devices — a hearing aid, a toothbrush, an implant, a sensor in a wet or explosive environment. No connector means no path for water and no spark.
- High connect-disconnect cycles — a warehouse robot docking a hundred times a day would destroy a connector.
- Convenience where the loss does not matter — a phone on a bedside pad overnight.
Where it is worse:
- Efficiency. 70 to 85% against 95% wired. A phone charged wirelessly every day for three years wastes a few kilowatt-hours — trivially small in absolute terms, and multiplied by a billion phones it is not.
- Heat, which shortens battery life for the reasons Chapter 9.3 gave.
- Speed. Wired reaches 100 W; wireless standards stop at 15.
- Cost and weight — two coils, ferrite, and control electronics on both sides.
The realistic summary. Wireless charging is a convenience feature whose real engineering justification is sealed enclosures and high cycle counts, and whose consumer popularity is about not handling a connector. It will not replace wired charging for anything that needs speed or efficiency, and it has permanently replaced it for anything that needs to be waterproof.
Chapter 9.5 closes Part 9 with the circuits that convert power between forms — the switching converters that made everything in this Part practical, and that turned a half-kilogram charger into something that fits in a plug.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Wireless power
k=\frac{M}{\sqrt{L_1L_2}}
f_0=\frac{1}{2\pi\sqrt{LC}}, \qquad Q=\frac{\omega_0L}{R}
The figure of merit:
k\sqrt{Q_1Q_2}
\eta_{max}=\frac{k^2Q_1Q_2}{\left(1+\sqrt{1+k^2Q_1Q_2}\right)^2}
| k\sqrt{Q_1Q_2} | \eta_{max} |
|---|---|
| 1 | 17% |
| 10 | 67% |
| 30 | 85% |
| 100 | 95% |
Poor coupling is compensated by high Q, which is why resonance makes loosely coupled transfer practical.
k\propto\frac{1}{d^3} \ \text{beyond the coil radius}
Skin depth, which sets Litz strand size:
\delta=\sqrt{\frac{\rho}{\pi f\mu}}
At 150 kHz in copper: 0.17 mm, so strands must be under about 0.34 mm.
Foreign object detection:
P_{tx}-P_{rx}\gt\text{threshold} \;\Rightarrow\; \text{stop}
What the next chapter fixes
Every supply so far delivers whatever voltage its source happens to provide. Chapter 9.5 covers the circuits that change it — turning 12 V into 5 V, or DC into mains AC — by switching fast rather than by dissipating the difference, which is why a modern charger is the size it is.