What determines wireless power transfer efficiency?

The coupling coefficient k between transmitter and receiver coils, and each coil's Q factor, which measures how low its losses are. With resonant coupling, maximum efficiency is η = k²Q₁Q₂/(1+√(1+k²Q₁Q₂))², rising with the product kQ. 90% needs kQ ≈ 19 and 95% kQ ≈ 39.

How much does efficiency drop as coils separate?

For coils of radius 0.15 m, k falls steeply: 0.26 at 5 cm, 0.084 at 15 cm, 0.024 at 30 cm. With Q = 300, efficiency is 97%, 92% and 76%; with Q = 100, 93%, 79% and 45%. Once spacing exceeds the radius, coil Q dominates efficiency.

With small coupling, reactive current from coil self-inductance becomes large, raising losses so little power gets through. Resonating each coil with a capacitor cancels the inductive effect, allowing efficient transfer even with weak coupling. EV charging uses frequencies near 85 kHz; the 88 μH coil here resonates with about 40 nF.

Wireless Power Transmission

Category: Electromagnetic Analysis – Power Electronics | Revised 2026-10-01

Wireless Power Transfer

Theory: coupling coefficient and Q

Overview

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Wireless EV charging is said to exceed 90% efficiency with coils more than 10 cm apart. How?

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As coils separate, the fraction of transmitter flux linking the receiver — the coupling coefficient k — becomes quite small, far weaker than a transformer's k of 0.99 or more. Efficiency is still achievable by resonating both coils with capacitors and using low-loss (high-Q) coils. Maximum efficiency depends on the product of k and Q, so high Q compensates for small k. Design obtains k from electromagnetic analysis of coil shape and spacing, and estimates Q from winding and ferrite losses.

Maximum efficiency

$$ \eta_{max} = \frac{k^2 Q_1 Q_2}{\left(1 + \sqrt{1 + k^2 Q_1 Q_2}\right)^2},\qquad k = \frac{M}{\sqrt{L_1 L_2}},\qquad Q = \frac{\omega L}{R} $$

Efficiency with both sides resonant and optimal load R_L = R₂√(1 + k²Q₁Q₂). M is mutual inductance and R coil resistance (loss).

Mutual inductance of coaxial circular coils

$$ M = N_1 N_2\,\mu_0 a\left[\left(\frac{2}{\kappa} - \kappa\right)K(\kappa) - \frac{2}{\kappa}E(\kappa)\right],\qquad \kappa^2 = \frac{4a^2}{4a^2 + d^2} $$

a is coil radius, d spacing, K and E complete elliptic integrals of the first and second kind. Self-inductance L ≈ N²μ₀a(ln(8a/r) − 2) (wire radius r = 1.5 mm) is 88.3 μH for 10 turns at 0.15 m radius.

Coffee Break Trivia

From Tesla's dream to WiTricity

The idea of sending power without wires goes back to Tesla's experiments in the late 19th century. Short-range inductive charging has long powered devices like electric toothbrushes, but in 2007 Kurs and colleagues at MIT lit a 60 W bulb about 2 m away with two resonant coils, showing the potential of magnetic resonance. SAE J2954 later standardised the 85 kHz band for EV charging, with practical systems from a few to tens of kilowatts.

Worked examples

Example 1: spacing, coupling and maximum efficiency (radius 0.15 m, 10 turns)

Spacing	Spacing/radius	Coupling k	Efficiency (Q = 100)	Efficiency (Q = 300)
5 cm	0.33	0.264	92.7%	97.5%
10 cm	0.67	0.141	86.8%	95.4%
15 cm	1.00	0.084	78.8%	92.4%
20 cm	1.33	0.053	68.8%	88.2%
30 cm	2.00	0.024	44.6%	75.9%
45 cm	3.00	0.009	15.5%	49.7%

Example 2: required kQ and resonance

Item	Value
kQ for 90% efficiency	19.0
kQ for 95% efficiency	39.0
Resonant capacitor at 85 kHz	39.7 nF
Coil resistance for Q = 300	0.157 Ω
Coil resistance for Q = 100	0.472 Ω

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To keep 92% rather than 79% at 15 cm, coil resistance must be held to about 0.16 Ω.

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Yes. At 85 kHz the skin effect crowds current to the conductor surface, so a thick solid wire's resistance is several times its DC value; Litz wire, twisted fine insulated strands, is used instead. A ferrite plate behind the coil concentrates flux, raising k and reducing leakage into the vehicle body. But ferrite has losses, and eddy currents in body steel lower Q. The simple formulas in the table omit these effects, so electromagnetic analysis of the real geometry is used to obtain k and component losses. Tolerance to misalignment matters too: a coil whose k changes greatly with a few centimetres of lateral offset is hard to use.

Design and analysis workflow

Set target kQ from power, spacing and misalignment range.
Vary coil shape and ferrite layout, computing k by electromagnetic analysis.
Derive Q from Litz-wire AC resistance, ferrite loss and eddy losses in surrounding metal.
Check resonant capacitors and load matching with circuit simulation.
Confirm leakage fields meet human-exposure limits.

Coffee Break Trivia

“Efficiency dropped once mounted on the car”

A receiver coil reached 93% at 15 cm on the test bench but dropped to 88% once mounted on the vehicle, and its surroundings heated up. The floor-pan steel was the cause: part of the coil's flux entered it, inducing eddy currents that lowered Q. Electromagnetic analysis including the steel reproduced the extra loss and efficiency drop. Enlarging the ferrite plate and adding an aluminium shield to keep flux off the steel restored efficiency to 92%.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Q from DC resistance	Efficiency overestimated	Use AC resistance
Omitting nearby metal	Eddy losses missed	Include the body
Ignoring misalignment	Real-world efficiency drop	Compute k with offset
Ignoring load matching	Peak efficiency not reached	Match optimal load
No leakage-field check	Exposure limits exceeded	Compute surrounding field

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