What is the kQ figure of merit?
For two resonant coils with coupling coefficient k and unloaded quality factors Q₁ = ωL₁/R₁ and Q₂ = ωL₂/R₂, the link figure of merit is
With both sides tuned to resonance and the load chosen optimally, the best coil-to-coil efficiency the link can reach is
This is the standard result for a two-coil resonant inductive link. Zargham and Gulak analyse the maximum achievable efficiency of near-field links in detail, and Ohira discusses why the kQ product, rather than k or Q alone, is the quantity that matters. The value of η_max does not depend on which side is series- or parallel-tuned once the load is optimal; what changes is the load value that achieves it.
How efficient is a link for a given k and Q?
| k | Q₁ = Q₂ | U | η_max | R_L,opt / R₂ |
|---|---|---|---|---|
| 0.01 | 100 | 1 | 17.2 % | 1.41 |
| 0.05 | 100 | 5 | 67.2 % | 5.1 |
| 0.1 | 100 | 10 | 81.9 % | 10 |
| 0.2 | 100 | 20 | 90.5 % | 20 |
| 0.1 | 300 | 30 | 93.6 % | 30 |
| 0.3 | 300 | 90 | 97.8 % | 90 |
Two consequences are worth remembering. First, with Q = 100 per coil, reaching 90 % needs only k = 0.19, which is why loosely coupled resonant links work at all; Kurs et al. demonstrated mid-range transfer with strongly resonant high-Q coils on this principle. Second, the curve is very flat at high U: going from 94 % to 98 % takes more than three times the U, so beyond a point losses elsewhere in the chain dominate.
Why does coil Q matter as much as coupling?
Q = ωL/R compares the energy a coil stores with the energy it loses per cycle. In a resonant link the transmitter current circulates many times before the receiver absorbs the energy, so every ohm of coil resistance is paid for repeatedly. The geometry that sets k is often fixed by the product (a phone, a vehicle floor, an implant), whereas Q can still be improved by conductor choice, winding layout and frequency.
Coupling and quality factor come from different places, which is why it pays to treat them separately. The coupling coefficient is a property of geometry alone for air-core coils: coil shapes and sizes, the gap and the alignment. AirInduct Sim computes it, together with M, L_tx and L_rx, from closed-form Biot–Savart and Neumann expressions for eight coil topologies, and its sweeps and optimizer search the geometry for the highest k within your size limits. Quality factor depends on the conductor and its losses at the operating frequency: wire diameter, skin and proximity effect, litz construction, and any nearby lossy material. Those losses are best measured on a prototype or computed with a model that resolves the current distribution in the conductor. Combining the simulated k with a measured or estimated Q in U = k√(Q₁Q₂) gives a realistic ceiling on the link efficiency before any power electronics are designed.
How do skin depth and litz wire affect Q?
At high frequency, current crowds towards the conductor surface. The skin depth, the depth at which current density falls to 1/e of its surface value, is
| Frequency | Copper skin depth δ |
|---|---|
| 85 kHz (SAE J2954 nominal) | 0.227 mm |
| 100 kHz | 0.209 mm |
| 360 kHz (Qi2 MPP) | 0.11 mm |
| 6.78 MHz (ISM band) | 25.4 µm |
In a multi-turn coil the proximity effect adds to this: the field of neighbouring turns pushes current to one side of each conductor, and in tightly wound coils it can cost more than the skin effect itself. Litz wire addresses both by dividing the conductor into many insulated strands, each thinner than the skin depth, twisted so that every strand occupies every position in the bundle. It works well in the tens to hundreds of kilohertz. At 6.78 MHz the skin depth is only about 25 µm, strands would have to be impractically fine, and many designs use solid wire, copper tube or wide PCB traces instead. Sullivan gives a cost-aware method for choosing strand number and diameter.
Worked example: estimating Q and η for a 40 mm coil pair
Two circular coils, radius 40 mm, 10 turns of 1 mm solid copper wire, 10 mm apart, operated at 125 kHz. All values are computed on this page.
| Quantity | Value |
|---|---|
| Self-inductance L (thin-wire formula) | 18.11 µH |
| Coupling coefficient k at 10 mm | 0.417 |
| Wire length per coil | 2.51 m |
| DC resistance | 55.17 mΩ |
| Skin depth at 125 kHz | 0.187 mm |
| Resistance with skin effect only | 90.76 mΩ |
| Q upper bound (skin effect only) | 157 |
| η_max if Q₁ = Q₂ = 100 | 95.3 % |
| η_max if Q₁ = Q₂ = 50 | 90.9 % |
The skin-only resistance treats the current as flowing in a surface layer of thickness δ, which is a fair approximation when the wire diameter is several skin depths. It ignores proximity effect between turns, so the real Q of this coil will be lower than 157; measure it with an impedance analyser before relying on a number. Even at Q = 50 the link’s ceiling is high because k is large; the same coils at a 40 mm gap would need much higher Q for the same efficiency. Try other values in the efficiency calculator, and see coupling coefficient for how k falls with distance.
References
- M. Zargham and P. G. Gulak, “Maximum achievable efficiency in near-field coupled power-transfer systems,” IEEE Trans. Biomedical Circuits and Systems 6(3), 228–245 (2012). doi.org/10.1109/TBCAS.2011.2174794
- G. A. Covic and J. T. Boys, “Inductive power transfer,” Proceedings of the IEEE 101(6), 1276–1289 (2013). doi.org/10.1109/JPROC.2013.2244536
- A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007). doi.org/10.1126/science.1143254
- T. Ohira, “What in the world is Q?,” IEEE Microwave Magazine 17(6), 42–49 (2016). doi.org/10.1109/MMM.2016.2538512
- C. R. Sullivan, “Optimal choice for number of strands in a litz-wire transformer winding,” IEEE Trans. Power Electronics 14(2), 283–291 (1999). doi.org/10.1109/63.750181
- J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999), ISBN 978-0-471-30932-1. www.wiley.com/en-us/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321