In Part 1 we built the field equations and the $k \cdot Q$ efficiency rule. Here we turn to the two variables that dominate real product design: distance and frequency.
3. The distance–power scaling law
Substituting the field decay from §1.1 into power flow ($P \propto B^2$) gives the axial power retention:
$$\eta_{space}(d) = \left(\frac{r^2}{r^2 + d^2}\right)^{3}$$
Computed for three transmitter diameters (normalized to $d \to 0$):
| Gap $d$ | Ø200 mm | Ø100 mm | Ø80 mm |
|---|---|---|---|
| 25 mm | −0.8 dB (83%) | −2.9 dB (51%) | −4.3 dB (37%) |
| 50 mm | −2.9 dB (51%) | −9.0 dB (12.5%) | −12.3 dB (6%) |
| 100 mm | −8.5 dB (14%) | −17.8 dB (1.7%) | −21.6 dB (0.7%) |
At the same 50 mm gap, a Ø200 coil delivers 8× the power of a Ø80 coil. This is the fundamental reason consumer products moving from "contact charging" to "power across air" must re-engineer their coil systems from scratch.
4. Frequency engineering: every kHz has a price
4.1 Skin effect and Litz wire
Alternating current crowds toward the conductor surface, with skin depth:
$$\delta = \sqrt{\frac{\rho}{\pi f \mu_0}}$$
For copper at 220 kHz, $\delta \approx 0.14\ \mathrm{mm}$ — the entire core of a thick wire is wasted. Litz wire keeps each strand at $d_s \le 2\delta$ and transposes the bundle so AC resistance approaches DC resistance; it is the only correct answer for high-$Q$ windings. Our fixed ranking for winding selection: multi-strand Litz > served (yarn-wrapped) Litz > plain enamelled wire.
4.2 Eddy-current loss: the frequency-squared penalty
Any conductor sitting in an alternating field develops eddy currents. The loss density in a thin plate:
$$P_{eddy} = \frac{\pi^2 t^2 f^2 B_{max}^2}{6\rho} \quad [\mathrm{W/m^3}]$$
Loss scales with frequency squared — the invisible ceiling on frequency choice. In our levitation products the floating permanent magnet (sintered NdFeB, $\rho \approx 1.5\ \mu\Omega\cdot m$) and the drive core sit directly in the transmit field. Measured on the same transmitter: bare coil input 0.75 W; installed in the levitation mechanism, over 2 W — the difference becomes metal heating. The engineering toolbox against eddy loss: push frequency down, shape the field geometry, use high-resistivity magnetic materials, and copper shielding that is transparent to the static field.
4.3 The engineering convergence of the frequency window
Weighing AM broadcast avoidance, device losses, EMC certification paths, and human exposure limits, the industry has converged on 110–205 kHz (the WPC Qi band) up to roughly 230 kHz. Our production anchor values:
$$L \cdot C \approx 500\ \mathrm{\mu H \cdot nF} \quad \Longleftrightarrow \quad f_0 \approx 220\ \mathrm{kHz}$$
The Wireless Power Transfer Engineering Series — Part 1 · Part 2 · Part 3 · Part 4