Wireless power transfer (WPT) is, at its core, a controlled alternating-magnetic-field energy channel between two detached LC resonant cavities. This series starts from the field equations and builds the complete mathematical chain — coupling coefficient, quality factor, transfer efficiency — that we use in production: levitated lighting, cross-material power delivery, long-range sensor power, and cable-free decorative lighting.
1. How a magnetic field crosses space
1.1 The on-axis field of a circular coil
Integrating the Biot–Savart law, an $N$-turn circular coil of radius $r$ carrying current $I$ produces, at axial distance $d$, a flux density of:
$$B(d) = \frac{\mu_0 N I r^2}{2\left(r^2 + d^2\right)^{3/2}}$$
with $\mu_0 = 4\pi \times 10^{-7}\ \mathrm{H/m}$. Hidden in this formula is the first engineering law of WPT:
$$\frac{B(d)}{B(0)} = \left(\frac{r^2}{r^2 + d^2}\right)^{3/2}$$
The field's decay rate with distance is set by the coil radius alone. Double the radius and the field retention at the same distance jumps a tier. This is the mathematical basis for our design rule: we trade coil area for distance — we never push voltage to brute-force it.
1.2 Induction and mutual inductance
The EMF induced in a receiving coil follows Faraday's law:
$$\mathcal{E}_2 = -N_2 \frac{d\Phi_{21}}{dt} = -j\omega M I_1$$
where the mutual inductance $M$ is defined by the Neumann formula:
$$M = \frac{\mu_0}{4\pi} \oint_{C_1} \oint_{C_2} \frac{d\vec{l}_1 \cdot d\vec{l}_2}{|\vec{r}_1 - \vec{r}_2|}$$
In engineering we normalize this as the coupling coefficient:
$$k = \frac{M}{\sqrt{L_1 L_2}}, \quad 0 \le k \le 1$$
2. Resonance: why two cavities must sing at the same frequency
2.1 The resonance condition
A single-turn air-core coil presents only a few ohms of inductive reactance at tens of kilohertz — driving it directly is hopelessly inefficient. The fix is to make both transmitter and receiver into LC resonant cavities, tuned to the same frequency:
$$f_0 = \frac{1}{2\pi\sqrt{L_1 C_1}} = \frac{1}{2\pi\sqrt{L_2 C_2}}$$
At resonance the imaginary part of the loop impedance vanishes:
$$Z(\omega_0) = R + j\left(\omega_0 L - \frac{1}{\omega_0 C}\right) = R$$
Energy sloshes losslessly between electric and magnetic fields, and the circulating current is amplified by the quality factor $Q$:
$$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R}$$
2.2 The exact solution for link efficiency
For a weakly coupled pair of resonant cavities, define the dimensionless coupling figure of merit:
$$U = k\sqrt{Q_1 Q_2}$$
The maximum link efficiency then has a closed-form solution:
$$\eta_{link}^{max} = \frac{U^2}{\left(1 + \sqrt{1 + U^2}\right)^2}$$
With $Q_1 = Q_2 = 80$ (typical for Litz wire plus NPO capacitors), the sensitivity of efficiency to coupling is immediate:
| Coupling $k$ | $U$ | $\eta_{link}^{max}$ |
|---|---|---|
| 0.02 | 1.6 | 31% |
| 0.05 | 4.0 | 61% |
| 0.10 | 8.0 | 78% |
| 0.20 | 16.0 | 88% |
Efficiency is set not by frequency, but by the product $k \cdot Q$. This yields the design philosophy that runs through the whole series: every structural technique — larger coils, flux guides, alignment, shorter gaps — serves $k$; every material technique — Litz wire, C0G/NPO capacitors, low-loss formers — serves $Q$.
2.3 Critical coupling and frequency splitting
When $k > k_c = 1/\sqrt{Q_1Q_2}$ the system enters the over-coupled region and the resonance peak splits in two:
$$f_{\pm} = \frac{f_0}{\sqrt{1 \pm k}}$$
The double peak is the physical root of "closer distance, lower efficiency" — and the reason near-range applications need detuning protection: bring the coils too close and the resonant voltage on the receiver can spike high enough to kill downstream components. Every "it burned when I placed it too close" failure in the industry traces back to this.
The Wireless Power Transfer Engineering Series — Part 1 · Part 2 · Part 3 · Part 4