Wireless Power Transfer: Sending Energy Through Empty Space

The governing physics: mutual inductance and Faraday's law

Strip WPT to its core and it is a transformer with an air gap. A primary coil driven by an alternating current I₁ produces an oscillating magnetic field; the fraction of that field's flux threading a secondary coil couples the two through the mutual inductance M. Faraday's law then gives the voltage induced in the receiver:

• EMF₂ = −M · dI₁/dt — the induced voltage is proportional to how fast the primary current changes.
• For a sinusoidal drive I₁ = I₀ sin(ωt), the peak induced EMF is ωMI₀, so higher frequency ω directly buys more voltage.

The mutual inductance factorizes cleanly through the dimensionless coupling coefficient k:

• M = k√(L₁L₂), with 0 ≤ k ≤ 1.
• k = 1 is a perfect iron-cored transformer (all flux shared). A phone on a charging pad sits at k ≈ 0.3–0.7; the MIT resonance experiment across 2 m ran at k ≈ 0.0001–0.01 — almost no shared flux at all.

This is why naive induction dies with distance: for two coaxial coils of radius a separated by R ≫ a, the field is a magnetic dipole falling off as 1/R³, so k collapses roughly as (a/R)³. Double the gap and the coupling drops eight-fold. The genius of resonance is that it lets a vanishing k still deliver real power.

Resonance: how a coupling of 0.001 still moves kilowatts

The trick discovered by Aristeidis Karalis, John Joannopoulos and Marin Soljačić at MIT (Kurs et al., Science, 2007) is strongly-coupled magnetic resonance. Add a capacitor to each coil so it becomes an LC resonator tuned to the same angular frequency ω₀ = 1/√(LC). Now the relevant comparison is not k against 1, but the coupling rate κ against the loss rates.

Coupled-mode theory reduces the whole system to two damped oscillators exchanging energy at rate κ, each leaking at rate Γ. The energy-transfer efficiency is governed by a single figure of merit:

• U = κ/√(Γ₁Γ₂) = k√(Q₁Q₂), where Q = ω₀L/R = ω₀/2Γ is the resonator quality factor.
• The maximum DC-to-DC efficiency is η_max = U²/(1 + √(1+U²))².

The lesson: efficiency depends on the product kQ, not k alone. The MIT self-resonant copper coils reached Q ≈ 950. Even with k ≈ 0.01 across 2 m, kQ ≈ 9.5 — well into the strongly-coupled regime (U > 1), yielding roughly 40% wall-to-load. Pull the coils apart and κ falls, but as long as κ stays above √(Γ₁Γ₂) the power keeps flowing; only when the system crosses into the weakly-coupled regime does efficiency fall off a cliff. It is exactly the physics of resonant energy exchange between coupled pendulums — the beat between two nearly-degenerate normal modes.

The controlling variables: Q, frequency, and skin depth

Everything in near-field WPT is a fight to keep Q high, because kQ is the whole game. Q = ω₀L/R, so you want low resistance R at the operating frequency — and that is subtle, because R is not the DC resistance.

• Skin effect. At frequency f, AC current crowds into a surface layer of thickness δ = √(2ρ/ωμ₀). For copper (ρ = 1.68×10⁻⁸ Ω·m): δ ≈ 206 μm at 100 kHz, but only 25 μm at 6.78 MHz. Thicker wire barely helps above a few skin depths, so designers use litz wire — hundreds of insulated strands thinner than δ — to keep AC resistance near DC.
• Frequency is a two-edged sword. Higher ω raises the induced EMF (∝ ω) and can raise Q, but skin and proximity losses climb, and above ~30 MHz the coil starts to radiate, dumping energy as escaping waves. Practical near-field bands cluster in the ISM slots: 87–205 kHz (Qi phones), 6.78 MHz (AirFuel/resonant), 13.56 MHz (RFID/NFC).
• Coupling geometry. k depends on coil size, alignment, and gap. Coaxial coils of comparable diameter to the gap couple well; lateral misalignment or a large gap-to-diameter ratio kills k. This is why an EV charging pad the size of a dinner plate can push 11 kW across 15 cm, while a phone coil the size of a coin needs near-contact.

A worked feel: charging a phone through the back glass

Take a Qi charger running at f = 150 kHz (ω = 2π·150 kHz ≈ 9.4×10⁵ rad/s). Suppose the transmit coil has L₁ = 24 μH and the phone coil L₂ = 15 μH, with k ≈ 0.5 across the 3 mm glass-plus-air gap.

• Mutual inductance: M = k√(L₁L₂) = 0.5·√(24·15) μH ≈ 9.5 μH.
• Induced open-circuit EMF for a 1 A peak primary current: EMF₂ = ωMI₀ = (9.4×10⁵)(9.5×10⁻⁶)(1) ≈ 8.9 V — comfortably above the ~5 V the phone rectifies to.
• Quality factor: if the coils' AC resistance is ~0.3 Ω, Q = ωL/R ≈ (9.4×10⁵)(24×10⁻⁶)/0.3 ≈ 75. Then kQ ≈ 37 — deep in the efficient regime, which is why aligned Qi pads hit 75–80% DC-to-DC.

Now separate the coils to 20 mm and k plunges toward ~0.05; M drops ten-fold, the induced EMF falls below the rectifier threshold, and — crucially — the reflected load impedance the transmitter sees collapses, so it can no longer couple power in. That non-linear cliff is why your phone stops charging the instant it slides a centimetre off the pad, even though the field is still there.

The far field: beaming power as radiation

Cross the boundary R ≫ λ and the physics changes character entirely. The reactive near field has faded; what remains is a propagating electromagnetic wave carrying real Poynting flux S = E × H away from the source. Now power is radiated, and the game is aiming, not coupling. The bookkeeping is the Friis transmission equation:

• P_r/P_t = G_t · G_r · (λ / 4πR)², where G_t and G_r are transmit and receive antenna gains and λ is the wavelength.
• Gain buys directivity: a large aperture of diameter D radiates a beam of half-angle θ ≈ 1.22 λ/D (the diffraction limit), so bigger dishes and shorter wavelengths make tighter beams.

The physics that dooms an omnidirectional far-field link is the same inverse-square spreading that dims a distant star — but a focused beam beats it. A 5.8 GHz microwave link (λ ≈ 5.2 cm) through a 1 m aperture spreads to a spot radius of only ~63 m at 1 km, so with a matching receiver almost all the flux is captured. This is solar-power-satellite territory: NASA/DOE studies of the 1970s and JAXA's roadmap propose a geostationary array beaming gigawatts at 2.45 GHz to a kilometre-scale ground rectenna. Lasers push further — William Brown's 1975 JPL demo relayed 30 kW of microwave power across 1.5 km at 84% collection efficiency, and in 2015 Mitsubishi and JAXA beamed 1.8 kW over 55 m and Japan's national institute demonstrated ~54% end-to-end over shorter hops.

The rectenna and where the energy actually goes

Far-field WPT needs a receiver that turns oscillating fields back into DC. The device is the rectenna (rectifying antenna), invented by William C. Brown at Raytheon in the 1960s: an antenna feeding a Schottky-diode rectifier, one element per half-wavelength, tiled across the collection area. Well-designed 2.45 GHz rectennas convert incident microwaves to DC at 80–90% RF-to-DC efficiency.

The end-to-end efficiency is a product of several stages, and that is where reality bites:

• DC → RF in the transmitter (magnetron or solid-state amplifier): 70–90%.
• Beam collection (fraction of radiated power hitting the receiver, set by aperture and range): highly variable, near 100% only for tightly focused links.
• RF → DC at the rectenna: 80–90%.

Multiply these and even the best microwave demonstrations land near 50–55% wall-to-wall. That is the honest ceiling that separates a lab curiosity from a copper wire (which runs >99% over the same span). A vital safety point: at the beam intensities proposed for space solar power (~1 kW/m² at the ground, comparable to sunlight), and at microwave frequencies below the ionizing threshold, the mechanism is dielectric heating, not ionization — the same physics as a microwave oven, deliberately kept to non-hazardous flux densities.

Subtleties and misconceptions

• “Resonance transmits through a special channel.” No new field is created. Resonance simply raises the circulating current (and stored energy) in each coil by a factor of Q, amplifying the ordinary magnetic dipole field so a tiny coupling still exchanges usable power. It is impedance matching in disguise, not a new force.
• “Near-field power radiates and wastes energy.” In the pure near field (R ≲ λ/2π) the fields are reactive — energy oscillates back and forth without net radiation, like the sloshing energy in an LC tank. Uncoupled energy returns to the source rather than escaping, which is why an unloaded WPT transmitter draws little power.
• “Wireless is inefficient by nature.” Aligned inductive links reach 90–95% — comparable to a wired supply. The losses come from misalignment, gap, and low Q, not from the absence of a wire.
• “Tesla already did this over kilometres.” Nikola Tesla's Wardenclyffe Tower (1901–1917) aimed to use the Earth-ionosphere cavity as a resonator, but never demonstrated efficient long-range power transfer; the 1/R³ near-field decay and radiation losses defeat the scheme. The modern lineage traces to Brown's rectenna (1960s) and MIT's resonance work (2007).
• “Higher frequency is always better.” Frequency raises induced EMF and can raise Q, but skin/proximity losses and unwanted radiation grow too — every band is a compromise, which is why the standards cluster at specific ISM frequencies.

The two regimes of wireless power: near-field magnetic coupling versus far-field radiative beaming

Property	Near-field (inductive/resonant)	Far-field (microwave/laser)
Range vs wavelength	R ≲ λ/2π (sub-wavelength)	R ≫ λ (many wavelengths)
Mechanism	Non-radiating reactive B (or E) field	Propagating EM wave (Poynting flux)
Falls off as	~1/R³ (dipole field), or slower if resonant	~1/R² (inverse-square, if focused)
Typical frequency	87 kHz – 13.56 MHz	2.45 / 5.8 GHz, or 1064 nm laser
Distance / power	mm–m / mW–100 kW (EV pads)	km–orbit / kW–GW (proposed SBSP)
Efficiency achieved	75–95% (aligned), ~40% at 2 m (MIT)	~54% DC-DC over 1.5 km (2015 Japan)