Frequency Splitting in Resonant Wireless Power Transfer

Coupling & Quality

Coupling coefficient k 0.080

Loaded Q factor 120

Drive frequency ω/ω₀ 1.000

Live Readouts

Efficiency η(ω)

0.0%

Critical k_c = 1/Q

0.0083

Coupling regime

Undercoupled

Split frequencies ω±/ω₀

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How it works

Two identical resonant coils (natural frequency ω₀, loss rate Γ = ω₀/2Q, mutual coupling rate κ = kω₀/2) obey the coupled-mode equations:

da₁/dt = (iω₀-Γ)a₁ - iκa₂ + s(t)
da₂/dt = (iω₀-Γ)a₂ - iκa₁

Solving for the steady-state response to a sinusoidal drive s(t)=e^(iωt) gives the transfer efficiency to the load resonator:

η(ω) = Γ|a₂|² / (Γ|a₁|² + Γ|a₂|²)
a₁,a₂ from (iΔ+Γ)² + κ² in the denominator, Δ = ω-ω₀

Undercoupled (k < k_c = 1/Q): a single efficiency peak at ω = ω₀, efficiency below the achievable maximum.
Critically coupled (k = k_c): the single peak reaches its highest value — the classic "critical coupling" impedance-match point used in real WPT design.
Overcoupled (k > k_c): the single resonance bifurcates — the spectrum splits into two peaks at ω± = ω₀ ± √(κ²−Γ²), each still near-maximal efficiency, but driving exactly at ω₀ now under-performs both split peaks. This is the well-known "frequency splitting" phenomenon strongly-coupled magnetic-resonance WPT systems (e.g. the MIT WiTricity work) must actively track or avoid.

Move the coupling slider past k_c with the drive fixed at ω₀ and watch the spectrum dip in the middle as the two side peaks grow — the same behaviour measured on a network analyzer for real overcoupled resonant WPT links.

Transmitter coil Receiver coil

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