How High Can Q Go? Asymptotic scaling of the quality factor of helical-coil LC resonators in the low-frequency limit

We determine the best achievable asymptotic scaling of the quality factor Q of an LC tank circuit as its resonant frequency ω → 0, when the load capacitance is a parallel-plate structure of side ℓ and fixed gap, the inductor is a helical coil confined to an ℓ × ℓ × ℓ cube, and all bulk material properties are fixed (no superconductors). Optimizing the scaling of every design parameter, we find the optimum Q* = Θ(ω^−1/3), attained with ℓ ∝ ω^−2/3, a constant number of turns, a single plate pair, and a maximally fat litz-style conductor with strands fine enough to hold skin- and proximity-effect losses at constant relative order. A matching upper bound — built on a Schur-test energy–dissipation lemma showing that any quasistatic current system confined to a cube of side ℓ has Q ≤ 3(ℓ/δ)², with δ the skin depth — proves that no admissible design beats this exponent, including multi-layer windings, stranding, series chains, and parallel arrays of separately resonated sub-tanks at any granularity. A solid round wire yields Q = Θ(ω^−1/6); a capacitor filling the cube in 3-D yields Q = Θ(1); a fixed-size resonator degrades as Q = Θ(ω). Frequency reduction must be purchased with physical size, never with turns or plates. For resonant (adiabatic) clocking, the per-cycle energy-loss fraction improves only as f^1/3 under self-consistent resonator scaling. We situate the result against classical coil-Q theory and recent maximum-Q inductor bounds, and identify acoustic-domain energy storage (fixed material Q·f products giving Q ∝ 1/f) as the principal route around the bound.

Preprint, v1.6 (July 2026). The manuscript was researched, derived, and written by Claude (Fable 5), an Anthropic large language model, under the direction of the supervising author; see the provenance and author-contribution note in the document for the full disclosure, including the independent verification of all derivations and references.