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arxiv: 2604.20497 · v1 · submitted 2026-04-22 · ⚛️ physics.optics

Maximum Q-factor of planar inductors

Mohamed Ismail Abdelrahman , Matteo Ciabattoni , Francesco Monticone This is my paper

Pith reviewed 2026-05-09 23:27 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords inductorsanalysisdesignfactorareaboundinductanceinductor
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The pith

An analytical upper bound on the maximum Q-factor of electrically small planar inductors is derived as a function of design area, accounting for ohmic and radiation losses via EM analysis and convex optimization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

On-chip inductors are essential for radio-frequency integrated circuits but consume significant chip space because their inductance and quality factor are tied to physical size. This work applies detailed electromagnetic modeling combined with convex optimization to calculate the highest possible Q-factor achievable for a given footprint when the inductor is electrically small. The resulting bound is expressed analytically and further analyzed using modal techniques to reveal how performance scales with area and material conductivity. Both resistive losses inside the conductor and radiation losses to the surroundings are included in the model, with radiation becoming more relevant at larger sizes. Existing inductor designs from published literature are compared to this theoretical limit to identify which ones are near-optimal and which have room for improvement. The study also considers the addition of kinetic inductance as a way to potentially increase both density and Q-factor beyond conventional limits. The overall goal is to provide a clear benchmark that can guide future designs toward more compact and higher-performance RF systems.

Core claim

we employ rigorous electromagnetic analysis together with convex optimization techniques to derive a fundamental bound on the maximum achievable Q-factor of electrically-small planar inductors as a function of the available design area

Load-bearing premise

The derivation relies on the electrically-small approximation and on the ability of convex optimization to capture all relevant loss mechanisms without unstated modeling simplifications in the EM formulation.

Figures

Figures reproduced from arXiv: 2604.20497 by Francesco Monticone, Matteo Ciabattoni, Mohamed Ismail Abdelrahman.

Figure 1
Figure 1. Figure 1: Illustration of the problem of establishing upper bounds on the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (A) Upper bound Qmmax (normalized to electrical size ka) for inductors confined to a square optimizable area of radius a, plotted as a function of electrical size and metal conductivity. The bound is computed using both convex optimization (SDR) and modal analysis (CMT) methods, as discussed in the Appendix B and C respectively, showing virtually perfect agreement. We have verified that the numerical solut… view at source ↗
Figure 3
Figure 3. Figure 3: Upper bound Qmmax (blue curve) numerically evaluated for rectangular design areas with a fixed width, W = λ/50, as a function of axial ratio. The results show that only modest growth is obtained as the length L is increased. The corresponding optimal current distributions at different axial ratios ζ = L/W are shown in the insets (not to scale for clarity). The optimal current solution maintains a wide elli… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical results for the generalized bound, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

On-chip inductor design plays a critical role in the advancement of radio-frequency integrated circuits (RFICs). Inductors typically occupy a substantial portion of the chip area as their performance metrics, namely, inductance density and Quality factor ($Q$-factor), are fundamentally tied to the available footprint, thereby limiting miniaturization. To better understand and quantify these limitations, we employ rigorous electromagnetic analysis together with convex optimization techniques to derive a fundamental bound on the maximum achievable $Q$-factor of electrically-small planar inductors as a function of the available design area. The analysis yields analytical expressions for the bound and, via modal analysis techniques, identifies and interprets operational regimes and scaling trends with respect to design area and material conductivity. The analysis accounts for both ohmic and radiation losses, with the latter becoming significant as the inductor size increases. A broad set of state-of-the-art inductor designs from the literature is evaluated against the established $Q$-factor upper bound, identifying designs that approach the theoretical limit as well as those with potential for further improvement. The study is extended to include the effect of kinetic inductance, which offers a promising avenue toward next-generation inductors with higher inductance densities and $Q$-factors. By establishing this benchmark, this work aims to guide and inspire the design of more efficient and compact planar inductors for high-performance RF systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to derive a fundamental upper bound on the maximum Q-factor of electrically-small planar inductors as a function of available design area. This is obtained via rigorous electromagnetic analysis combined with convex optimization, explicitly including ohmic and radiation losses. Modal analysis yields analytical expressions, identifies operational regimes, and extracts scaling trends with area and conductivity. The bound is tested against a collection of state-of-the-art designs from the literature, and the analysis is extended to incorporate kinetic inductance.

Significance. If the bound holds, the work supplies a useful theoretical benchmark for on-chip inductor design in RFICs, with direct implications for miniaturization. The combination of first-principles EM analysis and convex optimization to produce the limit, together with the closed-form scaling expressions and modal interpretation of regimes, is a clear strength. The explicit treatment of radiation losses at larger sizes and the forward-looking discussion of kinetic inductance add practical value. Direct comparison with published designs helps distinguish near-optimal realizations from those with remaining headroom.

minor comments (3)
  1. [Abstract] Abstract: the statement that 'a broad set of state-of-the-art inductor designs' is evaluated would be strengthened by stating the selection criteria and the exact number of designs considered.
  2. [Modal analysis] Modal analysis: the transition points between the identified operational regimes should be tied explicitly to the relevant equations or parameter thresholds so that readers can reproduce the regime boundaries.
  3. [Optimization formulation] Optimization formulation: confirm that the convex program remains convex once the electrically-small approximation is relaxed at the upper end of the area range considered.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We sincerely thank the referee for the positive and accurate summary of our manuscript, as well as for recognizing its significance in providing a theoretical benchmark for planar inductor design in RFICs. The recommendation for minor revision is noted, and we appreciate the emphasis on the combination of EM analysis, convex optimization, modal interpretations, and comparisons to literature designs. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; bound derived from first-principles EM and convex optimization

full rationale

The central derivation applies standard electromagnetic modal analysis to the inductor geometry, formulates a convex optimization problem over the available design area to maximize Q while accounting for ohmic and radiation losses, and obtains analytical scaling expressions in the electrically-small limit. These steps are internally consistent, rely on Maxwell's equations and convex programming rather than data fitting or self-referential definitions, and do not reduce any claimed bound to its own inputs by construction. No load-bearing self-citation chain, uniqueness theorem imported from the authors' prior work, or ansatz smuggling is required for the result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are detailed beyond standard EM assumptions and optimization framework.

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