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Maximum Q-factor of planar inductors

This paper establishes a fundamental theoretical upper bound on the maximum achievable Q-factor for electrically-small planar inductors as a function of design area by combining rigorous electromagnetic analysis with convex optimization, while also evaluating state-of-the-art designs against this benchmark and exploring the potential of kinetic inductance for next-generation improvements.

Original authors: Mohamed Ismail Abdelrahman, Matteo Ciabattoni, Francesco Monticone

Published 2026-04-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohamed Ismail Abdelrahman, Matteo Ciabattoni, Francesco Monticone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build the most efficient, high-performance "energy storage tank" for electricity, but you are forced to build it flat on a tiny piece of silicon, like drawing a circuit on a postage stamp. This is the challenge of designing on-chip inductors for modern electronics.

In the world of radio-frequency chips (the brains behind your phone's Wi-Fi and 5G), these inductors are crucial. They act like the flywheels in a car engine, storing energy in magnetic fields to smooth out power and filter signals. But there's a catch: the smaller you make them, the more energy they waste as heat or radiate away into space. This waste is measured by something called the Q-factor. A high Q-factor means the inductor is efficient and "pure"; a low Q-factor means it's leaky and sloppy.

For years, engineers have been guessing how small they can make these components before they become too inefficient. They've been trying to find the "perfect shape" by trial and error, often wasting time and computer power trying to design something that physics says is impossible.

This paper is like a physics-based rulebook that finally answers the question: "What is the absolute best performance we can possibly get for a flat inductor of a specific size?"

Here is the breakdown of their discovery, using some everyday analogies:

1. The "Speed Limit" of Efficiency

Think of the Q-factor as the fuel efficiency of a car. You want to go as far as possible on a tank of gas (energy).

The Problem: Engineers have been trying to design faster, smaller cars, but they didn't know the theoretical speed limit. They kept building engines that were too heavy or aerodynamic shapes that didn't work, only to find out they were hitting a wall they couldn't see.
The Solution: The authors used advanced math (called Convex Optimization) to calculate the "speed limit" for these flat inductors. They didn't just guess; they proved mathematically what the maximum possible efficiency is for any given size. Now, if an engineer designs a chip and their inductor is only 50% as efficient as this new "speed limit," they know they have plenty of room to improve. If it's 99%, they know they've reached the peak and shouldn't waste time trying to tweak it further.

2. The Shape of the Perfect Loop

The researchers discovered that the "perfect" shape for an inductor changes depending on how big it is, much like how a runner's stride changes with speed.

The Tiny Regime (The Single Loop): When the inductor is very small, the best shape is a single, wide loop. Imagine a hula hoop. If you make the hoop too thin (a skinny wire), it gets hot (resistance). If you make it too thick, you lose magnetic power. The math showed that a specific, moderately wide loop is the sweet spot.
The Medium Regime (The Double Helix): As the inductor gets slightly bigger, something interesting happens. A single big loop starts to "leak" energy into the air (radiation loss), like a radio antenna broadcasting a signal you didn't want. To stop this, the optimal design suddenly splits into two smaller loops carrying current in opposite directions.
- Analogy: Think of two people spinning a jump rope. If they spin it in the same direction, the rope flies everywhere (radiation). If they spin it in opposite directions, the rope stays tight and contained. This "two-loop" shape traps the energy better, even though it uses a bit more wire (which adds a little heat). It's a trade-off to stop the energy from flying away.

3. The "Square" Rule

The paper also looked at whether a long, skinny rectangle is better than a square.

The Finding: A square (or circle) is almost always the winner. Stretching the inductor into a long rectangle doesn't help much.
Analogy: Imagine trying to fill a bucket with water. A square bucket holds water efficiently. If you stretch it into a long, thin trough, you might get a little more water, but the surface area increases, causing more evaporation (loss). The math showed that the "square" shape minimizes the waste for a given amount of space.

4. The "Super Material" Cheat Code

Finally, the paper looked at what happens if we use special materials, like graphene or superconductors, instead of regular copper.

The Concept: Regular inductors store energy in a magnetic field (like a magnet). But in these special materials, the electrons themselves have "inertia" (they are heavy and hard to speed up). This creates Kinetic Inductance.
Analogy: Imagine pushing a shopping cart.
- Regular Inductor: You are pushing a light cart, but the wheels are loose, and it wobbles (magnetic field).
- Kinetic Inductor: You are pushing a cart filled with lead bricks. The cart is heavy and hard to get moving (inertia), but once it's moving, it carries a massive amount of momentum.
The Result: By using these "heavy electron" materials, you can break the old speed limits. You can get a much higher Q-factor without making the chip bigger. This is the key to the next generation of tiny, super-efficient electronics.

Why This Matters

Before this paper, designing these chips was like trying to climb a mountain in the fog. Engineers were guessing how high the peak was.

Now: They have a map. They know exactly where the peak is.
The Benefit: This saves time and money. If a design is close to the limit, engineers stop trying to improve it and move on. If a design is far from the limit, they know exactly where to focus their efforts. It also tells them that if they want to go beyond the limit, they can't just change the shape; they need to change the material (like using graphene).

In short, this paper provides the ultimate benchmark for flat inductors, telling us the absolute best we can do with current materials, and showing us the "cheat codes" (new materials) we need to go even further.

1. Problem Statement

On-chip inductors are critical components in Radio-Frequency Integrated Circuits (RFICs), wireless power transfer, and quantum computing systems. However, their performance is fundamentally limited by the available chip area. The Quality factor ( $Q$ -factor), defined as the ratio of stored energy to energy dissipated per cycle, is intrinsically tied to the physical footprint of the inductor.

The Gap: While significant research exists on lower bounds for the $Q$ -factor of antennas (to maximize bandwidth), the complementary problem of determining the fundamental upper bound ( $Q_{max}$ ) for magnetic systems (inductors) has been largely overlooked.
The Challenge: Designing high- $Q$ inductors with small footprints is difficult because reducing size increases ohmic losses and radiation losses. Current design methods often rely on trial-and-error or specific geometric optimizations without knowing if a design is approaching the theoretical physical limit.

2. Methodology

The authors employ a rigorous combination of electromagnetic theory and mathematical optimization to derive fundamental limits independent of specific geometric layouts.

Method of Moments (MoM): The problem is discretized using Rao-Wilton-Glisson (RWG) basis functions to represent surface currents. The electromagnetic interactions are modeled via an impedance matrix $Z = R + jX$, where $R$ accounts for ohmic and radiation losses, and $X$ represents stored magnetic and electric energy.
Convex Optimization (SDR): The maximization of the $Q$ -factor is formulated as a Quadratically Constrained Quadratic Program (QCQP). Since this is non-convex and prone to local optima, the authors apply Semidefinite Relaxation (SDR). This transforms the problem into a convex semidefinite program. Theoretical guarantees confirm that for this specific problem structure, the relaxed solution yields the same global optimum as the original non-convex problem.
Modal Analysis (CMT): To validate the optimization results, the authors use Characteristic Mode Theory (CMT) to solve a generalized eigenvalue problem, identifying the specific current modes that maximize the ratio of inductive stored energy to total power dissipation.
Analytical Derivation: Based on the numerical results, the authors derive closed-form analytical expressions for the $Q$ -factor bound, modeling the optimal current distribution as a circular loop for small sizes.

3. Key Contributions

Derivation of Fundamental Upper Bounds: The paper establishes the first rigorous analytical and numerical upper bounds for the $Q$ -factor of electrically small planar inductors as a function of design area and material conductivity.
Identification of Operational Regimes: The analysis reveals three distinct regimes based on the electrical size ($ka$) and conductivity:
- Small Size Regime: The optimal current is a single, wide loop. $Q$ scales linearly with area. Ohmic losses dominate.
- Intermediate Regime: As size increases, radiation losses become significant. The optimal current narrows to reduce radiation, but this increases ohmic resistance.
- Large Size Regime: To mitigate rapidly increasing radiation losses (scaling as $a^4$ ), the optimal current distribution splits into two counter-rotating loops. This configuration minimizes the net magnetic dipole moment, reducing radiation at the cost of slightly higher ohmic losses.
Geometry Optimization: The study confirms that square or circular design areas yield the highest $Q$ -factor density. Elongating the geometry (rectangular shapes) provides only marginal gains in absolute $Q$ but significantly reduces $Q$ -factor density.
Inclusion of Kinetic Inductance: The framework is extended to account for kinetic inductance (inertial effects of electrons), which is significant in superconductors and 2D materials (e.g., graphene). This allows the $Q$ -factor to exceed the bounds derived for conventional metals.

4. Key Results

Analytical Bound Formula: For small planar inductors dominated by magnetic inductance, the upper bound is approximated by:
$Q_{max} \approx 0.15 \frac{ka}{R_s/\eta_0}$
Where $k$ is the wavenumber, $a$ is the radius, $R_s$ is surface resistance, and $\eta_0$ is free-space impedance. This bound depends only on area, frequency, and material, not on the specific layout.
Rectangular Limits: For rectangular areas with high aspect ratios, the bound saturates and is governed by the smallest dimension ( $W$ or $L$ ), leading to the modified bound:
$Q_{max} \approx 0.125 \frac{k \cdot \min(W, L)}{R_s/\eta_0}$
Benchmarking: The authors evaluated state-of-the-art inductor designs from literature against these bounds. Many designs are far from the limit, indicating significant room for improvement. For example, a 200 $\mu$ m graphene inductor achieved a $Q$ of 12, while the bound predicts a potential $Q$ of 46.
Kinetic Inductance Impact: When kinetic inductance is included (e.g., in graphene or superconductors), the bound shifts to:
$Q_{max} \approx 0.15 \frac{ka}{R_s/\eta_0} + \omega\tau$
This second term ( $\omega\tau$ ) is geometry-independent, suggesting that for very small devices, material properties can override geometric limitations.

5. Significance

Design Guidance: The derived bounds serve as a "North Star" for RFIC designers, allowing them to assess whether a specific inductor design is near-optimal or if further geometric optimization is futile.
Resource Efficiency: By defining achievable limits, inverse design algorithms can avoid wasting computational resources searching for non-existent solutions that violate physical laws.
Next-Generation Materials: The inclusion of kinetic inductance highlights the potential of 2D materials (like intercalated graphene) to break the performance barriers of traditional copper inductors, particularly for miniaturized quantum and RF systems.
Miniaturization: The findings provide a theoretical roadmap for maximizing inductance density and $Q$ -factor in the limited footprint of modern integrated circuits.

In conclusion, this work bridges the gap between theoretical electromagnetic limits and practical inductor design, providing a rigorous framework to quantify and push the boundaries of on-chip inductor performance.