[Paper Review] Self inductance of a wire loop as a curve integral

R. P. Dengler|arXiv (Cornell University)|Apr 4, 2012

Magnetic Properties and Applications2 references17 citations

TL;DR

This paper derives a new curve integral formulation for the self-inductance of a thin wire loop by excluding singularities at close-proximity points and evaluating their contribution as a constant term dependent on current distribution. The method yields a self-inductance expression accurate to first order in wire radius for loops with sharp corners and second order for smooth loops, with the correction term proportional to wire length and a distribution-dependent constant Y.

ABSTRACT

It is shown that the self inductance of a wire loop may be written as a curve integral akin to the Neumann formula for the mutual inductance of two wire loops. The only difference is that contributions where the two integration variables get too close to each other must be excluded from the curve integral and evaluated in detail. The contributions of these excluded segments depend on the distribution of the current in the cross section of the wire. They add to a simple constant proportional to the wire length. The error of the new expression is of first order in the wire radius if there are sharp corners and of second order in the wire radius for smooth wire loops.

Motivation & Objective

• To derive a closed-form expression for the self-inductance of a thin wire loop using a curve integral approach.
• To resolve the singularity in the standard Neumann formula when self-inductance is formally applied to a single loop.
• To quantify the contribution from close-proximity current elements that are excluded from the curve integral.
• To determine how the current distribution in the wire cross-section affects the self-inductance correction term.
• To establish error bounds for the approximation in terms of wire radius and loop geometry.

Proposed method

• Formal derivation begins with the magnetic energy integral involving current density, leading to a Neumann-type expression.
• The self-inductance is split into two parts: a curve integral over distant points (|s - s′| > b) and a short-range contribution (|s - s′| < b).
• The short-range contribution is evaluated by modeling the wire as a cylinder of radius a and length 2b, using cylindrical symmetry.
• For straight segments, the short-range integral is computed analytically using known results from the appendix (equation A.4).
• For curved segments, the distance between points is expanded in powers of curvature, and the leading-order correction is derived using series expansion.
• The curve integral contribution is corrected by including the O(b²/R²) term from curvature, which cancels the corresponding term in the short-range integral, ensuring consistency.

Experimental results

Research questions

• RQ1 Can the self-inductance of a thin wire loop be expressed as a curve integral analogous to the Neumann formula for mutual inductance?
• RQ2 What is the correct way to handle the singularity that arises when the two integration variables coincide in the self-inductance integral?
• RQ3 How does the current distribution in the wire cross-section affect the self-inductance correction term?
• RQ4 What is the order of magnitude of the error in the curve integral approximation for loops with sharp corners versus smooth curves?
• RQ5 Can the short-range contribution be analytically evaluated and expressed as a constant term proportional to wire length?

Key findings

• The self-inductance of a wire loop can be written as a curve integral with a cutoff at |s - s′| > a/2, excluding singularities.
• The excluded short-range contribution adds a constant term proportional to wire length l, specifically (μ₀/4π)lY, where Y depends on current distribution: Y = 0 for surface current, Y = 1/2 for uniform current.
• The error of the formula is O(μ₀a) for loops with sharp corners and O(μ₀a²/l) for smooth loops, with a = wire radius.
• The curvature correction term O(b²/R²) from the short-range integral cancels exactly with a corresponding term from the curve integral, ensuring consistency.
• For straight segments, the short-range integral is analytically evaluated and contributes a logarithmic term in the self-inductance expression.
• The method is exact in the limit a ≪ l, with the only approximation arising from the cutoff scale b = √(aR), which is consistent across all terms.