Air Gap / Magnetic Circuit Calculator

Reluctance, magnetomotive force and core flux

Calculation

Mode:

Reluctance \(\mathcal{R}_m\) from \(l,\mu_r,A\)

Magnetomotive force \(\Theta\) from \(\Phi\) and \(\mathcal{R}_m\)

Core flux \(\Phi\) from \(N,I,l,\mu_r,A\)

Magnetic path length l [mm] Relative permeability μr [-] Core area A [cm²]

Quick Introduction

The magnetic circuit is the magnetic analogy to electrical circuits: reluctance acts like magnetic resistance, magnetomotive force \(\Theta=N\cdot I\) is the driving quantity, and flux \(\Phi\) is the resulting magnetic flow variable.

Air gaps strongly increase total reluctance and reduce flux. This is critical in electromagnets, transformers, and energy-storage inductors.

Sizing schema:

1) define geometry and material (l, A, μr)

2) compute reluctance (including air-gap contribution)

3) use \(\Theta=N\cdot I\) to determine \(\Phi\) and check \(B\)

Formulas (MathJax)

\[\mathcal{R}_m=\frac{l}{\mu_0\mu_rA}\]

\[\Theta=\Phi\cdot\mathcal{R}_m\]

\[\Theta=N\cdot I\]

\[\Phi=\frac{\Theta}{\mathcal{R}_m}\quad,\quad B=\frac{\Phi}{A}\]

Symbol Legend

\(\mathcal{R}_m\): reluctance [A/Wb]
\(\Theta\): magnetomotive force [A]
\(\Phi\): magnetic flux [Wb]
\(\mu_0\): permeability of free space
\(\mu_r\): relative permeability
\(A\): area [m²]
\(l\): magnetic path length [m]

Examples

Example 1: \(l=180\,mm\), \(\mu_r=2000\), \(A=2.5\,cm^2\) ⇒ \(\mathcal{R}_m\approx2.864\times10^6\,A/Wb\).

Example 2: \(\Phi=0.8\,mWb\), \(\mathcal{R}_m=2.864\times10^6\,A/Wb\) ⇒ \(\Theta\approx2291\,A\).

Example 3: \(N=250\), \(I=1.2\,A\) ⇒ \(\Theta=300\,A\), leading to lower \(\Phi\) than example 2.

Detailed Documentation & Summary

The magnetic-circuit model is a powerful engineering approximation for magnetic component predesign. It maps familiar circuit principles into magnetics: driving force (MMF), opposition (reluctance), and resulting flow (flux).

In real components, total reluctance is composed of multiple sections (core segments, air gap, leakage paths). The air gap often dominates because \(\mu_r\approx1\), reducing flux but increasing energy storage capability, which is beneficial in many inductor topologies.

Accurate design must also account for nonlinear B-H behavior, saturation, temperature, fringing at air-gap edges, tolerances, and frequency-dependent effects. This calculator provides robust first-order sizing support.

Summary

Computes reluctance, MMF, and core flux
Directly links geometry, material, and winding data
Well suited for early-phase magnetic-circuit sizing with/without air gap

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