Magnetizing Inductance⚠ unverified
Electrical / Power Electronics · Compute the magnetizing inductance of a magnetic core
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| N | N | — | 1.0 | Number of turns (dimensionless) |
| Ae | Ae | m^2 | 1.0 | Effective cross-sectional area of the core |
| mu | μ | H/m | 1.0 | Magnetic permeability of the core material |
| le | le | m | 1.0 | Effective magnetic path length |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | L | H | Magnetizing inductance, in henries (H). Returns 0.0 when ``le`` is not positive |
The science & history
Understanding the Parameters
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$N$ — inductance scales as $N^2$: doubling turns quadruples $L_m$ (and multiplies induced voltage by 2 for the same $dB/dt$).
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$A_e$ — larger core area means more flux $\Phi = B A_e$ for a given $B$, hence more flux linkage per amp.
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$\mu$ — $\mu = \mu_r\mu_0$. Ferrites have high $\mu_r$ but gap or powder cores use a much lower effective $\mu$ dominated by the air gap. Using bulk $\mu_r$ without the gap underestimates reluctance and overestimates $L_m$.
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$\ell_e$ — longer path → higher reluctance $\mathcal{R} = \ell_e/(\mu A_e)$ → lower inductance.
- $L_m$ — high $L_m$ means small magnetizing current (good for transformers); lower $L_m$ (gapped) is intentional in flyback/boost inductors to store energy without saturation.
Derivation (Approaching a Proof)
Ampère's law for a closed magnetic path (mean length $\ell_e$, uniform $H$): $N I = H\,\ell_e$, so $H = N I/\ell_e$. In a linear medium $B = \mu H$. Flux through the core is $\Phi = B A_e$, and flux linkage $\lambda = N\Phi$. Inductance is defined by $\lambda = L I$, so
$$L = \frac{N\Phi}{I} = \frac{N\,(\mu H)\,A_e}{I} = \frac{N\,\mu\,(N I/\ell_e)\,A_e}{I} = \frac{\mu N^2 A_e}{\ell_e}.$$
Equivalently, reluctance $\mathcal{R} = \ell_e/(\mu A_e)$ and $L = N^2/\mathcal{R}$. Energy in the field is $\tfrac12 L I^2$, consistent with Inductor Stored Energy.
Gapped cores: $\mathcal{R}_{total} \approx \ell_g/(\mu_0 A_g) + \ell_c/(\mu_c A_c)$; the gap term usually dominates, so effective $\mu$ drops and $L$ is set mostly by gap geometry.
History
Magnetic-circuit analogues (reluctance, MMF) date to the late 19th century and remain the standard transformer/inductor design framework. Core catalogs list $A_e$, $\ell_e$, and $A_L$ (nH/turn$^2$), where $L = A_L N^2$ is exactly this formula with $A_L = \mu A_e/\ell_e$.
Related Concepts: Inductor Stored Energy, Inductor Energy, Transformer Turns Ratio, Core Loss Steinmetz
Notes: Registry calculator magnetizing-inductance (unverified). Linear, uniform-field magnetic
circuit; no leakage, no saturation. SI units: $\mu$ in H/m ($\mu_0 = 4\pi\times 10^{-7}$), $A_e$ in
m$^2$, $\ell_e$ in m.