Air gaps and the inductance factor

Intent

• Compute the inductance factor of a gapped core from its dimensions, and the inductance from the turns.
• Explain why a gap makes the inductance depend on the gap rather than on the core material.
• Explain how a gap lowers the flux density that an inductor reaches at a given current.

Theory

Magnetic circuits

A magnetic path behaves like a resistive circuit: the winding’s magnetomotive force, its turns times its current, drives the flux through reluctances in series, here the core’s and the gap’s [Erickson & Maksimović 2020, Ch. 10]. The core’s reluctance is its path length divided by its permeability and area; the gap’s is the gap length divided by the permeability of free space and the area. Because the core’s relative permeability is large, even a short gap can dominate the total.

The inductance factor

The inductance per turn squared, the inductance factor $A_L$, is the inverse of the total reluctance [Dixon 2001], [Erickson & Maksimović 2020, Ch. 10]:

$$A_L \approx \frac{\mu_0 A_e}{l_g + \frac{l_e}{\mu_i}}$$

$A_L$

inductance factor (per turn squared)

$A_e$

effective core area

$l_e$

effective magnetic path length

$l_g$

air-gap length

$\mu_i$

initial relative permeability

$\mu_0$

permeability of free space

Inductance factor of a gapped core✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• linear (unsaturated) magnetic material
• no fringing flux at the gap
• flux density uniform over A_e

Notes

fringing makes the real $A_L$ larger than this estimate; use the manufacturer's $A_L$ for a specified gap

Source

Dixon 2001; Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: magnetic circuits)

Derivation

step-by-step sympy derivation

Equation id

mag.AL_gap · equations.yaml entry

$$L = A_L N^{2}$$

$L$

inductance

$A_L$

inductance factor (per turn squared)

$N$

number of turns

Inductance from the inductance factor✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• linear (unsaturated) magnetic material

Source

Dixon 2001; Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory)

Derivation

step-by-step sympy derivation

Equation id

mag.L_from_AL · equations.yaml entry

When the gap term dominates, $A_L$ depends on the gap length and the area and hardly on $\mu_i$: the gap makes the inductance insensitive to the spread of the core material [Erickson & Maksimović 2020, Ch. 10].

Why power inductors are gapped

The core material saturates at a given flux density, and a gap does not change that limit. It lowers the inductance factor, however, so a given inductance needs more turns, and at a given current the flux density falls (peak flux density): the gapped inductor carries more current before it saturates.

Worked example

An example gapped core:

Gapped core example

Symbol	Value	How it is obtained
$A_e$ effective core area	100 mm²	given
$l_e$ effective magnetic path length	50 mm	given
$l_g$ air-gap length	0.5 mm	given
$\mu_i$ initial relative permeability	2000	given
$N$ number of turns	20	given
$A_L$ inductance factor (per turn squared)	239 nH	Inductance factor of a gapped core
$L$ inductance	95.7 µH	Inductance from the inductance factor

With the given gap, the inductance factor is 239 nH, and 20 turns give 95.7 µH. The gap’s reluctance is much larger than the core’s here, so the result barely depends on $\mu_i$.

Try it

▶ Open in the equation explorer: Inductance factor of a gapped core
Preset from the example gapped-core

Sweep $l_g$ on its logarithmic axis: for small gaps the core material sets $A_L$, for large gaps the gap does, and $A_L$ then falls in inverse proportion to the gap length. Halve $\mu_i$ at a small gap and at a large gap to see where the material still matters.

Bench exercise

Measure the inductance of a winding on a pair of ungapped ferrite core halves. Then place thin non-magnetic spacers of known thickness, such as plastic film, between the halves and measure again. Compare the change with the equation, counting every place where the flux path crosses a spacer. The measured inductance is somewhat larger than predicted, because the flux fringes around the gap [Dixon 2001].

Gotchas

• Fringing makes the real $A_L$ larger. The flux bulges out around the gap, which widens its effective area, so the simple equation underestimates the inductance [Dixon 2001].
• Use the manufacturer’s $A_L$ for a specified gap. The manufacturer’s $A_L$ is measured, while fringing is hard to calculate; the equation serves to understand a gap and to estimate a custom one.
• Turns multiply as a square. Doubling the turns on the same core quadruples the inductance.

Go deeper

• Book Magnetics Design Handbook (SLUP132) — L. H. Dixon, Texas Instruments (Unitrode) · retrieved 2026-09-23
  Lloyd Dixon's magnetics design handbook, reproduced by TI from the 2001 Unitrode edition. The ti.com copy requires a free myTI login.
• Book chapter Basic Magnetics Theory (Ch. 10) — R. W. Erickson, D. Maksimović · retrieved 2026-09-23
  Magnetic circuits, the transformer model with magnetizing and leakage inductance, and core and winding loss mechanisms.
• Course Magnetics for Power Electronic Converters — University of Colorado Boulder (Coursera) · retrieved 2026-09-23
  Magnetic circuits, core and winding losses, and inductor and transformer design.

Quiz

1. A core has $A_L = 250\,\mathrm{nH}$ per turn squared. How many turns give $100\,\mu\mathrm{H}$?

   1. 4
   2. 20
   3. 40
   4. 400
   Answer and explanation

   Correct answer: B The turns squared are the inductance divided by $A_L$, 400; their square root is 20.

2. The turns on a given core are doubled. What happens to the inductance?

   1. It doubles
   2. It quadruples
   3. It is unchanged
   4. It halves
   Answer and explanation

   Correct answer: B The inductance is $A_L$ times the square of the turns.

3. Why does a gap make the inductance nearly independent of the core material?

   1. The gap's reluctance dominates the total, and it depends only on the gap length and the area
   2. The gap raises the material's permeability
   3. The gap removes hysteresis
   4. The gap shortens the magnetic path
   Answer and explanation

   Correct answer: A With a large relative permeability, the core's reluctance is small compared with even a short gap's.

4. Compared with the simple equation, how does fringing flux around a gap change the real $A_L$?

   1. It makes it smaller
   2. It makes it larger
   3. It has no effect
   4. It makes it depend on the current
   Answer and explanation

   Correct answer: B Fringing widens the gap's effective area, lowering its reluctance and raising $A_L$.

5. A core has an effective area of $1\,\mathrm{cm^2}$ and a total gap of $1\,\mathrm{mm}$; its own reluctance is negligible. What is $A_L$, ignoring fringing?

   1. $12.6\,\mathrm{nH}$
   2. $126\,\mathrm{nH}$
   3. $1.26\,\mu\mathrm{H}$
   4. $12.6\,\mu\mathrm{H}$
   Answer and explanation

   Correct answer: B $\mu_0$ times the area divided by the gap: $4\pi \times 10^{-7} \times 10^{-4}/10^{-3}$.

6. At the same inductance and current, how does a gapped core compare with an ungapped one of the same size?

   1. It reaches a lower flux density, because it needs more turns for the same inductance
   2. It reaches a higher flux density
   3. It saturates at a higher flux density because the gap raises the material's limit
   4. There is no difference
   Answer and explanation

   Correct answer: A The peak flux density is the inductance times the current divided by the turns and the area; the gap does not change the material's limit, but more turns lower the flux density.