Leakage inductance

Intent

• Explain leakage inductance as the flux that links one winding but not the other, stored mostly between the windings.
• Relate the coupling coefficient, the short-circuit inductance and the leakage inductances of the transformer model.
• Estimate the leakage of a winding arrangement, and what it costs in a flyback.

Theory

Coupled windings

Two windings on one core share most of their flux, but not all of it. At their terminals they behave as two self-inductances, $L_{11}$ and $L_{22}$, and a mutual inductance $L_{12}$ [Erickson & Maksimović 2020, Ch. 10], [Alexander & Sadiku 2017, Ch. 13]. How much of the flux they share is the coupling coefficient:

$$k = \frac{L_{12}}{\sqrt{L_{11} L_{22}}}$$

$k$

coupling coefficient

$L_{11}$

inductance of winding 1, winding 2 open

$L_{12}$

mutual inductance

$L_{22}$

inductance of winding 2, winding 1 open

Coupling coefficient of two windings✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• definition

Notes

1 for perfect coupling; the leakage inductances make it smaller

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: the transformer model; coupled inductors); Alexander & Sadiku 2017, Ch. 13 (Magnetically Coupled Circuits)

Equation id

xfmr.k · equations.yaml entry

The short-circuit test

Shorting the secondary holds its voltage, the rate of change of its flux linkage, at zero: its current then cancels most of the flux the two windings share, and the primary sees mainly the two windings’ leakage (derived here):

$$L_\mathrm{sc} = L_{11} \left(1 - k^{2}\right)$$

$L_\mathrm{sc}$

inductance with the other winding shorted

$L_{11}$

inductance of winding 1, winding 2 open

$k$

coupling coefficient

Inductance of a winding with the other winding shorted✓ verified by 8 tests

Assumptions, convention and source

Assumptions

• two linearly coupled windings without resistance

Notes

the short holds the second winding's voltage, the rate of change of its flux linkage, at zero, so its current cancels most of the flux the two windings share; what is left is mainly the two windings' leakage (`xfmr.L_sc_T`): 1 − k² is small for a well-coupled transformer

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: the transformer model; coupled inductors); derived here; Alexander & Sadiku 2017, Ch. 13 (Magnetically Coupled Circuits)

Derivation

step-by-step sympy derivation

Equation id

xfmr.L_sc · equations.yaml entry

With $k$ close to 1 this is a small fraction of $L_{11}$: an impedance meter at the primary, with the secondary shorted, measures it directly.

The transformer model

The transformer model splits each winding’s inductance into a leakage inductance and the magnetizing inductance $L_M$ the two share, with an ideal 1:$n$ transformer between them [Erickson & Maksimović 2020, Ch. 10]. Seen from the primary with the secondary shorted:

$$L_\mathrm{sc} = L_{\ell 1} + \frac{L_{\ell 2}' L_M}{L_{\ell 2}' + L_M}$$

$L_\mathrm{sc}$

inductance with the other winding shorted

$L_M$

magnetizing inductance (primary)

$L_{\ell 1}$

primary leakage inductance

$L_{\ell 2}'$

secondary leakage inductance, referred to the primary

Short-circuit inductance in the transformer model✓ verified by 7 tests

Assumptions, convention and source

Assumptions

• two linearly coupled windings without resistance
• transformer modelled by its magnetizing inductance and one leakage inductance per winding (T model), referred to the primary

Notes

with $L_M$ much larger than the leakage inductances, $L_\mathrm{sc} \approx L_{\ell 1} + L_{\ell 2}'$: the short-circuit test measures the total leakage referred to the primary, not how it divides between the windings

Source

Erickson & Maksimović 2020, Ch. 10 (Basic Magnetics Theory: the transformer model with magnetizing and leakage inductances); derived here

Derivation

step-by-step sympy derivation

Equation id

xfmr.L_sc_T · equations.yaml entry

Since $L_M$ is much larger than the leakage inductances, the short-circuit test measures the sum of the two leakage inductances referred to the primary, the leakage $L_\mathrm{lk}$ that a flyback’s clamp has to absorb. It does not tell how that sum divides between the windings.

Where the leakage flux lives

The leakage flux runs through the winding window. Its energy follows from the magnetomotive force across the window: it rises through the primary’s layers, stays at its peak across the spacing, and falls back to zero through the secondary (leakage, P-S) [Erickson & Maksimović 2020, Ch. 10]. Splitting the primary around the secondary halves the peak and brings the leakage down to about a quarter (leakage, P-S-P). Only the winding geometry and the permeability of free space enter these equations: the core material and the gap set the magnetizing inductance, not the leakage.

What leakage costs

When a flyback’s switch turns off, the energy stored in the leakage inductance (leakage energy) cannot reach the secondary. It goes to a clamp or a snubber in every period (leakage power) [Erickson & Maksimović 2020, Ch. 4]; see snubbers and clamps.

Worked example

An example pair of windings, as an impedance meter reads them:

Coupled windings example

Symbol	Value	How it is obtained
$L_{11}$ inductance of winding 1, winding 2 open	404 µH	given
$L_{22}$ inductance of winding 2, winding 1 open	25.3 µH	given
$L_{12}$ mutual inductance	100 µH	given
$L_M$ magnetizing inductance (primary)	400 µH	given
$L_{\ell 1}$ primary leakage inductance	4 µH	given
$L_{\ell 2}'$ secondary leakage inductance, referred to the primary	4 µH	given
$n$ turns ratio, secondary/primary	0.25	given
$k$ coupling coefficient	0.99	Coupling coefficient of two windings
$L_\mathrm{sc}$ inductance with the other winding shorted	7.96 µH	Inductance of a winding with the other winding shorted
${L_\mathrm{sc}}_{\mathrm{T}}$ inductance with the other winding shorted	7.96 µH	Short-circuit inductance in the transformer model
$V_2/V_1$ open-circuit voltage ratio	0.248	Open-circuit voltage ratio of a transformer

The coupling coefficient is 0.9901. With the secondary shorted, the primary reads 7.96 µH. The transformer model with 400 µH magnetizing inductance and 4 µH and 4 µH of leakage at the primary and the secondary, both referred to the primary, gives the same value, 7.96 µH: nearly the sum of the two. With the secondary open, the voltage ratio is 0.2475, just below the turns ratio of 0.25 (see measuring magnetics).

Try it

▶ Open in the equation explorer: Inductance of a winding with the other winding shorted
Preset from the example coupled-pair

Sweep $k$ from 0.9 towards 1: the short-circuit inductance falls with $1 - k^2$. A small change in $k$ near 1 changes the leakage a lot.

Bench exercise

Measure a transformer’s primary inductance with the secondary open, then with it shorted by a short, wide link, with an LCR meter reading the series inductance. Choose a test frequency well below the self-resonance and high enough that the shorted reading no longer changes with frequency (see Measuring magnetics). The ratio of the two readings gives $k$ through the short-circuit equation. Then connect the two windings in series, aiding and opposing: the difference of the two readings is four times the mutual inductance [Alexander & Sadiku 2017, Ch. 13]. Compare the measured leakage with the estimate from the winding geometry.

Gotchas

• The short adds its own inductance. A long or thin shorting wire adds to the reading; use a short, wide link.
• Too low a test frequency reads high. With the secondary shorted, the magnetizing inductance’s reactance must be much larger than the secondary’s resistance referred to the primary; otherwise it carries part of the current and the reading rises towards the open-circuit inductance.
• The test gives the sum. The short-circuit test measures the primary’s and the secondary’s leakage together, referred to the primary.
• A gap does not remove leakage. Gapping the core lowers $L_M$ and $k$, but the leakage inductance itself is set by the windings.

Go deeper

• 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.
• Book chapter Transformer Design (Ch. 12) — R. W. Erickson, D. Maksimović · retrieved 2026-09-24
  Designing a transformer when core loss, rather than saturation, limits the ac flux density, with copper and core loss traded against each other.
• 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.

Quiz

1. Two windings read $400\,\mu\mathrm{H}$ and $25\,\mu\mathrm{H}$, each with the other open, and their mutual inductance is $99\,\mu\mathrm{H}$. What is their coupling coefficient?

   1. 0.25
   2. 0.99
   3. 4
   4. 0.0099
   Answer and explanation

   Correct answer: B The mutual inductance over the square root of the product of the two inductances: 99 over 100.

2. A primary reads $1\,\mathrm{mH}$ with the secondary open, and the coupling coefficient is 0.99. What does it read with the secondary shorted?

   1. $10\,\mu\mathrm{H}$
   2. $19.9\,\mu\mathrm{H}$
   3. $990\,\mu\mathrm{H}$
   4. $1.99\,\mu\mathrm{H}$
   Answer and explanation

   Correct answer: B The open-circuit inductance times one minus the coupling coefficient squared: 1 mH times 0.0199.

3. What does the short-circuit test measure?

   1. The primary's and the secondary's leakage together, referred to the primary
   2. The magnetizing inductance
   3. The primary's leakage alone
   4. The winding capacitance
   Answer and explanation

   Correct answer: A With the magnetizing inductance much larger than the leakage, the reading is nearly the sum of the two leakage inductances referred to the primary.

4. Splitting the primary around the secondary (P-S-P) changes the leakage roughly how?

   1. To about half
   2. It doubles
   3. To about a quarter
   4. Not at all
   Answer and explanation

   Correct answer: C The peak MMF in the window halves, and the stored energy goes with its square.

5. A core's gap is made twice as long. What happens to the leakage inductance?

   1. It halves
   2. It barely changes: the winding geometry sets it
   3. It doubles
   4. It vanishes
   Answer and explanation

   Correct answer: B The gap lowers the magnetizing inductance; the leakage is set by the field between the windings, which the gap hardly touches.

6. Where does a flyback's leakage energy go when the switch turns off?

   1. To the output
   2. Into the core, where it is lost as core loss
   3. To the clamp or snubber, every period
   4. It stays in the leakage inductance
   Answer and explanation

   Correct answer: C The leakage flux does not link the secondary, so its energy cannot reach the output; a clamp or snubber must take it.