# Leakage inductance

Leakage inductance derives from the electrical property of an imperfectly-coupled transformer whereby each winding behaves as a self-inductance in series with the winding's respective ohmic resistance constant. These four winding constants also interact with the transformer's mutual inductance. The winding leakage inductance is due to leakage flux not linking with all turns of each imperfectly-coupled winding.

Leakage reactance is usually the most important element of a power system transformer due to power factor, voltage drop, reactive power consumption and fault current considerations.[1][2]

Leakage inductance depends on the geometry of the core and the windings. Voltage drop across the leakage reactance results in often undesirable supply regulation with varying transformer load. But it can also be useful for harmonic isolation (attenuating higher frequencies) of some loads.[3]

Leakage inductance applies to any imperfectly-coupled magnetic circuit device including motors.[4]

## Leakage inductance and inductive coupling factor

Fig. 1 LPσand LSσ are primary and secondary leakage inductances expressed in terms of inductive coupling coefficient ${\displaystyle k}$  under open-circuited conditions.

The magnetic circuit's flux that does not interlink both windings is the leakage flux corresponding to primary leakage inductance LPσ and secondary leakage inductance LSσ. Referring to Fig. 1, these leakage inductances are defined in terms of transformer winding open-circuit inductances and associated coupling coefficient or coupling factor ${\displaystyle k}$ .[5][6][7]

The primary open-circuit self-inductance is given by

${\displaystyle L_{oc}^{pri}=L_{P}=L_{M}+L_{P}^{\sigma }}$  ------ (Eq. 1.1a)

where

${\displaystyle L_{P}^{\sigma }=L_{P}\cdot {(1-k)}}$  ------ (Eq. 1.1b)
${\displaystyle L_{M}=L_{P}\cdot {k}}$  ------ (Eq. 1.1c)

and

• ${\displaystyle L_{oc}^{pri}=L_{P}}$  is primary self-inductance
• ${\displaystyle L_{P}^{\sigma }}$  is primary leakage inductance
• ${\displaystyle L_{M}}$  is magnetizing inductance
• ${\displaystyle k}$  is inductive coupling coefficient

Measuring basic transformer inductances & coupling factor

Transformer self-inductances ${\displaystyle L_{P}}$  & ${\displaystyle L_{S}}$  and mutual inductance ${\displaystyle M}$  are, in additive and subtractive series connection of the two windings, given by,[8]

${\displaystyle L_{ser}^{+}=L_{P}+L_{S}+2M}$ , and,
in subtractive connection,
${\displaystyle L_{ser}^{-}=L_{P}+L_{S}-2M}$
such that these transformer inductances can be determined from the following three equations:[9][10]
${\displaystyle L_{ser}^{+}-L_{ser}^{-}=4M}$
${\displaystyle L_{ser}^{+}+L_{ser}^{-}=2\cdot (L_{P}+L_{S})}$
${\displaystyle L_{P}=a^{2}.L_{S}}$ .

The coupling factor is derived from the inductance value measured across one winding with the other winding short-circuited according to the following:[11][12][13]

Per Eq. 2.7,
${\displaystyle L_{sc}^{pri}=L_{S}\cdot {(1-k^{2})}}$  and ${\displaystyle L_{sc}^{sec}=L_{P}\cdot {(1-k^{2})}}$
Such that
${\displaystyle k={\sqrt {1-{\frac {L_{sc}^{pri}}{L_{S}}}}}={\sqrt {1-{\frac {L_{sc}^{sec}}{L_{P}}}}}}$

The Campbell bridge circuit can also be used to determine transformer self-inductances and mutual inductance using a variable standard mutual inductor pair for one of the bridge sides.[14][15]

It therefore follows that the open-circuit self-inductance and inductive coupling factor ${\displaystyle k}$  are given by

${\displaystyle L_{oc}^{sec}=L_{S}=L_{M2}+L_{S}^{\sigma }}$  ------ (Eq. 1.2), and,
${\displaystyle k={\frac {\left|M\right|}{\sqrt {L_{P}L_{S}}}}}$ , with 0 < ${\displaystyle k}$  < 1 ------ (Eq. 1.3)

where

${\displaystyle L_{S}^{\sigma }=L_{S}\cdot {(1-k)}}$
${\displaystyle L_{M2}=L_{S}\cdot {k}}$

and

• ${\displaystyle M}$  is mutual inductance
• ${\displaystyle L_{oc}^{sec}=L_{S}}$  is secondary self-inductance
• ${\displaystyle L_{S}^{\sigma }}$  is secondary leakage inductance
• ${\displaystyle L_{M2}=L_{M}/a^{2}}$  is magnetizing inductance referred to the secondary
• ${\displaystyle k}$  is inductive coupling coefficient

The electric validity of the transformer diagram in Fig. 1 depends strictly on open-circuit conditions for the respective winding inductances considered. More generalized circuit conditions are as developed in the next two sections.

## Inductive leakage factor and inductance

A nonideal linear two-winding transformer can be represented by two mutual inductance-coupled circuit loops linking the transformer's five impedance constants as shown in Fig. 2.[6][16][17][18]

Fig. 2 Nonideal transformer circuit diagram

where

• M is mutual inductance
• ${\displaystyle R_{P}}$  & ${\displaystyle R_{S}}$  are primary and secondary winding resistances
• Constants ${\displaystyle M}$ , ${\displaystyle L_{P}}$ , ${\displaystyle L_{S}}$ , ${\displaystyle R_{P}}$  & ${\displaystyle R_{S}}$  are measurable at the transformer's terminals
• Coupling factor ${\displaystyle k}$  is defined as
${\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}$ , where 0 < ${\displaystyle k}$  < 1 ------ (Eq. 2.1)

The winding turns ratio ${\displaystyle a}$  is in practice given as

${\displaystyle a={\sqrt {L_{P}/L_{S}}}=N_{P}/N_{S}\approx v_{P}/v_{S}\approx i_{S}/i_{P}=}$  ------ (Eq. 2.2).[19]

where

• NP & NS are primary and secondary winding turns
• vP & vS and iP & iS are primary & secondary winding voltages & currents.

The nonideal transformer's mesh equations can be expressed by the following voltage and flux linkage equations,[20]

${\displaystyle v_{P}=R_{P}\cdot i_{P}+{\frac {d\Psi {_{P}}}{dt}}}$  ------ (Eq. 2.3)
${\displaystyle v_{S}=-R_{S}\cdot i_{S}-{\frac {d\Psi {_{S}}}{dt}}}$  ------ (Eq. 2.4)
${\displaystyle \Psi _{P}=L_{P}\cdot i_{P}-M\cdot i_{S}}$  ------ (Eq. 2.5)
${\displaystyle \Psi _{S}=L_{S}\cdot i_{S}-M\cdot i_{P}}$  ------ (Eq. 2.6),
where
• ${\displaystyle \Psi }$  is flux linkage
• ${\displaystyle {\frac {d\Psi }{dt}}}$  is derivative of flux linkage with respect to time.

These equations can be developed to show that, neglecting associated winding resistances, the ratio of a winding circuit's inductances and currents with the other winding short-circuited and at open-circuit test is as follows,[21]

${\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}\approx {\frac {L_{sc}}{L_{oc}}}\approx {\frac {L_{sc}^{sec}}{L_{P}}}\approx {\frac {L_{sc}^{pri}}{L_{S}}}\approx {\frac {i_{oc}}{i_{sc}}}}$  ------ (Eq. 2.7),
where,
• ioc & isc are open-circuit and short-circuit currents
• Loc & Lsc are open-circuit and short-circuit inductances.
• ${\displaystyle \sigma }$  is the inductive leakage factor or Heyland factor[22][23][24]
• ${\displaystyle L_{sc}^{pri}}$  & ${\displaystyle L_{sc}^{sec}}$  are primary and secondary short-circuited leakage inductances.

The transformer inductance can be characterized in terms of the three inductance constants as follows,[25][26]

${\displaystyle L_{M}=a{M}}$  ------ (Eq. 2.8)
${\displaystyle L_{P}^{\sigma }=L_{P}-a{M}}$  ------ (Eq. 2.9)
${\displaystyle L_{S}^{\sigma }=L_{S}-{M}/a}$  ------ (Eq. 2.10) ,

where,

Fig. 3 Nonideal transformer equivalent circuit
• c is magnetizing inductance, corresponding to magnetizing reactance XM
• LPσ & LSσ are primary & secondary leakage inductances, corresponding to primary & secondary leakage reactances XPσ & XSσ.

The transformer can be expressed more conveniently as the equivalent circuit in Fig. 3 with secondary constants referred (i.e., with prime superscript notation) to the primary,[25][26]

${\displaystyle L_{S}^{\sigma \prime }=a^{2}L_{S}-aM}$
${\displaystyle R_{S}^{\prime }=a^{2}R_{S}}$
${\displaystyle V_{S}^{\prime }=aV_{S}}$
${\displaystyle I_{S}^{\prime }=I_{S}/a}$ .

Fig. 4 Nonideal transformer equivalent circuit in terms of coupling coefficient k[27]

Since

${\displaystyle k=M/{\sqrt {L_{P}L_{S}}}}$  ------ (Eq. 2.11)

and

${\displaystyle a={\sqrt {L_{P}/L_{S}}}}$  ------ (Eq. 2.12),

we have

${\displaystyle aM={\sqrt {L_{P}/L_{S}}}\cdot k\cdot {\sqrt {L_{P}L_{S}}}=kL_{P}}$  ------ (Eq. 2.13),

which allows expression of the equivalent circuit in Fig. 4 in terms of winding leakage and magnetizing inductance constants as follows,[26]

Fig. 5 Simplified nonideal transformer equivalent circuit
${\displaystyle L_{P}^{\sigma }=L_{S}^{\sigma \prime }=L_{P}\cdot (1-k)}$  ------ (Eq. 2.14 ${\displaystyle \equiv }$  Eq. 1.1b)
${\displaystyle L_{M}=kL_{P}}$  ------ (Eq. 2.15 ${\displaystyle \equiv }$  Eq. 1.1c).

The nonideal transformer in Fig. 4 can be shown as the simplified equivalent circuit in Fig. 5, with secondary constants referred to the primary and without ideal transformer isolation, where,

${\displaystyle i_{M}=i_{P}-i_{S}^{'}}$  ------ (Eq. 2.16)
• ${\displaystyle i_{M}}$  is magnetizing current excited by flux ΦM that links both primary and secondary windings
• ${\displaystyle i_{P}}$  is the primary current
• ${\displaystyle i_{S}'}$  is the secondary current referred to the primary side of the transformer.

## Refined inductive leakage factor

Refined inductive leakage factor derivation

a. Per Eq. 2.1 & IEC IEV 131-12-41 inductive coupling factor ${\displaystyle k}$  is given by

${\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}$  --------------------- (Eq. 2.1):

b. Per Eq. 2.7 & IEC IEV 131-12-42 Inductive leakage factor ${\displaystyle \sigma }$  is given by

${\displaystyle \sigma =1-k^{2}=1-{\frac {M^{2}}{L_{P}L_{S}}}}$  ------ (Eq. 2.7) & (Eq. 3.7a)

c. ${\displaystyle {\frac {M^{2}}{L_{P}L_{S}}}}$  multiplied by ${\displaystyle {\frac {a^{2}}{a^{2}}}}$  gives

${\displaystyle \sigma =1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}}$  ----------------- (Eq. 3.7b)

d. Per Eq. 2-8 & knowing that ${\displaystyle a^{2}L_{S}=L_{S}^{\prime }}$

${\displaystyle \sigma =1-{\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}$  ---------------------- (Eq. 3.7c)

e. ${\displaystyle {\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}}$  multiplied by ${\displaystyle {\frac {L_{M}.L_{M}}{L_{M}^{2}}}}$  gives

${\displaystyle \sigma =1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{\prime }}{L_{M}}}}}}$  ------------------ (Eq. 3.7d)

f. Per Eq. 3.5 ${\displaystyle \approx }$  Eq. 1.1b & Eq. 2.14 and Eq. 3.6 ${\displaystyle \approx }$  Eq. 1.1b & Eq. 2.14:

${\displaystyle \sigma =1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}$  --- (Eq.3.7e)

All equations in this article assume steady-state constant-frequency waveform conditions the ${\displaystyle k}$  & ${\displaystyle \sigma }$  values of which are dimensionless, fixed, finite & positive but less than 1.

Referring to the flux diagram in Fig. 6, the following equations hold:[28][29]

Fig. 6 Magnetizing and leakage flux in a magnetic circuit[30][28][31]
σP = ΦPσM = LPσ/LM[32] ------ (Eq. 3.1 ${\displaystyle \approx }$  Eq. 2.7)

In the same way,

σS = ΦSσ'M = LSσ'/LM[33] ------ (Eq. 3.2 ${\displaystyle \approx }$  Eq. 2.7)

And therefore,

ΦP = ΦM + ΦPσ = ΦM + σPΦM = (1 + σPM[34][35] ------ (Eq. 3.3)
ΦS' = ΦM + ΦSσ' = ΦM + σSΦM = (1 + σSM[36][37] ------ (Eq. 3.4)
LP = LM + LPσ = LM + σPLM = (1 + σP)LM[38] ------ (Eq. 3.5 ${\displaystyle \approx }$  Eq. 1.1b & Eq. 2.14)
LS' = LM + LSσ' = LM + σSLM = (1 + σS)LM[39] ------ (Eq. 3.6 ${\displaystyle \approx }$  Eq. 1.1b & Eq. 2.14),

where

• σP & σS are, respectively, primary leakage factor & secondary leakage factor
• ΦM & LM are, respectively, mutual flux & magnetizing inductance
• ΦPσ & LPσ are, respectively, primary leakage flux & primary leakage inductance
• ΦSσ' & LSσ' are, respectively, secondary leakage flux & secondary leakage inductance both referred to the primary.

The leakage ratio σ can thus be refined in terms of the interrelationship of above winding-specific inductance and Inductive leakage factor equations as follows:[40]

${\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}=1-{\frac {L_{M}^{2}}{L_{P}L_{S}{^{'}}}}=1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{'}}{L_{M}}}}}=1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}$  ------ (Eq. 3.7a to 3.7e).

## Applications

Leakage inductance can be an undesirable property, as it causes the voltage to change with loading.

High leakage transformer

In many cases it is useful. Leakage inductance has the useful effect of limiting the current flows in a transformer (and load) without itself dissipating power (excepting the usual non-ideal transformer losses). Transformers are generally designed to have a specific value of leakage inductance such that the leakage reactance created by this inductance is a specific value at the desired frequency of operation. In this case, actually working useful parameter is not the leakage inductance value but the short-circuit inductance value.

Commercial and distribution transformers rated up to say 2,500 kVA are usually designed with short-circuit impedances of between about 3% and 6% and with a corresponding ${\displaystyle X/R}$  ratio (winding reactance/winding resistance ratio) of between about 3 and 6, which defines the percent secondary voltage variation between no-load and full load. Thus for purely resistive loads, such transformers' full-to-no-load voltage regulation will be between about 1% and 2%.

High leakage reactance transformers are used for some negative resistance applications, such as neon signs, where a voltage amplification (transformer action) is required as well as current limiting. In this case the leakage reactance is usually 100% of full load impedance, so even if the transformer is shorted out it will not be damaged. Without the leakage inductance, the negative resistance characteristic of these gas discharge lamps would cause them to conduct excessive current and be destroyed.

Transformers with variable leakage inductance are used to control the current in arc welding sets. In these cases, the leakage inductance limits the current flow to the desired magnitude.

Transformer leakage reactance has a large role in limiting circuit fault current within the maximum allowable value in the power system.[2]

## References

1. ^ Kim 1963, p. 1
2. ^ a b Saarbafi & Mclean 2014, AESO Transformer Modelling Guide, p. 9 of 304
3. ^ Irwin 1997, p. 362.
4. ^ Pyrhönen, Jokinen & Hrabovcová 2008, Chapter 4 Flux Leakage
5. ^ The terms inductive coupling factor and inductive leakage factor are in this article as defined in International Electrotechnical Commission Electropedia's IEV-131-12-41, Inductive coupling factor and IEV-131-12-42, Inductive leakage factor.
6. ^ a b Brenner & Javid 1959, §18-1 Mutual Inductance, pp. 587-591
7. ^ IEC 60050 (Publication date: 1990-10). Section 131-12: Circuit theory / Circuit elements and their characteristics, IEV 131-12-41 Inductive coupling factor
8. ^ Brenner & Javid 1959, §18-1 Mutual Inductance - Series connection of Mutual Inductance, pp. 591-592
9. ^ Brenner & Javid 1959, pp. 591-592, Fig. 18-6
10. ^ Harris 1952, p. 723, fig. 43
11. ^ Voltech, Measuring Leakage Inductance
12. ^ Rhombus Industries, Testing Inductance
13. ^ This measured short-circuit inductance value is often referred to as the leakage inductance. See for example are, Measuring Leakage Inductance,Testing Inductance. The formal leakage inductance is given by (Eq. 2.14).
14. ^ Harris 1952, p. 723, fig. 42
15. ^ Khurana 2015, p. 254, fig. 7.33
16. ^ Brenner & Javid 1959, §18-5 The Linear Transformer, pp. 595-596
17. ^ Hameyer 2001, p. 24
18. ^ Singh 2016, Mutual Inductance
19. ^ Brenner & Javid 1959, §18-6 The Ideal Transformer, pp. 597-600: Eq. 2.2 holds exactly for an ideal transformer where, at the limit, as self-inductances approach an infinite value ( ${\displaystyle L_{P}}$  → ∞ & ${\displaystyle L_{S}}$  → ∞ ), the ratio ${\displaystyle L_{P}/L_{S}}$  approaches a finite value.
20. ^ Hameyer 2001, p. 24, eq. 3-1 thru eq. 3-4
21. ^ Hameyer 2001, p. 25, eq. 3-13
22. ^ Knowlton 1949, pp. §8–67, p. 802: Knowlton describes The Leakage Factor as "The total flux which passes through the yoke and enters the pole = Φm = Φa + Φe and the ratio Φma is called the leakage factor and is greater than 1." This factor is evidently different from the inductive leakage factor described in this Leakage inductance article.
23. ^ IEC 60050 (Publication date: 1990-10). Section 131-12: Circuit theory / Circuit elements and their characteristics, IEV ref. 131-12-42: "Inductive leakage factor
24. ^ IEC 60050 (Publication date: 1990-10). Section 221-04: Magnetic bodies, IEV ref. 221-04-12: "Magnetic leakage factor - the ratio of the total magnetic flux to the useful magnetic flux of a magnetic circuit." This factor is also different from the inductive leakage factor described in this Leakage inductance article.
25. ^ a b Hameyer 2001, p. 27
26. ^ a b c Brenner & Javid 1959, §18-7 Equivalent Circuit for the nonideal transformer, pp. 600-602 & fig. 18-18
27. ^ Brenner & Javid 1959, p. 602, "Fig. 18-18 In this equivalent circuit of a (nonideal) transformer the elements are physically realizable and the isolationg property of the transformer has been retained."
28. ^ a b Erickson & Maksimovic, Chapter 12 Basic Magnetic Theory, §12.2.3. Leakage inductances
29. ^ Kim 1963, pp. 3-12, Magnetice Leakage in Transformers; pp. 13-19, Leakage Reactance in Transformers.
30. ^ Hameyer 2001, p. 29, Fig. 26
31. ^ Kim 1963, p. 4, Fig. 1, Magnetic field due to current in the inner winding of a core-type transformer; Fig. 2, Magnetic field due to current in the outer winding of Fig. 1.
32. ^ Hameyer 2001, pp. 28, eq. 3-31
33. ^ Hameyer 2001, pp. 28, eq. 3-32
34. ^ Hameyer 2001, pp. 29, eq. 3-33
35. ^ Kim 1963, p. 10, eq. 12
36. ^ Hameyer 2001, pp. 29, eq. 3-34
37. ^ Kim 1963, p. 10, eq. 13
38. ^ Hameyer 2001, pp. 29, eq. 3-35
39. ^ Hameyer 2001, pp. 29, eq. 3-36
40. ^ Hameyer 2001, p. 29, eq. 3-37