Leakage inductance

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Leakage inductance derives from the electrical property of an imperfectly-coupled transformer whereby each winding behaves as a self-inductance constant in series with the winding's respective ohmic resistance constant, these four winding constants also interacting with the transformer's mutual inductance constant. 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[edit]

Fig. 1 LPσand LSσ are primary and secondary leakage inductances expressed in terms of inductive coupling coefficient 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 .[5][6][7]

The primary open-circuit self-inductance is given by

------ (Eq. 1.1a)

where

------ (Eq. 1.1b)
------ (Eq. 1.1c)

and

  • is primary self-inductance
  • is primary leakage inductance
  • is magnetizing inductance
  • is inductive coupling coefficient

Measuring basic transformer inductances & coupling factor

Transformer self-inductances & and mutual inductance are, in additive and subtractive series connection of the two windings, given by,[8]

in additive connection,
, and,
in subtractive connection,
such that these transformer inductances can be determined from the following three equations:[9][10]
.

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,
and
Such that

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 are given by

------ (Eq. 1.2), and,
, with 0 < < 1 ------ (Eq. 1.3)

where

and

  • is mutual inductance
  • is secondary self-inductance
  • is secondary leakage inductance
  • is magnetizing inductance referred to the secondary
  • 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[edit]

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
  • & are primary and secondary winding resistances
  • Constants , , , & are measurable at the transformer's terminals
  • Coupling factor is defined as
, where 0 < < 1 ------ (Eq. 2.1)

The winding turns ratio is in practice given as

------ (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]

------ (Eq. 2.3)
------ (Eq. 2.4)
------ (Eq. 2.5)
------ (Eq. 2.6),
where
  • is flux linkage
  • 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]

------ (Eq. 2.7),
where,
  • ioc & isc are open-circuit and short-circuit currents
  • Loc & Lsc are open-circuit and short-circuit inductances.
  • is the inductive leakage factor or Heyland factor[22][23][24]
  • & 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]

------ (Eq. 2.8)
------ (Eq. 2.9)
------ (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]

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

Since

------ (Eq. 2.11)

and

------ (Eq. 2.12),

we have

------ (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
------ (Eq. 2.14 Eq. 1.1b)
------ (Eq. 2.15 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,

------ (Eq. 2.16)
  • is magnetizing current excited by flux ΦM that links both primary and secondary windings
  • is the primary current
  • is the secondary current referred to the primary side of the transformer.

Refined inductive leakage factor[edit]

Refined inductive leakage factor derivation

a. Per Eq. 2.1 & IEC IEV 131-12-41 inductive coupling factor is given by

--------------------- (Eq. 2.1):

b. Per Eq. 2.7 & IEC IEV 131-12-42 Inductive leakage factor is given by

------ (Eq. 2.7) & (Eq. 3.7a)

c. multiplied by gives

----------------- (Eq. 3.7b)

d. Per Eq. 2-8 & knowing that

---------------------- (Eq. 3.7c)

e. multiplied by gives

------------------ (Eq. 3.7d)

f. Per Eq. 3.5 Eq. 1.1b & Eq. 2.14 and Eq. 3.6 Eq. 1.1b & Eq. 2.14:

--- (Eq.3.7e)

All equations in this article assume steady-state constant-frequency waveform conditions the & 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 Eq. 2.7)

In the same way,

σS = ΦSσ'M = LSσ'/LM[33] ------ (Eq. 3.2 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 Eq. 1.1b & Eq. 2.14)
LS' = LM + LSσ' = LM + σSLM = (1 + σS)LM[39] ------ (Eq. 3.6 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]

------ (Eq. 3.7a to 3.7e).

Applications[edit]

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 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]

See also[edit]

External links[edit]

IEC Electropedia links:

References[edit]

  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 ( → ∞ & → ∞ ), the ratio 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, p. §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

Bibliography[edit]

  • Brenner, Egon; Javid, Mansour (1959). "Chapter 18 – Circuits with Magnetic Coupling". Analysis of Electric Circuits. McGraw-Hill. pp. esp. 586–617.
  • Didenko, V.; Sirotin, D. (2012). "Accurate Measurement of Resistance and Inductance of Transformer Windings" (PDF). XX IMEKO World Congress – Metrology for Green Growth. Busan, Republic of Korea, September 9−14, 2012.
  • Erickson, Robert W.; Maksimovic, Dragan (2001). "Chapter 12: Basic Magnetics Theory (Instructor slides only for book)" (PDF). Fundamentals of Power Electronics (2nd ed.). Boulder: University of Colorado (slides) / Springer (book). pp. 72 slides. ISBN 978-0-7923-7270-7.
  • "Electropedia: The World's Online Electrotechnical Vocabulary". IEC 60050 (Publication date: 1990-10). Archived from the original on 2015-04-27.
  • Hameyer, Kay (2001). Electrical Machines I: Basics, Design, Function, Operation (PDF). RWTH Aachen University Institute of Electrical Machines.
  • Harris, Forest K. (1952). Electrical Measurements (5th printing (1962) ed.). New York, London: John Wiley & Sons.
  • Heyland, A. (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ. pp. 561–564. Missing or empty |url= (help)
  • Heyland, A. (1906). A Graphical Treatment of the Induction Motor. Translated by George Herbert Rowe; Rudolf Emil Hellmund. McGraw-Hill. pp. 48 pages.
  • Irwin, J. D. (1997). The Industrial Electronics Handbook. A CRC handbook. Taylor & Francis. ISBN 978-0-8493-8343-4.
  • Khurana, Rohit (2015). Electronic Instrumentation and Measurement. Vikas Publishing House. ISBN 9789325990203.
  • Kim, Joong Chung (1963). The Determination of Transformer Leakage Reactance by Using an Inpulse Driving Function. 57 pages: University of Oregon.
  • Knowlton, A.E., ed. (1949). Standard Handbook for Electrical Engineers (8th ed.). McGraw-Hill. p. 802, § 8–67: The Leakage Factor.
  • MIT-Press (1977). "Self- and Mutual Inductances". Magnetic circuits and transformers a first course for power and communication engineers. Cambridge, Mass.: MIT-Press. pp. 433–466. ISBN 978-0-262-31082-6.
  • Pyrhönen, J.; Jokinen, T.; Hrabovcová, V. (2008). Design of Rotating Electrical Machines. p. Chapter 4 Flux Leakage.
  • "Mutual Inductance" (PDF). Rhombus Industries Inc. 1998. Retrieved 4 August 2018.
  • Saarbafi, Karim; Mclean, Pamela (2014). "AESO Transformer Modelling Guide" (PDF). Calgary: AESO - Alberta Electric System Operator (prepared by Teshmont Consultants LP). pp. 304 pages. Retrieved August 6, 2018.
  • Singh, Mahendra (2016). "Mutual Inductance". Electronics Tutorials. Retrieved 6 January 2017.
  • "Measuring Leakage Inductance" (PDF). Voltech Instruments. 2016. Retrieved 5 August 2018.