Leakage inductance

Leakage inductance is that property of an electrical transformer that causes a winding to appear to have some pure inductance in series with the mutually-coupled transformer windings.

Usually, this is an undesirable property, but it is sometimes deliberately introduced into a transformer that is used as a ballast for a gas discharge lamp such as a fluorescent lamp, or in a transformer used for arc welding. In this case, the leakage inductance limits the current flow to the desired magnitude.

Leakage inductance is primarily controlled by the design of the windings and the geometry of the magnetic core used to form the transformer.

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.

Leakage inductance applies to any imperfectly-coupled magnetic circuit device including motors; it reduces efficiency (ratio of power out to power in) of transformers and other inductive devices.

Definition and measurement of leakage inductance

Leakage inductance can be estimated during design of a transformer from the dimensions of the windings on the core. More precise estimates can be obtained from solution of the magnetic field around the core, using computerized methods.

Several measurement methodologies are employed to assess transformer inductance. The primary one is measurement of various quantities at the transformer terminals, with one coil driven and the other coil open-circuited. Another methodology employs both open-circuit measurements as well as short-circuit measurements, where one coil is short circuited and the other driven. A different methodology is based on assessing magnetic flux of each winding in a "window" of the core where the coil windings pass through it. The resulting assessments of leakage inductance differ because different kinds of losses are taken into account by the methodologies.

The inductance (including mutual inductance and incident leakage inductances) of a magnetic core transformer changes with frequency and current due to dissipative losses which affect the magnetic permeability of the core. At high frequencies and currents the effect is nonlinear and significant.

Depending on the application of the transformer, the leakage inductance may be expressed in units of henrys; for power transformers the inductance may be more usefully expressed as a per cent impedance based on the transformer's rated power.

Transformer operation

A transformer works because a magnetic circuit in the core which transfers energy couples its electric windings, which are inductors. A magnetic circuit is an electromagnet, a magnet that is magnetized only when electricity flows in its coil. But not all of the electric energy can be transferred to magnetic energy in the core (and not all of the magnetic energy in the core can be transferred back to another electric circuit) – some becomes a magnetic field in the air. The "wasted" energy in that magnetic field is called leakage flux. However, leakage flux is not lost in the same way as electrical energy which is dissipated in the form of heat. Leakage flux remains as part of the energy in the electric circuit. Its effect is to add some incremental "resistance" to the flow of electricity in that circuit. That "resistance" is called leakage inductance.

Inductance of a transformer

The inductance of a transformer is a composite of various inductances, which in the form of inductive reactances, will behave in a manner according to Kirchoff's voltage and current laws. The inductance can therefore be decomposed into its constituent parts. Some of those constituent inductances have well-known relationships which hold in an ideal transformer, and also in the inductive transformer model (below), so their values can be determined from measured quantities. Leakage inductances are remainder inductances, which means after the other inductances are accounted for, some is left over, so a value can then be assigned to them. The principal inductances of a two-winding transformer which are found in an ideal transformer are primary self-inductance, secondary self-inductance, and mutual inductance. Though mutual inductance is a single entity, because it results from shared magnetic flux, it is synthetically divided into two abstract pieces which are mathematically defined: magnetizing inductance referred to the primary, and magnetizing inductance referred to the secondary. The division is not an arithmetic operation, it is a geometric one, and it is not arbitrary – it is in accord with Faraday's Law, which governs the relation between current in an electric circuit and flux in a magnetic circuit. In an ideal transformer, self-inductance is magnetizing inductance, and the inductances are usually referred to simply as primary inductance and secondary inductance. But in the transformer inductance model (and in real transformers), self-inductance of each inductor is a composite of magnetizing inductance and leakage inductance. Leakage inductances are in series with the mutual inductance of the transformer (or its magnetizing inductance components).

Inductive circuit model of transformer

Leakage inductances cannot be studied in isolation – they are an integral part of a transformer's inductance. Real transformers have losses, nonlinearties, and complex fields that do not lend themselves to convenient analysis; ideal transformers have nice qualities, but lack the elements of interest, in this case leakage inductances. So an ideational model is constructed, one which takes an ideal transformer and incorporates leakage inductances and related inductances which are part of a real transformer, but does not include other elements of a real transformer, notably DCR. An ideational transformer model does not perform like an ideal transformer, nor like a real transformer. Such a model cannot be built or tested in a laboratory (or it would come up against all the other excluded elements of a real transformer), so a gedankenexperiment, (thought experiment), is conducted which assigns appropriate voltages, currents, etc. to the inputs of the model, and assesses resulting relevant quantities like leakage inductances. The results of the model can not be assigned to any real transformer: the model must incorporate many other elements of real transformers like DCR and AC resistance, interwinding capacitance, and core losses before the results of the model approach the measurements of a real transformer. The purpose of the model is to elucidate the concept of leakage inductance and its relationship to the operation of a transformer. Fig. 1 below is a schematic of an inductive ideational model of a transformer.

Model definitions

The fundamental quantities, those that are measured or assumed and used as input to the model, are primary and secondary self-inductances, mutual inductance and count of coils in each winding (usually reported as a single number, the winding ratio), and if the model includes them, primary and secondary DC resistances. The output of the model is primary and secondary leakage inductance, and two derived ratios, coupling factor and inductive leakage factor. These leakage inductances are defined in terms of measured transformer winding open-circuit inductances.

The winding turns ratio '$a$' is defined as

$a:\Leftrightarrow N_{P}/N_{S}$ ------ (Eq. 1.0)

where

• $N_{P}$ is the number of turns in the primary winding
• $N_{S}$ is the number of turns in the secondary winding

The actual physical turns ratio is not a usual specification of real transformers - the coils will be wound to whatever ratio they need to be to obtain the specified working voltage ratios. While there is no methodology for indirectly measuring the actual turns ratio, a more useful ratio is a measured one which approximates the actual ratio; it incorporates some degree of losses including leakage inductance which would otherwise have to be somehow assessed independently. The following is one way of using measured quantities to approximate the ratio:

$a'={\sqrt {L_{P}/L_{S}}}$------ (Eq. 1.1) (usually written simply as '$a$' and referred to inexactly as the 'turns ratio'.)

where $a'\approx a$. The ratio is reasonably accurate only when leakage inductances are small and proportional to the size (inductance) of the coils, which is the usual case. As leakage inductances approach zero, the ratio approaches the winding ratio of an ideal transformer which would be a count of the number of turns. Another name for the measured ratio is the 'transformer ratio'.

$M:\Leftrightarrow {\sqrt {m1\cdot m2}}$ ------ (Eq. 1.2) is mutual inductance

where

• $m1$ is magnetizing inductance referred to the primary
• $m2$ is magnetizing inductance referred to the secondary

$L_{P}$ is primary winding self-inductance

$L_{S}$ is secondary winding self-inductance

$\sigma 1$ is primary leakage inductance

$\sigma 2$ is secondary leakage inductance

Fundamental relations

The self-, leakage, and magnetizing inductances are not all independent quantities; certain relations must hold between them. For each winding, the leakage inductance and magnetizing inductance component of mutual inductance are in series; they may be considered as additive. They are also in series with the DCR of the winding (not included in the model, but very important in real transformer measurement). These must be combined as complex impedances. The magnetizing inductance components of mutual inductance are related according to the transformer ratio. These relationships are expressed precisely as,

Fig. 2 L_P^σand L_S^σ are primary and secondary open-circuit leakage inductances (denoted $\sigma 1$ and $\sigma 2$ in the text).

$L_{P}=\sigma 1+m1$ for the primary ------ (Eq. 2.1)

$L_{S}=\sigma 2+m2$ for the secondary ------ (Eq. 2.2)

$m1=a^{2}\cdot m2$ ------ (Eq. 3.1)

Leakage inductance specification

.

There are three specifications associated with leakage inductance: a scalar absolute value of inductance in Henries for each winding, and two unitless ratios indicative of transformer performance, inductive leakage factor and inductive coupling factor.

The leakage inductances are,

$\sigma 1=L_{P}-a{M}$ ------ (Eq. 4.1)

$\sigma 2=L_{S}-{M}/a$ ------ (Eq. 4.2)

The coupling factor, denoted k, is defined as,

$k:\Leftrightarrow {\frac {\left|M\right|}{\sqrt {L_{P}L_{S}}}}$, where 0 < k < 1 ------ (Eq. 4.3)

The inductive leakage factor, denoted $\sigma$, is defined as,

$\sigma :\Leftrightarrow 1-k^{2}$, where 1 > σ > 0 ------ (Eq. 4.4)

$k$ may be viewed as the useful proportion of the total inductance (the geometric mean of primary self-inductance and secondary self-inductance), and $\sigma$ as the unusable proportion of inductance, though they aren't addable like arithmetic proportions.

Short-circuit inductance

See also: Resonant inductive coupling

Measurement of short-circuit inductance

Short-circuit inductance of a real linear two-winding transformer is inductance measured across the primary or secondary winding when the other winding is short-circuited. Short-circuit measurement is used as an alternative to or complement of open-circuit measurements of transformers. Which measurement methodology is most relevant depends on the application.　

Equivalent circuit

Measured primary and secondary short-circuit inductances may be considered as constituent parts of primary and secondary self-inductances. They are related according to the coupling factor as,

$L_{\mathrm {Psc} }=(1-k^{2})\cdot L_{P}\,$ ------ (Eq. 4.5)

$L_{\mathrm {Ssc} }=(1-k^{2})\cdot L_{S}\,$ ------ (Eq. 4.6)

Where

• $k$ is coupling coefficient
• $L_{P}$ is primary self-inductance
• $L_{S}$ is secondary self-inductance

Short-circuit inductance measurement is used in conjunction with open-circuit inductance measurements to obtain various derived quantities like $k$, the inductive coupling factor and $\sigma$, the inductive leakage factor. $k$ is derived according to :

$k={\sqrt {1-{\frac {L_{\text{sc}}}{L_{\text{oc}}}}}}$

where

• $L_{\text{sc}}$ is the short-circuit measurement of primary or secondary inductance
• $L_{\text{oc}}$ is the corresponding open-circuit measurement of primary or secondary inductance

Other transformer parameters like leakage inductance and mutual inductance which cannot be directly measured may be defined in terms of k.

Short-circuit inductance is one of the parameters that determines the resonance frequency of the magnetic phase synchronous coupling in a resonant transformer and wireless power transfer. Short-circuit inductance is the main component of the current-limiting parameter in leakage transformer applications.

Reducing leakage inductance in transformers

Leakage flux (and incident leakage inductance of the electric circuits) occurs when the magnetic field of each turn of each winding fails to intersect ("couple" with) the magnetic field of each turn of the other winding. The magnetic core which is geometrically concentric vis-a-vis each winding is a conduit of winding coupling: the magnetic flux of the core is thousands of times the flux of the windings, so it is by far the most important conduit of magnetic field lines of force. The core is the only conduit of coupling when the windings are not centroid, i.e.share the same leg or spindle of the core (though if the windings are close enough to each other, some coupling will occur through the air). Centroid windings bring the turns of each coil into close proximity, so that magnetic field lines of force intersect via proximity to each other, in addition to their proximity to the core. It is fairly common to construct transformers so the primary winding is adjacent to the core, and secondary windings are layered around the circumference of it.

However, the number of magnetic field lines of force varies inversely with the cube of the distance from the electric conductor, so even small distances between layers of one coil and those of another can result in noticeably reduced coupling. In high quality transformers, the layers of centroid windings are interleaved alternately by layer. The layers are wound in the same direction, so that the current in the primary and secondary coils will run in the same direction.

In layer-interleaved windings the turns of one coil near one end of the coil are still some distance from the turns on the opposite end of the other coil, and the same inverse cube rule applies to longitudinal distance between turns. So some transformer coils are bifilar wound, which means that wire from the primary and secondary coils is slightly twisted together, and wound as if the twisted pair was a single winding. The current of each circuit will run in parallel through this kind of winding. This minimizes geometric distance in both transverse and longitudinal directions between turns of the coils.

The geometry of the cross-section of the coil - its effective radius versus its height- also affects leakage inductance.

The geometry of the magnetic core also affects leakage inductance. A toroidal core which has a closed magnetic circuit (i.e. the magnetic circuit is completely within the core), dramatically reduces leakage inductance.

While centroid windings, layer-interleaving, and bifilar winding all increase coupling and therefore decrease leakage inductance, they also progressively increase interwinding capacitance. Capacitance is in parallel with the mutual inductance.

Applications

Leakage inductance is often treated as an undesirable property, as it causes the voltage to change with loading.

But in some 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, the parameter actually working effectively is short-circuit inductance.

High leakage transformer

Large commercial transformers are usually designed with a short-circuit leakage reactance impedance of between 3% and 10%. If the load is resistive and the leakage reactance is small (<10%) the output voltage will not drop by more than 0.5% at full load, ignoring other resistances and losses.

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.

See also

• Blocked rotor test
• Circle diagram
• Open circuit test
• Short circuit test
• Steinmetz equivalent circuit

Bibliography

• Brenner, Egon; Javid, Mansour (1959). "Chapter 18 – Circuits with Magnetic Coupling". Analysis of Electric Circuits. McGraw-Hill. pp. esp. 586–617. {{cite conference}}: Invalid |ref=harv (help); Unknown parameter |booktitle= ignored (|book-title= suggested) (help)
• 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. {{cite conference}}: Invalid |ref=harv (help)CS1 maint: location (link)
• Harris, Forest K. (1952). Electrical Measurements (5th printing (1962) ed.). New York, London: John Wiley & Sons. {{cite book}}: Invalid |ref=harv (help)
• Heyland, A. (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ. pp. 561–564. {{cite web}}: 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. {{cite book}}: Invalid |ref=harv (help)
• Khurana, Rohit (2015). Electronic Instrumentation and Measurement. Vikas Publishing House. ISBN 9789325990203. {{cite book}}: Invalid |ref=harv (help)
• Knowlton, A.E., ed. (1949). Standard Handbook for Electrical Engineers (8th ed.). McGraw-Hill. p. 802, § 8–67: The Leakage Factor. {{cite book}}: Invalid |ref=harv (help)
• 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. {{cite conference}}: Invalid |ref=harv (help)
• Pyrhönen, J.; Jokinen, T.; Hrabovcová, V. (2008). Design of Rotating Electrical Machines. p. Chapter 4 Flux Leakage. {{cite book}}: Invalid |ref=harv (help)
• Singh, Mahendra (2016). "Mutual Inductance". Electronics Tutorials. Retrieved 6 January 2017. {{cite web}}: Invalid |ref=harv (help)