Leakage inductance derives from the electrical property of an imperfectly-coupled
transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
whereby each
winding
An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil (spiral or helix). Electromagnetic coils are used in electrical engineering, in applications where electric currents interact with magnetic fields, in de ...
behaves as a
self-inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
in
series
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* Caroline Series (born 1951), English mathematician, daughter of George Series
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with the winding's respective
ohmic resistance constant. These four winding constants also interact with the transformer's
mutual inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of th ...
. 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
Voltage drop is the decrease of electrical potential along the path of a current flowing in an electrical circuit. Voltage drops in the internal resistance of the source, across conductors, across contacts, and across connectors are undesirab ...
,
reactive power consumption and
fault current
In an electric power system, a fault or fault current is any abnormal electric current. For example, a short circuit is a fault in which a live wire touches a neutral or ground wire. An open-circuit fault occurs if a circuit is interrupted by a f ...
considerations.
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.
Leakage inductance applies to any imperfectly-coupled magnetic circuit device including
motor
An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy.
Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power g ...
s.
Leakage inductance and inductive coupling factor
![Coupling coefficient2](https://upload.wikimedia.org/wikipedia/commons/c/c1/Coupling_coefficient2.gif)
The magnetic circuit's flux that does not interlink both windings is the leakage flux corresponding to primary leakage inductance L
Pσ and secondary leakage inductance L
Sσ. Referring to Fig. 1, these leakage inductances are defined in terms of transformer winding
open-circuit inductances and associated
coupling coefficient or coupling factor
.
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,
::::in additive connection,
::::, and,
::::in subtractive connection,
::::
:::such that these transformer inductances can be determined from the following three equations:
::::
::::
::::.
The coupling factor is derived from the inductance value measured across one winding with the other winding short-circuited according to the following:
:::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.
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
:*
is the approximate turns ratio
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.
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).
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,
: ------ (Eq. 2.3)
: ------ (Eq. 2.4)
: ------ (Eq. 2.5)
: ------ (Eq. 2.6),
:where
:* is flux linkage
:* is derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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
The open-circuit test, or no-load test, is one of the methods used in electrical engineering to determine the no-load impedance in the excitation branch of a transformer.
The no load is represented by the open circuit, which is represented on th ...
is as follows,
: ------ (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
:* & are primary and secondary short-circuited leakage inductances.
The transformer inductance can be characterized in terms of the three inductance constants as follows,
: ------ (Eq. 2.8)
: ------ (Eq. 2.9)
: ------ (Eq. 2.10) ,
where,
:*LM 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 electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly ...
in Fig. 3 with secondary constants referred (i.e., with prime superscript notation) to the primary,
:
:
:
:.
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,
: ------ (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
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 131-12-42Inductive 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:
:σP = ΦPσ/ΦM = LPσ/LM ------ (Eq. 3.1 Eq. 2.7)
In the same way,
:σS = ΦSσ'/ΦM = LSσ'/LM ------ (Eq. 3.2 Eq. 2.7)
And therefore,
:ΦP = ΦM + ΦPσ = ΦM + σPΦM = (1 + σP)ΦM ------ (Eq. 3.3)
:ΦS' = ΦM + ΦSσ' = ΦM + σSΦM = (1 + σS)ΦM ------ (Eq. 3.4)
:LP = LM + LPσ = LM + σPLM = (1 + σP)LM ------ (Eq. 3.5 Eq. 1.1b & Eq. 2.14)
:LS' = LM + LSσ' = LM + σSLM = (1 + σS)LM ------ (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:
: ------ (Eq. 3.7a to 3.7e).
Applications
Leakage inductance can be an undesirable property, as it causes the voltage to change with loading.
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
In electrical engineering, particularly power engineering, voltage regulation is a measure of change in the voltage magnitude between the sending and receiving end of a component, such as a transmission or distribution line. Voltage regulation desc ...
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
Arc welding is a welding process that is used to join metal to metal by using electricity to create enough heat to melt metal, and the melted metals, when cool, result in a binding of the metals. It is a type of welding that uses a welding powe ...
sets. In these cases, the leakage inductance limits the current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
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.[
In addition, the leakage inductance of a HF-transformer can replace a series ]inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
in a resonant converter A resonant converter is a type of electric power converter that contains a network of inductors and capacitors called a "resonant tank", tuned to resonate at a specific frequency. They find applications in electronics, in integrated circuits.
The ...
. In contrast, connecting a conventional transformer
A transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer' ...
and an inductor in series results in the same electric behavior as of a leakage transformer, but this can be advantageous to reduce the eddy current losses in the transformer windings caused by the stray field.
See also
*Blocked rotor test A blocked rotor test is conducted on an induction motor. It is also known as short-circuit test (because it is the mechanical analogy of a transformer short-circuit test), locked rotor test or stalled torque test. From this test, short-circuit cur ...
*Circle diagram
First conceived by A.heyland in 1894 and B.A. Behrend in 1895, the circle diagram is the graphical representation of the performance of the electrical machine drawn in terms of the locus of the machine's input voltage and current. The circle di ...
*Mutual inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of th ...
* Steinmetz equivalent circuit
* Short-circuit inductance
*Short-circuit test
The purpose of a short-circuit test is to determine the series branch parameters of the equivalent circuit of a transformer.
Method
The test is conducted on the high-voltage (HV) side of the transformer where the low-voltage (LV) side (or the ...
*Voltage regulation
In electrical engineering, particularly power engineering, voltage regulation is a measure of change in the voltage magnitude between the sending and receiving end of a component, such as a transmission or distribution line. Voltage regulation desc ...
Notes
References
External links
IE
Electropedia
links:
Linked flux
Ideal voltage source
Inductance
Ideal current source
Coupling
Inductive coupling
Inductive coupling factor
Inductive leakage factor
Ideal transformer
Magnetic leakage factor
Self-inductance
Mutual inductance
Bibliography
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Electric transformers
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