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.
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 of some loads.
Leakage inductance applies to any imperfectly-coupled magnetic circuit device including motors.

Leakage inductance and inductive coupling factor

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.
The primary open-circuit self-inductance is given by
where
and

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,
The coupling factor is derived from the inductance value measured across one winding with the other winding short-circuited according to the following:
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
where
and
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

The winding turns ratio is in practice given as
where
The nonideal transformer's mesh equations can be expressed by the following voltage and flux linkage equations,
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,
The transformer inductance can be characterized in terms of the three inductance constants as follows,
where,
The transformer can be expressed more conveniently as the equivalent circuit in Fig. 3 with secondary constants referred to the primary,
Since
and
we have
which allows expression of the equivalent circuit in Fig. 4 in terms of winding leakage and magnetizing inductance constants as follows,
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,

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
b. Per Eq. 2.7 & Inductive leakage factor is given by
c. multiplied by gives
d. Per Eq. 2-8 & knowing that
e. multiplied by gives
f. Per Eq. 3.5 Eq. 1.1b & Eq. 2.14 and Eq. 3.6 Eq. 1.1b & Eq. 2.14:
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:
In the same way,
And therefore,
where

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:

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 without itself dissipating power. 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 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 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.