Leakage inductance

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 L_P^σ and secondary leakage inductance L_S^σ. These leakage inductances are defined in terms of transformer winding open-circuit inductances as well as the transformer's coupling coefficient or coupling factor k, the primary open-circuit self-inductance being given by

where

and

It therefore follows that the transformer secondary open-circuit self, magnetizing and leakage inductances are given by

Inductive coupling factor k is given as

where

and

The electric validity of the above transformer diagram depends strictly on open circuit conditions for the respective winding inductances considered, more generalized circuit conditions being as developed in the next two sections.

Inductive leakage factor and inductance

A real linear two-winding transformer can be represented by two mutual inductance coupled circuit loops linking the transformer's five impedance constants as shown in the diagram at right, where,

M is mutual inductance

L_P & L_S are primary and secondary winding self-inductances

R_P & R_S are primary and secondary winding resistances

Constants M, L_P, L_S, R_P & R_S are measurable at the transformer's terminals

Coupling factor k is given as absolute value of coupling coefficient

Winding turns ratio a is in practice given as

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

v P = R P i P + d Ψ P d t ------ (Eq. 2.3) v S = − R S i S − d Ψ S d t ------ (Eq. 2.4) Ψ P = L P i P − M i S ------ (Eq. 2.5) Ψ S = L S i S − M i P ------ (Eq. 2.6), where

ψ is flux linkage

dψ/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 no-load is as follows,

σ = 1 − M 2 L P L S = 1 − k 2 = L s c L o c = L s c s e c L P = L s c p r i L S = i o c i s c ------ (Eq. 2.7),

where,

σ is the inductive leakage factor or Heyland factor

i_oc & i_sc are no-load and short circuit currents

L_oc & L_sc are no-load and short circuit inductances.

The transformer can thus be further defined in terms of the three inductance constants as follows,

L M = a M ------ (Eq. 2.8) L P σ = L P − a M L S σ = L S − M / a ,

where,

L_M is magnetizing inductance, corresponding to magnetizing reactance X_M

L_P^σ & L_S^σ are primary & secondary leakage inductances, corresponding to primary & secondary leakage reactances X_P^σ & X_S^σ.

The transformer can be expressed more conveniently as the first shown equivalent circuit with secondary constants referred (i.e., with prime superscript notation) to the primary,

L S σ ′ = a 2 L S − a M R S ′ = a 2 R S V S ′ = a V S I S ′ = I S / a .

Since

k = M / L P L S ------ (Eq. 2.9)

and

a = L P / L S ------ (Eq. 2.10),

we have

a M = L P / L S ∗ k ∗ L P L S = k L P ------ (Eq. 2.11),

which allows expression as second shown equivalent circuit with winding leakage and magnetizing inductance constants as follows,

L P σ = L S σ ′ = L P ∗ ( 1 − k ) ------ (Eq. 2.12) L M = k L P ------ (Eq. 2.13).

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

i_M = i_P - i_S^' ------ (Eq. 2.14) i_M is magnetizing current excited by flux Φ_M that links both primary and secondary windings. i_S^' is the secondary current referred to the primary side of the transformer.

Refined inductive leakage factor

Referring to the flux diagram below at left, primary driven and the secondary open-circuited, the following equation holds:

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M ------ (Eq. 3.1)

In the same way,

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M ------ (Eq. 3.2)

And therefore,

Φ_P = Φ_M + Φ_P^σ = Φ_M + σ_PΦ_M = (1 + σ_P)Φ_M ------ (Eq. 3.3) Φ_S^' = Φ_M + Φ_S^σ' = Φ_M + σ_SΦ_M = (1 + σ_S)Φ_M ------ (Eq. 3.4) L_P = L_M + L_P^σ = L_M + σ_PL_M = (1 + σ_P)L_M ------ (Eq. 3.5) L_S^' = L_M + L_S^σ' = L_M + σ_SL_M = (1 + σ_S)L_M ------ (Eq. 3.6),

where

σ_P is primary inductive leakage factor

σ_S is secondary inductive leakage factor

Φ_M is mutual flux (main flux) / L_M is magnetizing inductance

Φ_P^σ is primary leakage flux / L_P^σ is primary leakage inductance

Φ_S^σ is secondary leakage flux / L_S^σ is secondary leakage inductance

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:

σ = 1 − k 2 = 1 − M 2 L P L S = 1 − a 2 M 2 L P a 2 L S = 1 − L M 2 L P L S ′ = 1 − 1 L P L M . L S ′ L M = 1 − 1 ( 1 + σ P ) ( 1 + σ S ) ------ (Eq. 3.7).

Leakage inductance in practice

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.

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.

Measurement of inductive coupling factor

The inductive 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, L s c p r i = L S ⋅ ( 1 − k 2 ) and L s c s e c = L P ⋅ ( 1 − k 2 ) . Therefore, the inductive coupling factor is given by k = 1 − L s c p r i L S = 1 − L s c s e c L P . ------ (Eq. 4.1)