Leakage inductance

Template:Short description 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.<ref>Template:Harvnb</ref><ref name=Saarbafi-9>Template:Harvnb</ref>

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.Template:Sfn

Leakage inductance applies to any imperfectly coupled magnetic circuit device including motors.<ref name=Pyrhonen>Template:Harvnb</ref>

Leakage inductance and inductive coupling factorEdit

File:Coupling coefficient2.gif

Fig. 1 L_P^σand L_S^σ are primary and secondary leakage inductances expressed in terms of inductive coupling coefficient <math>k</math> under open-circuited conditions.

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. Referring to Fig. 1, these leakage inductances are defined in terms of transformer winding open-circuit inductances and associated coupling coefficient or coupling factor <math>k</math>.<ref>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.</ref><ref name=18-1>Template:Harvnb</ref><ref>IEC 60050 (Publication date: 1990-10). Section 131-12: Circuit theory / Circuit elements and their characteristics, IEV 131-12-41 Inductive coupling factor</ref>

The primary open-circuit self-inductance is given by

<math>L_{oc}^{pri}=L_P=L_M+L_P^\sigma</math> ------ (Eq. 1.1a)

where

<math>L_P^\sigma=L_P\cdot{(1-k)}</math> ------ (Eq. 1.1b)

<math>L_M=L_P\cdot{k}</math> ------ (Eq. 1.1c)

and

<math>L_{oc}^{pri}=L_P</math> is primary self-inductance
<math>L_P^\sigma</math> is primary leakage inductance
<math>L_M</math> is magnetizing inductance
<math>k</math> is inductive coupling coefficient

Measuring basic transformer inductances & coupling factor

Transformer self-inductances <math>L_P</math> & <math>L_S</math> and mutual inductance <math>M</math> are, in additive and subtractive series connection of the two windings, given by,<ref name=Brenner1959-591>Template:Harvnb</ref>

in additive connection,

<math>L_{ser}^{+}=L_P+L_S+2M</math>, and,

in subtractive connection,

such that these transformer inductances can be determined from the following three equations:<ref>Brenner & Javid 1959, pp. 591-592, Fig. 18-6</ref><ref>Harris 1952, p. 723, fig. 43</ref>

<math>L_P=a^2.L_S</math>.

The coupling factor is derived from the inductance value measured across one winding with the other winding short-circuited according to the following:<ref name="Voltech">Template:Harvnb</ref><ref name="Rhombus">Template:Harvnb</ref><ref>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).</ref>

Per Eq. 2.7,

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.<ref>Harris 1952, p. 723, fig. 42</ref><ref>Khurana 2015, p. 254, fig. 7.33</ref>

It therefore follows that the open-circuit self-inductance and inductive coupling factor <math>k</math> are given by

<math>L_{oc}^{sec}=L_S=L_{M2}+L_S^\sigma</math> ------ (Eq. 1.2), and,

<math>k=\frac {\left | M\right|}{\sqrt{L_PL_S}}</math>, with 0 < <math>k</math> < 1 ------ (Eq. 1.3)

where

<math>L_S^\sigma=L_S\cdot{(1-k)}</math>

and

<math>M</math> is mutual inductance
<math>L_{oc}^{sec}=L_S</math> is secondary self-inductance
<math>L_S^\sigma</math> is secondary leakage inductance
<math>L_{M2}= L_M/a^2</math> is magnetizing inductance referred to the secondary
<math>k</math> is inductive coupling coefficient
<math>a \equiv \sqrt {\frac {L_p} {L_s}} \approx N_P/N_S</math>Template:Efn 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 inductanceEdit

Template:See also

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.<ref name=18-1/><ref name=18-5>Template:Harvnb</ref><ref>Template:Harvnb</ref><ref name="ElecTut">Template:Harvnb</ref>

File:Basic transformer circuits.jpg

Fig. 2 Nonideal transformer circuit diagram

where

M is mutual inductance
<math>R_P</math> & <math>R_S</math> are primary and secondary winding resistances
Constants <math>M</math>, <math>L_P</math>, <math>L_S</math>, <math>R_P</math> & <math>R_S</math> are measurable at the transformer's terminals
Coupling factor <math>k</math> is defined as

<math>k=\left | M\right |/\sqrt{L_PL_S}</math>, where 0 < <math>k</math> < 1 ------ (Eq. 2.1)

The winding turns ratio <math>a</math> is in practice given as

<math>a=\sqrt{L_P/L_S}=N_P/N_S\approx v_P/v_S \approx i_S/i_P=</math> ------ (Eq. 2.2).<ref>Template:Harvnb</ref>

where

N_P & N_S are primary and secondary winding turns
v_P & v_S and i_P & i_S are primary & secondary winding voltages & currents.

The nonideal transformer's mesh equations can be expressed by the following voltage and flux linkage equations,<ref>Template:Harvnb</ref>

<math>v_P=R_P \cdot i_P+\frac{d\Psi{_P}}{dt}</math> ------ (Eq. 2.3)

<math>v_S=-R_S \cdot i_S-\frac{d\Psi{_S}}{dt}</math> ------ (Eq. 2.4)

<math>\Psi_P=L_P \cdot i_P-M \cdot i_S</math> ------ (Eq. 2.5)

<math>\Psi_S=L_S \cdot i_S-M \cdot i_P</math> ------ (Eq. 2.6),

where

<math>\Psi</math> is flux linkage
<math>\frac {d \Psi}{d t}</math> 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,<ref>Template:Harvnb</ref>

<math>\sigma=1-\frac{M^2}{L_PL_S}=1-k^2\approx\frac{L_{sc}}{L_{oc}}\approx \frac{L_{sc}^{sec}}{L_P}\approx\frac{L_{sc}^{pri}}{L_S}\approx\frac{i_{oc}}{i_{sc}}</math> ------ (Eq. 2.7),

where,

i_oc & i_sc are open-circuit and short-circuit currents
L_oc & L_sc are open-circuit and short-circuit inductances.
<math>\sigma</math> is the inductive leakage factor or Heyland factor<ref>Template:Harvnb: Knowlton describes The Leakage Factor as "The total flux which passes through the yoke and enters the pole = Φ_m = Φ_a + Φ_e and the ratio Φ_m/Φ_a 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.</ref><ref>IEC 60050 (Publication date: 1990-10). Section 131-12: Circuit theory / Circuit elements and their characteristics, IEV ref. 131-12-42: "Inductive leakage factor</ref><ref>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.</ref>
<math>L_{sc}^{pri}</math> & <math>L_{sc}^{sec}</math> are primary and secondary short-circuited leakage inductances.

The transformer inductance can be characterized in terms of the three inductance constants as follows,<ref name="Hameyer, p. 27">Template:Harvnb</ref><ref name="Brenner 18-7">Template:Harvnb</ref>

<math>L_M=a{M}</math> ------ (Eq. 2.8)

<math>L_P^\sigma=L_P-a{M}</math> ------ (Eq. 2.9)

<math>L_S^\sigma=L_S-{M}/a</math> ------ (Eq. 2.10) ,

where,

File:TREQCCTHeyland.jpg

Fig. 3 Nonideal transformer equivalent circuit

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 equivalent circuit in Fig. 3 with secondary constants referred (i.e., with prime superscript notation) to the primary,<ref name="Hameyer, p. 27"/><ref name="Brenner 18-7"/>

<math>L_S^{\sigma\prime}=a^2L_S-aM</math>

<math>R_S^\prime=a^2R_S</math>

<math>V_S^\prime=aV_S</math>

<math>I_S^\prime=I_S/a</math>.

File:TREQCCTHeyland-to-k.jpg

Fig. 4 Nonideal transformer equivalent circuit in terms of coupling coefficient k<ref name="Brenner 18-18">Template:Harvnb</ref>

Since

<math>k=M/\sqrt{L_PL_S}</math> ------ (Eq. 2.11)

and

<math>a=\sqrt{L_P/L_S}</math> ------ (Eq. 2.12),

we have

<math>aM=\sqrt{L_P/L_S} \cdot k \cdot \sqrt{L_PL_S}=kL_P</math> ------ (Eq. 2.13),

which allows expression of the equivalent circuit in Fig. 4 in terms of winding leakage and magnetizing inductance constants as follows,<ref name="Brenner 18-7"/>

File:TREQCCTHeylandConverted.jpg

Fig. 5 Simplified nonideal transformer equivalent circuit

<math>L_P^\sigma=L_S^{\sigma\prime}=L_P \cdot (1-k)</math> ------ (Eq. 2.14 <math>\equiv</math> Eq. 1.1b)

<math>L_M=kL_P</math> ------ (Eq. 2.15 <math>\equiv</math> 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,

<math>i_M = i_P - i_S^'</math> ------ (Eq. 2.16)

<math>i_M</math> is magnetizing current excited by flux Φ_M that links both primary and secondary windings
<math>i_P</math> is the primary current
<math>i_S'</math> is the secondary current referred to the primary side of the transformer.

Refined inductive leakage factorEdit

Refined inductive leakage factor derivation

a. Per Eq. 2.1 & IEC IEV 131-12-41 inductive coupling factor <math>k</math> is given by

<math>k=\left | M\right | /\sqrt{L_PL_S}</math> --------------------- (Eq. 2.1):

b. Per Eq. 2.7 & IEC IEV 131-12-42 Inductive leakage factor <math>\sigma</math> is given by

<math>\sigma=1-k^2=1-\frac{M^2}{L_PL_S}</math> ------ (Eq. 2.7) & (Eq. 3.7a)

c. <math>\frac{M^2}{L_PL_S}</math> multiplied by <math>\frac{a^2}{a^2}</math> gives

<math>\sigma=1-\frac{a^2M^2}{L_Pa^2L_S}</math> ----------------- (Eq. 3.7b)

d. Per Eq. 2-8 & knowing that <math>a^2L_S=L_S^\prime</math>

<math>\sigma=1-\frac{L_M^2}{L_PL_S^\prime}</math> ---------------------- (Eq. 3.7c)

e. <math>\frac{L_M^2}{L_PL_S^\prime}</math> multiplied by <math>\frac{L_M.L_M}{L_M^2}</math> gives

<math>\sigma=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^\prime}{L_M}}</math> ------------------ (Eq. 3.7d)

f. Per Eq. 3.5 <math> \approx</math> Eq. 1.1b & Eq. 2.14 and Eq. 3.6 <math> \approx</math> Eq. 1.1b & Eq. 2.14:

<math>\sigma=1-\frac{1}{(1+\sigma_P)(1+\sigma_S)}</math> --- (Eq.3.7e)

All equations in this article assume steady-state constant-frequency waveform conditions the <math>k</math> & <math>\sigma</math> values of which are dimensionless, fixed, finite & positive but less than 1.

Referring to the flux diagram in Fig. 6, the following equations hold:<ref name=Erickson>Template:Harvnb</ref><ref name=Kim1963-3>Template:Harvnb</ref>

File:Main & leakage inductances.jpg

Fig. 6 Magnetizing and leakage flux in a magnetic circuit<ref>Template:Harvnb</ref><ref name=Erickson/><ref name=Kim1963-4>Template:Harvnb</ref>

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M<ref>Template:Harvnb</ref> ------ (Eq. 3.1 <math>\approx</math> Eq. 2.7)

In the same way,

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M<ref>Template:Harvnb</ref> ------ (Eq. 3.2 <math>\approx</math> Eq. 2.7)

And therefore,

Φ_P = Φ_M + Φ_P^σ = Φ_M + σ_PΦ_M = (1 + σ_P)Φ_M<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref> ------ (Eq. 3.3)

Φ_S^' = Φ_M + Φ_S^σ' = Φ_M + σ_SΦ_M = (1 + σ_S)Φ_M<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref> ------ (Eq. 3.4)

L_P = L_M + L_P^σ = L_M + σ_PL_M = (1 + σ_P)L_M<ref>Template:Harvnb</ref> ------ (Eq. 3.5 <math> \approx</math> Eq. 1.1b & Eq. 2.14)

L_S^' = L_M + L_S^σ' = L_M + σ_SL_M = (1 + σ_S)L_M<ref>Template:Harvnb</ref> ------ (Eq. 3.6 <math>\approx</math> Eq. 1.1b & Eq. 2.14),

where

σ_P & σ_S are, respectively, primary leakage factor & secondary leakage factor

Φ_M & L_M are, respectively, mutual flux & magnetizing inductance

Φ_P^σ & L_P^σ are, respectively, primary leakage flux & primary leakage inductance

Φ_S^σ' & L_S^σ' 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:<ref>Template:Harvnb</ref>

<math>\sigma=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}=1-\frac{L_M^2}{L_PL_S{^'}}=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^'}{L_M}} =1-\frac{1}{(1+\sigma_P)(1+\sigma_S)}</math> ------ (Eq. 3.7a to 3.7e).

ApplicationsEdit

Leakage inductance can be an undesirable property, as it causes the voltage to change with loading.

File:Kvglr.jpg

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 <math>X/R</math> 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.<ref name=Saarbafi-9/>

In addition, the leakage inductance of a HF-transformer can replace a series inductor in a resonant converter.<ref>Template:Cite conference</ref> In contrast, connecting a conventional 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.