Leakage inductance

Fig. 1 L_P^σand L_S^σ are primary and secondary leakage inductances expressed in terms of inductive coupling coefficient ${\displaystyle k}$ 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 ${\displaystyle k}$.[5][6][7]

The primary open-circuit self-inductance is given by

${\displaystyle L_{oc}^{pri}=L_{P}=L_{M}+L_{P}^{\sigma }}$ ------ (Eq. 1.1a)

where

${\displaystyle L_{P}^{\sigma }=L_{P}\cdot {(1-k)}}$ ------ (Eq. 1.1b)

${\displaystyle L_{M}=L_{P}\cdot {k}}$ ------ (Eq. 1.1c)

and

• ${\displaystyle L_{oc}^{pri}=L_{P}}$ is primary self-inductance
• ${\displaystyle L_{P}^{\sigma }}$ is primary leakage inductance
• ${\displaystyle L_{M}}$ is magnetizing inductance
• ${\displaystyle k}$ is inductive coupling coefficient

It therefore follows that the open-circuit self-inductance and inductive coupling factor ${\displaystyle k}$ are given by

${\displaystyle L_{oc}^{sec}=L_{S}=L_{M2}+L_{S}^{\sigma }}$ ------ (Eq. 1.2), and,

${\displaystyle k={\frac {\left|M\right|}{\sqrt {L_{P}L_{S}}}}}$, with 0 < ${\displaystyle k}$ < 1 ------ (Eq. 1.3)

where

${\displaystyle L_{S}^{\sigma }=L_{S}\cdot {(1-k)}}$

${\displaystyle L_{M2}=L_{S}\cdot {k}}$

and

• ${\displaystyle M}$ is mutual inductance
• ${\displaystyle L_{oc}^{sec}=L_{S}}$ is secondary self-inductance
• ${\displaystyle L_{S}^{\sigma }}$ is secondary leakage inductance
• ${\displaystyle L_{M2}=L_{M}/a^{2}}$ is magnetizing inductance referred to the secondary
• ${\displaystyle k}$ is inductive coupling coefficient
• ${\displaystyle a\equiv {\sqrt {\frac {L_{p}}{L_{s}}}}\approx N_{P}/N_{S}}$ [lower-alpha 1] 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.

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.[6][16][17][18]

Fig. 2 Nonideal transformer circuit diagram

where

• M is mutual inductance
• ${\displaystyle R_{P}}$ & ${\displaystyle R_{S}}$ are primary and secondary winding resistances
• Constants ${\displaystyle M}$, ${\displaystyle L_{P}}$, ${\displaystyle L_{S}}$, ${\displaystyle R_{P}}$ & ${\displaystyle R_{S}}$ are measurable at the transformer's terminals
• Coupling factor ${\displaystyle k}$ is defined as

${\displaystyle k=\left|M\right|/{\sqrt {L_{P}L_{S}}}}$, where 0 < ${\displaystyle k}$ < 1 ------ (Eq. 2.1)

The winding turns ratio ${\displaystyle a}$ is in practice given as

${\displaystyle a={\sqrt {L_{P}/L_{S}}}=N_{P}/N_{S}\approx v_{P}/v_{S}\approx i_{S}/i_{P}=}$ ------ (Eq. 2.2).[19]

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,[20]

${\displaystyle v_{P}=R_{P}\cdot i_{P}+{\frac {d\Psi {_{P}}}{dt}}}$ ------ (Eq. 2.3)

${\displaystyle v_{S}=-R_{S}\cdot i_{S}-{\frac {d\Psi {_{S}}}{dt}}}$ ------ (Eq. 2.4)

${\displaystyle \Psi _{P}=L_{P}\cdot i_{P}-M\cdot i_{S}}$ ------ (Eq. 2.5)

${\displaystyle \Psi _{S}=L_{S}\cdot i_{S}-M\cdot i_{P}}$ ------ (Eq. 2.6),

where

• ${\displaystyle \Psi }$ is flux linkage
• ${\displaystyle {\frac {d\Psi }{dt}}}$ 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 open-circuit test is as follows,[21]

${\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}\approx {\frac {L_{sc}}{L_{oc}}}\approx {\frac {L_{sc}^{sec}}{L_{P}}}\approx {\frac {L_{sc}^{pri}}{L_{S}}}\approx {\frac {i_{oc}}{i_{sc}}}}$ ------ (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.
• ${\displaystyle \sigma }$ is the inductive leakage factor or Heyland factor[22][23][24]
• ${\displaystyle L_{sc}^{pri}}$ & ${\displaystyle L_{sc}^{sec}}$ are primary and secondary short-circuited leakage inductances.

The transformer inductance can be characterized in terms of the three inductance constants as follows,[25][26]

${\displaystyle L_{M}=a{M}}$ ------ (Eq. 2.8)

${\displaystyle L_{P}^{\sigma }=L_{P}-a{M}}$ ------ (Eq. 2.9)

${\displaystyle L_{S}^{\sigma }=L_{S}-{M}/a}$ ------ (Eq. 2.10) ,

where,

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,[25][26]

${\displaystyle L_{S}^{\sigma \prime }=a^{2}L_{S}-aM}$

${\displaystyle R_{S}^{\prime }=a^{2}R_{S}}$

${\displaystyle V_{S}^{\prime }=aV_{S}}$

${\displaystyle I_{S}^{\prime }=I_{S}/a}$.

Fig. 4 Nonideal transformer equivalent circuit in terms of coupling coefficient k[27]

Since

${\displaystyle k=M/{\sqrt {L_{P}L_{S}}}}$ ------ (Eq. 2.11)

and

${\displaystyle a={\sqrt {L_{P}/L_{S}}}}$ ------ (Eq. 2.12),

we have

${\displaystyle aM={\sqrt {L_{P}/L_{S}}}\cdot k\cdot {\sqrt {L_{P}L_{S}}}=kL_{P}}$ ------ (Eq. 2.13),

which allows expression of the equivalent circuit in Fig. 4 in terms of winding leakage and magnetizing inductance constants as follows,[26]

Fig. 5 Simplified nonideal transformer equivalent circuit

${\displaystyle L_{P}^{\sigma }=L_{S}^{\sigma \prime }=L_{P}\cdot (1-k)}$ ------ (Eq. 2.14 ${\displaystyle \equiv }$ Eq. 1.1b)

${\displaystyle L_{M}=kL_{P}}$ ------ (Eq. 2.15 ${\displaystyle \equiv }$ 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,

${\displaystyle i_{M}=i_{P}-i_{S}^{'}}$ ------ (Eq. 2.16)

• ${\displaystyle i_{M}}$ is magnetizing current excited by flux Φ_M that links both primary and secondary windings
• ${\displaystyle i_{P}}$ is the primary current
• ${\displaystyle i_{S}'}$ is the secondary current referred to the primary side of the transformer.

Referring to the flux diagram in Fig. 6, the following equations hold:[28][29]

Fig. 6 Magnetizing and leakage flux in a magnetic circuit[30][28][31]

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M[32] ------ (Eq. 3.1 ${\displaystyle \approx }$ Eq. 2.7)

In the same way,

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M[33] ------ (Eq. 3.2 ${\displaystyle \approx }$ Eq. 2.7)

And therefore,

Φ_P = Φ_M + Φ_P^σ = Φ_M + σ_PΦ_M = (1 + σ_P)Φ_M[34][35] ------ (Eq. 3.3)

Φ_S^' = Φ_M + Φ_S^σ' = Φ_M + σ_SΦ_M = (1 + σ_S)Φ_M[36][37] ------ (Eq. 3.4)

L_P = L_M + L_P^σ = L_M + σ_PL_M = (1 + σ_P)L_M[38] ------ (Eq. 3.5 ${\displaystyle \approx }$ Eq. 1.1b & Eq. 2.14)

L_S^' = L_M + L_S^σ' = L_M + σ_SL_M = (1 + σ_S)L_M[39] ------ (Eq. 3.6 ${\displaystyle \approx }$ 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:[40]

${\displaystyle \sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}=1-{\frac {L_{M}^{2}}{L_{P}L_{S}{^{'}}}}=1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{'}}{L_{M}}}}}=1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}}$ ------ (Eq. 3.7a to 3.7e).

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

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 ${\displaystyle X/R}$ 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.[2]

In addition, the leakage inductance of a HF-transformer can replace a series inductor in a resonant converter.[41] 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.

IEC Electropedia links:

• Brenner, Egon; Javid, Mansour (1959). "Chapter 18 – Circuits with Magnetic Coupling". Analysis of Electric Circuits. McGraw-Hill. pp. esp. 586–617.
• 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}}: CS1 maint: location (link)
• Erickson, Robert W.; Maksimovic, Dragan (2001). "Chapter 12: Basic Magnetics Theory (Instructor slides only for book)" (PDF). Fundamentals of Power Electronics (2nd ed.). Boulder: University of Colorado (slides) / Springer (book). pp. 72 slides. ISBN 978-0-7923-7270-7.
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• Harris, Forest K. (1952). Electrical Measurements (5th printing (1962) ed.). New York, London: John Wiley & Sons.
• Heyland, A. (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ. 15: 561–564.
• 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.
• Khurana, Rohit (2015). Electronic Instrumentation and Measurement. Vikas Publishing House. ISBN 9789325990203.
• Kim, Joong Chung (1963). The Determination of Transformer Leakage Reactance by Using an Inpulse Driving Function. University of Oregon.
• Knowlton, A.E., ed. (1949). Standard Handbook for Electrical Engineers (8th ed.). McGraw-Hill. p. 802, § 8–67: The Leakage Factor.
• 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.
• Pyrhönen, J.; Jokinen, T.; Hrabovcová, V. (2008). Design of Rotating Electrical Machines. p. Chapter 4 Flux Leakage.
• "Mutual Inductance" (PDF). Rhombus Industries Inc. 1998. Retrieved 4 August 2018.
• Saarbafi, Karim; Mclean, Pamela (2014). "AESO Transformer Modelling Guide" (PDF). Calgary: AESO - Alberta Electric System Operator (prepared by Teshmont Consultants LP). pp. 304 pages. Retrieved August 6, 2018.
• Singh, Mahendra (2016). "Mutual Inductance". Electronics Tutorials. Retrieved 6 January 2017.
• "Measuring Leakage Inductance". Voltech Instruments. 2016. Retrieved 5 August 2018.