Talk:Leakage inductance

I think the figure titled "Magnetic flux of the transformer" at http://en.wikipedia.org/wiki/File:Transformer_flux.gif) has a material error.

The error lies in showing the leakage flux passing through the core for some of its path and through air for the remainder of the path. This is incorrect. The path of leakage flux in a transformer is entirely external to the core material.

If an expert on the subject can confirm this, I recommend that the figure be corrected.

The figure can be corrected by widening the representation of the windings to make a little more air space between the core and the windings, and moving the dashed loops that represent leakage flux out of the core area and into that space.

Trabuco (talk) 22:32, 1 February 2010 (UTC)

A figure is needed for the industrial leakage inductance as well as explicit relationship between the academic and industrial leakage inductances.Ee011 (talk) 15:17, 12 July 2010 (UTC)

Industrial vs academic

I've never seen this distinction anywhere but on Wikipedia, and I've witnessed testing on large power transformers. You can calculate what the leakage inductance should be, and you can measure it...but it's the same phenomenon in either case. 'calculated' vs. 'measured' I have seen, but nobody says 'industrial leakage reactance'. --Wtshymanski (talk) 15:51, 19 December 2010 (UTC)

Removed image from Chinese standard

In general this seems to be a decent article, I really like the first drawing, which is quite nice. It might have shown the mutual-flux windings as intermingled, but it's pretty nice as it is. I may be able to provide some references for this article, too.

But I have no idea what that image was in the article. There was an awful formula that might have been its purpose, but without explanation, it had to go.

The way one determines leakage inductance is via the short-circuited primary and then flip the transformer around and do the same for the secondary. I'm trying to remember the formulas... the page needs a physical model of a transformer to make the point. Nah, can't remember. here's the reference. It's very good, and gives all the necessary formulas and their derivations. After I've studied it I'll work out the equations here. See Chapter 3 Pulse Transformers and Delay Lines pages 64-82.

• Jacob Millman, and Herbert Taub, 1965, Pulse, Digital and Switching Waveforms: Devices and circuits for their generation and processing, McGraw-Hill Book Company, New York, Library of Congress Catalog Card Number 64-66293.

An excellent drawn model (on page 204) of all the parasitics including the various capacitances (primary, mutual, secondary), ohmic losses (winding resistances and core losses), and inductances, can be found in Chapter 5 Transformers and Iron-Cored Inductors pages 199-253, with a fantastic set of references (about 100, no kidding), on pages 252-253:

• F. Langford-Smith, editor, 1953, Radiotron Designer's Handbook, 4th Edition, Wireless Press for Amalgamated Wireless Valve Company PTY, LTD, Sydney, Australia together with EectronTube Division of the Radio Corporation of America, Harrison, N. J. No LCCCN.

Bill Wvbailey (talk) 16:32, 24 December 2010 (UTC)

Proposed article revamp

Transformer Talk discussion 'Leakage induction' section moved to this replace 'Proposed article revamp' section herein ofLeakage induction Talk pages.Cblambert (talk) 15:36, 5 May 2013 (UTC)

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As advertized, unexplained gaps in trying to get from 'Real transformer deviations from ideal' to 'Equivalent circuit' sections can probably best be realized via enhancements to Leakage inductance article as proposed in draft herein.

Since you are discussing chanimg the Leakage inductance article, this discussion should be on the talk page of that article and only there.Constant314 (talk) 14:39, 5 May 2013 (UTC)

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Leakage inductance derives from the electrical property of an imperfectly-coupled transformer whereby each winding behaves as a self-inductance constant in series with the winding's respective ohmic resistance constant, these four winding constants also interacting with the transformer's mutual inductance constant. These winding self and leakage inductance constants are due to leakage flux not linking with all turns of each imperfectly-coupled winding.

The leakage flux alternately stores and discharges magnetic energy with each electrical cycle acts as an inductor in series with each of the primary and secondary circuits.

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.

Although discussed exclusively in relation to transformers in this article, leakage inductance applies to any imperfectly-coupled magnetic circuit device including especially motors.

Leakage factor and inductance

Real transformer circuit diagram

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 coefficient k is given as

$k=M/{\sqrt {L_{P}L_{S}}}$, with 0 < k < 1

• Winding turns ratio a is in practice given as

$a=N_{P}/N_{S}=v_{P}/v_{S}=i_{S}/i_{P}={\sqrt {L_{P}/L_{S}}}$.

The two circuit loops can be expressed by the following voltage and flux linkage equations:

$v_{P}=R_{P}i_{P}+{\frac {d\Psi {_{P}}}{dt}}$

$v_{S}=-R_{S}i_{S}-{\frac {d\Psi {_{S}}}{dt}}$

$\Psi _{P}=L_{P}i_{P}-Mdi_{S}$

$\Psi _{S}=L_{S}i_{S}-Mi_{P}$

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 the secondary circuit's short circuit and no-load inductances and currents is as follows,

$\sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}={\frac {L_{sc}}{L_{oc}}}={\frac {i_{oc}}{i_{sc}}}$,

where,

• σ is the 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.

Real transformer equivalent circuit

Real transformer equivalent circuit in terms of coupling coefficient k

Simplified real transformer equivalent circuit

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

$L_{M}=a{M}$

$L_{P}^{\sigma }=L_{P}-a{M}$

$L_{S}^{\sigma }=L_{S}-a{M}$,

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.

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

$R_{S}^{'}=a^{2}R_{S}$

$V_{S}^{'}=aV_{S}$

$I_{S}^{'}=I_{S}/a$.

Since

$k=M/{\sqrt {L_{P}L_{S}}}$

and

$a={\sqrt {L_{P}/L_{S}}}$,

we have

$aM={\sqrt {L_{P}/L_{S}}}*k*{\sqrt {L_{P}L_{S}}}=kL_{P}$,

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

• $L_{P}^{\sigma }=L_{S}^{\sigma }{'}=L_{P}*(1-k)$
• $L_{M}=kL_{P}$.

Expanded leakage factor

Magnetizing and leakage flux in a magnetic circuit

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

• i_M = i_P - i_S^'
• i_M is magnetizing current excited by flux Φ_M that links both primary and secondary windings.

Referring to the flux diagram at right, the leakage factor can be defined as follows,

• σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M
• σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M
• Φ_P = Φ_M + Φ_P^σ = (1 + σ_P)Φ_M
• Φ_S^' = Φ_M + Φ_S^σ' = (1 + σ_S)Φ_M
• L_P = L_M + L_P^σ = (1 + σ_P)L_M
• L_S^' = L_M + L_S^σ' = (1 + σ_S)L_M,

where

• σ_P is primary leakage factor
• σ_S is secondary leakage factor
• Φ is magnetic flux.

The leakage factor σ can thus be expanded in terms of the interrelationship of above winding-specific inductance and leakage factor equations as follows:

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

I haven't heard of the Heyland factor before, but it seems simply related to the coupling coefficient. If you are going to add this stuff, I'd rather that you use coupling coef as its name describes its function.Constant314 (talk) 16:38, 4 May 2013 (UTC)

Heyland factor is probably used more often in Europe, or even not given any formal name as such, but the derivation is as shown and has been very common for over a century. The 'Heyland factor', or it's unidentified equivalent, is how you derive transformer or motor from short-circuit and open-circuit measurements, which is a classical approach to doing it and which Leakage inductance article has been trying to describe incorrectly for several years. This stuff is proposed in Leakage inductance article, not in Transformer article proper. I am not proposing anything major in difference in Transformer article proper. Right now, everything about Leakage inductance is nonsense, so the alternative is to scrap Leakage inductance article. But there can be no doubt about the substance of the derivation per se. One may quibble with the presentation and the naming of this stuff but not with the substance. I will address these issues in proposed changes to Leakage inductance article. You are the one who suggested to start with ideal transformer and to first with linear (cum real) transformer. So back to the question: How do you fill the gaps between ideal and real? That is what the 'Heyland' derivation answers.Cblambert (talk) 20:03, 4 May 2013 (UTC)

Just so we are clear. Coupling coefficient k is one thing. Heyland factor σ is another. I will put the emphasis on leakage factor σ instead of Heyland factor σ.Cblambert (talk) 20:14, 4 May 2013 (UTC)

I agree that the Leakage inductance article needs work. But please mve this discussion to that talk page and remove it from the Transformer article talk page. The discusssion of changes to the Leakage inductance article needs to be on its talk page only.Constant314 (talk) 14:56, 5 May 2013 (UTC)

Revamp implemented.Cblambert (talk) 17:08, 5 May 2013 (UTC)

Applications of leakage inductance

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.

Leakage reactance is also used for some negative resistance devices, 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.

Bibliography

• Brenner, Egon (1959). "Chapter 18 - Circuits with Magnetic Circuits". Analysis of Electric Circuits. McGraw-Hill. pp. esp. 586–602. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Dixon, Lloyd (2001). "Power Transformer Design" (PDF). Magnetics Design Handbook. Texas Instruments. pp. 4-1 to 4-12. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help)
• Hameyer, Kay (2001). "§3 Transformer". Electrical Machines I: Basics, Design, Function, Operation. RWTH Aachen University Institute of Electrical Machines. pp. 23–52. {{cite conference}}: |access-date= requires |url= (help); Unknown parameter |booktitle= ignored (|book-title= suggested) (help)
• Heyland, A. (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ. pp. 561–564. {{cite web}}: Missing or empty |url= (help)
• Pyrhönen, J. "§4. Flux Leakage". {{cite web}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

See also

• Circle diagram
• Steinmetz equivalent circuit

de:Streufluss#Streuinduktivität

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Cblambert (talk) 06:29, 2 May 2013 (UTC)

Proposed Leakage inductance text updated to reflect leakage factor σ = 1 - k², where k = coupling coefficient.

Cblambert (talk) 02:35, 5 May 2013 (UTC)

Article title

Nice CBLambert! I believe this technical stuff will be good in this article. Any layperson looking for transformer information won't get swamped with too much formulae input right away in the Transformer article. Having said that there needs to be a place for this higher-tech stuff and it would be a shame to waste your technical research, knowhow and ambition. This looks great to me, as anybody looking for Leakage inductance probably wants the nitty-gritty that you seem to be good at. I do have a suggestion about this article... the name be changed to Transformer leakage inductance so that readers typing in Transformer will get the auto-complete showing the article name. High-tech type readers looking for lots of EE stuff will see it immediately before even making a selection of another article. As well the usual quickie mention in [[[Transformer]] and a few other places with a link here is great. I can't make thus happen, as an IP, and you are the big contributor here now. This process could be done with other aspects of Transformer as well, keeping the main article a little more simple and prose oriented. It's a bigger field than meets the eye, at first. Best! 174.118.142.187 (talk) 15:29, 6 May 2013 (UTC)

I also agree with you. I think that title Leakage inductance (Transformer) is better.--Neotesla (talk) 14:04, 18 May 2013 (UTC)

Leakage inductance is generic to any magnetic circuit, not to transformer only.Cblambert (talk) 18:08, 18 May 2013 (UTC)

No. Better to leave article as is. Better to show, eventually, that leakage inductance is also important for motor generally and induction motors in particular. The point is not to be technical per se as much as to show that the threads of interest, the interrelationship between different aspect of electromagnetism. If you look at the German, Polish and Japan versions of Leakage inductance, all are close to correct, which only counts in horseshoes game. Japan version is closest to being right but is still wrong. Why would any encyclopedia not want to be 100% right.Cblambert (talk) 17:54, 6 May 2013 (UTC)

Please take care so that it may become explanation not much complicated for beginners.

L_e1 and L_e2 are the leakage inductance

This model is very easy to understand for them. So do not remove it. The description of Japan wikipedia has not mistaken about the definition of leakage inductance, but most of the recognition of transformer industry has mistaken. [1] What do you think about this description, which was corrected by someone. However, it is the fact. Different opinion(definition? recognition?) exists for each different industry. They claims that Le or Lsc is leakage inductance, respectively. Example is follows. [2][3] They think that the leakage inductance is Lsc. I asked this matter to several universities. (The University of Tokyo, The University of electro-Communications, Tokyo Institute of Technology and others.) They professor and associate professor said that we did not notice such a important indications until now.　However, we can not decide this matter at Wikipedia’s closed discussion.　Well, leakage inductance theory should be described summary first. It should not be described in complex. There is no relation between the resistance and leakage inductance. So resistance should not be described in summary(headlines). And the resistance should be described as detailed below terms. --Neotesla (talk) 13:11, 18 May 2013 (UTC)

The direction of the leakage fluxΦ_S^σ' is different from the actual measurements. It is reverse.

Magnetizing and leakage flux in a magnetic circuit

Of course, It can be added a negative formula by reversing the arrow, but it differs from the intuition. --Neotesla (talk) 13:34, 18 May 2013 (UTC)

Image and description that you provide is incorrect, contradicts and duplicates rest of article and is not provided with citations and is therefore unaccepable.Cblambert (talk) 17:28, 18 May 2013 (UTC)

Re: . . . not much complicated for beginners. Cblambert comment: Better to start correct and simplify from there.

Re: This model is very easy to understand for them. Cblambert comment: But wrong.

Re: The description of Japan wikipedia has not mistaken about the definition of leakage inductance, but most of the recognition of transformer industry has mistaken. Cblambert comment: This statemene does not make sense to me.

Re: There is no relation between the resistance and leakage inductance. Cblambert comment: There is a relation - their in series.

Re: Rest of Neotesla comment. Cblambert comment: There can be no 'winging' or wishfull thinking about leakage inductance. Transformer and motor industies are over 100 years in the making. Everything about leakage inductance has been proven 100%. The derivation from basic principles is exactly as shown in 'Leakage factor and inductance' and 'Expanded leakage factor' section. Challenge is to simplify from this derivation as needed. The whole thing has to be consistent. It is not sufficient to try to patch old and new part of article together arbitrarily.

Re: The direction of the leakage fluxΦ_S^σ' is different from the actual measurements. Cblambert comment: The direction is arbitrary starting point reference. Could be changed to show 'right' direction.

Cblambert (talk) 18:08, 18 May 2013 (UTC)

Simple description of leakage inductance from basic principles

Real transformer equivalent circuit in terms of coupling coefficient k

The nub of leakage inductance is as shown in TREQCCTHeyland-to-k.jpg here.

Open circuit neglecting resistance is

Self-inductance L_P =

Leakage inductance L_P*(1-k) + magnetizing inductance L_P*k

And not L_P = L_P*(1-k) + M*k!!

Short circuit is not equal to leakage inductance.

The best that can be said about short circuit is that, neglecting resistances, the ratio of the short-circuit & open circuit inductances and curents is as follows:

$\sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-k^{2}={\frac {L_{sc}}{L_{oc}}}={\frac {i_{oc}}{i_{sc}}}$

Everything about this relationship is consistent:

• Coupling coefficient is defined in Wikipedia and countless other references as

$k={\frac {M}{\sqrt {L_{P}*L_{S}}}}$

• Hence
• $1-k^{2}={\frac {L_{sc}}{L_{oc}}}={\frac {L_{sc}}{L_{P}}}$
• Therefore, $L_{sc_{sec}}=L_{P}*(1-k^{2})$
• It follows that a short circuit of the primary side, but NOT of the secondary side, yields $L_{sc_{pri}}=L_{S}*(1-k^{2})$

Since we know that leakage inductances are by definition

• $L_{P}^{\sigma }=L_{P}*(1-k)=L_{P}*(1-aM)$
• $L_{S}^{\sigma }=L_{S}*(1-k)=L_{S}*(1-aM)$
• $L_{S}^{\sigma \prime }=L_{P}*(1-k)=L_{S}*a^{2}-aM$

and that

• $a={\frac {N_{P}}{N_{S}}}={\sqrt {\frac {L_{P}}{L_{S}}}}$

transformer leakage inductances L_P^σ, L_s^σ and L_s^σ' are known in terms of M, L_P, L_S, k, a, i_oc and i_oc as outlined in article's real transformer image, real transformer, and this article's three equivalent circuits.

Nothing complicated about this. And fundamentally correct, as has been for a century.Cblambert (talk) 20:20, 18 May 2013 (UTC)

• (About the serial resistance)

Your formula is not wrong, but it can be not found a description of the resistance in your formula. In addition, my figure do not contradict with yours. It means that there is no relation between the resistance. You proves it yourself in your formula. The resistance in the figure interfere with the understanding of beginners. So, resistance should be removed from the ｆigure in the first step explanation.

"Re: There is no relation between the resistance and leakage inductance. Cblambert comment: There is a relation - their in series."

You seem to really believe that relation exists with the series resistance and leakage inductance. It is very awkward to convince you. But there is no relation with resistance. In addition, measurement of the coupling coefficient can be precisely to using the impedance analyzer (which is called resonance method) except to using the traditional short circuit method. We had been measuring many transformers until now. The coupling coefficient parameter was the same at all to be measured by either method. And, this results applied to the formula you shows, it is concluded that there is no relation between the resistance and the leakage inductance.--Neotesla (talk) 05:04, 19 May 2013 (UTC)

Real transformer has leakage inductance in series with leakage inductance, which can be neglected when doing short-cicuit and open-circuit testing but the can be no doubt about the series relationship of winding resistance and inductance windings. Read in particular transformer and many other references. It would be a mistake to necessarily assume that you know more about this article than I do. Please stick to hard facts pro and con. Please no opinions. No harm is done by leaving real transformer in winding resistance and inductance in series. This practice is quite common. One works out inductances neglecting resistance for testing simplication purposes but leaves the resistances in for inclusion in equivalent circuit. For example, the resistance constant in an induction motor's rotor secondary winding is critical important to the starting and operation behavior of the motor. See Steinmetz equivalent circuit.Cblambert (talk) 05:54, 19 May 2013 (UTC)

All documents are old and poor in practical use. You do not understand it? It does not pragmatically. So, It should be modified slightly. Therefore, we have to correct not so much description, according to the formula of the old literature. We need to extract the simple formula from there. It does not conflict at all with strong fact that you say. I also developing many solution using the new(simplify) formula already. There is no problem at all.--Neotesla (talk) 06:14, 19 May 2013 (UTC)

Also, Wikipedia article audience targets the non-expert, which is different than that of the beginner.

There were too many symbolic errors in Leakage inductance article and inconsistencies with the critical transformer and Induction motor articles. Also, proposed changes were posted in my personal Talk page, Transformer Talk page and Leakage inductance Talp page. Every effort was made to accommodate other editors. There is no question of going back to article ridden with errors and inconsistencies with critical Transformer and Induction motor articles.Cblambert (talk) 06:32, 19 May 2013 (UTC)

"There were too many symbolic errors in Leakage inductance article" that you say. But there was no symbolic error(s) in current description. It is only described simplify magnetic theories. So, there was no clearance for an error enters there. In addition, this article is the Leakage inductance. So you should simply describe only that of the leakage inductance. About the real transformer matter should be described in the article of the transformer. For reference, this book is selling a lot and new.[4]Toroidal core utilization encyclopedia No one would buy this book when description wrong. Series resistance is not written in this one. This is the current standard.--Neotesla (talk) 07:24, 19 May 2013 (UTC)

Link provided does not work for me and points to search page full of Chinese document title. I would not be surprised each document hiding a virus. It is obvious that this article cannot be discussed without getting into all the other related transformer impedance constants. Indeed, how does anyone know what the abstraction 'leakage inductance' is? The article's title is whatever is best for Wikipedia readers. Note added early today in 'See also' section of Blocked rotor test, Open circuit test and Short circuit test, which helps differentiate between various transformer and induction motor impedances including of course 'leakage inductance'. But the 'leakage inductance' impedance constant cannot be measured in isolation. Even winding resistance gets in the way in trying to define 'leakage inductance'. I have even toyed with the idea of renaming the article 'Transformer equivalent circuit' or 'Transformer impedance constants' . . . Hence, these is no need to worry too much about inclusion of winding resistance. Best to leave resistance in to 'round out the picture'.Cblambert (talk) 17:24, 19 May 2013 (UTC)

Are you not able to distinguish between Chinese and Japanese? Those descriptions are Japanese. And I am a Japanese.

• "It is obvious that this article cannot be discussed without getting into all the other related transformer impedance constants."

Description about the impedance is not required. In this article, it has to be explained the relation with the leakage flux and the leakage inductance. So this description should be only explained about inductance.

• "But the 'leakage inductance' impedance constant cannot be measured in isolation."

Because of current phase vector is different by 90 degrees, it will be measured separately without interfering with each other thereof. What is the means that the impedance analyzer exists?

• "Indeed, how does anyone know what the abstraction 'leakage inductance' is?"

At least the author of the documents I showed knows all about the leakage inductance.　Resonant type leakage transformer that I have invented is to be used for many billion all over the world as LCD backlight. Maybe you have used LCD monitor, notebook and LCD TV. Those are designed based on the formula of the simplified magnetic model and have been commercialized. Indeed, nothing is written about only leakage inductance alone in the old literature. That is only written a little description of all other together. Please believe a little new technical literature that I showed.

• "Hence, these is no need to worry too much about inclusion of winding resistance."

Why do you stick to the resistance?

• "Best to leave resistance in to round out the picture"

There is no means to leave the resistance.--Neotesla (talk) 06:03, 20 May 2013 (UTC)

L_P^σand L_S^σ are the leakage inductance

I rewrite this figure for adapt to your descriptions. However, it is not so completely symmetrical.--Neotesla (talk) 14:44, 20 May 2013 (UTC)

Q: Are you not able to distinguish between Chinese and Japanese?

A: Evidently not. Sorry, but it is all Chinese to me. Why do did you send raw search document page?!

Q: Description about the impedance is not required etc.

A: I disagreed. One can't fully treat k and leakage in isolation.

Q: What is the means that the impedance analyzer exists?

A: Analyzer is only one aspect.

Q: At least the author of the documents I showed knows all about the leakage inductance. . . Please believe a little new technical literature that I showed.

A: Two many superlative claims in this point. Please stick to the facts of the article proper. It has all been said long ago, inductance-wise. Please make specific positive points.

Q: Why do you stick to the resistance?

A: Because we want to relate impedances as part of real transformer.

Q: There is no means to leave the resistance.

A: I don't agree. The article hangs together just fine generally as is.

Q: I rewrite this figure for adapt to your descriptions.

A: Figure should:

• Not show L_M2 but should leave as L_M/a²
• Not show '1' dimension on either side as this is not electrical convention
• Show L_P . (1 - k) and L_S . (1 - k)
• Show L_P . k and L_S . k

We don't want the figure to be symmetrically. We want to emphasize that magnetizing inductance is equivalent, by 'referring', no matter which winding side is considered.Cblambert (talk) 21:12, 20 May 2013 (UTC)

Minimizing leakage flux

This article could use a section on minimizing leakage inductance. Common centroid, interleaved layers and bifilar winding come to mind.Constant314 (talk) 21:54, 31 May 2014 (UTC)

--

A possible source for minimizing leakage inductance in multilayer audio or power transformers is The Radiotron Designer's Handbook, pages 217ff. A footnote says that it follows the treatment of Crowhurst, N. H. in Electronic Eng. 21.254 (April 1949) 129. (Ref C28). Radiotron says that "the insulation between the sections is the limiting factor" and offers a formula for this limit that is eventually reached when adding more interleavings:

a/(3*N^2) < c, where a is the total thickness of all the winding sections, N is the number of leakage flux areas, c is the thickness of each insulation section.

Radiotron goes on to give an example of how to calculate leakage inductance in a multilayer transformer; it requires the use of a chart + some complicated nomographs.

In my past as an EE I ran headlong into this issue of the "thickness of the insulation section"; it not only included the thickness of the enamel insulation but more significantly the multiple wraps of Nomex "paper" that insulated each layer of the interleaving; the thickness becomes a quite serious matter if you have to build to IEC standards that require double insulation. In my professional life even a "gate-drive" pulse transformer (see below) wound on a ferrite toroid had to be wound (single layer bifilar) with special double-insulated wire with a minimum thickness that met the IEC standards (I can't remember the thickness but it was more than the diameter of the wire).

A problem introduced when interleaving (i.e. multilayer designs) is more interwinding capacitance (cf ref C29) in Radiotron.

Also, for pulse-transformer design: the use of twisted-pair windings (a sort of bifilar winding) and coax-cable "transmission-line" windings (an extension of the twisted-pair idea). One rudimentary reference for this is Millman and Taub, 1965, Pulse, Digital and Switching Waveforms, McGraw-Hill, Inc. Cf section 3-20 "Transmission-Line" pulse transformers(pages 106ff). This section is followed by a number of references re "nanosecond pulse transformers". A google search of these words coughs up some of the references but you can't get them unless you're affiliated with (or in) a university library using library computers (which I'm not). For example, here's the abstract of Winningstad, C.N.: Nanosecond Pulse Transformers, IRE Trans. Nucl. Sci. Vol. NS-6, pp. 26-31, March 1959:

"The transmission-line approach to the design of transformers yields a unit with no first-order rise-time limit since this approach uses distributed rather than lumped constants. The total time delay through the transmission-line-type transformer may exceed the rise time by a large factor, unlike conventional transformers. The extra winding length can be employed to improve the low-frequency response of the unit. Transformers can be made for impedance matching, pulse inverting, and dc isolation within the range of about 30 to 300 ohms with rise times of less than 0.5 × 10-9 seconds, and magnetizing time constants in excess of 5 × 10-7 seconds. Voltage-reflection coefficients of 0.05 or less, and voltage-transmission efficiencies of 0.95 or better can be achieved."

Yes, everything that you do to decrease leakage tends to increase inter-winding capacitance, which is usually something that you do not want. It sounds like you have a lot of relevant information, but don't copy too much verbatim because of copyright issues.Constant314 (talk) 00:02, 3 June 2014 (UTC)

If I encounter any more info I'll add it here; there's something I read somewhere(Ferroxcube manual?) that said one should "fill the winding area" e.g. in a ferrite "pot core". But I don't know why this should be the case. Bill Wvbailey (talk) 15:01, 2 June 2014 (UTC)

I believe it means keep increasing the size of wire until you fill the window to minimize copper loss.Constant314 (talk) 00:02, 3 June 2014 (UTC)

---

The following reference is really interesting. Their results are remarkable, and the paper is nice because it gives some actual numbers: "The coupling coefficient of the transformer is measured with eq. (1). The sandwich winding transformer is 0.9897505 and the coaxial cable transformer is 0.9999448." [Compare this to my O.R. values, see below]. They use the formula

k = (L_add - L_oppose)/(4*sqrt(L₁*L₂ ))

Do-Hyun Kim, Joung-Hu Park, High Efficiency Step-Down Flyback Converter Using Coaxial Cable Coupled-Inductor, Journal of Power Electronics, Vol. 13, No. 2, March 2013. http://koreascience.or.kr/article/ArticleFullRecord.jsp?cn=E1PWAX_2013_v13n2_214

Abstract: "This paper proposes a high efficiency step-down flyback converter using a coaxial-cable coupled-inductor which has a higher primary-secondary flux linkage than sandwich winding transformers. The structure of the two-winding coaxial cable transformer is described, and the coupling coefficient of the coaxial cable transformer and that of a sandwich winding transformer are compared [etc]"

[The following is O.R. but it gives a point of reference to the numbers above: I wound three small coils with 3 layers on small formers with no inter-layer insulation (for Ferroxcube RM5 cores) and got the following results: bifilar twisted (10.5:10.5) yielded 0.9979. Normal sloppy winding (27:9) yielded 0.9963, interleaved (p:s:p 14:9:13) yielded 0.9979. I.e. the improvement was tiny. To get these values I "resonated" the windings in primary-open-circuited and short-circuited conditions (and then flipped the primary-secondary sense of the transformer resonated them again and then averaged the results); see more at Millman and Taub page 69ff; the formula to get k from the resonant frequencies is simple and easily derived, but I don't have a source for it, yet.] Bill Wvbailey (talk) 12:42, 3 June 2014 (UTC)

--- See especially the 2nd paragraph:

Hang-Seok Choi, [2003? 06/11/12?], AN-4140: Transformer Design Consideration for Offline Flyback Converters Using Fairchild Power Switch (FPS™), Power Supply Group / Fairchild Semiconductor Corporation

"(3) Minimization of Leakage Inductance

The winding order in a transformer has a large effect on the leakage inductance. In a multiple output transformer, the secondary with the highest output power should be placed closest to the primary for the best coupling and lowest leakage. The most common and effective way to minimize the leakage inductance is a sandwich winding . . ..

"Secondary windings with only a few turns should be spaced across the width of the bobbin window instead of being bunched together, in order to maximize coupling to the primary. Using multiple parallel strands of wire is an additional technique of increasing the fill factor and coupling of a winding with few turns . . .."

Multiple parallel wires also act as Litz wire to reduce the effects of skin effect; skin effect is a serious problem in high power transformers (especially those working at >15 KHz). Bill Wvbailey (talk) 14:06, 5 June 2014 (UTC)

Other uses of leakage inductance

To my knowledge the section in the article is correct: in the arc-welding industry, in particular, you encounter both neon-sign transformers (for high-freq stabilized welding) with leakage inductance introduced by winding on two separate "cores" with a slot cut in the separation, and low-cost arc welders and battery chargers designed with E-I laminations and an air-gap introduced between the E and I. Another trick is to wind two windings not overlapped but separated on long EƎ laminations, maybe slots etc.

Inrush mitigation: There's another built-in problem with iron-core transformers on AC lines -- the inrush "surge" that occurs when AC power is first applied (the amplitude depends on exactly when the switch closes, and the residual magnetization of the iron); we encountered this phenomenon in rudimentary arc-welders and fancy ones that had to "charge up" capacitors. This inrush can be very severe (100's to 1000's of amps) and it will cause improperly-specified fuses to "blow"; one way to mitigate it is to introduce leakage inductance (another approach we used to good effect was to introduce resistance in the AC line and then switch it out).

Tuned oscillator circuits: Millman and Taub (page 616) remark that "the only essential difference between the tuned oscillator and blocking oscillator is in the tightness of coupling between the transformer windings" [typically one wants tight coupling in a blocking oscillator circuit]. Bill Wvbailey (talk) 15:41, 2 June 2014 (UTC)

The concept of "ratio of magnetic flux = inductance ratio" is incorrect

I removed the following formula.

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M

But, Jim1138 told me that it should be explain the reasons. So I stated as follows. The reason is that the Leakage factor is the ratio of the magnetic flux, but it is not the ratio of the inductance. The concept of "ratio of magnetic flux = inductance ratio" is incorrect. When the system current increases or decreases the mutual flux is proportional to the voltage across the mutual inductance and the leakage flux is proportional to the current in the secondary winding. Even if "coupling coefficient = mutual inductance / self inductance" is unchanged, the leakage flux changes greatly in proportion to the current flowing through the secondary winding, and the coupling factor value varies. Please look into this point again in detail. you should investigation into this point again in detail. 153.227.36.195 (talk) 08:20, 27 December 2016 (UTC)

I have to agree that there is something wrong with those equations. The ratio of the primary leakage flux to Mutual flux is a variable that depends on both primary and secondary currents, whereas the ratio of the inductance is a constant. But what is the meaning of σ_P if it is a variable depending currents that vary moment by moment? Constant314 (talk) 04:13, 28 December 2016 (UTC)

After more thought, the equation σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M makes senses if it is measured with the secondary open circuited

and σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M makes sense if it is measured with the primary open circuited.

Does it make sense that we just need to add some conditions tot he equations? Constant314 (talk) 06:59, 28 December 2016 (UTC)

If you discript the condition that "other windings is opened", it will be correct. In order to prevent the reader's misunderstanding, it is necessary to indicate at the same time that the flux ratio substantial changes when the secondary winding current is flowing in that case.153.227.36.195 (talk) 20:54, 30 December 2016 (UTC)

I'm of the opinion that the best way out of this conundrum is

σ_P = L_P^σ/L_M

σ_S = L_S^σ'/L_M

In other words, just drop the flux terms. I am convinced that the flux terms would apply under the condition that the other windings were open circuited and the flux terms themselves are RMS values or phasor amplitudes, but I don't have a reference to back it up. I have discussed this with the person that originally posted the material. He is sure that he accurately reflected what was in the reference(Hameyer, Kay (2001)), but there was probably some explanation that made it make sense. Unfortunately, the reference was an on line version of a very expensive book that is no longer on line or available where I can find it. So, the material is attributed to a book that is unavailable to anybody in the current discussion and I cannot even find it for sale.

So, at this point, I think we agree that the section is subject to multiple interpretations and is and will remain confusing unless it is clarified. However, we do not have access to the source reference to get that clarifiaction. I'm not sure that we even need the section. It doesn't make sense to me as is and I've designed signal transformers and flyback transformers from 60 Hz to 30 MHz (not all in one transformer). Maybe it makes sense to big equipment and big transformer designers who already know the material. I have several references on magnetics, rotating machines, transformers and power transmission. Leakage factor does not appear in any of them. Maybe it's a translated term from another language. Constant314 (talk) 22:19, 3 January 2017 (UTC)

Also note that i_M = i_P - i_S^' implies the primary and secondary have the same number of turns. Constant314 (talk) 02:54, 4 January 2017 (UTC)

Kay Hameyer's credentials FYI:

Kay Hameyer (Senior MIEEE, Fellow IET) received the M.Sc. degree in electrical engineering from the University of Hannover, Germany. He received the Ph.D. degree from University of Technology Berlin, Germany. After his university studies he worked with the Robert Bosch GmbH in Stuttgart, Germany, as a design engineer for permanent magnet servo motors and automotive board net components.

In 1988 he became a member of the staff at the University of Technology Berlin, Germany. From November to December 1992 he was a visiting professor at the COPPE Universidade Federal do Rio de Janeiro, Brazil, teaching electrical machine design. In the frame of collaboration with the TU Berlin, he was in June 1993 a visiting professor at the Universite de Batna, Algeria. Beginning in 1993 he was a scientific consultant working on several industrial projects. He was a guest professor at the University of Maribor in Slovenia, the Korean University of Technology (KUT) in South-Korea. Currently he is guest professor at the University of Southampton, UK in the department of electrical energy. 2004 Dr. Hameyer was awarded his Dr. habil. from the faculty of Electrical Engineering of the Technical University of Poznan in Poland and was awarded the title of Dr. h.c. from the faculty of Electrical Engineering of the Technical University of Cluj Napoca in Romania. Until February 2004 Dr. Hameyer was a full professor for Numerical Field Computations and Electrical Machines with the K.U.Leuven in Belgium. Currently Dr. Hameyer is the director of the Institute of Electrical Machines and holder of the chair Electromagnetic Energy Conversion of the RWTH Aachen University in Germany (http://www.iem.rwth-aachen.de/). Next to the directorship of the Institute of Electrical Machines, Dr. Hameyer is the dean of the faculty of electrical engineering and information technology of RWTH Aachen University. Currently he is elected member and evaluator of the German Research Foundation (DFG). In 2007 Dr. Hameyer and his group organized the 16th International Conference on the Computation of Electromagnetic Fields COMPUMAG 2007 in Aachen, Germany.

His research interests are numerical field computation, the design and control of electrical machines, in particular permanent magnet excited machines, induction machines and numerical optimisation strategies. Since several years Dr. Hameyer's work is concerned with the magnetic levitation for drive systems. Dr. Hameyer is author of more than 180 journal publications, more than 350 international conference publications and author of 4 books.

Dr. Hameyer is an elected member of the board of the International Compumag Society, member of the German VDE, a senior member of the IEEE, a Fellow of the IET and a founding member of the executive team of the IET Professional Network Electromagnetics.Cblambert (talk) 14:40, 4 January 2017 (UTC)

Knowlton's Standard Handbook for Electrical Engineers says in §8-67 The Leakage Factor. 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 and is greater than 1.Cblambert (talk) 15:01, 4 January 2017 (UTC)

Re Also note that iM = iP - iS' implies the primary and secondary have the same number of turns.: The S' means by definition secondary referred the primary, as detailed in various other Wikipedia articles include the Transformer article.Cblambert (talk) 15:56, 4 January 2017 (UTC)

The first 4 equations

σ_P = Φ_P^σ/Φ_M

σ_S = Φ_S^σ'/Φ_M

Φ_P = Φ_M + Φ_P^σ = Φ_M + Φ_PΦ_M = (1 + σ_P)

Φ_S^' = Φ_M + Φ_S^σ' = Φ_M + Φ_SΦ_M = (1 + σ_S)

are interrelated, which is why it not only makes no sense to remove the first 2 equations but shows the 3rd & 4th equation more fully expanded needed correction. The whole section, Refined leakage factor, hangs together seamlessly. The onus is on editors knowledgeable in the matter to show in detail why any one part of this section is not consistent with the end result and with the rest of this article. Cblambert (talk) 16:42, 4 January 2017 (UTC)

Hameyer's credentials are not being disputed. It's just that no one seems to be able to put their eyes on the source reference document. I appreciate you clarifying the meaning of iS'. Could you clarify whether the currents and fluxes are instantaneous values, RMS values, phasor values or phasor amplitudes? When I see currents written with lower case "i" I generally presume that means instantaneous (functions of time) values. Constant314 (talk) 21:50, 4 January 2017 (UTC)

I have sent Hameyer an e-mail asking for permission to get a copy of 2001 course notes document. My sense is that the flux equations are in general for steady-state frequency conditions (especially power frequency conditions) although the the flux linkage equations are generalized in terms of the derivative with respect to time, which is consistent with your sense that lc "i" is instantaneous. The key question as to the validity of Leakage factor being equal to the ratio of the inductance, would support the notion that the "i" is instantaneous for steady-state power frequency conditions with the end-result leakage factor derived from RMS flux equations.Cblambert (talk) 07:22, 5 January 2017 (UTC)

It is important to get to the bottom of this issue as is affect several interrelated articles including the Transformer, Induction motor, Circle diagram, Steinmetz equivalent circuit and other Wikipedia articles.Cblambert (talk) 07:28, 5 January 2017 (UTC)

The cause of the confusion was understood. The leakage factor is a term used for the core material, and it has not to be explained along with the leakage flux. And there was a cause to further deepen the confusion. The meaning of the leakage flux defined in the electromagnetics literature and the meaning of the "magnetic flux leaking from the core material" used in the explanation of the core material catalog are different. So it is wrong to explain the leakage factor in the secsion in the leakage inductance, and this explanation should be independent as a leakage factor of the core material.153.227.36.195 (talk) 17:42, 5 January 2017 (UTC)

So, is this article about "leakage flux" or "magnetic flux leaking from the core material"?Constant314 (talk) 18:11, 5 January 2017 (UTC)

This article is "Leakage inductance", is not it? This is a discussion of electromagnetism. Why is discussed about the "leakage factor" which is the term of core material?153.227.36.195 (talk) 19:19, 5 January 2017 (UTC)

Author of this talk section started saying that ratio of magnetic flux = inductance ratio" is incorrect and he now suddenly casts doubt on the whole article!? Is this discussion section grasping at straws? This article, Leakage inductance, has a long history which revolves around why it is that any magnetic device involves "Leaking inductance" which is imperfect as with any transformer or motor. The article has been used, and/or copied, extensively in/from corresponding Polish, Ukrainian, Japanese and other languages. Leakage inductance is used to explain in amazing simple terms the meaning of the coupling coefficient in terms of primary & secondary inductance, primary & secondary self-inductance, primary & secondary leakage inductance, magnetizing inductance, winding turns ratio and resistance and inductance referred to the primary. The leakage factor explains how it can be derived from also amazingly simple open-circuit and short-circuit tests. The leakage inductances can be used to derive a simplified equivalent circuit for magnetic devices. Leakage inductance provides the key to explaining many things about imperfect magnetic devices, which unavoidably requires dabbling in magnetic relationships. There is nothing wrong with magnetic relationships.Cblambert (talk) 22:42, 5 January 2017 (UTC)

It is correct that the magnetic flux ratio and the inductance ratio are not equal. Please refer to the following description.

leakage flux is proportional to the load current

So electromagnetically defined leakage flux is also zero when the load current is zero. However, the term "magnetic flux leaking from the core material" in the core material term is not zero. We need to be aware that terms which are similar to events of different meanings are used.153.227.36.195 (talk) 01:22, 6 January 2017 (UTC)

Dear 153.227.36.195, You have to show in detail what exactly you find is not supported in terms of sources by the article. It is not enough to point to a Google search with several book entries. If we all have a stomach for complex, tedious electro-magnetic relationships, we could always consider merging the Leakage inductance article into the Inductance article.Cblambert (talk) 02:57, 6 January 2017 (UTC)

By load current, do you mean secondary current? Constant314 (talk) 20:24, 6 January 2017 (UTC)

Yes it is. In the Electromagnetism, the leakage flux is proportional to the current of the secondary winding. This secondary winding current is the load current. There is a description in the document on the indicated link. The definition of leakage flux in a transformer described in the Electromagnetism is "The flux interlinks to the windings only one side and traverses paths not interlinks with other windings." So the leakage flux is zero, when the secondary winding current is zero.[5] But the definition of leakage flux in the Magnetism is "The magnetic flux which does not follow the special purpose path in a magnetic circuit."[6] There is no concept of load current here. So these two leakage fluxes are similar but different. The value of the formula also differs. If we made an article called The "Leakage flux", we have to describe these differences carefully. In some cases, the same technical term is used in different meanings in different fields of expertise. Leakage flux is a typical example of it. And the Electromagnetism and the Magnetism should not be confused. Cblambert said "in § 8-67 The Leakage Factor. The total flux which passes through the yoke". Here, total flux and yoke are the terms of the Magnetism. And the term leakage factor is the same. Here is the confusion between the Electromagnetism and the Magnetism. I found a literature very close to the description he seems to have quoted.[7] Here is yoke, total flux and leakage factor. If so then the leakage flux referred to here is that of magnetics, so it is different from that of a transformer. This is the cause of confusion. And there is one more problem. The following formula revived, is it correct?

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M

The decisive thing is that the dimensions are different on the left and right sides of the equal sign.[8] The dimensions of the magnetic flux are as follows,

kg m² s^-2 A^-1

The dimension of the inductance is as follows,

kg m² s^-2 A^-2

If connect with equal sign, the notion of the electric current A is insufficient. Since this is a physical quantity, it can be objectively understand where there is a misunderstanding.153.227.36.195 (talk) 06:57, 8 January 2017 (UTC)

Maybe this is correct.

$\sigma _{P}={\frac {\phi _{P}^{\sigma }}{\phi _{M}}}={\frac {L_{P}^{\sigma }i_{P}}{L_{M}i_{M}}}$

$\sigma _{S}={\frac {\phi _{S}^{\sigma }}{\phi _{M}}}={\frac {L_{S}^{\sigma }i_{S}}{L_{M}i_{M}}}$

This is consistent with the description of the textbook and many literature. Besides, I think that it is reasonable that the dimensions on both sides of the equal sign match. If you can agree, it should be considered carefully and adopt this formula.153.227.36.195 (talk) 11:00, 8 January 2017 (UTC)

Maybe this is correct?! This is not serious discussion. Concrete, defendable, constructive proposals are needed.

This is consistent with the description of the textbook and many literature? Do you mean, This is consistent with descriptions of textbooks and the literature? If so, which description(s)? From which textbook(s)? In which literature?

This article starts from coupling factor in top section, progressing to leakage factor in the 2nd section in order to arrive at the equivalent circuit of a nonlinear transformer in 3rd section and refinement of leakage factor in 4th section. Both factors (coupling and leakage) are dimensionless as they are described in terms of the ratio of inductance and flux, so the zero-current argument invoked is meaningless. If there are any holes to pick in the article, they need to start at the top of the article, not at the end, as the whole article is clearly built with scientific logic from top to bottom.

This talk discussion definitely needs to stop grasping at straws.Cblambert (talk) 20:18, 8 January 2017 (UTC)

I find these two equations to be nonsense without further explanation

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M

The right hand expression is clearly a constant. Φ_S^σ'/Φ_M clearly depends on both primary and secondary current. In particular, secondary current can be zero and thus Φ_S^σ' can be zero. This expression can be saved, if there is some condition can be put on the currents, such as the primary is driven by a voltage source at the transformer's rated vlotage and the the secondary is connected to a resistive load such that the transformer is delivering its fully rated secondary current. Further, there is the question as to whether Φ_M is an instantaneous, phasor or RMS quantity. I rather doubt that it is an instantaneous value because if it were, it could be instantaneously zero and the expression Φ_S^σ'/Φ_M would contain a divide by zero. Until these simple definitions are clarified, the rest of the section is hopelessly undecipherable. If we cannot clarify the meanings of these symbols then I think the entire section should be removed to the talk page until the symbols are clarified. Constant314 (talk) 21:01, 8 January 2017 (UTC)

Refined leakage factor

Magnetizing and leakage flux in a magnetic circuit

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.3.1)

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.

Referring to the flux diagram at right, the winding-specific leakage ratio equations can be defined as follows,

σ_P = Φ_P^σ/Φ_M = L_P^σ/L_M ------ (Eq.3.2)

σ_S = Φ_S^σ'/Φ_M = L_S^σ'/L_M ------ (Eq.3.3)

Φ_P = Φ_M + Φ_P^σ = Φ_M + σ_PΦ_M = (1 + σ_P)Φ_M ------ (Eq.3.4)

Φ_S^' = Φ_M + Φ_S^σ' = Φ_M + σ_SΦ_M = (1 + σ_S)Φ_M ------ (Eq.3.5)

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 leakage factor
• σ_S is secondary leakage factor
• Φ_M is mutual flux (main flux).
• Φ_P^σ is primary leakage flux.
• Φ_S^σ is secondary leakage flux.

The leakage ratio σ can thus be refined in terms of the interrelationship of above winding-specific inductance and leakage factor equations as follows:

$\sigma =1-{\frac {M^{2}}{L_{P}L_{S}}}=1-{\frac {a^{2}M^{2}}{L_{P}a^{2}L_{S}}}=1-{\frac {L_{M}^{2}}{L_{P}L_{S}^{\prime }}}=1-{\frac {1}{{\frac {L_{P}}{L_{M}}}.{\frac {L_{S}^{\prime }}{L_{M}}}}}=1-{\frac {1}{(1+\sigma _{P})(1+\sigma _{S})}}$ ------ (Eq.3.7).Cblambert (talk) 00:08, 9 January 2017 (UTC)Cblambert (talk) 20:24, 10 January 2017 (UTC)

I disagree completely with the previous discussion. Cblambert (talk) 00:23, 9 January 2017 (UTC)

I was thinking whether this problem could be solved or not. A hint was obtained from considerations of dimension. It is established the right and left sides of the equation under a limited electric current condition. So what is limited electric current condition? It is when the current values of the denominator and the molecule are equal. That is,

$\sigma _{P}={\frac {\phi _{P}^{\sigma }}{\phi _{M}}}={\frac {L_{P}^{\sigma }i_{P}}{L_{M}i_{M}}}$

$\sigma _{S}={\frac {\phi _{S}^{\sigma }}{\phi _{M}}}={\frac {L_{S}^{\sigma }i_{S}}{L_{M}i_{M}}}$

Although I delayed, the basis of the above formula is,[9]

$\phi =LI$

Here do the following,

$i_{P}=i_{M}$

${\frac {\phi _{P}^{\sigma }}{\phi _{M}}}={\frac {L_{P}^{\sigma }}{L_{M}}}$

Then,

$i_{M}=i_{P}-i_{S}$

That means,

$i_{S}=0$

So,

$\sigma _{S}=0$

This was initially suggested by Constant314. I also agree that. So, I think that such a limited conditions which related to the current are described somewhere in Hameyer.--153.227.36.195 (talk) 03:02, 10 January 2017 (UTC)

We can conjecture on the talk page, but for the article, we need the actual definitions and conditions from the reliable source. That being said, and strictly for the purpose of discourse, I note, that under normal conditions (primary driven by a voltage source, secondary load mostly resistive) that i_M lags i_S by almost 90 degrees. So how are we to take the meaning of i_S/i_M? Constant314 (talk) 21:50, 10 January 2017 (UTC)

This equal sign is not established when current flows through the secondary winding. In other words, it is only established under the condition that the secondary winding is open. At first, Cblambert brought the term and formula of the Magnetism which called the "Leakage factor" to the Leakage inductancethe of the Electromagnetism article. At first glance it seemed like a reckless challenge. However, under the limited conditions it is possible to match the equations of electromagnetism and magnetism. I am just trying to respect the intention of the writer as much as possible. But as you pointed out, there is no clear source yet.--121.2.169.188 (talk) 23:52, 10 January 2017 (UTC)

The case for Hameyer's Refined leakage factor equations

The logic of Hameyer's Refined leakage factor equations: 1. We know that $\sigma$ is a fixed, finite