Talk:Leakage inductance

Error in figure
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)

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)

--- 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


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
 * LP & LS are primary and secondary winding self-inductances
 * RP & RS are primary and secondary winding resistances
 * Constants M, LP, LS, RP & RS are measurable at the transformer's terminals
 * Coupling coefficient k is given as
 * $$k=M/\sqrt{L_PL_S}$$, 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:
 * $$v_P=R_Pi_P+\frac{d\Psi{_P}}{dt}$$
 * $$v_S=-R_Si_S-\frac{d\Psi{_S}}{dt}$$
 * $$\Psi_P=L_Pi_P-Mdi_S$$
 * $$\Psi_S=L_Si_S-Mi_P$$
 * 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,
 * $$\sigma=1-\frac{M^2}{L_PL_S}=1-k^2=\frac{L_{sc}}{L_{oc}}=\frac{i_{oc}}{i_{sc}} $$,

where,
 * σ is the leakage factor or Heyland factor
 * ioc & isc are no-load and short circuit currents
 * Loc & Lsc are no-load and short circuit inductances.

The transformer can thus be further defined in terms of the three inductance constants as follows,
 * $$L_M=a{M}$$
 * $$L_P^\sigma=L_P-a{M}$$
 * $$L_S^\sigma=L_S-a{M}$$,

where,
 * LM is magnetizing inductance, corresponding to magnetizing reactance XM
 * LPσ & LSσ are primary & secondary leakage inductances, corresponding to primary & secondary leakage reactances XPσ & XSσ.

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.
 * $$L_S^{\sigma'}=a^2L_2-aM$$
 * $$R_S^'=a^2R_S$$
 * $$V_S^'=aV_S$$
 * $$I_S^'=I_S/a$$.

Since
 * $$k=M/\sqrt{L_PL_S}$$

and
 * $$a=\sqrt{L_P/L_S}$$,

we have
 * $$aM=\sqrt{L_P/L_S}*k*\sqrt{L_PL_S}=kL_P$$,

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
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,
 * iM = iP - iS'
 * iM is magnetizing current excited by flux ΦM that links both primary and secondary windings.

Referring to the flux diagram at right, the leakage factor can be defined as follows,
 * σP = ΦPσ/ΦM = LPσ/LM
 * σS = ΦSσ'/ΦM = LSσ'/LM
 * ΦP = ΦM + ΦPσ = (1 + σP)ΦM
 * ΦS' = ΦM + ΦSσ' = (1 + σS)ΦM
 * LP = LM + LPσ = (1 + σP)LM
 * LS' = LM + LSσ' = (1 + σS)LM,

where
 * σP is primary leakage factor
 * σS 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:
 * $$\sigma=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}=1-\frac{L_M^2}{L_PL_S{^'}}=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^'}{L_M}} =1-\frac{1}{(1+\sigma_P)(1+\sigma_S)}$$.
 * 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.

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.[[Image:Coupling Coefficient.gif|350px| thumb|right|Le1 and Le2 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. 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.  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 fluxΦSσ' is different from the actual measurements. It is reverse. [[Image:Main & leakage inductances.jpg|175px|thumb|right|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 fluxΦSσ' 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
The nub of leakage inductance is as shown in TREQCCTHeyland-to-k.jpg here.
 * Open circuit neglecting resistance is
 * Self-inductance LP =
 * Leakage inductance LP*(1-k) + magnetizing inductance LP*k
 * And not LP = LP*(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:
 * $$\sigma=1-\frac{M^2}{L_PL_S}=1-k^2=\frac{L_{sc}}{L_{oc}}=\frac{i_{oc}}{i_{sc}} $$
 * 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 LPσ, Lsσ and Lsσ' are known in terms of M, LP, LS, k, a, ioc and ioc 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.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)


 * 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 LM2 but should leave as LM/a2
 * Not show '1' dimension on either side as this is not electrical convention
 * Show LP . (1 - k) and LS . (1 - k)
 * Show LP . k and LS . 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 Ã 10-9 seconds, and magnetizing time constants in excess of 5 Ã 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 = (Ladd - Loppose)/(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.
 * σP = ΦPσ/ΦM = LPσ/LM
 * σS = ΦSσ'/ΦM = LSσ'/LM

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 σP 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 σP = ΦPσ/ΦM = LPσ/LM makes senses if it is measured with the secondary open circuited
 * and σS = ΦSσ'/ΦM = LSσ'/LM 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
 * σP = LPσ/LM
 * σS = LSσ'/LM
 * 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 iM = iP - iS' 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 = Φm = Φa + Φe and the ratio Φm/Φa 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
 * σP = ΦPσ/ΦM
 * σS = ΦSσ'/ΦM
 * ΦP = ΦM + ΦPσ = ΦM + ΦPΦM = (1 + σP)
 * ΦS' = ΦM + ΦSσ' = ΦM +  ΦSΦM = (1 + σS)

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.
 * |The 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. 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." 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. 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?
 * σP = ΦPσ/ΦM = LPσ/LM
 * σS = ΦSσ'/ΦM = LSσ'/LM
 * The decisive thing is that the dimensions are different on the left and right sides of the equal sign. 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
 * σP = ΦPσ/ΦM = LPσ/LM
 * σS = ΦSσ'/ΦM = LSσ'/LM

The right hand expression is clearly a constant. ΦSσ'/ΦM clearly depends on both primary and secondary current. In particular, secondary current can be zero and thus ΦSσ' 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 ΦM 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 ΦSσ'/ΦM 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
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,
 * iM = iP - iS' -- (Eq.3.1)
 * iM is magnetizing current excited by flux ΦM that links both primary and secondary windings.
 * iS' 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,
 * σP = ΦPσ/ΦM = LPσ/LM -- (Eq.3.2)
 * σS = ΦSσ'/ΦM = LSσ'/LM -- (Eq.3.3)
 * ΦP = ΦM + ΦPσ = ΦM + σPΦM = (1 + σP)ΦM -- (Eq.3.4)
 * ΦS' = ΦM + ΦSσ' = ΦM +  σSΦM = (1 + σS)ΦM -- (Eq.3.5)
 * LP = LM + LPσ = LM + σPLM = (1 + σP)LM -- (Eq.3.6)
 * LS' = LM + LSσ' = LM + σSLM = (1 + σS)LM -- (Eq.3.7),

where
 * σP is primary leakage factor
 * σS is secondary leakage factor
 * ΦM is mutual flux (main flux).
 * ΦPσ is primary leakage flux.
 * ΦSσ 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:
 * $$\sigma=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}=1-\frac{L_M^2}{L_PL_S^\prime}=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^\prime}{L_M}} =1-\frac{1}{(1+\sigma_P)(1+\sigma_S)}$$ -- (Eq.3.8).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,
 * $$\phi=L I$$
 * 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 iM lags iS by almost 90 degrees. So how are we to take the meaning of iS/iM? 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.--153.227.36.195 (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 can be recast along the follow lines:
 * a) We know from article's Eq. 2.1 & IEC IEV 131-12-41 that the coupling factor $$k$$ (a value that is dimensionless, fixed, finite, positive & less that 1) is given by the following:
 * $$k=M/\sqrt{L_PL_S}$$ -- (Eq. 1).


 * b) We also know from article Eq. 2.7 & IEC IEV 131-12-42 that the leakage factor $$\sigma$$ can be derived in terms of coupling factor k to read as follows:
 * $$\sigma=1-k^2=1-\frac{M^2}{L_PL_S}$$ -- (Eq. 2)
 * Note that $$\sigma$$ is therefore also a value that is dimensionless, fixed, finite, positive & less that 1.


 * c) Above Eq. 3.8 can therefore be given by the following:
 * c.1) by multiplying the $$\frac{M^2}{L_PL_S}$$ term by $$\frac{a^2}{a^2}$$, we get
 * $$\sigma=1-k^2=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}$$ -- (Eq. 3)


 * c.2) According to Eq. 2-8, $$L_M=aM$$, and given that by definition $$a^2L_S=L_S^\prime$$, we have:
 * $$\sigma=1-k^2=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}=1-\frac{L_M^2}{L_PL_S^\prime}$$ -- (Eq. 4)


 * c.3) By multiplying the $$\frac{L_M^2}{L_PL_S^\prime}$$ term by $$\frac{L_M.L_M}{L_M^2}$$, Eq. 3.8 becomes:
 * $$\sigma=1-k^2=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}=1-\frac{L_M^2}{L_PL_S^\prime}=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^\prime}{L_M}}$$ -- (Eq. 5)


 * c.4) Expressed in terms of currents and inductances only, Eq. 3.1, Eq. 3.2, Eq. 3.3, Eq. 3.6 & Eq. 3.7 can be restated as follows:
 * iM = iP - iS' -- (Eq.3.1)
 * σP = LPσ/LM -- (Eq.3.2)
 * σS = LSσ'/LM -- (Eq.3.3)
 * LP = LM + LPσ = LM + σPLM = (1 + σP)LM -- (Eq.3.6)
 * LS' = LM + LSσ' = LM + σSLM = (1 + σS)LM -- (Eq.3.7)
 * such that, '''as Hameyer shows, above Eq. 3.8 is thus proven to be equal to:
 * $$\sigma=1-k^2=1-\frac{M^2}{L_PL_S}=1-\frac{a^2M^2}{L_Pa^2L_S}=1-\frac{L_M^2}{L_PL_S^\prime}=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^\prime}{L_M}} =1-\frac{1}{(1+\sigma_P)(1+\sigma_S)}$$ -- (Eq.3.8)'''


 * c.5) Regarding development of Eq. 3.1 to Eq. 3.4 in terms of flux relationships, it is known that, far from being nonsense, σP = ΦPσ/ΦM & σS = ΦSσ'/ΦM must necessarily be values that are dimensionless, fixed, finite, positive & less that 1. Further, since in sinusoidal steady-state conditions and at rated primary voltage ΦM (and magnetizing current) can be represented as a fixed non-zero phasor (or RMS) value, it follows that ΦSσ' (and current) must by definition be a fixed non-zero phasor (or RMS) value. Brenner and Javid show on pp. 591-593 that a transformer can be shown in additive and subtractive series connection of the two windings to be equal to:
 * Additive connection = $$L_{ser}^+=L_P^\sigma+L_S^\sigma-2M$$ -- (Eq. 5)
 * Subtractive connection = $$L_{ser}^-=L_P^\sigma+L_S^\sigma+2M$$ -- (Eq. 6)
 * such that winding inductances can be determined form the 3 equations:
 * $$L_{ser}^{+}-L_{ser}^{-}=4M$$ -- (Eq. 7) and $$L_{ser}^{+}+L_{ser}^{-}=2*(L_P^\sigma+L_S^\sigma)$$ -- (Eq. 8) and $$a=\sqrt{L_P/L_S}$$ -- (Eq. 2.10).


 * c.6) The case for Hameyer's equations for the refined leakage factor is therefore conclusively proven and will be modified in this sense to restore the Refine leakage factor section of the Leakage inductance article.Cblambert (talk) 05:01, 11 January 2017 (UTC)
 * Since there are now no flux ratios or current ratios there is no longer a need to define them. I am satisfied with what you have now.  The variables in Eq 3.1 don't seem to show up anywhere else.  You might want to leave that out.Constant314 (talk) 05:28, 11 January 2017 (UTC)
 * Is σ in here the same as σ of this page ?
 * $$\sigma=\frac{\Phi_t}{\Phi_g}$$
 * --153.227.36.195 (talk) 10:33, 11 January 2017 (UTC)


 * I don't think so. That appears to be about flux at an intentional air gap.  At a gap, some of the flux goes where you want it to go and some of it fringes around.  That leakage factor is the ratio of the total flux to the flux that stays in the desired gap. Constant314 (talk) 18:45, 11 January 2017 (UTC)
 * I don't think so either. This article defines leakage factor in terms of the coupling factor per Eq. 1 and Eq. 2 above.Cblambert (talk) 21:57, 11 January 2017 (UTC)
 * I suggest that the flux and current relationships in flux diagram and in Eq. 3.1 to Eq. 3.8 all be maintained as it is important to the understanding of the Leakage inductance article to tie together the flux diagram Main_%26_leakage_inductances.jpg to all the Hameyer equations.Cblambert (talk)
 * I have no problem with flux and current as long as they have a clear, sensible definition and conditions under which they apply are explicit. I would rather that those definitions and conditions come from the source, but if you can reconstruct them from memory and we can agree that they are clear and consistent then I will have no objection.  I do object to using them without definition.  If I look elsewhere, I may find the same symbols used in a similar situation but with a different meaning, as we have seen the use "leakage factor" with the same symbol used to mean something else. Constant314 (talk) 00:21, 12 January 2017 (UTC)


 * You have to be kidding. I would be interested in knowing about any such instances of different meaning for similaar situation because I have taken greats pains to other this. We are all in this together. Right?Cblambert (talk) 00:36, 12 January 2017 (UTC)
 * Then the meaning of σ here and the σ of the link shown by me will be different. Is that no problem? Then the meaning of σ here and the σ of the link shown by me will be different. Is that a problem? By the way, at in § 8-67 The Leakage Factor. The total flux which passes through the yoke", is it possible to see what was written about leakage factor? Also, the number of documents describing leakage inductance and leakage factor in the same field is extremely little. I tried searching for "search "leakage inductance leakage factor" but I can not find anything other than this page of Wikipedia and its quotations.--153.227.36.195 (talk) 02:01, 12 January 2017 (UTC)

I am not saying that the leakage factor you describe is wrong, but the leakage factor in this article is a very special definition and not general. So I recommend that you should add an annotation that the same term is used in other fields so that the reader will not misunderstand the leakage factor. At the same time, I found out that the leakage inductance has the same problem, did you know? For example, Measuring Leakage Inductance,To summarize,
 * $$L_{s/c}=a^2L_\text {inductance of the short-circuit impedance}$$

I think that this should also be annotated.--153.227.36.195 (talk) 20:11, 12 January 2017 (UTC)

For a reference, this definition is more general for the leakage factor and the leakage inductance.--153.227.36.195 (talk) 20:59, 12 January 2017 (UTC)

What is the general definition of the leakage factor?
We also need to suggest readers about its general definition.--153.227.36.195 (talk) 20:11, 12 January 2017 (UTC)

"The magnetic leakage factor" for Magnetic materials and components and "The inductive leakage factor" for the Circuit theory [Http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-42] are really the same thing? The symbol σ is described on one side and the symbol is not described on the other side. If they are different, it should be highlighted for readers.--153.227.36.195 (talk) 11:04, 13 January 2017 (UTC)

Definition of leakage inductance in other industrial fields
I found that there is a definition different from the definition of this article like Measuring Leakage Inductance or, To summarize them,
 * $$L_{s/c}=a^2L_\text {inductance of the short-circuit impedance}$$

I think that this should be annotated.

They are saiing that they define the inductance of the leakage impedance as leakage inductance or define the primary side conversion value of it as so. I think that we should need to suggest them for readers.--153.227.36.195 (talk) 20:11, 12 January 2017 (UTC)

The essential problem with the Refined inductive leakage factor section remains
We still have these equations with no description about the conditions under which they apply.
 * σP = ΦPσ/ΦM = LPσ/LM -- (Eq. 3.1)
 * σS = ΦSσ'/ΦM = LSσ'/LM -- (Eq. 3.2)

It is obvious that they don't apply under all conditions. I believe that I can guess the conditions which is this: Eq. 3.1 applies when the primary is driven and the secondary is open. Under this condition, iM = iP which means ΦPσ and ΦM are produced by the same current and their ratio is the same as LPσ/LM. Eq. 3.2 applies when the secondary is driven and the primary is open circuited. I now consider this to be obvious. If the three of us agree that it is obvious, then lets put it in and be done. Constant314 (talk) 21:10, 13 January 2017 (UTC)
 * Exactly. Under those condition, those formula will be hold. I would like you to adopt it as a result of careful consideration.--153.227.36.195 (talk) 22:38, 13 January 2017 (UTC)
 * Anyway I studied valuable thing this time in here. I started by biting that formula in a short-temper without knowing the inductive leakage factor at all. I knew only the leakage factor which is the magnetic leakage factor. The outcome of this finding is the result of Cblambert's efforts and thanks for making many links to references. I had been barking like a wolf all the while. Please forgive me for many overstuffing behavior.--153.227.36.195 (talk) 23:08, 13 January 2017 (UTC)

Coupling factor and coupling coefficient
Although these two terms are very similar, I think that their relation may be follows,
 * $$\text{Coupling factor}=\left | \text{Coupling coefficient} \right | $$

So the coupling coefficient is fit for the former (Eq. 2.1), but the coupling factor is fit for the latter (Eq. 2.1). Then, according to IEC IEV 131-12-41, the latter (Eq. 2.1) should be dscripted as follows,
 * $$k=\left | M\right | /\sqrt{L_PL_S}$$ - (Eq. 2.1)

--153.227.36.195 (talk) 04:46, 15 January 2017 (UTC)


 * First of all, let's read and examine Measuring Leakage Inductance before the discussion. And how to obtain the coupling coefficient. --153.227.36.195 (talk) 23:59, 16 January 2017 (UTC)

Old url for Hameyer
Old url for Hameyer is http://materialy.itc.pw.edu.pl/zpnis/electric_machines_I/ForStudents/Script_EMIHanneberger.pdf/ As in for Hameyer, Kay (2001). "Electrical Machine I: Basics, Design, Function, Operation" (PDF). RWTH Aachen University Institute of Electrical Machines. Retrieved 11 January 2013.page=133 in Induction motor.Cblambert (talk) 12:44, 17 January 2017 (UTC)

Refined inductive leakage factor derivation
The Refined inductive leakage factor derivation box below, taken verbatim from article, is obtained from basic principles, which happens to agreed with Haymeyer (and with Constant314). The onus is on others to disprove this derivation.

Refined inductive leakage factor derivation

a. Per Eq. 2.1 & IEC IEV 131-12-41 inductive coupling factor k is given by
 * $$k=\left | M\right | /\sqrt{L_PL_S}$$ - (Eq. 2.1)

b. Per Eq. 2.7 & IEC IEV 131-12-42 Inductive leakage factor $$\sigma$$ is given by
 * $$\sigma=1-k^2=1-\frac{M^2}{L_PL_S}$$ -- (Eq. 2.7) & (Eq. 3.7a)

c. $$\frac{M^2}{L_PL_S}$$ multiplied by $$\frac{a^2}{a^2}$$ gives
 * $$\sigma=1-\frac{a^2M^2}{L_Pa^2L_S}$$ - (Eq. 3.7b)

d. Per Eq. 2-8 & knowing that $$a^2L_S=L_S^\prime$$
 * $$\sigma=1-\frac{L_M^2}{L_PL_S^\prime}$$ -- (Eq. 3.7c)

e. $$\frac{L_M^2}{L_PL_S^\prime}$$ multiplied by $$\frac{L_M.L_M}{L_M^2}$$ gives
 * $$\sigma=1-\frac{1}{\frac{L_P}{L_M}.\frac{L_S^\prime}{L_M}}$$ -- (Eq. 3.7d)

f. Per Eq. 2.14, Eq. 3.1, Eq. 3.2, Eq. 3.5 & Eq. 3.6
 * $$\sigma=1-\frac{1}{(1+\sigma_P)(1+\sigma_S)}$$ --- (Eq.3.7e)'''

All equations in this article assume steady-state constant-frequency waveform conditions the k & $$\sigma$$ values of which are dimensionless, fixed, finite & positive but less that 1. Cblambert (talk) 17:18, 17 January 2017 (UTC)


 * My only objection is that σP and σS are elsewhere set equal to the ratios of inadequately defined fluxes. Constant314 (talk) 21:05, 17 January 2017 (UTC)
 * Do you have a proposal on how to close this gap? One of us could always invest US$15 to get, through IEEE Explore, MIT-Press's Self- and Mutual Inductance, which conveniently covers the following topics: The Coupled-Circuit Equations, Coefficient of Coupling and Leakage Coefficient, Measurement of the Parameters, Leakage Inductance, Equivalent Circuits for Two-Winding Transformers, Summary, Problems.Cblambert (talk) 00:37, 18 January 2017 (UTC)
 * I have added a citation-footnote reading: ' U. of Colorado - Magnetics, slide 32: It can be seen by inspection that, for steady-state sinusoidal waveform conditions applied to a real, linear transformer, σP is equal to the leakage-to-magnetizing ratios of both inductances and fluxes. With both U. of Colorado and Hameyer agreeing, I think the onus is indeed on others to disprove the derivation.Cblambert (talk) 05:48, 18 January 2017 (UTC)


 * The Voltech document is about measuring LPσ + LSσ' under the conditions of a short circuited secondary. The article is about how to correct the measurement in the case where the short circuit is not quite a perfect short circuit.  It says nothing about fluxes and nothing about the ratio of leakage flux to magnitizing flux.  There is nothing apparent about flux ratios on the U of Colorado slide.  As draw on slide 32, magnetizing flux is a function of both primary current and secondary current and therefor ΦPσ/ΦM and ΦSσ'/ΦM are not constants.  For example I can make ΦSσ'/ΦM = 0 by open circuiting the secondary.  My proposal for fixing this is to remove all reference to ΦPσ, ΦM and  ΦSσ' in equations 3.1 and 3.2 and delete equations 3.3 and 3.4 until we get a source that defines the fluxes.  The rest of the math deals with constants that are independent of the fluxes and the final result and all the intermediate steps are unchanged. Constant314 (talk) 17:38, 18 January 2017 (UTC)
 * If Cblambert needs formula for deriving (Eq. 1.4) I will write it.--153.227.36.195 (talk) 19:31, 18 January 2017 (UTC)
 * Re Voltech: I know exactly what Voltech is doing, but the document does say as that in a perfect world leakage inductance is as quoted. Voltech is invoked here because sentence used to refer to open-circuit conditions only and circuiting is a good approximation in transformer of a certain rating.Cblambert (talk) 01:37, 19 January 2017 (UTC)
 * Re flux, the fluxes are shown in Eq. 3.1 & Eq. 3.2 as a ratio, which does not vary.Cblambert (talk) 01:37, 19 January 2017 (UTC)
 * Hameyer pp. 28-29, eq. 3-31 thru eq. 3-36 is for now until disproved a valid reference.Cblambert (talk) 01:37, 19 January 2017 (UTC)
 * Re U. of Colorado: I have deleted the footnote.Cblambert (talk) 01:43, 19 January 2017 (UTC)
 * Re former Eq. 1.4, any equation needs to be consistent with the rest of the article in terms of notations, and otherwise melding seamlessly and logically. Any eq. should be excluded from in 1st sub-section as it deals with the k diagram.Cblambert (talk) 02:10, 19 January 2017 (UTC)

You have not adopt no matter how I show many sources. The term Leakage inductance is not just for Magnetism persons. Also it should be considered the leakage inductance for Electromagnetism persons and Electronics engineers. If you are an engineer, you would think how to obtain those specific values if you wanted to use those values for applications not only in theoretical. I already showed one of the derivative methods.[| formula (22)] And I show another way to derive even more easily those values as follows, In the figure on the right, short-circuit the secondary winding and find the short-circuited inductance value of the primary side.
 * $$L_{sc}^{pri}=L_P(1-k)+\frac{1}{\frac{1}{kL_P}+\frac{1}{L_P(1-k)}}=L_P-kL_P+\frac{1}{\frac{L_P-kL_P+kL_P}{kL_P^2-k^2L_P^2}}=L_P-kL_P+\frac{kL_P^2-k^2L_P^2}{L_P}$$
 * $$=L_P-kL_P+kL_P-k^2L_P=L_P-k^2L_P=L_P(1-k^2)$$
 * $$\frac{L_{sc}^{pri}}{L_P}=1-k^2$$
 * $$k^2=1-\frac{L_{sc}^{pri}}{L_P}$$
 * $$k=\sqrt{1-\frac{L_{sc}^{pri}}{L_P}}$$
 * $$k=\sqrt{1-\frac{L_{sc}^{pri}}{L_{oc}^{pri}}}$$

This result is consistent with the source which I showed.--153.227.36.195 (talk) 02:47, 19 January 2017 (UTC)
 * Thanks. This may well be the case (I need to take time to check this). The point that there already exists a good simple solution including in terms of a drawings. Adding an alternative solution adds to the confusion, which is evidently quite complex to deal with now. A compelling reason is needed to warrant adding this extra complexity. I make no different in principle between magnetism, electromagnetism & electronics. There used to be a section of Leakage inductance in practice which someone went missing. I will restore it.Cblambert (talk) 04:31, 19 January 2017 (UTC)

Validity of sources
I would like to comment about the vadility of sources such Hameyer's 2001 course document, which was available only until at least 2013 and which, some are of the opinion, that since no longer available online should be considered as not a valid source. I for example happen to have a copy of Knowlton's 1949 Standard Handbook for Electrical Engineers as we;; as copies of a number of 1960s vintage textbooks, which I refer to extensively. Does that mean that if I cite these books in articles in these sources, which later no longer becomes available from me for any particular reason that references in these no longer a valid citations/ What happens to all the broken links in articles for papers, articles and other lesser publications that later only become available by purchasing? If the answer to these questions are that sources need to be only generally available, Wikipedia has a huge problem that greatly threatens the quality of some, in some cases, inherently high quality articles.Cblambert (talk) 05:27, 19 January 2017 (UTC) An IEEE Explore search shows Kay Hameyer figuring between 1994 amd 2016 in 159 conference publications, 126 journal & magazine articles and in 5 early access articles. 94 of these documents being published by the IEEE Transactions on Magnetics.Cblambert (talk) 23:50, 20 January 2017 (UTC)
 * The standard for scholarly citation is publication in a peer-reviewed professional journal with a reputation for accuracy. "Hameyer's 2001 course document" fails on all counts. You might want to review WP:RS, or better yet, WP:MEDRS. Sbalfour (talk) 02:10, 28 November 2017 (UTC)
 * I agree. Even if Hameyer is an authority, there is no guarantee that he was rigorous in his course notes.  He might have made approximations or simplifications appropriate for his audience.  Part of the information might have been verbal and not part of the notes.  Yes, I agree that Hameyer is not a RS, although it seams that the standard has fallen as more content finds its way on to the internet.  Constant314 (talk) 04:52, 28 November 2017 (UTC)

Need for explicit IEC explanation endnote
Endnote is felt necessary due to counter repeated recently expressed view by some that IEC is "an industry association".Cblambert (talk) 17:48, 21 February 2017 (UTC)

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Notes / copyedit
The amount of discussion belies the amount of real work done on this article. It appears that outside EE's possibly associated with research institutions, rather than seasoned WP editors, have been in charge here.

Lead

 * 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. The winding self-inductance constant and associated leakage inductance is due to leakage flux not linking with all turns of each imperfectly-coupled winding. What?
 * Leakage flux... acting as an inductance in series with...?? Consider the electric circuit equivalent of that statement:  (some kind of)current... acting as an inductance in series with [a resistor]. That makes no sense at all:  current isn't inductance and doesn't act like inductance, and neither does flux.   Inductance is a derivative of a magnetic vector field at a fixed point; flux is a quantization of the field.  We don't want to go into vector calculus here, but we should be very clear for the purpose, what the terms actually represent.  Maybe we just want to say in the second phrase (since the first phrase referred to leakage flux), "...; the resultant leakage inductance is a series inductance." (inferred to be in series with the self- or mutual inductance).
 * Although discussed exclusively in relation to transformers in this article, leakage inductance applies to any imperfectly-coupled magnetic circuit device including motors. Wordy -> "Leakage inductance applies to any imperfectly-coupled magnetic circuit device."
 * last para of lead - we don't do references in text like that - move to footnote

Section 1

 * This section is a lengthy juxtaposition of two disjoint topics and should be split into separate sections
 * it says: The magnetic circuit's flux that does not interlink both windings is the leakage flux No! No! No! It is the magnetic flux NOT traveling in the magnetic circuit that interlinks both windings, which is the leakage flux.
 * eq. 1.3, 2.1 and 2.9 are duplicates; eq. 2.1 is replicated in the sidebar
 * In eq.1.1, substituting in for the right-hand side, I get $$L_P = L_P$$; when I do the same in eq.1.2, I get $$L_S = L_S$$. Are these supposed to be definitions?...axioms?...identities?
 * At the bottom of section 1 and again in the sidebox, $$a^2$$ is used, but '$$a$$' is not defined
 * Section 1 sidebar uses terms, additive and subtractive series connection. This procedure is not described, and is not patently obvious
 * In section 1 sidebox, Campbell bridge is not described, but used as if familiar. No reference to it can be found in the encyclopedia.
 * In eq. 1.3, $$\left | M \right |$$ appears, but '$$M$$' is undefined; below, $$L^{sec}_{sc}$$ is defined but not used
 * In Inductive coupling factor subsection, it says, Per Eq. 2.7, but eq. 2.7 hasn't been seen yet

Section 2

 * This section is a lengthy juxtaposition of two disjoint topics and should be split into separate sections (see also section 1)
 * In section 2, it says five impedance constants as shown in the diagram at right, but I see 13 symbols, presumptively constants, in that diagram
 * It appears that five impedance constants may refer to $$L_P$$ and $$L_S$$; these constants are inductances, not impedances; their associated impedances must be their inductive reactances, $$X_{L_P}$$ and $$X_{L_S}$$
 * Eq. 2.2 uses symbol '$$a$$', but the text uses symbol 'a' - just distracting; anyway, the turns ratio is more conventionally and mneumonicly designated '$$\mathcal n$$'.
 * The term mesh equations in section 2 isn't described.
 * Eq. 2.10 is a duplicate of eq. 2.2
 * eq. 2.14-teeny, tiny little text - use "math /math"
 * $$a=N_P/N_S=v_P/v_S=i_S/i_P=\sqrt{L_P/L_S}$$ -- (Eq. 2.1) I don't like this at all: only one of these things defines the turns ratio; it may be approximately inferred from the other relations, which are measured quantities, and such inferred ratios are likely to differ from each other and from the defined value.

Section 3

 * justification for this whole section is absent: "refined" with respect to what? Why do we need a refined factor, and how much better is it?
 * I note that this section was deleted in its entirety once, and my gut feel is that it should be so again, or the whole section moved to a footnote
 * Refs for this section are a college course handout, no longer available and not commercially published; these are not valid citations per WP:RS
 * the stuff in eq. 3.7 duplicates the derivation in the sidebox
 * most of the reductions in eq. 3.7 are superfluous anyway - move the algebra to a footnote, the encyclopedia is not a math textbook
 * Section 3: teeny, tiny little font - use "math /math"
 * Section 3 sidebox references eq. 3.7a-e, but there are no such equations.

Section 5

 * It says, iM is magnetizing current excited by flux ΦM. Figure 5 in that section is of the electric circuit. Without a corresponding diagram of the magnetic circuit, we really don't know where flux ΦM flows.  We're mixing models here and the result is uncertainty.

Applications section

 * In applications section, it says Leakage inductance can be an undesirable property... In many cases it is useful. That's an untenable clash in semantics.

Overview
This article needs some serious work in technical diction/accessibility, organization, presentation, formatting and copyediting. Sbalfour (talk) 02:50, 27 November 2017 (UTC)
 * can't discriminate knowns from unknowns in equations, or constants from variables
 * what's important? If I pick up any random numbered equation, I don't have any sense of its relevance
 * no discussion of techniques to reduce leakage inductance, like interleaved winding layers, bifilar windings, toroidal cores, etc
 * article uses two different methodologies, open-circuit and short-circuit to compute the various measures, without any discussion of their validity, trade-offs or applicability. We never use transformers in short or open circuit, so it casts doubt on the whole article.
 * there is no definition of leakage inductance, coupling factor, leakage inductance factor, or the 'inductance constants' (we compute something for them but it doesn't matter what we compute if that doesn't correspond to what we want assessed; and what is that?
 * The symbol $$=$$ in many, maybe most, cases doesn't represent relational equality, but a whole variety of other relationships like assignment, 'is defined as', 'implies', 'may be inferred from', 'is congruent to', 'may be represented as', and etc. The ambiguity renders equations as written confusing and just plain inaccurate.

Refined inductive linkage factor removed

 * Rubbish all of it -- single source, unreliable source & A major contributor to this section appears to have a close connection with its subject. Cblambert (talk) 20:37, 12 August 2018 (UTC)

Notes / structure
In the current article, we can't see the forest for the trees. Most of the definitional type equations were (and should be again) annotations in the circuit diagrams. That'd leave the content of the article essentially 4 (or 5) diagrams. Then we need to write the article around them. I propose the following restructure of the article:

Lead Definitions - things we'd find in an engineering specification of a transformer or other device

What goes here aren't equations, but descriptions of the concepts and relationships. In an engineering spec, they'd eventually be assigned some number derived from measurement. The concepts tell us what we'd like to assess. Nonideal linear two-winding transformers
 * Coupling coefficient & coupling factor
 * Leakage flux
 * Leakage inductance
 * Leakage inductance factor
 * etc
 * Transformer operation (meta description - there shouldn't be much math here)
 * Magnetic and electric circuits
 * Self-Inductance and mutual inductance


 * Next section is in two parts: first are definitions of measured or measurable quantities (since we can't actually measure things like leakage flux or leakage inductance). These would be in instructions to a lab tech and come back as a set of numbers on a datasheet.  These are contextual constants (since once measured, they can't change), so may never appear on the left-hand side of an assignment. The second part is derived quantities and the mathematics (probably vector & integral calculus) to obtain them.  For clarity in the article, derived and measured quantities could/should be distinguished by different fonts or colors.  It would help here if the relational equality operator '$$=$$' were distinguished from the assignment operator '$$:=$$', the transitive operator '$$\rightarrow$$', the 'is defined as' operator '$$:\Leftrightarrow$$', the 'implies' operator '$$\Rightarrow$$', the 'may be inferred from' operator '$$\Leftarrow$$' and in some cases the congruence operator '$$\cong$$ '.

Other transformers and devices Sbalfour (talk) 15:40, 29 November 2017 (UTC)
 * Inductance models
 * Terminal models
 * Open circuit
 * Open+short circuit
 * Duality models
 * Equivalent circuits w/annotations (most of the existing article would go in this section)
 * Advanced theory (operational factors influencing magnetic reluctance of core)
 * Eddy currents
 * Soft saturation
 * Reducing and increasing leakage inductance
 * Concentric windings
 * bifilar windings
 * interleaved layers
 * doped windings
 * air-gapped cores
 * toroidal cores
 * air-core transformers (explanation of magnetic circuit, inductance and leakage inductance in absence of magcore)
 * Multiwinding transformers
 * Nonlinear transformers (maybe for separate article; leakage inductance for these is a hyperbolic topic)
 * Inductors (leakage inductance just becomes part of self-inductance)
 * Induction motors and generators
 * Synchronous & asynchronous motors (i.e. inductive vs capacitative reactance and losses)
 * Leakage inductance devices (ones designed to exploit leakage inductance)

Core of the article
What this article is about is simple. There's only three things on the output side: leakage inductance, and two ratios, coupling factor and leakage factor. We might need to distinguish primary and secondary leakage inductance in some of the math. There are only 4 things on the input side, which we measure: voltage, current, DCR and winding ratio. Again we might need to distinguish these for primary and secondary. Everything else is cruft. So, we need three definitions, one for leakage and two for the ratios (these last are one-liners). We describe a procedure for measurement (put in a sidebox), do some math, and at the bottom, three (possibly 4) equations whose left-hand sides are leakage inductance, coupling factor and leakage factor, and right-hand sides are functions of the measured quantities. And that's it. 3 (maybe 4) equations, no more. Anything else, goes in a footnote, sidebox, link to other article, etc.

That's the core. The core needs context, so we pack it on, above and below, but we keep it out of the core.

Sbalfour (talk) 00:46, 1 December 2017 (UTC)

Symbol notation conventions
There are several confounding "styles" of symbol notation in the article. For example, designating various inductances on the primary side:

There are only 5 referred to quantities, yet it seems necessary to denote two of them with super- and sub- scripts? And as shown, two more with triple super- sub- scripts?? We could easily designate the 5 quantities P, S, M, l1, and l2. Their semantics are very clear even though I've not defined them here. We don't need L to tell the reader these are inductances - everything in the article is an inductance. If we need to reference reactance, it could be written $$X_P$$, etc. We also have the components of $$L_P$$: $$_M$$ and $$\sigma$$, one designated by a subscript $$L_M$$ and the other by a superscript $$L_P^\sigma$$. In addition, $$M$$ and $$L$$ are both used to represent inductance; the logical $$L_M$$ for mutual inductance is used for something else, doubly confounding. $$L_M$$ (and $$L_{M2}$$) don't have the same kind of existence as $$L_P$$ (and $$L_S)$$; they're placeholders for a mathematical computation. They could be represented like this $$L_{p \leftarrow M}$$ or $$L_{p/M}$$ or maybe $$L_{p \subset M}$$. That frees $$L_M$$ to represent mutual inductance. The relationship of these symbols could be denoted clearly by using the identity $$L_M$$ = $$L_{p \leftarrow M} \circ L_{s \leftarrow M}$$. "Short-circuit inductance" ($$L_{sc}^{pri}$$) is often referred to inexactly as leakage inductance; it's almost an alternate definition. $$L_P^{\sigma'}$$ might be a plausable way to represent it. The prime isn't very prominent, so maybe $$L_P^{\sigma+}$$ is more noticeable.
 * $$L_P$$ primary self-inductance
 * $$L_M$$ magnetizing inductance referred to the primary
 * $$L_{sc}^{pri}$$ primary short-circuit inductance
 * $$L_S^{\sigma'}$$ leakage inductance from secondary referred to the primary (formerly; deleted from article but still used in diagram)

Sbalfour (talk) 20:50, 3 December 2017 (UTC)

Inductive relationships, independent variables and equations
Starting with the first three equation 2.1 and two following unnumbered equations,
 * $$L_P=L_P^\sigma+L_M$$
 * $$L_P^\sigma=L_P\cdot{(1-k)}$$
 * $$L_M=L_P\cdot k$$

like any mathematician, I substituted the value of k from equation 4.3 below into the equations expecting to elucidate some fundamental relation:
 * $$L_P^\sigma=L_P\cdot{(1-\frac {M}{\sqrt {L_P \cdot L_S}})}$$
 * $$L_M=L_P\cdot{\frac {M}{\sqrt {L_P \cdot L_S}}}$$
 * $$\rightarrow L_P = L_P\cdot{(1-\frac {M}{\sqrt {L_P \cdot L_S}})} + L_P\cdot{\frac {M}{\sqrt {L_P \cdot L_S}}}$$
 * $$\rightarrow L_P = L_P - M\cdot \sqrt {\frac{L_P}{L_S}} + M \cdot \sqrt {\frac {L_P}{L_S}}$$
 * $$\rightarrow L_P = L_P + \sqrt {\frac{L_P}{L_S}} \cdot (-M + M)$$
 * $$\rightarrow L_P = L_P + 0$$
 * $$\rightarrow L_P = L_P$$

After that monumental effort, the result is a tautology. The reason is two-fold: the third equation above isn't independent- it follows directly from the other two. And k itself is twice substituted into an equation defining $$L_P$$, which is a constituent part of the definition of $$k$$ itself. $$k$$ is not actually an independent variable here - it's an alternate name, a shorthand for. The relationship is thus: $$k \cong$$

There are 20 equations in the article not counting sec. 5. There are as I can see, only 5 independent equations (defining 5 independent quantities): $$L_P$$, $$L_S$$, $$L_P^\sigma$$, $$L_S^\sigma$$, and $$M$$. There is also one inductive relationship not defined, or derivable from any of those, which is missing from the presentation. The rest of the equations are just shuffling between representations; no new relationships are thereby established. It's cruft - those who have or need different inputs to the model can do the math. This article is about elucidation of relationships, not representations. I'm going to shrink it down dramatically to just a set of independent equations/variables, in terms of the inputs to the inductive transformer model.
 * It's a total waste of effort. I wouldn't trust an equation in WP to convert from Fahrenheit to Celsius.  Even if you sweat out the maths typesetting and make it match some credible reference, WP:Randy from Boise will shortly be along and put his own unique stamp on it.  Delete all the equations and just explain it in general terms; anyone who really needs to know quantitatively what's going on, should pay for a professionally edited textbook. --Wtshymanski (talk) 00:58, 4 December 2017 (UTC)
 * Hmmm... I'd not suspect you'd want to trim everything. Some editor will come along an drop it back in, because "the article didn't have it". I'm already sort of down to the definitions... there's no conversion math, except one equation, and I'll delete it forthwith.  I hated sifting through all that math, to make sure I didn't omit anything. It was cruft... (Stay tuned.) I was imagining you sitting in the background laughing your @ss off while I sweated bullets. Cheers, Sbalfour (talk) 04:44, 4 December 2017 (UTC)

Article rewritten; archive?
The article has essentially been rewritten, and talk page discussion for all sections above Notes / Structure is now irrelevant and closed. I'm considering archiving all dead topics so editors have a clear picture of active topics. I'm aware, though that another editor has been accused of vandalism for using [hide] templates and archiving to conceal active topics. That is not my intention; the content referred to in dead topics no longer exists. Yes, it can be restored, with necessary accompanying citations, and an editor so doing can start a new topic for the justification. However, given the new structure of the article, I just don't think that's going to happen. Sbalfour (talk) 19:27, 4 December 2017 (UTC)
 * I would not object to archiving. Constant314 (talk) 01:16, 6 December 2017 (UTC)
 * And I would like to stop seeing those reference to Hameyer at the end. Not only is it not a WP:RS it is not available.Constant314 (talk) 12:50, 6 December 2017 (UTC)
 * Point taken - I, too, object to the Hameyer sources. They now belong to deleted sections that are reproduced only here, on the talk page.  I can confine them to the referencing sections, so that when archiving is complete, they'll be off the page.  It's perhaps not fair to vanish them altogether, but they CAN'T go back into the article. Sbalfour (talk) 21:16, 6 December 2017 (UTC)

Breath of fresh air
For a technical article (try magnetic reactance for the counter-example), this article is actually invigorating! It's free of jargon, differential, integral or algebraic math, wacky symbols like $$\int, \Sigma, \sqrt, dy/dx$$ (yeh, I fudged and left in a couple of sqrts... I'm thinking about that), discussions of magnetic circuits (most people are comfortable with electric circuits, but not magnetic ones) and vector fields, cryptic circuit diagrams and symbol names, and concepts that aren't defined in the article. The phraseology is a bit colloquial, but I don't think the technical accuracy is affected much. Sbalfour (talk) 04:24, 5 December 2017 (UTC)
 * 'most people are comfortable with electric circuits, but not magnetic one'??!! Really?!. I call this a heroic assumption.Cblambert (talk) 19:43, 10 August 2018 (UTC)
 * Unbelievable! Someone purporting to have a strong interest in the Leakage inductance article suggesting that magnetism is not needed to explain Leakage inductance. There is something very, very wrong with this notion. I can't believe that a major contributor to this article would be willing to put his thumb on the balance to bias things in favor of electric aspects to the disadvantage of magnetism aspects!!?? Just so we are clear, EE301 – MAGNETISM AND TRANSFORMERS: "A transformer is a magnetically coupled circuit, whose operation is governed by Faraday’s Law. Right? Cblambert (talk) 22:49, 10 August 2018 (UTC)

The definition of "a".
This is from Brenner and Javid. Do what you want with it.

The upper circuit has an arbitrary parameter a which is the turns ratio of the ideal transformer in the circuit. The circuit is equivalent to the actual non-ideal transformer. It works no matter what value you choose for a. If you choose a such that a2 = L1/L2, then the top circuit reduces to the bottom circuit. L1 and L2 are the open circuit actual terminal inductance for each side of the transformer. The important point is that a is not the turns ratio, although it is close and often said to be the turns ratio but it the square root of the inductance ratio and reduces to the turns ratio as the leakage inductance reduces to zero. Constant314 (talk) 01:13, 6 December 2017 (UTC)
 * Hmmmm...I'm aware of this little anomaly. The difference between deterministic 'a' in an ideal transformer and 'a' as derived by measured quantities in a real transformer, is due to inexactitudes in measurement.  The measured quantities include a little of this and that, so that 'a' does too.  I've got a real transformer I use to check my sanity, and it's a pud*ucking example of why things are so hard.  Even if I did count the turns, which I'm not going to do, it won't be the 'a' which makes everything else come out 'pretty close'. Sbalfour (talk) 21:29, 6 December 2017 (UTC)
 * P.S. Thanks, and stick around. I need some help on this article. It's been ~50 years since I worked with this stuff.Sbalfour (talk) 21:35, 6 December 2017 (UTC)
 * The equation $$L_P = a^2 \cdot L_S $$ used to be in the article, it was sourced by B&J with a footnote comment almost like yours. It is only reliable if the leakage inductances are small (less than a few percent) and proportional to the self-inductances of the the coils. Fortunately, that is the case much of the time.  It doesn't work very well at all for my bench transformer, because it has large and assymmetrical leakage losses (I designed it that way).  So I deleted the equation from the article. I don't know that there's any way to indirectly determine the physical turns ratio in a real transformer.  Transformers aren't spec'ed  with turns ratios, but with voltage ratios. These aren't open-circuit ratios - they're voltages at rated load and won't correspond to the turns ratio.  So, maybe it matters that we spell out some way of measuring $$a$$. The transformer model of the article postulates '$$a$$', as if we're told it.  But note that we haven't spelled out how to measure $$k$$ or $$\sigma$$, either. Sbalfour (talk) 00:12, 8 December 2017 (UTC)

Short-circuit inductance
This subsection has been incorporated from what was originally an article (really little more than a dictionary definition) of the same name. There was originally a partial derivation of short-circuit inductance in the Leakage inductance article as part of a section titled abstrusely, Inductive leakage factor and inductance. It had a bunch of abstract flux equations rather lacking context, and no discussion of what role short-circuit inductance plays in the measurement of a transformer. I was unable, based on that, to incorporate short-circuit inductance into the redrafted article at that time.

Short-circuit inductance is one of two complementary methodologies, which together are called the "terminal measurement model". In contrast is the "flux duality model", which is based on assessment of flux paths in a window of the core where the windings pass through. This model ignores terminal measurements. The two models naturally give somewhat differing results, on account of incorporating different kinds of losses, and measurement and estimation errors.

For derivation of Leakage inductance, there are only four relevant measurable quantities: open- and short-circuit primary and secondary inductances (or alternatively voltages and currents). Both sets of measurements are required, and both are part of standard laboratory procedure (I must state here that naively shunting across the terminals of a transformer and switching it on with the hope of measuring something, is extremely hazardous and likely to result in destruction of the transformer. This is not what is done in the laboratory.)

Justification for the validity of short-circuit measurements (that is, what do they actually mean) unfortunately relies on flux-path analysis, hence the original cryptic equations.

I do not think either laboratory procedures or magnetic circuit analysis (w/integral calculus) to be meaningful in the context of the article, since these confound the article's accessibility. Some more intuitive notion of what sounds like a dubious procedure is needed, which I have yet to formulate.

Sbalfour (talk) 17:05, 8 December 2017 (UTC)


 * I agree to move short-circuit inductance to this article. But generally, those who prefer theoretical writing don't like this being mentioned in the same article. That's because the short-circuit inductance is the measurement value, while the other parameters are theoretical amount. Therefore I thought that it would be better to make it independent and made another article. Originally "short-circuit inductance" was in this article. Please refer to the past description as well. (Leakage_inductance&oldid=401166149) Also, measurement of short-circuit inductance is simple, only measuring with LCR meter. It does not measure with actual specification power. Leakage inductance has been repeatedly rewritten largely from various standpoints. The most practical description is to describe only the results formula in the viewpoint of the electronics circuit designer. Then supplement the theory of the background according to the Electrical engineering textbook. Although the academic theoretical description method has a method described from the engineering viewpoint of the transformer and another from the magnetism viewpoint, and the previous description was biased to the magnetism viewpoint. Despite describing from either perspective the results are the same, it is very interesting matter for me. And why Europe like the "inductive coupling factor" and supports only positive values, whereas the textbook of engineering in other countries' transformer engineers like the "coupling coefficient" and also supports the positive and negative values. I understood the meaning at same time. --Discharger1016 (talk) 22:26, 8 December 2017 (UTC)
 * Open-circuit measurement, short-circuit measurement, direct current measurement of resistance, tank-circuit measurement, frequency response measurement and others are all complementary methodologies, some or all of which play a role in defining the specifications of a transformer including leakage inductance. Real world measurement is a complex and error-prone undertaking, which is not in most cases a topic of encyclopedia scholarly articles.  The article as it existed when I found it was nearly inaccessible.  In particular, algebra and higher math scares people off.  Leakage inductance is an ephemeral topic - it can't be seen or measured directly.  Bringing that into the understanding of a high school graduate without substantive scientific background (the average American) is the challenge.  Short-circuit measurement is worth a paragraph, because as you note, stick an LCR meter on the terminals, and you're done (however, if the little battery can't magnetize the core, your measurements will be meaningless). Encyclopedia Britannica doesn't have an article on Leakage inductance; it's part, if anywhere, of the article on transformers.   Maybe we should cop a clue - it's not possible actually to write a feature-length article on it, or short-circuit inductance.  That's why I brought them together. Sbalfour (talk) 19:47, 9 December 2017 (UTC)

Impedance
Regarding in Notes section: This kind of "resistance" in an AC circuit is a related quantity properly called inductive reactance, or more generally, impedance.

Impedance Z of a resistance in series with an inductive reactance, such as in a transformer winding, is equal to $$\sqrt{R^2+X^2}$$, where R is resistance and X in inductive reactance!Cblambert (talk) 02:12, 7 August 2018 (UTC)

Kay Hameyer's research while affiliated with RWTH Aachen University and other places
[https://researchgate.net/scientific-contributions/1005192_Kay_Hameyer Kay Hameyer, Dr.-Ing. Dr. h. c. dr hab. - Research contributions (367 Publications)]

IEEExplore search for Hameyer show 318 documents Source: https://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=hameyer

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.

Biography source: See link at http://info-optim.ro/hameyer.phpCblambert (talk) 15:34, 8 August 2018 (UTC)

Validity of sources II
Revision as of 19:35,27 November 2017 provided maintenance tags for Refined inductive leakage factor section including in terms of single source tag template.

Attributing a single source tag for Refined inductive leakage factor section, which relies on Hameyer 2001 source is not justified because the accompanying Refined inductive leakage factor derivation infobox shows how using multiple sources other than Hameyer 2001 source the two pats of the same section derivations arrive at the same end result. That is, the single-source attribution is in fact a multiple source section proving that the section as a whole stands together in a robust manner. The onus is on other editors to disprove the two streams of derivation of the section.

In DNA analysis this proof of ancient ancestor's origin based on living DNA testing is called triangulation. This is what happened with Refined inductive leakage factor section - Multiple-sourcing by triangulation.Cblambert (talk) 05:26, 8 August 2018 (UTC)


 * Its been awhile. Isn't the Hameyer 2001 a citation to course notes? Constant314 (talk) 05:36, 8 August 2018 (UTC)
 * Yes. I was busy on heavy-duty project which pre-occupied me. Hameyer 2001 was an extensive unpublished book-quality document targeting a student audience available online, which document has been cited in a number of papers based on a superficial online search. I have provided Hameyer's academic & research credentials in Talk section immediately above.Cblambert (talk) 05:55, 8 August 2018 (UTC)
 * If it is available on-line, that would be great. Can you provide the link.  It is not obvious. Constant314 (talk) 15:53, 8 August 2018 (UTC)
 * As mentioned in previous Talk section, it is unfortunately no longer available online. I will call Hameyer to see if I can get a copy of the pdf that used to be posted online. Be that is it may Hameyer 2001 is a valid source because there must be millions of sources on Wikipedia that were once available but though no longer readily available are still considered to be valid sources.Cblambert (talk) 20:16, 8 August 2018 (UTC)
 * In any case, Hameyer 2001 is shown to be valid by triangulation reflected in Refined inductive leakage factor derivation infobox.Cblambert (talk) 20:26, 8 August 2018 (UTC)
 * I have retrieved the Transformer article change in which the link to pdf http://materialy.itc.pw.edu.pl/zpnis/electric_machines_I/ForStudents/Script_EMIHanneberger.pdf isremoved from Biblio reference: 06:11, 17 March 2013 (diff | hist) . . (+270) . . m Transformer (Clean up Hameyer reference)Cblambert (talk) 21:25, 8 August 2018 (UTC)
 * It appears to be a dead link. Constant314 (talk) 22:12, 8 August 2018 (UTC)


 * Yes. It is dead link, which is why it was removed from the Transformer article change mentioned in my last comment here.


 * Also, making allowance for fact that WP articles must not contain original research, the write.com link at Data Triangulation: How the Triangulation of Data Strengthens Your Research provides interesting insights into data triangulation applicable for certain WP articles.Cblambert (talk) 22:24, 8 August 2018 (UTC)
 * See also Old url for Hameyer section above signed Cblambert (talk) 12:44, 17 January 2017 (UTC) - - - Sign up for this comment: Cblambert (talk) 23:00, 8 August 2018 (UTC)
 * Data triangulation seems to be about data credibility, not reference credibility, although there is no reason to argue that Hameyer himself is not an esteemed author.
 * The long and short of it is that Hameyer 2001 was significantly cited in connection with Transformer GA article as well as for Induction motor and Leakage inductance articles using old url, the pdf of which is no longer available online.Cblambert (talk) 23:25, 8 August 2018 (UTC)
 * If it is no longer available anywhere, then it doesn't serve the purpose of a reliable source as it it cannot be checked by other editors.Constant314 (talk) 01:52, 9 August 2018 (UTC)
 * Can you access Knowlton 1949? I can but the vast majority of WP editors can't? Yet the source is reliable. You can access Brenner & Javid but the vast majority of WP editors can't? Yet Brenner & Javid is a reliable source. You have not reacted to my triangulation argument about Refined inductive leakage factor derivation infobox, which by definition is valid. I any case I will phone Hameyer.Cblambert (talk) 03:25, 9 August 2018 (UTC)
 * Wikipedia says:
 * "The term "published" is most commonly associated with text materials, either in traditional printed format or online. However, audio, video, and multimedia materials that have been recorded then broadcast, distributed, or archived by a reputable party may also meet the necessary criteria to be considered reliable sources. Like text sources, media sources must be produced by a reliable third party and be properly cited. Additionally, an archived copy of the media must exist. It is convenient, but by no means necessary, for the archived copy to be accessible via the Internet." Hameyer 2001 is available from reliable 3rd party -- RWTH Aachen University.Cblambert (talk) 03:36, 9 August 2018 (UTC)
 * Until this issue is clarified to other WP editors' satisfaction, I have just a few minutes earlier:
 * Removed Refined leakage factor section
 * Retained Refined leakage factor infobox
 * Deleted Item f & Eq. 3.7e from Refined leakage factor infobox
 * The ball is in my court.Cblambert (talk) 04:03, 9 August 2018 (UTC)
 * Do you have access to Hameyer 2001?Constant314 (talk) 19:17, 9 August 2018 (UTC)
 * I unfortunately never downloaded a copy of the pdf. I have called Hameyer with no answer. Will try again but earlier in the day but he may be on vacation . . .Cblambert (talk) 23:04, 9 August 2018 (UTC)
 * 'Well! I'll be darned!' It turns out that the Hameyer 2001 (cum Hameyer 2004) document is actually available at the link https://web.archive.org/web/20130210003139/http://materialy.itc.pw.edu.pl/zpnis/electric_machines_I/ForStudents/Script_EMIHanneberger.pdf. I came across in reviewing the Hameyer 2001 reference cited in the Induction motor article. I have duly downloaded a copy of the errant pdf link and would encourage others to do the same before the link disappears again. This is really good news but what a do so and waste of time and keystrokes. I will soon be restoring the Refined leakage factor section & Retained Refined leakage factor sidebox to the way it was yesterday. There is surely a lesson to be learned by Wikipedia from this experience including in terms of a pressing need to create of a new Data/DNA/etc./etc. triangulation article. 'Hooray!'Cblambert (talk) 16:07, 10 August 2018 (UTC)
 * Note that the single source issue invoked so emphatically in this Leakage inductance article clearly seems to be a red herring if one considers Wikipedia's treatment in Articles with a single source. That is, there can be no question that the 'Magnetizing and leakage flux in a magnetic circuit' diagram and associated equations can somehow be derived from a source other then Hameyer 2004.Cblambert (talk) 18:20, 10 August 2018 (UTC)
 * We're still not out of woods. I notice that this new link dated 2004 shows improperly formatted math symbols throughout the document. I have sent an e-mail to IEM RWTH Aachen University's Petra Jonas-Astor & Kay Hameyer asking them to provide a properly formatted document, which is known to have existed in the 2001 version of the document.Cblambert (talk) 21:44, 10 August 2018 (UTC)
 * Yes, I noticed that also. Constant314 (talk) 23:14, 10 August 2018 (UTC)
 * In meantime, I have added Erickson & Maksimovic 2001 and Kim 1963 references in bibliography and associated footnoted citations to Refined inductive leakage factor section's Fig. 6 caption and 1st sentence to support Hayemer footnoted citationsCblambert (talk) 21:13, 12 August 2018 (UTC)
 * Latest changes to Refined inductive leakage factor section & Refined inductive leakage factor derivation sidebox make it clear there is enough triangulation of sources and equations to make it clear that Hameyer 2001 is not actually needed to prove Eq. 3.7a to Eq. 3.7e.Cblambert (talk) 00:22, 14 August 2018 (UTC)

I think equation 2.2 where several definitions of "a" are given, needs to be addressed.


 * $$a=N_P/N_S=v_P/v_S=i_S/i_P=\sqrt{L_P/L_S}$$ -- (Eq. 2.2)

These are only approximately equal and it is the last expression that must be used to make the math work out exactly. The other expressions are exact, I think, only when the leakage inductance is zero.


 * $$a=\sqrt{L_P/L_S} \simeq N_P/N_S \simeq v_P/v_S \simeq i_S/i_P$$

The expression for turns ratio under the transformer in fig. 4 sort of implies this, but suggests turns ratio determines inductance ratio whereas it is actually inductance ratio that determines $$ N_P/N_S $$. : If you want to keep using :$$a= N_P/N_S $$ Then explain that $$ N_P/N_S $$ is not the literal turns ratio, but is the actually an adjusted turn ratio required so that $$a=\sqrt{L_P/L_S} $$.

In fact, I would replace $$ N_P/N_S $$ everywhere with $$  a $$ except equation 2.2. That would include fig.3 and 4. The expression for turns ratio under the transformer in fig. 4 should be replaced with $$N_P/N_S \overset{\underset{\mathrm{def}}{}}{=} \sqrt{L_P/L_S}$$ which, if I got the right symbol says that the turns ratio is defined as the square root of the inductance ratio. I can redraw the figures if needed and address equation 2.2 if you would prefer.Constant314 (talk) 12:23, 14 August 2018 (UTC)
 * I have changed Eq. 2.2 to read:
 * $$a=\sqrt{L_P/L_S} \approx N_P/N_S\approx v_P/v_S \approx i_S/i_P=$$ -- (Eq. 2.2).
 * The footnoted citation for Eq. 2.2 continues to read:
 * Quoting from Brenner & Javid, I have further added footnoted citation for Fig. 4 that reads:
 * "Fig. 18-18 In this equivalent circuit of a (nonideal) transformer the elements are physically realizable and the isolationg property of the transformer has been retained."
 * There is in my view no need to change nonideal equivalent circuit in Fig. 4 with the addition of this quoted footnoted citation exactly as taken from Brennan & Javid's Fig. 18-18. Cblambert (talk) 15:59, 14 August 2018 (UTC)
 * Come to think of it, strictly speaking, for Wikipeida audience consumption, a turn ratio should be called the 'ratio of the primary to secondary turns'. Erickson & Maksimovic 2001 may do well to refer on p. 33 to effective turns ratio as $$n_e=\sqrt{L_P/L_S}$$. Trouble is Erickson & Maksimovic 2001 seem to be the only source using this turns ratio distinction.Cblambert (talk) 17:01, 14 August 2018 (UTC)
 * Also, as mentioned in the next section below: 'When the adjusted turns ratio N is equal to the ratio of the system-rated voltages, the ratio is called nominal, and the transformer is omitted from the single-line diagram in a per-unit system. When the adjusted turns ratio N is not equal to the ratio of the system rated voltages, it is said the transformer has an off-nominal turns ratio.' This is a not insignificant turns ratio distinction.Cblambert (talk) 17:49, 14 August 2018 (UTC)
 * I propose Eq. 2.2 to read:
 * $$a=\sqrt{L_P/L_S} = N_P/N_S\approx v_P/v_S \approx i_S/i_P=$$ -- (Eq. 2.2).
 * That's a good step. I think that there needs to be note regarding figure 3 that the circuit is a correct model no matter what value is used for a, so long as the transformer in the circuit is an ideal transformer.  And there needs to be a note that fig 4 is derived from fig 3 by assuming $$a=\sqrt{L_P/L_S}$$. Constant314 (talk) 17:26, 16 August 2018 (UTC)
 * Fig.3 & Fig.4 are exactly the same as Brenner & Javid show it in their figures. There is a difference between 'equivalent for all values of the "number" a' and 'no matter what value is used for a '. The note that Fig. 4 is derived from Fig. 3 by assuming $$a=\sqrt{L_P/L_S}$$ IS shown from Eq. 2.12:
 * $$a=\sqrt{L_P/L_S}$$ -- (Eq. 2.12).
 * It remains that in Fig. 3 $$a=N_P/N_S$$ and in Fig. 4 $$\sqrt{L_P/L_S}=N_P/N_S$$, $$a=\sqrt{L_P/L_S}$$ being simply used to derive Fig. 4 in the same terms as are used in Fig. 1. It also remains that Brenner & Javid say "The most popular value chosen for the "number" a is the turns ratio which the actual transformer has."  Cblambert (talk) 05:28, 17 August 2018 (UTC)
 * It should be clear from these last few comments that the equivalent circuits in Fig. 3, Fig. 4 & Fig. 5 are all equivalent to each other, with:
 * the ideal transformer in Fig. 3 being captioned $$a=N_P/N_S$$ because the other elements of the circuit are expressed in terms of $$a$$.
 * the ideal transformer in Fig. 4 being captioned $$\sqrt{L_P/L_S}=N_P/N_S$$ because the other elements of the circuit are expressed in the same terms as are used in Fig. 1.
 * no ideal transformer in Fig. 5 is order to depict the circuit in its simplest possible terms.
 * But the ideal transformer in Fig. 3 & Fig. 4 could just as well be captioned $$a=N_P/N_S=\sqrt{L_P/L_S}$$, this captioning being implicitly understood in Fig. 5 as well.Cblambert (talk) 15:24, 23 August 2018 (UTC)
 * But the ideal transformer in Fig. 3 & Fig. 4 could just as well be captioned $$a=N_P/N_S=\sqrt{L_P/L_S}$$, this captioning being implicitly understood in Fig. 5 as well.Cblambert (talk) 15:24, 23 August 2018 (UTC)

Turns ratio II
AESO Tranaformer Modelling Guide makes 37 mentions the term 'turns ratio', the second mention of which says "Therefore, a transformer is typically described by its rated voltage $$V_H$$ and $$V_X$$, which gives both the limits and the turns ratio."

The above section The_definition of "a" says "I don't know that there's any way to indirectly determine the physical turns ratio in a real transformer. Transformers aren't spec'ed with turns ratios, but with voltage ratios. These aren't open-circuit ratios - they're voltages at rated load and won't correspond to the turns ratio.

This is partly misleading, partly wrong.

Partly misleading because the term 'turns ratio' is often used.

Partly wrong because rated voltage is often based on no-load conditions whereby one starts with rated voltage (no-load voltage). This is associated with the voltage regulation of constant-potential transformer which Knowlton 1949 says "is the change in secondary voltage, expressed in per cent of rated secondary voltage, which occurs when the rated kVA output at a specified power factor is reduced to zero, with the primary impressed terminal voltage maintained constant."

Some even say "Rated voltage, secondary: The voltage which is generated for the transformer’s secondary line terminals with or without load (depending on the standard)."Cblambert (talk) 01:05, 9 August 2018 (UTC)
 * Knowlton further says:
 * The ratio of a transformer is the turn ratio of the tranformer, unless otherwise specified.
 * The voltage ratio of a transformer is the ratio of the rms primary terminal voltage to the rms secondary terminal voltage, under specified conditions of load.
 * The turn ratio of a trasformer is thr ratio of the number of turns of high-voltage winding to that in the low-voltage winding.
 * Note: in the case of a constant-potential transformer having taps for changing its voltage ratio, the turn ratio is based on the number of turns corresponding to the normal rated voltage of the respective windings, unless otherwise specified.
 * The true ratio of a current or a potential transformer is the ratio of rms primary current or voltage as the case may be, to the secondary current or voltage under specified conditions.
 * The marked ratio of a current or a potential transformer is the ratio of the primary current or voltage as the case may be, tothe secondary current or voltage, as given on the rating plate.


 * This is complicated by the fact that terminology is crucial for modelling of transformers for power system analysis purposes.Cblambert (talk) 01:42, 9 August 2018 (UTC)
 * AESO Transformer Modelling Guide says
 * "When the adjusted turns ratio N is equal to the ratio of the system-rated voltages, the ratio is called nominal, and the transformer is omitted from the single-line diagram in a per-unit system. When the adjusted turns ratio N is not equal to the ratio of the system rated voltages, it is said the transformer has an off-nominal turns ratio. It should be noted that the off-nominal turns ratio is a definition used for modelling, and it is not to be confused with the physical transformer nominal ratio that may appear on the nameplate of the transformer. The off-nominal turns ratio can be a real number or a complex number. If is is a complex numer, the transformer is called a phase-shifting transformer. In such a case, the voltages on the two sides of the transformer differ in phase as well as in magnitude."Cblambert (talk) 04:20, 9 August 2018 (UTC)
 * All to say that as Saarbafi says "$$E_H/E_X=N_H/N_X=I_X/I_H$$ This is the basic equation for all types of transformers" (Empahis added). It is utterly useless to discuss Transformer and Leakage inductance articles unless one is 100% clear about the symbiotic dependence of turns ratio with winding voltages and currents.Cblambert (talk) 01:00, 10 August 2018 (UTC)
 * According to Megger's Ohlen 2010,
 * §2.2 Definitions, §2.2.1 Turn ratio and voltage ratio: The turn ratio of a transformer is the ratio of the number of turns in a higher voltage winding to that in a lower voltage winding. The voltage ratio of a transformer is the ratio of the rms terminal voltage of a higher voltage winding to the rms terminal voltage of a lower voltage winding under specified conditions of load. For all practical purposes, when the transformer is on open circuit, its voltage and turns ratios may be considered equal.
 * §3.4 Tolerances for ratio, §3.4.2 IEEE Std 62-1995 (field testing): The turn ratio tolerance should be within 0.5% of the nameplate specification for all windings.
 * (Italics emphasis added) Cblambert (talk) 20:59, 22 August 2018 (UTC)


 * References
 * Cblambert (talk) 01:00, 10 August 2018 (UTC)
 * Cblambert (talk) 20:59, 22 August 2018 (UTC)

Redundant symbols
We have $$L_{oc}^{pri}=L_P$$ and $$L_{oc}^{sec}=L_S$$ and since $$L_{oc}^{pri}$$ and $$L_{oc}^{sec}$$ do not appear in any figures, it looks like we could eliminate $$L_{oc}^{pri}$$ and $$L_{oc}^{sec}$$ and have two fewer symbols. It might improve the readability. Constant314 (talk) 22:13, 22 August 2018 (UTC)
 * I don't agree. $$L_{oc}^{pri}$$ and $$L_{oc}^{sec}$$ are measurements that are equal to the respective inductances. This is significant. Maybe it should be expla'ned but deleting terms to improve readability is not a good enough reason.Cblambert (talk) 15:09, 23 August 2018 (UTC)
 * I propose simplifying by combining to $$L_{oc}^{pri}=L_P$$ is primary self-inductance & $$L_{oc}^{sec}=L_S$$ is secondary self-inductance as now shown in article.Cblambert (talk)
 * A factor at play in Fig. 1 is that the image is used in WP article of other languages.

The use of the term "Magnetizing inductance" is misleading
In figure 1 the term $$L_M$$ is defined as "magnetizing inductance", I think a better term would be "coupling inductance". The present definition suggests that the magnetizing inductance changes if the coupling coefficient $$k$$ changes. In real transformers, when voltage is applied to the primary and the secondary is left open circuit, there is a current that flows from the voltage source through the transformer primary. This current is commonly called the magnetizing current. The magnitude of the magnetizing current depends upon the voltage applied and the Primary inductance, $$L_M+L_P^\sigma$$, not just $$L_M$$. If the coupling coefficient went to 0 (remove the secondary) it implies the inductance responsible for magnetizing also goes to zero and so the magnetizing current should be infinite for any applied voltage. Clearly this is not the case. In fact, the coupling coefficient has no impact on magnetizing current.

Instead of calling $$L_M$$ "magnetizing inductance", if it were called "coupling inductance", I think this would avoid the potential confusion. I think it is also more descriptive. Leakage inductance is the fraction of winding inductance that does not couple to other windings, coupling inductance is the fraction that does, and the coupling coefficient defines the portion.

This is not to be confused with mutual inductance. The two terms are related but not equal.

If we replaced "magnetizing" with "coupling" everywhere in the article I think it will make the article more clear.

--IflyHG (talk) 18:49, 4 November 2022 (UTC)


 * I double checked. It looks correct.  By "coupling inductance" did you mean "mutual inductance" which has the symbol $$M$$?
 * I think that you might be referring to equation 1.1c which I have copied here
 * $$L_M=L_P\cdot{k}$$ -- (Eq. 1.1c)
 * You would expect the magnetizing inductance to be constant and not a function of k. That is more or less correct.  What the equation tells you is that decrease k, you must increase $$L_p$$, the inductance measured at the terminals. Constant314 (talk) 02:27, 5 November 2022 (UTC)