Leakage Inductance Determination For Transformers With Interleaving of Windings

 

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Leakage Inductance Determination for Transformers with Interleaving of Windings R. Doebbelin Doebbelin and A. Lindema Lindemann nn

Institute of Electric Power Systems, Otto-von-Guericke-University Magdeburg, Germany

Abstract—   A

diversity of formulas is given in literature for leakage inductance calculation of transformers transformers with interlea interleaving ving of windings. windings. In the paper, the meaning of inclu included ded terms and details of the application for concentric and pie windings transformers shall be discussed. Considering the limitation of common formulas to magnetically symmetric transformers an alternative approach shall be demonstrated, which enables leakage inductance prediction also in the case of  transformers with magnetically asymmetric arrangements of windings.

1. INTRODUCTION INTRODUCTION

Several aspects of the operation of power-ele Several power-electron ctronic ic circuits circuits with transformers transformers are significant significantly ly influenced by the leakage inductance of the respective transformer. For instance, it concerns power transf tra nsfer er capabi capabilit lity y of the cir circui cuitt and power power semico semicondu nducto ctorr stress stress parameter parameters. s. The tenden tendency cy to higher switching frequencies of power-electronic apparatuses results in an increasing relevance of  low transformer leakage inductance values. Mostly, the so-called interleaving of windings (i.e., the fragmentation of the primary and the secondary windings into a certain number of sub-windings and alternate interleaving of primary and secondary sub-windings) is used to meet this requirement. an appropriate approximation method it isbasis possible to determine the leakage inductance of aUsing transformer already in the design phase on the of predicted geometry parameters. Thus, circuit simulation can be used to shorten the development process of power-electronic apparatuses. Formulas which are given in literature generally rely on an approximation method for leakage inductance determination which has been established by Rogowski [1]. It is based on the consideration of  the energy of the leakage magnetic field and applies to a transformer design with windings arranged on the same leg. The basic formula of this method is LL  =  µ 0 · N 2 · lm  · λ · kσ

 

(1)

with  L L  — leakage inductance,  µ 0  — absolute permeability,  N  — number of turns of the winding to which which the leakage leakage inductance inductance refers,   lm   — mean length per turn for whole arrangement of  windings,  λ  — relative leakage conductance (depending on geometry parameters and on the degree of interleaving of windings),  k σ  — Rogowski factor (also depending on geometry parameters). The formulas which can be found in contemporary technical literature mostly represent simplified variants of the mentioned basic formula. Usually the Rogowski factor (whose value is nearly 1 in mostinarrangements) is omitted the formula the permeability relative leakage is included the main formula. Often,and constant factorsdescribing like absolute andconductance numerical values are merged into one coefficient, so that the physical background of the formula is not recognizable anymore, anym ore, e.g., in [2]. In some cases different different formulas are given for a transf transformer ormer design design without interleaving of windings and for transformer arrangements with interleaving of windings, e.g., in [3]. The degree of interleaving is considered in literature by the number of the couples of sub-windings [4, 5] or by the number number of interface interfacess (insulating (insulating interspaces interspaces)) between between the single sub-winding sub-windingss of an arrangemen arran gementt of windings windings [6, 7]. Furthermore urthermore,, the aim to present present a universal universal formula formula which is valid for transformer versions with concentric and also with pie windings arrangements often causes some confusion. confus ion. It also results results from the diversit diversity y of terms which which are used for the dimensions dimensions of windings windings (e.g., breadth, width, traverse, build, height, length) comparing the formulas and their descriptions which are given in different books and papers. 2. LEAKAGE LEAKAGE INDUCTANCE CALCU CALCULA LATION TION FOR MAG MAGNETICAL NETICALL LY SYMME SYMMETRIC TRIC ARRANGEMENTS

To enable a correct application of a formula for leakage inductance calculation both for transformers with wit h concen concentri tricc and transf transform ormers ers with with pie windin windings, gs, the attribu attribution tion of the chose chosen n symbols symbols for

 

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geometry parameters to the respective dimensions of the arrangement of windings has to be made quite clear. Therefore, based on variants which are valid for transformer versions without and with interleaving of windings, the following representation of the formula is proposed: lm · LL  =  µ 0 · N  · 2 nif   · X  par-lf  2

 

Σ

X  perp-lf 

3



  + Σ δ 

 

(2)

with ΣX  perp-lf   — sum of the dimensions of all sub-windings which are orientated perpendicular to the leakage flux,   X  par-lf    — dimension of the sub-windings which is orientated parallel to the leakage flux, Σδ  — sum of the thicknesses of all insulating interspaces between the sub-windings, nif  — number of insulating interspaces between the sub-windings (no consideration of Rogowski factor). Thus, in this formula the respective dimensions of the sub-windings are identified based on a comparison between their orientation and the orientation of leakage flux within the core window. The meaning of the geometry parameter symbols and the path of leakage flux Φ l  within the core window are illustrated in principle in Fig. 1 assuming the leakage flux to be concentrated in the insulating interspaces between the windings or sub-windings.

Figure 1: Illustratio Figure Illustration n of attr attributio ibution n of the dimensions dimensions of windings or sub-winding sub-windingss used in formula (2) and their orientation compared to the orientation of leakage flux Φ l   within the core window (left: concentr concentric ic windings, right: pie windings, only left hemisphere of arrangement displayed).

The proposed representation and also the variants of the formula which are given in literature enable a relatively uncomplicated approximate calculation of the leakage inductance of a transformer even in the case of interleaving of windings. However, it has to be stated that the scope is limited to magnetically symmetric arrangements of windings. These transformer versions are characterized by an odd number of primary sub-windings and an even number of secondary sub-windings or vice versa. ver sa. Furthermore urthermore,, the number number of turns of outer sub-windings sub-windings is half of the number number of turns of the inner sub-windings which belong to the same group of sub-windings (inner primary sub-windings if  the outer sub-windings belong to the primary winding; inner secondary sub-windings if the outer sub-windin sub-w indings gs belong to the secondary secondary winding). winding). In connection connection with this, it can be assumed that the outer sub-windings show a dimension perpendicular to the leakage flux which is half of the same dimension of an inner sub-winding which belongs to the same group of sub-windings. 3. UNIVERSAL UNIVERSAL LEAKAGE LEAKAGE INDUCTANCE CALCU CALCULA LATION TION

In the majorit ma jority y of applications applications magnetically magnetically symmetric symmetric transformer transformer versions versions are used because they show the lowest leakage inductance values considering a certain degree of interleaving of windings. Howe Ho weve ver, r, also also magneti magnetical cally ly asymme asymmetri tricc designs designs of windin windings gs are used in praxis praxis.. For instan instance ce in transformers for capacitor discharge welding machines a magnetically asymmetric design of windings is often applied to enable a variation of the turns ratio by means of alteration between series and parallel parall el connection of the primary sub-windings. sub-windings. Also transformer transformer versions, versions, whose primary primary subwindings have identical numbers of turns independent of the location of the respective sub-windings in inner or outer position, represent a typical example of magnetically asymmetric transformers. In formula (2) and its variants which are given in literature it is not possible to consider the real design and arrangement of sub-windings of a magnetically asymmetric transformer because only

 

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the sum value of the relevant dimension of the single sub-windings and the sum of thicknesses of  all insulating interspac interspaces es between the sub-windings sub-windings can be taken taken into account. account. Therefore, Therefore, in this case the mentioned formulas will deliver a result less than the real leakage inductance value which can only represent a rough approximation. Prediction of leakage inductance with a higher degree of accuracy even in the case of magnetically asymmetric transformers is possible based on the calculation of the individual leakage inductance values of all existing couples of sub-windings of the transformer arrangement which have finally to be merged together to the total leakage inductance of the transformer. In [8], Petrov described an abstract abstr act method to establish establish the required required combination combination formulas. formulas. This approach shall gs be of illustrated by means of leakage determination different arran arrangemen gements ts of windings windin an experimental experimen tal transformer. transform er. inductance The basic elements element s of thefor windings of  the transformer transformer are single pie-shaped pie-shaped coils (number (number of turns: turns: 11). Furthermore urthermore the transformer transformer (displayed in Fig. 2 (left)) is characterized by the following features: Core: PM 114/93 [9], material  N 2277 Primary winding: Consisting of 3 sub-windings (W 1111,   W 1122,  W 1133) realized using 8 coils Total number of turns:   N 1  = 88 Different Differe nt arrangemen arrangements ts of sub-windings sub-windings and distribution distribution of turns considered considered Series connection of the sub-windings and the coils forming the sub-windings Secondary winding: Consisting of 2 sub-windings (W 2211,   W 2222), realized using 8 coils Each sub-winding consisting of 4 coils connected in parallel Sub-windings connected in series 21  = 11;  W 2 22 2  :  N 2 22 2  = 11) Total number of turns:   N 2  = 22 (W 2211  :  N 21 Thickness Thic kness of insulating insulating interspaces interspaces between between the sub-winding sub-windings: s: 0.1 mm In Fig. 2 (right), the location of the sub-windings is illustrated. Due to the height of the outer primary sub-windings (W 1111   and  W 1133) which is half of the height of the inner primary sub-winding displayed ed version version represents represents a magnetically magnetically symmetric arrangement. arrangement. This classification classification (W 1122) the display is founded by the curve of the magnetomotive force (mmf) between the center leg and the outer leg of the core which shows the same maximum values in positive and negative range. According to the existence of 3 primary sub-windings and 2 secondary sub-windings in the considered consid ered transformer transformer we call it a 3-2 version. version. Applying Applying the method described in [8] we derived derived the following formula which enables the calculation of the total leakage inductance of a transformer with a 3-2 arrangement of its windings whose sub-windings are connected in series at the primary

x

W13

W22

W12

W21

W11

 

mmf 

Figure 2: Left: Experimental transformer. Right: Illustration of the location of sub-windings within the core window (only right hemisphere of the transformer displayed) and curve of the magnetomotive force (mmf) across the core window.

 

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side and at the secondary side as well:

LL total total  =

  N 12 2 N ref 

  N 13   N 12 N 11 13 ·N 21 21 12 ·N 21 21 11 ·N 21 21 N 1 ·N 2 · LL11/21 + N 1 ·N 2 · LL12/21 + N 1 ·N 2 · LL13/21   N 13   N 12 N 11 22 13 ·N 22 22 12 ·N 22 22 11 ·N 22 N 1 ·N 2 · LL11/22 + N 1 ·N 2 · LL12/22 + N 1 ·N 2 · LL13/22 21 ·N 22 22 12 ·N 13 13 11 ·N 13 13 11 ·N 12 12 · LL12/13 −   N 21 · LL11/13 −   N 12 · LL11/12 −   N 11 − N 11 N 22 N 12 N 12 N 12

  · +

· LL21/22

  (3)

N rref  ef    is the number of turns to which the leakage inductance values of the couples of subleakage inductance inductance values values windings refer (advantageously   N rref  ef   = 1 should be chosen). The single leakage

of the couples of sub-windings have been calculated using the basic formula of the method of  Rogowski (1) omitting the Rogowski factor. Due to the number of primary turns in the coefficient preceding prece ding the term in parenthese parentheses, s, the resulting total leak leakage age induc inductance tance refers to the primary side of the transfor transformer mer.. As it can be seen seen compar comparing ing the formu formulas las (2) and (3), in contrast contrast to leak leakage inductance calculation for magnetically symmetric transformers it is a more complex procedure in the case of magneti magnetical cally ly asymme asymmetri tricc ve versi rsions ons.. Howe Howeve ver, r, today today the calcul calculati ation on of the single leakage inductance values of the existing couples of sub-windings of the transformer arrangement and merging them together using the respective combination formula can be realized conveniently implem imp lemen entin tingg the algorithm algorithmss in into to a PC [10]. [10]. The values values of total total leak leakage ind induct uctanc ancee have have been calculated using formula (3) for the magnetically symmetric version and for different magnetically asymmetric arrangements of the windings of the experimental transformer. To enable an evaluation of the accuracy accuracy of leakage leakage inductance inductance prediction, prediction, also measuremen measurements ts have been carried out. So, concerning the realized transformer versions the respective short-circuit inductance values have been determined using an RLC meter (inductance measurement at the primary side during shortcircuiting circu iting the output terminals terminals of the secondary secondary side, measuring frequency frequency 50 Hz). Due to the given transformer design showing a closed core (no air gap intended) and hence a very high magnetizing inductance, the measured short-circuit inductance values will nearly coincide with the respective leakage leak age inductance inductance values. values. Therefore, Therefore, the measured measured values are considered considered to be the sought leakage inductance values. The results are presented in Table 1. Table 1: Comparison of calculated and measured leakage inductance values.

Calc Calcul ulat atio ion n Meas Me asur urem emen entt

Arrangement of the primary sub-windings W 1111 ,   W 12 12 ,  W 13 13   (N 11 11 –N 12 12 –N 1 13 3) 22 22–4 –44– 4–22 22 33 33–2 –22– 2–33 33 11 11–6 –66– 6–11 11 22 22–2 –22– 2–44 44 67 67,8 ,888 µ H 117,69 µ H 117,69 µ H 167,5 µ H 74,2 74,2 µ H 116,0 µ H 121,4 µ H 159,4 µ H

In the case of the magnetically symmetric arrangement of the primary sub-windings with the numbers of turns 22–44–22 the calculated result is identical to the value which is delivered by formula form ula (2). The leakage leakage inductance values values calculated calculated by means of formula (3) show show an average average dev deviati iation onthe from frosub-winding m theindings measured measu val uesthe of about 4.5%. Innthe case of thick thi.cker er ins insula ulatin tingg in inter terspa spaces ces between betw een sub-w s red (0.4values mm) deviation deviation is even eve slightly sligh tly lower. lower 4. CONCLUSIONS CONCLUSIONS

The calculation of leakage inductance of transformers with interleaving of windings has been discussed. cusse d. The origin of certain certain terms in approximati approximation on formulas for magnetically magnetically symmetric transtransformers and the attribution of these terms to the geometry of concentric and pie windings arrangements men ts has been clarified. clarified. An approac approach h which which enables enables leak leakage ind induct uctanc ancee predic predictio tion n also also in the case of magnetically asymmetric arrangements of windings has been demonstrated yielding good coincidence with results of measurement in the considered example. REFERENCES

1. Rogowski, W., “Ueber das Streufeld und den Streuinduktionskoeffizienten eines Transformators mit Scheibenwicklung und geteilten Endspulen,” Dissertation, Mitteilung ueber Forschungsarbeiten auf dem Gebiet des Ingenieurwesens, VdI, 1909. 2. McLyman, W. T.,  Transformer and Inductor Design Handbook , Marcel Dekker Inc., New York, 1988. 3. Flanagan, W. M.,   Handbook of Transformer Design and Applications , McGraw-Hill, 1992.

 

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4. Mecke, H., “Transformatoren f¨ur ur das Lichtbogenschweissen,” Dissertation B, TH Magdeburg, 1978. 5. Schuelting, L., “Optimierte Auslegung induktiver Bauelemente f¨ur ur den Mittelfrequenzbereich,” Dissertation, RWTH Aachen, 1993. 6. Snelling, E. C.,  Soft Ferrites , Butterworth & Co., 1988. 7. Tarter, R. E.,  Solid-state Power Conversion Handbook , Wiley-Interscience, 1993. 8. Petrov, Petrov, G. N., “Allgemeine “Allgemeine Methode der Berechnung Berechnung der Streuung von Transfor Transformatore matoren,” n,” Elektrotechnik und Maschinenbau , Vol. 51, No. 25, 345–350, 1933. 9. Data Sheet, “PM 114/93 Core and accessories,” EPCOS, 2006. 10. leakage Doebbelin Doebbelin, , R., C. T eicher eichert, t, transformers,” M. Benec Benecke ke,, and A. Lindem Lin demann ann, , “Compu “Co mputer ized d calcul calculatio ation n of  inductance values of  PIERS Online  , Vol. 5, No. 8,terize 721–726, 2009.