Rogowski Factor

 

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010

 

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

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 interleavi interleaving ng of windi windings. ngs. In the paper, the meaning meaning of included included 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. INTRODU INTRODUCTION CTION

Several aspects of the operation of power-ele Several power-electroni ctronicc circuits circuits with transformers transformers are significan significantly tly influenced by the leakage inductance of the respective transformer. For instance, it concerns power transfer trans fer capability capability of the circuit and pow p ower er semiconductor semiconductor stress stress parameters. parameters. The tendency tendency to higher switching frequencies of power-electronic apparatuses results in an increasing relevance of  low transformer transformer leakage leakage inductance inductance values. values. Mostly, Mostly, the so-called so-called interlea interleaving ving of windings 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. Using an appropriate approximation method it is possible to determine the leakage inductance of a transformer already in the design phase on the basis 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 determination which which has been established established by Rogowski Rogowski [1]. It is based on the consideratio consideration n 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   LL  — leakage inductance,   µ0  — absolute permeability,  N  — number of turns of the winding to which the leakage 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),   kσ  — 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 most arrangements) is omitted and the formula describing the relative leakage conductance is included in the main formula. Often, constant factors like absolute permeability and numerical values are merged into one coefficient, so that the physical background of the formula is not recognizable anymore, any more, e.g., in [2]. In some cases different different formulas formulas are given for a transf transformer ormer design without 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 of interface interfacess (insulating interspace interspaces) s) between the single sub-windings sub-windings of an arrangemen arran gementt of windings [6, 7]. Furthermore urthermore,, the aim to present present a univ universal ersal formula formula which is valid for transformer versions with concentric and also with pie windings arrangements often causes some confusion. It also results from the diversity of terms which are used for the dimensions of 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 INDUCT INDUCTANCE ANCE CALCULA CALCULATION TION FOR MAGNETICALL MAGNETICALLY Y SYMMETRIC ARRANGEMENTS

To enable a correct application of a formula for leakage inductance calculation both for transformers with concentri concentricc and transf transform ormers ers with pie windin windings, gs, the attribu attribution tion of the ch chose osen n sym symbols bols for geometry parameters to the respective dimensions of the arrangement of windings has to be made

 

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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 · 2 · nif   · X  par-lf 

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: Illustration of attribution of the dimensions of windings or sub-windings used in formula ( 2) and their orientation compared to the orientation of leakage flux Φ l   within 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. Furthermore, the number of turns of outer sub-windings is half of the 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-winding sub-w indingss 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 LEAKAGE INDUCTAN INDUCTANCE CE CALCULATION CALCULATION

In the ma jority jority 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. Howeve How ever, r, also magnetically magnetically asymmetric asymmetric designs of windings windings are used in praxis. For instance 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 parall parallel el connection conne of the primary sub-windings. sub-wind ings. Also transfor mer of versions, versio ns, whose sub-windings primary primary subwindings havection identical numbers of turns independent of transformer the location the respective in inner or outer position, represent a typical example of magnetically asymmetric transformers. In formula (2 (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 the sum value of the relevant dimension of the single sub-windings and the sum of thicknesses of 

 

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010

 

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all insulating interspaces interspaces between between the sub-winding sub-windingss 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 method to establish the required combination formulas. This approach shall be illustrated by means of leakage inductance determination for different arrangemen arran gements ts of windings windings of an experimental experimental transformer. transformer. The basic elements elements of the 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 12 12 ,  W 1 13 3 ) realized using 8 coils Total number of turns:   N 1  = 88 Different arrangements of sub-windings and distribution of turns considered Series connection of the sub-windings and the coils forming the sub-windings Secondary winding: Consisting of 2 sub-windings (W 2211,   W 22 22 ), realized using 8 coils Each sub-winding consisting of 4 coils connected in parallel Sub-windings connected in series Total number of turns:   N 2  = 22 (W 2211  :  N 2211  = 11;   W 2222  :  N 2222  = 11) Thic Thicknes kness of insulating interspace interspaces s between betw een the sub-windin sub-w indings: gs: 0.1Due mmto the height of the outer In Fig. 2 s(right), the location of the sub-windings is illustrated. primary sub-windings (W 1111   and  W 1133) which is half of the height of the inner primary sub-winding (W 1122) the displayed displayed version represents represents a magnetically magnetically symmetric symmetric arrang arrangemen ement. t. This classification classification 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 described in [8] we 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 side and at the secondary side as well:

LL total total   =



  N 12 · 2 N ref 

  N 1133 ·N 2211   N 12 N 11 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 2222 12 ·N 22 22 11 ·N 22 22 · LL13/22 · LL12/22 +   N N 11313 ··N  · LL11/22 +   N N 12 + N N 11 2 1 ·N 2 1 ·N 2   N 1122 ·N 1133   N 11 N 11 11 ·N 13 13 11 ·N 12 12 N 12 N 12 N 12 LL12/13 LL11/13 LL11/12

 −

·

·

 −

·

 −

  −

  N 2211 ·N 2222 N 22

· LL21/22

(3)



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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N ref  ref    is the number of turns to which the leakage inductance values of the couples of subwindings refer (advantageously   N ref  leakage age inductance inductance values values ref   = 1 should be chosen). The single leak

of the couples of sub-windings have been calculated using the basic formula of the method of  Rogowski (1 (1) omitting the Rogowski factor. Due to the number of primary turns in the coefficient preceding the term in parentheses, the resulting total leakage inductance refers to the primary side of the transf transform ormer. er. As it can be seen seen compar comparing ing the formula formulass (2) and (3 (3), in contrast to leakage inductance calculation for magnetically symmetric transformers it is a more complex procedure in the case of magneti magnetical cally ly asymme asymmetri tricc versi versions ons.. Howe Howeve ver, r, today today the calculati calculation 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 enting ting the algorit algorithms hms into a PC [10]. [10]. The values values of total total leak leakage inductanc inductancee ha have ve been calculated using formula (3 (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 measurement measurementss 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 of the secondary secondary side, measuring 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 inductance values. Therefore, the measured values are 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 1122 ,  W 13 13   (N 1 11 1 –N 12 12 –N 1 13 3) 22–4 22–44– 4–22 22 33–2 33–22– 2–33 33 11–6 11–66– 6–11 11 22–2 22–22– 2–44 44 67,8 67,888 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 (2 (2). The leakage leakage inductance inductance values values calculated calculated by means of formula formula (3 (3) show an average deviation deviat ion from the measured measured values values of about 4.5%. In the case of thicker thicker insulating insulating interspace interspacess between betw een the sub-winding sub-windingss (0.4 mm) the deviation deviation is even even slightly lower. lower. 4. CONCLUSI CONCLUSIONS ONS

The calculation of leakage inductance of transformers with interleaving of windings has been discussed. cusse d. The origin of certain certain terms in approximation approximation formulas formulas for magnetically magnetically symmetric symmetric transformers forme rs and the attribution of these terms to the geometry of concentric concentric and pie windings arrangearrangements men ts has been clarified. clarified. An approa approach ch which enables enables leakage leakage induct inductanc ancee predic prediction tion 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. 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.

 

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010

 

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8. Petrov Petrov,, G. N., “Allgemeine “Allgemeine Methode der Berechnung Berechnung der Streuung Streuung von Transformator ransformatoren,” en,” Elektrotechnik und Maschinenbau , Vol. 51, No. 25, 345–350, 1933. 9. Data Sheet, “PM 114/93 Core and accessories,” EPCOS, 2006. 10. Doebbelin, R., C. Teichert, M. Benecke, and A. Lindemann, “Computerized calculation of  leakage inductance values of transformers,”  PIERS Online , Vol. 5, No. 8, 721–726, 2009.