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Computerized Calculation of Leakage Inductance Values ofTransformers

R. Doebbelin, C. Teichert, M. Benecke, and A. LindemannInstitute of Electric Power Systems, Otto-von-Guericke-University Magdeburg, Germany

Abstract— A low-effort prediction of leakage inductance values of transformers is requiredin a lot of transformer applications in various fields of electrical engineering. The article de-scribes given approximation methods enabling leakage inductance calculation which have beenimplemented in Matlab R© based computer programs. Application range and accuracy of com-puterized leakage inductance calculation are illustrated. Further possibilities of computerizedtransformer investigation are considered.

1. INTRODUCTION

Power transfer and its limitation caused by internal losses or voltage drops are important aspects inall applications of power transformers. This is true in the case of mains-frequency operated trans-formers and is even more relevant in medium or high-frequency transformers. As is well-known, thedesign of a transformer with special focus on the windings has a big impact on internal losses andvoltage drops of the transformer. Copper losses are influenced by the cross-section and (becauseof the skin and proximity effect with influences of frequency and field distribution) also by the ar-rangement of windings. Imperfect magnetic coupling occurring in each real transformer is expressedby the term “leakage” and represented by leakage inductances in equivalent circuits of transform-ers. This way, the operation of electrical equipment including transformers can be investigated bymeans of calculation and simulation. Whereas in most power transformer applications particularlylow leakage inductance values are desirable, there are certain power electronic arrangements (e.g.,resonant topologies and power sources for laser beam generation [1]), which require first of all adefined leakage inductance value of the applied transformer.

In the case of existing transformers, leakage inductance can be determined by measurements.However, more often it is necessary to predict the leakage inductance of a transformer in thedesign phase, especially if circuit simulation is intended [2]. This can be done applying certainapproximation methods in which geometry parameters of the transformer are used to calculateits leakage inductance. Which approximation method has to be applied depends on the givenarrangement of windings. If the couple of coils, whose leakage inductance is to be calculated,is arranged on the same leg of a transformer, the method of Rogowski [3] and the method ofPetrov [4] can be used considering further aspects, which are explained in the following paragraph.If the coils are arranged on different legs, the method of Lebedev [5] can be applied for leakageinductance calculation.

Concerning the used terms “winding” and “coil”, the term “winding” is considered to be of highergrade, in general. Windings are realized as coils. However, in the case of fundamental transformerversions with one primary and one secondary winding, both terms can be used synonymously. Inthe case of transformers with interleaving of windings the primary winding and/or the secondarywinding consist of a certain number of single coils.

2. LEAKAGE INDUCTANCE CALCULATION FOR COILS ARRANGED ON THE SAMELEG

If a couple of coils is arranged on the same leg (e.g., in the case of shell-type transformers withall windings on the center leg, Fig. 1, left), prediction of leakage inductance can be obtained usingthe approximation methods of Rogowski and Petrov which have been established in the firsthalf of the last century. The most common method is the method of Rogowski which is based onthe consideration of the energy of the leakage magnetic field. The formulas for leakage inductancecalculation which are given in contemporary technical literature are usually predicated on thismethod but often slightly simplified resulting in a decrease of accuracy. As a precondition for theapplication of the method of Rogowski [3] the coils of the considered transformer should havethe same height in the case of concentric windings and the same width in the case of pie windings.

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Figure 1. Transformer design enabling leakage inductance calculation using the method of Rogowskileft: shell-type transformer, right: dimensions for calculation of the relative leakage conductance λ accordingto formula (2).

According to deviations in these dimensions, inaccuracies of the calculation results occur. Theleakage inductance (related to the primary side of the transformer) is calculated using the formula

LL = µ0 · w21 · lm · λ · kσ (1)

with LL — leakage inductance, µ0 — absolute permeability, w1 — number of primary turns,lm — mean length per turn for whole arrangement of coils, λ — relative leakage conductance,kσ — Rogowski factor. The relative leakage conductance has to be calculated related to thespecific arrangement of coils (see Fig. 1, right) using formula (2). The Rogowski factor represents acorrection factor concerning the length of leakage flux lines which is also geometry dependent andhas to be calculated as described in [3]. In most cases its value is between 0.9 and 1.

λ =1b

(δ +

a1 + a2

3

)(2)

The method of Petrov [4], which alternatively can be used, is even suitable if the dimensionsconcerning height and width of the coils are different, what makes the method more universal.According to the following formula, two leakage inductance constituents have to be calculated,which have to be summed up to the final leakage inductance (what here is related to the primaryside of the transformer again).

LLx=

µ0

2π· w2

1 · lmx· ln

(KCx

· g212

g1 · g2

)(3)

x = in — for portion inside core windowx = out — for portion outside core windowLLx — leakage inductance portionKCx — correction factor for consideration of core influence on leakage inductanceg12 — mean geometric distance between the cross-sections of the coils 1 and 2g1, g2 — mean geometric distance of the cross-section of a coil to itself

To determine the KC-factors related to the portions of the coils inside and outside the corewindow, the cross-sections of the coils have to be reflected at the adjacent or surrounding coresurface lines. The calculation of the KC-factors is based on the method of mean geometric distances(established by J. C. Maxwell [6]) considering the cross-sections of the coils and their mirror images(Fig. 2).

The methods of Rogowski and Petrov are explained in some more detail in [7].Leakage inductance calculation is a complex procedure which can be very time consuming.

Therefore, the algorithms of both methods have been implemented into a Matlab R© based cal-culation program. Using this program it is possible to determine the leakage inductance betweentransformer coils within the short computational time of a PC. In Fig. 3 a template of the realizedcomputer program is displayed as an example. It shows input parameter fields for core dimensionsand parameters describing the position of the coils as well as calculated values according to themethod of Petrov. At the bottom of the figure the position of the coils within the core windowis displayed together with the mirror images of the coils.

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Figure 2. Formation of mirror images of the windings required for determination of the correction factor KC

for the leakage inductance portion inside core window (left) and for the leakage inductance portion outsidecore window (right).

Figure 3. Template of the realized computer program for leakage induction calculation (example for calcu-lation using method of Petrov).

3. LEAKAGE INDUCTANCE CALCULATION FOR COILS ARRANGED ONDIFFERENT LEGS

Though shell-type transformers are very widespread, in certain cases the core-type transformerdesign (Fig. 4) has an advantage over the shell-type design of transformers. If coils are arrangedon different legs, like in the case of core-type transformers, leakage inductance can be calculatedby means of an approximation method which has been established by Lebedev [5]. As a basicassumption of this method, the leakage field can be considered to consist of three leakage fluxcomponents (Fig. 4). Resulting from this, the leakage inductance is assumed to be the sum of threeparts which are related to the single flux components

LL = LLin+ LLy

+ LLeq(4)

with inner leakage inductance LLinrelated to the leakage flux ΦLin

belonging to the section ofthe perimeter of the coils which is located within the core window (calculated using method ofRogowski or method of Petrov), leakage inductance component LLy

which is related to the

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Figure 4. Components of leakage flux of a core-type transformer.

yoke leakage flux ΦLyand the so-called leakage inductance of the equivalent transformer LLeq

related to the flux component ΦLeq.

The calculation of the single leakage inductance components and further aspects of the appli-cation of the method of Lebedev are explained in [8]. As well as the methods of Rogowski andPetrov, also the method of Lebedev has been implemented into a Matlab R© based computerprogram.

4. APPLICATION EXAMPLES

The mentioned approximation methods enable the direct calculation of the leakage inductancerelated to one couple of coils of a transformer. In the case of transformers with a complex designof interleaved windings, which is often used to minimize the total leakage inductance, generally theleakage inductance values of the respective couples of single coils have to be merged together intothe total transformer leakage inductance using certain combination formulas, e.g.,

LLtotal=

w21

w2ref

·(

w11

w1· L11/2 +

w12

w1· L12/2 −

w11 · w12

w21

L11/12

)(5)

for one of the simplest transformer versions with interleaving of windings (primary winding dividedinto two series-connected coils with — usually equal — number of turns w11 and w12 and thesecondary winding arranged between these primary coils). Further quantities: w1 — total numberof primary turns, Lx/y — leakage inductance values of the respective couples of coils, wref —number of turns to which the leakage inductance values of the couples of coils refer. The setupof such combination formulas for transformers with interleaving of windings is described in [9]. Inthe case of transformer versions with a higher degree of interleaving of windings and therefore ahigher number of single coils, the combination formulas are more complex and much more leakageinductance values of couples of coils have to be determined increasing the demand for computerizedcalculation.

In the performed investigation the application of combination formulas was necessary to enablethe comparison of the results of leakage inductance calculation with measurements concerningtransformer versions with different degree of interleaving of windings.

Concerning leakage inductance determination in the case of arrangements with all coils on thesame leg, shell-type transformer versions based on an identical core design but differing in therealization of windings have been considered. If the windings are interleaved, the primary coils areconnected in series and the secondary coils in parallel. As an example, a transformer variant with4 primary and 3 secondary coils (degree of interleaving shortly described as 4-3) is displayed inFig. 5.

Beside computerized calculation due to the considered approximation methods, computer-basednumerical finite elements method (FEM) calculations can be used for leakage inductance determi-nation. Especially in the case of shell-type transformers a rotation-symmetric FEM model with

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identical parameters of core area and mean length per turn compared to the real transformer canbe used. Despite the simplifications of this model characterized by a reduction of the effort forthe input of the geometry as well as the calculation expenditure [10], the performed FEM sim-ulations yielded a relatively good coincidence of their results with the leakage inductance valuesobtained from measurement. Concerning the approximation calculation methods, it can be statedthat the method of Petrov shows a higher accuracy only in the case of the initial (magneticallyasymmetric) 1-1 variant whereas the method of Rogowski gives results which are closer to themeasurements in the considered variants with interleaving of windings which are magnetically sym-metric because the width of the outer (primary) coils is half of the width of the inner (primary)coils (Fig. 6).

In addition to the comparison between the predicted and measured values, Fig. 6 illustrates theleakage inductance decreasing effect of interleaving of windings. So, in the case of the 5-4 variant,the leakage inductance is reduced to about 4% of the initial 1-1 variant.

Concerning leakage inductance determination in the case of arrangements with coils on differentlegs, several core-type transformer versions have been considered. The example in Fig. 7 with twocoils on each leg represents a 2-2 variant regarding interleaving of windings, if two coils are connectedin series which are arranged on different legs to form the primary and the secondary winding,respectively. Arrangements with a connection of the coils on the same leg are not considered inFig. 8, because in these cases no real interleaving of windings exists.

Figure 5. Shell-type transformer example with piewindings, degree of interleaving of windings 4-3,(w2 = 1).

1-1 2-1 3-2 4-3 5-4

50

100

150

200

250

300to

tal le

aka

ge

in

du

cta

nce

, L

L,t

ota

l [nH

] Petrov method calculation

Rogowski method calculation

FEM simulation

measurement

Figure 6. Predicted and measured total leakage in-ductance values of shell-type transformers with dif-ferent degree of interleaving of windings (wref = 1).

Figure 7. Core-type transformer examplewith concentric windings.

measurement

calculation, Lebedev and Rogowski

calculation, Lebedev and Petrov0

3

6

9

12

tota

l le

akage inducta

nce,

LL

,to

tal [n

H]

core-type transformer: N11+N12, N21+N22

core-type transformer: N11+N22, N12+N21(primary, secondary)

Figure 8. Predicted and measured total leakage in-ductance values of the considered core-type trans-former (wref = 1).

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Without interleaving of windings, leakage inductance is about 50 times higher than the valuesin Fig. 8, because of weak magnetic coupling between the coils of different legs of a core-typetransformer.

To predict the total leakage inductance of a core-type transformer with interleaving of windings,leakage inductance values have to be merged together, which belong to couples of coils which arelocalized on different legs (leakage inductance to be calculated due to the method of Lebedev)and other couples of coils which are arranged on the same leg. Concerning the last-mentioned case,alternatively the method of Rogowski or the method of Petrov can be used. As it can be seenin Fig. 8, the application of the method of Rogowski results in a higher degree of coincidencewith measured values.

5. CONCLUSIONS

The leakage inductance of transformers can be predicted applying approximation methods relatingto the respective transformer design. Using computerized calculation after implementing the algo-rithms of the approximation methods into appropriate calculation software (e.g., MATLAB R©), itis possible to determine the relevant leakage inductance values of a transformer within the shortcomputational time of a PC. Thus, efficient investigations of transformer concepts with differentgeometry of windings can be performed. Furthermore, FEM simulation can be used for the de-termination of leakage inductance values of transformers. Regarding the considered transformerversions both the results of approximate calculation methods and FEM simulation are comparableto measurements. Beyond the determination of leakage inductance, FEM simulation is a useful in-vestigation tool for transformers to clarify certain aspects of magnetic field distribution concerningcore and leakage paths and current density distribution within the conductors of the windings.

REFERENCES

1. Stadler, A., M. Albach, and S. Chromy, “The optimization of high frequency operated trans-formers for resonant converters,” Proceedings of 11th European Conference on Power Electron-ics and Applications (EPE2005), 77, Dresden, Germany, Sept. 2005.

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3. Rogowski, W., Ueber das Streufeld und den Streuinduktionskoeffizienten eines Transforma-tors mit Scheibenwicklung und geteilten Endspulen, (Dissertation), VdI, Mitteilung ueberForschungsarbeiten auf dem Gebiet des Ingenieurwesens, 1909.

4. Petrov, G. N., “Calculation of leakage of transformers (russ.),” Elektricestvo, No. 8, 3–8, 1935.5. Lebedev, V. K., “Calculation of the short-circuit resistance of welding transformers with yoke

leakage (russ.),” Automatic Welding [Avtomaticeskaja Svarka], Kiev, Vol. 11, No. 4, 37–44,1958.

6. Maxwell, J. C., Lehrbuch der Elektrizitaet und des Magnetismus, Vol. 2, Springer-Verlag,Berlin, 1883.

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8. Doebbelin, R., M. Benecke, and A. Lindemann, “Calculation of leakage inductance of core-typetransformers for power electronic circuits,” Proceedings of 13th International Power Electronics& Motion Control Conference EPE-PEMC, Poznan, Poland, September 1–3, 2008.

9. Petrov, G. N., “Allgemeine Methode der Berechnung der Streuung von Transformatoren,”Elektrotechnik und Maschinenbau, Vol. 51, No. 25, 345–350, 1933.

10. Schaetzing, W., FEM fuer Praktiker: Elektrotechnik, Vol. 4, Expert-Verlag, 2003.