Calculation of effective inductance ofgapped core assemblies

P.D. Evans. B.Sc.(Eng.). Ph.D., D.I.C.. A.C.G.I., and B.M. Saied, B.Sc.(Eng),M.Sc. Ph.D.

Indexing terms: Power electronics, Mathematical techniques, Magnetic devices and properties

Abstract: A simple general method of allowing for the effect of fringe fields around the airgaps in inductor coresis described. Experimental results for two standard E core shapes are used to demonstrate that the technique isreliable and necessary when calculating inductances. Three alternative methods of allowing for fringe fields arealso described and compared.

1 Introduction

Gapped cores are widely used in the construction of induc-tors in power electronics equipment. They are typically ofthe C or E type and are usually made either from a rangeof laminated steels, or, for high-frequency operation, froma suitable grade of ferrite material. Typical functions ofthese inductors include the snubber inductors in thyristorand transistor equipment, output filter components inforward convertor type switched-mode power supplies(SMPS) or the coupled inductor in a flyback convertortype of SMPS. In these applications the function of theinductor is to act to control the rate of change of current,or as an energy store in a filter, or both. One example ofthis combined duty is the inductor in the output line of theforward convertor. Here, the series inductance, in com-bination with the shunt capacitance across the output ter-minals, controls the quality of the DC output, by filteringout the higher frequency switching harmonics. However,this inductance also controls the rate of rise of current inthe main transistor in the SMPS, which switches theprimary of the high-frequency transformer. In this applica-tion, the inductor therefore affects the peak current ratingof the transistor that has to be used, which has implica-tions on the transistor base drive current and the capacityof the associated base drive power supply. For the reliabil-ity and security of the circuit therefore it is important thatthis inductor is well designed so that it does not saturateunder peak current conditions.

Although an inductor appears to be a relatively simplecomponent, a complete formal design would be complexand would include a number of aspects. The principal ofthese are:

(i) design of the magnetic circuit on a DC basis includ-ing the effect of fringe flux around the gap

(ii) calculation of the core losses in the magneticmaterial at the operating frequency and with the appropri-ate current waveform

(iii) design of the electric circuit, that is to say, thenumber of turns and the size of wire to achieve the desiredinductance without undue I2R loss

(iv) calculation of additional eddy current losses in theconductor due to skin and proximity effects where theseare significant

(v) calculation of the thermal conditions, so that thevarious losses are dissipated satisfactorily.

Precise solution of these problems is generally not necess-ary because of the empirical data provided by core manu-

Paper 4279B (P6), first received 27th January and in revised form 25th September1985Dr. Evans is with the School of Electrical Engineering, University of Bath, Claver-ton Down, Bath BA2 7AY, United Kingdom. Dr. Saied is with the University ofMosul, Mosul, Iraq

facturers in a convenient form. Typically, to meet anygiven specification the designer can choose from a range ofstandard core shapes and sizes. For each core, data such asinductance per (turn)2, temperature rise per watt of totalloss and against flux density and frequency are generallygiven.

Thus, for standard applications it is usually possible toarrive at a satisfactory design from these data. Also, ofcourse, the low cost of construction of inductors meansthat it is usually quicker to make several prototypes tosettle on a design close to the optimum than to undertakea complex design study.

However, in view of the increasing use of power elec-tronics equipment of all types, along with the variousinductive components required, it is of interest to probefurther into inductor design so that empirical data is lessvital, leaving the designer with greater flexibility andchoice.

The present paper addresses one of the simpler prob-lems, (i) above, which concerns the effect of fringe fluxaround the airgap, or airgaps, in an inductor. It developsthe work that has already been reported on this topic [1].

Numerous other methods have been proposed to handlethis problem and results from some of them [2, 3, 4] havebeen compared with the theoretical and experimentalresults reported in this paper.

2 Statement of problem

It is well known that flux crossing the airgap betweencores is not confined within a volume defined by the physi-cal cross section of the core, but extends into fringe fieldsaround the airgap. In large structures, where the ratio ofairgap length to the linear dimensions of the magneticcircuit area is small, the fringe field contributes such asmall amount to the total flux that it may be neglected formost practical purposes. In small components, such asferrite cores, however, the opposite conditions exist. Thatis to say, the airgap length is relatively large compared tothe linear dimensions of the core and accordingly the totalflux contained in the fringe fields may be of the same orderof magnitude as the main airgap flux.

To calculate inductance, magnetic loading of the coreand core losses, it is therefore necessary to estimate quitecarefully the flux contribution due to the fringe fields.

3 Theoretical method

Calculation of the fringe flux distribution around theairgap between inductor cores comes into the category of a3-dimensional magnetostatic problem. At present, there-fore, it would be solved relatively easily, either using acomputer package based on finite difference or finite-element methods, such as TOSCA [5], and these might be

IEE PROCEEDINGS, Vol. 133, Pt. B, No. I, JANUARY 1986 41

most suitable for special cases with unusual geometries, orby developing a purpose-made field computation program.However, these methods require special techniques and aregenerally expensive to undertake for the design of low-costcomponents. In addition, they may produce too muchinformation. They give the flux distribution, which is sum-mated along suitable boundaries to derive the total fluxcontained in the fringe fields. For the present purposes, itis sufficient to know only total flux and the field shape istherefore not of direct interest.

Early work by Carter [6] used a conformal transform-ation technique to calculate the fringe fields at the edge ofa pole in a dynamoelectric machine. This, and subsequentwork has been quite widely used in the design of electricalmachines [7, 8], but is probably best known through'Carter's coefficient' [9] which allows the effective distancebetween the slotted surfaces in the airgap of an electricalmachine to be calculated. The techniques of Carter havebeen well described by Binns and Lawrenson [10].

This work can also be applied to the present problem,however, because the magnetic geometries produced by thegapped limbs of C and E cores are substantially the sameas those considered by Carter [6]. These geometries areillustrated in Fig. 1. Fig. la represents the edge of an iso-lated pole which is at some magnetic potential with respect

! g \ .9

|Ag

_J L.

77777777///////

Fig. 1

c d

Geometries of magnetic boundaries

to the other iron surface and is separated from it by dis-tance, g. Conformal transformation requires that theseboundaries are equipotentials and consequently the mag-netic material is assumed to be infinitely permeable. It isalso assumed that all boundaries extend to infinity. Fig. Ib,which represents one edge of C or E core limbs separatedby a total airgap of length 2g, say, can be handled by thegeometry of Fig. la by exploiting symmetry in the usualway.

Whereas the geometry of Fig. la may be thought ofas a homopolar system, Fig. lc represents a heteropolarcase. The two poles, the edges of which are separated by adistance 2c, are at positive and negative potentials and areseparated from the zero potential surface by an airgap g.This magnetic geometry applies for the limbs of an induc-tor core that are separated by the winding, and again sym-metry can be used to apply it to a practical geometry ofthis type shown in Fig. Id.

For both these geometries, Carter's results are formu-lated in a convenient way. For the homopolar case, it is

assumed that the pole width can effectively be increased byan additional amount Xg, as shown in Fig. la where X is aconstant determined from Carter's work. It is assumedthat the flux density is zero outside this region, but withinit, it is the same as under the main pole. The pole ofincreased width is therefore in effect surrounded by aperfect magnetic insulator, so that magnetic circuit calcu-lations on the resulting geometry can be carried out usingsimple magnetic circuit calculations. This arrangementtherefore allows the total flux crossing the airgap to becalculated without considering explicitly the distribution inspace of the fringe fields. The distribution can, however, beobtained from Carter's [6] work and has been used suc-cessfully [8]. The area of interest in the present type ofwork where fringe field distribution might be of concern isin the calculation of eddy current loss in the conductors ofan inductor due to fringe flux incident upon them, in high-frequency components such as flyback convertor induc-tors, but this is given no further consideration here.

The factor X is most readily obtained from the curvesshown in Fig. 2. Curve (i) in Fig. 2 is taken from Fig. 3 in

10 20 50 curved)

0 5

Fig. 2 Fringe factors

10 , 15c/g

20 curve (ii)

Carter's paper [6] but with the ordinate rescaled so that X,as defined here, is plotted directly against x/g. Curve (ii) inFig. 2 comes directly from Table 3 in Carter's paper [6].The homopolar geometry uses curve (i). The abscissa ofthis curve is the normalised parameter x/g. To obtain thetotal fringe flux crossing the airgap up to some distance xaway from the edge of the pole it is necessary to increasethe pole width by X airgap lengths. The choice of x is notstraightforward, but neither is it very critical because of theway the curve flattens out for large values of x/g. Thechoice of the characteristic distance x is discussed below inthe context of the two examples considered.

For the heteropolar case, Fig. lc and d, this indetermi-nacy is not present because the fringe field is forced to zerobetween the two poles. In this case, therefore, the abscissais the normalised parameter c/g, where c is the semi-distance between the inner edges of the two poles, asshown in Fig. 1. Once more, each pole edge is increased inwidth by X airgap lengths.

4 Application of method

The proposed method of accounting for the fringe flux ininductor cores was tested with two practical examples offerrite E cores. For both cases, which are described indetail below, the inductance was calculated for a range ofairgap lengths and the predictions were compared withmeasurements.

42 IEE PROCEEDINGS, Vol. 133, Pt. B, No. 1, JANUARY 1986

Calculation of inductance was by means of simple mag-netic circuit theory, as stated above. In this case the simpledefinition of inductance may be used:

L = N(p/I (1)

This definition assumes that all the flux </> links all theturns N of the windings; that is to say, the proportion offringe flux that does not link with all the turns in thewinding is negligibly small. Attempts to allow for thiseffect in more detail would greatly increase the complexityof the calculations without a commensurate improvementin accuracy.

The magnetic circuit equation that derives fromAmperes Law is:

NI = (j>RT AT (2)

where RT is the overall reluctance of the magnetic circuit.Eqns. 1 and 2 combine in the usual way to give:

L = N2/R: H (3)

It is sometimes convenient to work in the normalised formof inductance per (turn)2 in which case

L = \/RT H/turn2 (4)

and this gives an expression for inductance in terms of thedimensions of the magnetic circuit. (The symbol AL is oftenused for inductance per (turn)2).

For the present work it has been assumed that the corematerial has infinite permeability and therefore makes nocontribution to the total reluctance of the magnetic circuit.This is closely the position in most practical cases. Onlywith very small airgaps does the permeability of the corematerial come into account, and under these conditionsthe fringe flux is generally not a significant quantity.

The overall reluctance of the magnetic circuit RT maytherefore be calculated from the airgap regions alone.Hence

RT = l/n0A (5)

where / is the total gap length, and A is the effective area,which includes an allowance for the fringe flux. Thisreduces the problem of calculating inductance to one ofcalculating effective physical dimensions, using the Afactors derived from Carter's work.

To establish the reliability of the proposed method, twoexamples were undertaken, using different ferrite E coredesigns. Both cores are of the type used in switched-modepower supplies and similar high-frequency switching cir-cuits. They are consequently relatively small in physicalsize and present a severe test for the proposed technique.

4.1 Example 1: Ferrite E core, Type E65/33/27The principal dimensions of this standard ferrite E core areshown in Fig. 3. The core has a generally simple rectilinear

tX

| 1

—65mm

32.8

.132.8mm

27.4 mm

A B I-* *-\ -*A [-*"I ->rwm' ^ "in.

design and the airgap regions have a rectangular crosssection. An inductor may be formed in the usual way byplacing two cores together with an airgap between themand with a coil around the centre limb.

The shaded areas indicated on Fig. 3b are the addi-tional airgap areas, calculated by the method describedabove, to allow for the fringe flux crossing the airgaps.Portions of the type identified by A are of the homopolartype discussed above, and were calculated using curve (i) inFig. 2. It is necessary to summate the fringe flux up tosome distance x away from the core and the distance usedfor this was a characteristic dimension of the core, asshown in Fig. 3. The choice of this distance is not criticalas X changes slowly at large values of x/g. Portions B areof the heteropolar type, calculated using curve (ii) in Fig. 2.Contribution from fringe flux at the corners of the limbshas been neglected. Although steps could be taken to allowfor it, it is thought that in practice this would be anattempt to achieve a precision which is neither required,nor justifiable from the assumptions made in the methodas a whole.

It is a simple matter to calculate the overall reluctanceRT of the magnetic circuit of the E core from the effectivereluctance of each airgap, i.e.

RT = Rc (6)

where Rce is the effective reluctance of the centre airgap,and Roe is the effective reluctance of the airgap betweenouter limbs.

Predicted and measured results for this E core geometryare presented in Fig. 4, where inductance per (turn)2 is

0.9 h

0.8

0.7

0.6

i 0.5

§0.4

0.2

0.1

\\

20mm 10.4mmFig. 3 Dimensions of ferrite core type E65/33/27

0 1 2 3 4 5airgap length, mm

Fig. 4 Calculated and measured values of inductance for E core typeE65/33/27

Calculations by proposed method• Measured valuesx Values published in data sheetO Values calculated by method of Snelling• Values calculated by methods of Slemon and Straughen and of Wood

Values calculated neglecting fringe flux

plotted against the airgap that separates the two E cores.It can be seen that there is generally good agreementbetween the two sets of results, a standard of agreementthat is usually adequate for design purposes in power elec-tronics equipment. The values of AL provided in the datasheet for the E65/33/27 are also plotted in Fig. 4, and itcan be seen that these too are in general agreement with

IEE PROCEEDINGS, Vol. 133, Pt. B, No. 1, JANUARY 1986 43

measured and calculated results. The inductance, calcu-lated assuming that there is no fringe flux, is also shown inFig. 4. It is clear that the inductance so calculated is con-siderably lower than observed in practice. For example,with a 5 mm airgap the actual inductance is approximatelytwice the value of inductance that is calculated with zerofringe flux. In this case, therefore, the total fringe flux isapproximately equal to the main flux. Even with a 2 mmairgap the fringe flux has the effect of making the induc-tance more than 50% greater than expected.

4.2 Example 2: Ferrite E core. Type ETD 44/22/15The shape of this E core is illustrated in Fig. 5. It is one ofa range of cores recently introduced for switched-mode

<

X —- *

. . J

1 I!2.3

122.3mm

Fig. 5 Dimensions of ferrite core type ETD44/22/15

power supply applications. It was chosen for considerationfor three reasons. First, as stated above it represents astate-of-the-art design of ferrite. Secondly, it is a morecomplex shape than the first example. Designed for eco-nomic transformer and inductor design, the centre limb iscircular in cross-section to minimise turn length for a givenenclosed ferrite area, and the outer limbs which togetherhave the same area as the centre limb, are shaped toaccommodate the bobbin which contains the windings.Thirdly, the ferrite area is considerably smaller than in thefirst example (the centre limb area is approximately550 mm2 for the E65/33/27 and approximately 180 mm2

for the ETD 44/22/15). The effect of fringe flux is thereforegreater with this core and this fact, together with thecomplex shape presents a more severe test for the pro-posed method.

0.3 -

.0.2

0.1

0 1 2 3 4 5airgap length, mm

Fig. 6 Calculated and measured values of inductance for E core typeETD44/22/15See Fig. 4 for key

The method of applying the effective additional airgapareas to this case is indicated in Fig. 5, and the use of thehomopolar and heteropolar results in portions A and Brespectively is again identified.

Measured and predicted results are shown in Fig. 6 andthere is good agreement between them. The calculatedvalues obtained by assuming fringe fields are absent arealso plotted in Fig. 6 and demonstrate again the large con-tribution made to the total flux by the fringe fields. With aspacing of 5 mm between the two E cores, for example, theactual inductance is around three times the inductancethat would be achieved if there was no fringe flux. In thiscase, therefore, the fringe flux contribution to the total fluxis greater than that of the main flux. Even with a 2 mmgap the fringe flux is slightly greater than the main flux.

5 Comparison with alternative methods

Numerous other methods have been used to make allow-ance for fringe fields around the airgaps of inductor coresand in similar conditions. Three of them [2, 3, 4] havebeen considered here for comparison with the proposedmethod.

The most common method, mentioned for example inSlemon and Straughen [2], is a general 'rule of thumb'which recommends that the linear dimensions of the corelimbs are increased by one gap length or, in the case of alimb with a circular cross section, that its diameter beincreased by one gap length. Results calculated by thismethod for the E65/33/27 and the ETD 44/22/15 cores areshown in Figs. 4 and 6, respectively. It can be seen that themethod gives a measure of improvement over the unmod-ified results, but in neither case does it give a full allowancefor the fringe flux. It seems probable, however, that thismethod arose as a greatly simplified version of the Cartermethod described in Sections 3 and 4 above and mayapply for a restricted range of geometries.

The method referred to by Snelling [3] is based on con-formal transformation but is the subject of private commu-nication between Astle and Snelling and the details of themethod do not appear to have been disclosed. Neverthe-less, using as it does conformal transformation techniquesit is likely to follow a similar procedure to that given byCarter. The result quoted by Snelling is that the semi-widths of a rectangular pole face, or the radius of a circularpole face, should be increased by the quantity:

0.241 GKUwhere b is the total width of the window of the core and gis the airgap length. The parameter b, the window usedhere, does not appear in the present work which usesCarter's results. This suggests that the geometry on whichthe conformal transformation is performed is different, andis probably the pot core geometry which is approximatelyaxisymmetric. This method, therefore, does not distinguishbetween the homopolar and heteropolar conditions causedby the interlinking of the windings and the core althoughSnelling mentions in the text that the winding has an effecton the fringe flux. Application of this method to the twocores under consideration gives the results which areplotted on Figs. 4 and 6. It can be seen that these resultsagree very closely with the results from the method pro-posed in this paper. For the E65/33/27 in Fig. 4 the differ-ence is negligibly small, whereas for the ETD 44/22/15 inFig. 6 Snelling's method underestimates the results by asmall margin, typically around 10%.

44 IEE PROCEEDINGS, Vol. 133, Pt. B, No. 1, JANUARY 1986

The method given by Wood [4], which appears to beanother 'rule of thumb' states that the airgap length shouldbe modified to an effective length la, such that:

la =abl

(a + b)(b + /)(7)

where a and b are the core dimensions and / is the physicalairgap length.

The other methods discussed so far have allowed forfringe flux by assuming that the gap length is unchangedwhile the effective airgap area increases. This reduces theeffective reluctance. In Wood's method the area isunchanged but the gap length is assumed to be reduced,thereby arriving at a reduced airgap reluctance by alterna-tive means. Both methods are equally valid. In the formermethod, modified dimensions predict the correct airgapflux density, whereas in the latter method the modifieddimensions predict the correct core flux density. This isdiscussed further in Section 6. Apart from this slightlydifferent approach, it appears that Wood's methodamounts to the same approximation as that of Slemonand Straughen. Evidently the airgap length, and hencethe reluctance, is to be reduced by the factor

la ab

l)(b(8)

where it can be seen that the numerator is the physicalairgap area and the denominator is the area of the airgapwith each linear dimension increased by one gap length.

Identical results would therefore be produced by themethods of Wood and of Slemon and Straughen, in spiteof their greatly differing formulations.

6 Discussion

It has been established that in gapped cores fringe fluxmay increase by a considerable margin the total flux cross-ing an airgap. This has been thought of in terms of areduction in the effective reluctance of the magnetic circuit,which produces an increase in inductance. Although thisappears to be an advantageous state of affairs, there aresome additional consequences which need to be taken intoaccount. The first results from the fact that the core fluxdensity will be greater than the airgap flux density becausethe core has to carry all the flux. The ratio of the airgapand core flux densities will clearly be related directly to theactual and effective airgap reluctances which have beendiscussed above in connection with inductance calcu-lations. It is important, therefore, from the point of view ofmagnetic conditions, that in calculating the maximumampere turns in the inductor, the core flux density ratherthan the airgap flux density should be considered so thatthe core is not run into saturation.

The second effect follows directly from this and con-cerns the calculation of core losses. The actual core fluxdensity which includes the fringe flux is greater than theairgap flux density needs to be calculated and used in theestimation of core losses.

The third effect concerns the total inductive energy thatmay be stored in a pair of cores, and whether this isaffected by the fringe flux insofar as it leads to a higherinductance. The answer to this is derived most convenient-ly by expressing the stored energy EL not in the most fam-iliar form

but in the alternative form

EL = W (10)

which applies for linear conditions.In this case, the total flux (/> is limited by the cross-

sectional area of the core and is therefore not affected byfringe flux, and the maximum current / is determined inany given application by heat loss and temperature rise.

7 Conclusions

The paper has investigated the effects of fringe flux aroundthe airgaps of inductor cores on the inductance that can beachieved. It has been shown that with the range of coresizes that is typically used in high-frequency applicationsthis effect is considerable, an inductance being twice thatwhich would be predicted if fringe flux were neglected notbeing unusual. A method of handling this problem hasbeen described which utilises the early work of Carter. Themethod has the advantage that, although being quick andsimple to implement by hand, it is based on sound physicalprinciples and produces results that agree well with mea-surement. Three alternative theoretical methods were alsostudied for comparison. It was found that two of themwere less precise than the proposed method. The thirdmethod, mentioned by Snelling, is also based on conformaltransformation, and appears to give results which are gen-erally good. The main reservations of it therefore are thatits derivation is unknown and that it does not seem todistinguish between the fringe field regions which areaffected by the winding that those which are not. It there-fore becomes a matter of applying a formula of unknownorigin with the potential attendant risks, especially forgeometries that are dissimilar to those considered above.

Additional consequences of fringe flux were noted con-cerning the core flux density which must be calculated cor-rectly so that it does not saturate, and so that core lossesmay be estimated reliably. It was also noted that althoughthe effective inductance is usually higher than expected dueto additional flux in the fringe fields, there is no increase inthe maximum energy storage capacity of the cores.

8 References

E, = \LI2 (9)

1 EVANS, P.D., and SAIED, B.M.: 'Calculation of the effective induc-tance of gapped ferrite cores'. IEE Conf. Publ. 234, 1984, pp. 103-106

2 SLEMON, G. R., and STRAUGHEN, A.: 'Electric machines'(Addison-Wesley, 1982), p. 61

3 SNELLING, E.C.: 'Soft ferrites' (Iliffe Books, London, 1969), p. 1754 WOOD, P.J.: 'Power switching convenors' (Van Nostrand Rhein-

hold, 1981), p. 2715 SIMKIN, J., and TROWBRIDGE, C.W.: 'Three dimensional nonlin-

ear electromagnetic field computations, using scalar potentials', IEEProc. B, Electr. Power Appi, 1980,127, (6), pp. 368-374

6 CARTER, F.W.: 'Note on airgap and interpolar induction', J. IEE,1900, 29, pp. 925-933

7 LAWRENSON, P.J., and GUPTA, S.K.: 'Fringe and permeancefactors for segmented rotor reluctance machines', Proc. IEE, 1971,118,(5), pp. 669-674

8 EVANS, P.D., and EASTHAM, J.F.: 'Slotless-disc alternator withAC-side excitation', IEE Proc. B., Electr. Power Appl. 1983, 130, (6),pp. 399-406

9 SAY, M.G.: 'Performance and design of alternating current machines'(Pitman, 1958)

10 BINNS, K.J. and LAWRENSON, P.J.: 'Analysis and computation ofelectric and magnetic field problems' (Pergamon, 1973, 2nd Edn.),Chap. 8

IEE PROCEEDINGS, Vol. 133, Pt. B, No. I, JANUARY 1986 45