SCIENCE

Calculation of self and mutual impedances for coilson ferromagnetic cores

D.J. Wilcox, MSc, PhD, CEng, MIEEM. Conlon, BSc, MEngSc, PhDW.G. Hurley, BE, MS

Indexing terms: Ferromagnetics, Electromagnetic theory

Abstract: By means of integral transform tech-niques, the paper establishes a new set of self- andmutual-impedance formulas relating to coils onferromagnetic cores of circular cross-section. If thecore is straight and infinitely long, the formulasare expressed in terms of convergent integrals thatmay be evaluated numerically. In the case ofclosed cores, e.g. toroidal cores, the formulas for-mally reduce to convergent series that may betruncated according to the degree of accuracyrequired. The formulas follow directly from thesolution of Maxwell's equations and thereforeoffer the ultimate in accuracy.

1 Introduction

Analytic solution of Maxwell's equations, satisfying rele-vant boundary conditions, seems the natural basis forderiving accurate formulas for the self and mutual imped-ances of coils. This approach has been successful for thecase of coils in air [1] but does not appear to have beensuccessful for the case of coils on ferromagnetic cores. Aswill be seen for the simple but important case treated inthe paper, rigorous formal analysis leads to solution inthe form of an integral of a complexity that appears todefy any realistic chance of analytic evaluation. Thus, ifthis integral had been derived in earlier times, which ispossible but not evident from a survey of the literature, itwould doubtless have been considered to be of no practi-cal value to classical analysts seeking formulas whichcould be readily evaluated using only elementary com-puting aids. Fortunately, this restriction no longerapplies. The integrand in question is, in fact, quite wellbehaved and, with any computer now available, the inte-gral may be readily evaluated to any required degree ofaccuracy using numerical quadrature techniques.

The basic formula derived in the paper gives self andmutual impedances for filamentary turns on an infinitely-long homogeneous core of circular cross-section. This isthen adapted to the more likely case of a magnetically-closed core of finite length. Further development thengives self and mutual impedances for coils rather than

Paper 6123A (S4, S8, S2), first received 9th December 1985 and inrevised form 11th July 1986Dr. Wilcox and Dr. Conlon are with the Faculty of Engineering, Uni-versity College, Galway, Eire and Mr. Hurley is with the National Insti-tute for Higher Education, Limerick, Eire

individual turns. It is felt that these formulas could, forexample, find immediate application in the detailedanalysis of the behaviour of transformer windings undersurge conditions.

Since the formulas are based on the rigorous solutionof Maxwell's equations, they necessarily take full accountof eddy currents induced into the core. In particular, fre-quency dependence owing to skin effect is properly rep-resented. Also, use of the formulas in circuit analysis willcause damping effects consistent with actual eddy-currentlosses.

2 Derivation of basic impedance formula

The physical arrangement under consideration is shownin Fig. 1. The ferromagnetic core, taken to be infinitely

medium 1 / \ J.

A (A

Fig. 1 Filamentary turns on an infinite ferromagnetic core

long, is treated as a homogeneous medium of conductivi-ty <J2

a n d permeability \ilr in the radial direction and /i22

in the axial direction, thus allowing for the possibility ofgrain orientation. The core radius is b. A filamentary turnof radius a encircles the core at z = 0, and carries a sinus-oidal current i^. This is represented in the ensuing fre-quency domain analysis by /^, i.e. i^t) = 1$ e"0' where cois the angular frequency.

The mutual impedance between the energising turnand a second (open) filamentary turn of radius r locatedat an axial separation z follows directly from the solutionfor the electric field intensity E^ = E^r, z). The same fieldsolution may also be applied, within reason, to the deter-mination of the self impedance of a nonfilamentary turnof small cross-section.

The following basic assumptions are made:(a) the field is quasi-stationary(b) all capacitive effects (i.e. displacement currents in

any rz plane) may be separately accounted for.

470 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 7, SEPTEMBER 1988

These are the usual assumptions and can be justified upto quite high frequencies. Note, in particular, that itfollows from the stated assumptions that, at any instantin time, the current is taken to have the same value at allpoints around the energising turn.

The following identities apply on the grounds of sym-metry :

BE.

H ^ o ^ = 0 ^ = 0d(f> d(f>

The relevant field equations, in cylindrical co-ordinates,then reduce to

dz

dr= -jconzHz (1)

where the Dirac functions <5(r — a) and d(z) locate theenergising current at the precise co-ordinates (a, 0) in ther — z plane.

Eliminating Hr and Hz from eqns. 1, using subscriptsto distinguish the two media, gives:

(a) Medium 1 (r ^ b):

J2E

dz2

J2£

dr2

F

dr

(b) Medium 2 (r

H2zd2E<t>2 _ d2E

b):

ldl?

dz: dr2

(2)

(3)

where it has been assumed that ax = 0 (i.e. that medium1 is totally nonconducting).

It is now a matter of solving eqns. 2 and 3 to satisfythe following field continuity requirements:

at r = b

at r = b (4)

0 asIt is also required that £0 2 = 0 at r = 0, E4

r -* oo, and E4>v E^2 -* 0 as z -* + oo.A particularly neat method of solution is to employ

the Fourier integral transform. Choosing p as the trans-form parameter, and using an asterisk to indicate a quan-tity transformed wrt. z, eqns. 2 and 3 transform to:

£5dr2

r

- a) (5)

(6)

respectively, where c = ii2Jn2r and m is defined by:

m = {jcon2za2}112 (7)

Eqn. 6 is recognised as Bessel's equation of the modifiedtype. Its general solution may be immediately writtendown as:

£ £ = AIx(Tr) + BKtfr) (8)

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where standard notation is used for the modified Besselfunctions Kx and lv A and B are constants wrt. r. Thecomplex quantity T is specified by:

T2 = cp2 + m2 = — P2 + jcofi2z a2 (9)

Applying the condition £^2 = 0 at r = 0 gives:

where E$b, the field intensity at the core surface, remainsto be determined.

Solution of eqn. 5 is rather more complicated. Therequired result, derived in the Appendix, is

£"* =

(11)

where the symbol ^ ^ is used to indicate that r and ashould be interchanged for the case r > a.

E$b is deduced from the remaining boundary conditionHZ1 = HZ2 at r = b. This corresponds to the requirement:

dr

1 1 d(rE*2)

jcofi2z r dr(12)

Applying eqn. 12 yields the result:

E%b = -J

h (13)

+

Inserting this result into eqns. 10 and 11 gives therequired solutions for E$t and E$2, respectively.

Solutions for E</)1 and E^2 are obtained by Fouriertransform inversion. The solution in medium 1 is of par-ticular interest:

M2* JL ^— (14)

Mutual impedance, represented by Z, between the indu-cing turn and the filamentary turn at co-ordinates (r, z)gives an induced voltage V = ZI^. This is identified withthe voltage:

v = —

Thus

r, z)rd<j> = -

rt z)

r, z)

(15)

The required mutual impedance formula follows fromthe above analysis as:

Z = jwLl + Ze

where

Ll =r oo r>a

J-oo

(16)

(17)

471

and

z, = /•

nx r i»2z P io 2 dp (18)

fi2z p Komi^rb))

As noted in the introduction, the formula is forbiddingfrom the point of view of analytical evaluation. It ishowever perfectly amenable to numerical evaluation.

Note that the same formula may also be used to deter-mine the self impedance of a turn of small circular cross-section by simply choosing r and z to coincide with apoint on the surface of the wire, e.g. r = a + a and z = 0,where a is the radius of the wire.

3 Special cases

Two special cases merit consideration:

3.1 Case of coils in airIn this case, n2r = \i2z = nx (= n0) and c2 = 0. We thenhave T = P, and it is easily verified that Zc = 0. Thisleaves only the term jo>Lx, where Lx, given by eqn. 17,may be alternatively specified as:

(19)

(20)

cos (pz) dp

The integral involved is known [2], giving the result

2

k

where K(k) and E(k) are complete elliptic integrals of thefirst and second kinds, respectively, and where

Aar

z2 + (a + r)'(21)

Eqn. 20 is a classical result, given by Maxwell [3], andconfirms the validity of the analysis, at least so far as thisspecial case is concerned. More importantly, an alterna-tive means, i.e. eqn. 20, has been established for comput-ing Lv This is helpful since, despite appearances,numerical evaluation of the integral in eqn. 17 is rathermore exacting than the numerical evaluation of the inte-gral in eqn. 18.

It is also apparent that the first term in eqn. 16 may beinterpreted as corresponding to the effects of a primaryenergising field completely in air whilst Zc, which weshall refer to as the correction impedance, may be inter-preted as corresponding to the effects of the reflected fieldcaused by the introduction of the ferromagnetic core.

3.2 Case of zero flux penetration into the coreIn the limiting case as co -*• 00, or a2 ~* °° (a perfectlyconducting core), we expect zero flux penetration into thecore. In this case, | F | -*• 00 and the correction-impedanceformula simplifies to —j(oL2, where L2 is frequency inde-pendent as given by the formula:

( 2 2 )

The integral does not appear to correspond to any stan-dard form, but may be evaluated numerically without dif-

ficulty to give Ly in the formula

l -L2) (23)

In certain work, notably in the analysis of surge pheno-mena in transformers, it is customary to assume negligi-ble flux penetration into the core. It is thereforenoteworthy that a rigorously based formula for this rela-tively simple case is not available in the literature. Thisdeficiency is rectified by the formula of eqn. 23 in con-junction with eqns. 20 and 22.

4 Numerical calculations

For numerical treatment, it is convenient to present thebasic impedance formula of eqn. 16 as:

(24)

(25)

x F(P) cos (Pz) dp

where Lx is given by eqn. 20 and

F(P)=jcon1

~ — /(D

9(P)

with the auxilliary functions defined by:

fix) = x

and

g(x) = xK0(xb)

K^xb)(26)

The functions in eqn. 26 are readily evaluated from stan-dard formulas [4] and are finite for finite x. Note thatonly one complex function is involved, namely/(F).

The integrand in eqn. 24 is bounded for all P and gen-erally converges rapidly with increasing p. Numericalevaluation of Z thus presents no particular difficulties.Our experience shows that an accuracy of better than 1part in 103 can be obtained using 16-point Gaussianquadrature.

5 Closed cores

In many electrical devices, for example transformers, thecore will form a closed magnetic circuit. In practice, theshape of the core is usually toroidal or rectangular.However, with some justification, we shall convenientlyside-step the issue of shape by imagining the core to havebeen cut open and straightened out to its length /. Areturn flux path of zero reluctance is then provided byplacing the straightened-out core between two infiniteplates of perfect magnetic material (/1 = 00, a — 0) asshown in Fig. 2. This effectively closes the magneticcircuit as required.

This simulation is designed, of course, to facilitateanalysis. Nevertheless, it is soundly based, and may beexpected to give good accuracy for turns reasonably closeto the core, provided that / > b (assuming a realistic coreshape). It is thought that this latter condition will applyin many practical cases of interest.

5.1 Derivation of closed-core formulaLet the energising turn be located halfway along thestraightened core. Note that this can always be arranged

472 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 7, SEPTEMBER 1988

by simply choosing to open the actual core at the loca-tion most remote from the turn involved. The field is thensymmetrical about the plane z = 0. In particular, the fieldis the same at both boundary plates and minimum.

Fig. 2 Simulation of closed magnetic core

Since the plates are infinitely permeable and noncon-ducting, it follows that the simulation corresponds to theboundary condition Hr = 0 at z = ± 1/2. Lines of fluxtherefore enter or leave the plates at right angles. Fur-thermore, a line of flux leaving at the right-hand platemay be legitimately considered to immediately re-enterfrom the corresponding position on the left-hand plate.

A little thought confirms that the required boundarycondition will be equivalently satisfied, relative to—1/2 ^ z ^ 1/2, if the boundary plates are removed andan infinite number of images introduced along an infinitecore in the manner shown in Fig. 3. Thus, writing the

Fig. 3 Use of image turns to satisfy boundary requirements

general formula of eqn. 16 as

Z = I °° G(a, r,= I °J—

the corresponding mutual-impedance formula for theclosed core is simply

Z= [°°G(a,r,/3) £ e*J— oo k = — oo

Noting that

dfi

' fc=-oo V 'fc = - o o * fc= - o o

eqn. 27 immediately reduces to:

' t = — rr.

where

2nk

I

(27)

(28)

(29)

(30)

Thus, far from complicating the formula, the closed-corecase formally converts the integrals of the infinite-core

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 7, SEPTEMBER 1988

case to straightforward summations. More precisely, forthe case of a closed core, we have:

(3D

(32)

Z = jcoLi + Zc

where, on the basis of eqn. 17

1 k=-co

and, on the basis of eqn. 18,

2rc «zc - , L

' fc= -

(33)

where F(P) is defined by eqn. 25.Doubtless, the result of eqn. 29 could have been

derived directly from Fourier analysis. However, thegiven derivation is particularly compact.

In practical work, the infinite series in eqns. 32 and 33will need to be truncated. Noting that single-sided sum-mation is justified, we then have:

cos(&z) (34)

andT V - 1

x F(Pk)ak cos (Pkz){

J

where the Lanczos sigma factor [5]

sin (kn/N)

(35)

(36)(kn/N)

has been introduced to suppress possible Gibbs' oscil-lations. The constant C is given by

C =jcob <—I mblo(mb)

(37)

where m is defined in eqn. 7.The choice of N in eqns. 34 and 35 depends, of course,

on the degree of accuracy required. Note that theFourier-series nature of the equations implies mini-misation of error in a least-squares sense.

It should, however be pointed out that the series ineqn. 34 tends to converge rather less rapidly than that ineqn. 35, especially if r = a. Given this circumstance, it isrecommended that, instead of using eqn. 34, Lx be calcu-lated using the formula of eqn. 20 applied to the arrange-ment of Fig. 3 (with the core removed). In practice, only afew of the images will need to be taken into account.

5.2 Illustrative resultsEqn. 31 yields a complex value, say A+jB, for themutual impedance between filamentary turns at (a, 0) and(r, z). Let this impedance be written in the form:

Z=Rm((o)+jcoLm(co) (38)

where the notation emphasises that Rm and Lm are, ingeneral, frequency dependent.

Results, intended simply to demonstrate the power ofthe formula, have been computed for the illustrative case

473

a = 0.2 m, b = 0.1 m, / = 1.4 m with \i2r = \i2z, H2z/Hi =

1000 and a2 = 100 O" 1 m"1 . Contours of constant Lm

are shown in Fig. 4 for the limiting case co -*• oo. Compa-

Fig. 4 Contours of constant mutual inductance with zero flux penetrat-ion into core (increments of 0.02 pH)

centre line

rative results at a> = 2n 106 rad/s and a> = 2n 100 rad/s,shown in Figs. 5 and 6, respectively, illustrate the effectsof flux penetration onto the core. In all cases, the con-

Fig. 5 Contours of constant mutual inductance with flux penetrationinto core at 1 MHz (increments of 0.02 fiH)

centreline

Fig. 6 Contours of constant mutual inductance with flux penetrationinto core at 100 Hz (increments of 0.02 pH outside core, 4 fiH within core)

centreline

tours are drawn for increments of 0.02 piH except withinthe core in Fig. 8 where, to avoid clutter, the increment is4 /iH. Close to the energising turn, contours pack veryclosely and have been omitted. Contours of constant Rm

are somewhat similar, but are not given.

6 Formulas for coils

In practice, we are usually interested in the self andmutual impedances of coils composed of many turnsrather than the self and mutual impedances of individual

turns of small cross-section. Here we shall specialise onthe usual case of coils of rectangular cross-section asillustrated in Fig. 7.

w2

°2

•L_LJ Jcore

air

turns

coil 1 N2 turns

coil 2

Fig. 7 Coils of rectangular cross-section on infinite core

6.1 Coils on an infinite coreIn this case, as in previous cases, the impedance formulais presented in the form:

Z = jcoLx + Zc (39)

As previously identified, the inductance Lx correspondsto the situation where the ferromagnetic core is removedand replaced by air, i.e. Lx corresponds to the classicalcase of coils in a homogeneous air (or similar) medium.This case has been studied extensively in the literature. Inparticular, Grover [1] presents useful formulas/tables.Alternatively, it is often satisfactory to represent each coilin Fig. 7 by one or two filamentary coils. Calculations arethen based on the formula of eqn. 20. Note that \i\Zc\ $>coLu which is frequently the case, it will not be necessaryto compute LY with great accuracy.

Surprisingly, derivation of an appropriate formula forZc is perfectly straightforward without recourse to fila-mentary approximations, etc.

On the reasonable assumption, indeed extremely rea-sonable given that we are now dealing with the effects ofa secondary field, that coil currents may be taken to beuniformly distributed over each coil cross-section, fullaccount of the sections in Fig. 7 is obtained by taking Zc

as

z. =rco rw2/2 rwi/2 fr2 p,2 |>aK (0a)K (0r)j

' \ \ \ ^777-;JO J-w2/2 J-wi/2 Jn Jai L ^l\P°)

x F(/?) cos [J?(z + T2 — T J ] rfa dr dti rft2 dfi (40)

The internal integrals are readily evaluated to give theformula:

Z =c hlh2wlw2

\PiWa2, Pav)Px{Pr2, firJQJo

F(fi) cos (fiz) (41)

474 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 7, SEPTEMBER 1988

where

p2

U(x, y) = ^2 [cos ( i ^ ) -

with

na.

(42)

(43)

where the modified Struve functions Lv(cc) are defined by

£ » = £(a/2)v+2m + 1

x [F(m + 3/2)r(v + m + 3/2)] (44)

Although the integrand of eqn. 41 is somewhat morecomplicated than for the filamentary case (eqn. 24),numerical computation per se is no more exacting.

For self-inductance calculations, say for coil 1 in Fig.7, the appropriate value of correction impedance is givenby simply putting N2 = Nu w2 = wx, h2 = hx and z = 0in eqn. 41.

6.2 Coils on closed coresCalculation of Lx for this case is the same as in the pre-vious case except that, in addition, account must betaken of coil images. In practice, it is entirely justified torepresent image coils on a filamentary basis as in Fig. 3.As noted previously, only a few images need be con-sidered.

The appropriate formula for Zc is given by presentingeqn. 41 in summation form, i.e.:

Zc = 7 NiN2\ C +h1h2w1w2

i

cos(pkz) (45)

where C is as previously specified in eqn. 37.For the purposes of illustration and comparison, Fig.

8 shows computed contours of constant mutual indue-

Fig. 8 Contours of constant mutual inductance with flux penetrationinto core at 1 MHz (increments of 0.02 pH)

centreline

tance for the case hx = wt = 0.1 m, h2 = xv2 = 0.1 m,a = 0.2 m, / = 1.4 m, /i2r = \i2z, A*22/^I = 1000, t = 0.1 m,a2 = 100 Q"1 m"1 and co = In 106 rad/s. Co-ordinates

correspond to the position of coil 2. Naturally, there areregions which coil 2 cannot physically occupy. The factorN1N2 has been suppressed and therefore Fig. 8 may becompared directly with Fig. 5.

Comparison of Figs. 5 and 8 gives an impression ofthe extent to which mutual impedance calculations basedon single filament approximation is justified. Note,however, that exact calculations are not much moredemanding of computer time than calculations based onapproximating filaments.

7 Conclusions

A new set of formulas has been established for calculatingthe self and mutual impedances of coils on homogeneousferromagnetic cores of circular cross-section.

The given formulas, which cover many of the casesthat are likely to arise in practice, are fully consistentwith Maxwell's equations and, as such, offer the ultimatein accuracy.

The formulas for an infinite core do call for numericalintegration. However, with available computers andestablished numerical quadrature techniques, this is nogreat obstacle.

For coils on closed cores, as simulated in the analysis,integration processes formally reduce to summation ofconvergent infinite series. Thus, in principle, it is possibleand straightforward, to attain any required degree ofaccuracy. Ultimately, accuracy depends on the extent towhich our closed core simulation is justified. This may beclarified, for example, by measurements on actual cores.It is not, however, expected that serious error will arise inpractice, subject to conditions specified in the text.

It is felt that the work could find particular applicationas a gauge by which simpler, approximate, formulas maybe properly accredited. Note, however, that 'exact' calcu-lations are not particularly time consuming.

Further theoretical or experimental work could bedone to establish the extent to which the formulas areapplicable to, or may be adapted to, the case of lami-nated cores and/or cores of noncircular cross-section.

8 References

1 GROVER, F.W.: 'Inductance calculations: working formulas andtables' (Dover Publications, 1946)

2 LUKE, Y.L.: 'Integrals of Bessel functions' (McGraw-Hill Book Co.Inc., 1962)

3 MAXWELL, J.C.: 'A treatise on electricity and magnetism' (Oxford,Clarendon Press, 1873)

4 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of mathemati-cal functions' (Dover Publications, 1965)

5 LANCZOS, C: 'Applied analysis' (Pitman, 1957)

9 Appendix

9.1 Solution of eqn. 5As a convenient device, we imagine medium 1 to bedivided into regions b ^ r ^ a~ and r ^ a+ by a thinboundary tube of mean radius a and thickness 2e. Thusa~ = a — e and a+ = a + e, where e is infinitesimallysmall. The RHS of eqn. 5 is identically zero in the twodefined regions, and therefore the following general solu-tions apply:

DKtfr) (46)

£J, =where we may immediately put F = 0 so as to satisfy therequirement E$1 -> 0 as r -* co.

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 7, SEPTEMBER 1988 475

If we now consider the flux flowing down the bound-ary tube in the z-direction, it is evident that this tends tozero as e -> 0, even at z = 0. It follows that E4>1 and hence££, must be continuous across the boundary. In particu-lar, since the functions involved are continuous, it followsthat:

Cl.iPa) + DKtfa) = GK^pa) (47)

A second useful identity follows from eqn. 46:

E*b = Cltfb) + DKtfb) (48)

A third identity is now required. This may be deduced byfirst transforming eqn. 1, as applicable to medium 1,using the Fourier transform prescribed in the text. Wethen have:

(49)

'I (50)

i) (51)

d(r - a) (52)

Integrating this equation across the boundary tube gives:

dr - [ # * (a+) - J5f» (<T)] = /* (53)

From eqns. 49 and 51:

P2 dH^'*' ~ ~dr~

rJa~

The first term on the LHS of eqn. 53 vanishes as ei.e. as the integration range tends to zero, leaving:

H*l(a+)-H*l(a-)=-I<t>

From eqn. 50

- 0 ,

(54)

dr

+ DKt(Pr)}ldr

Thus eqn. 54 becomes:

PGK0(Pa+) + a-) - pDK0(pa+) = -joouJ* (55)

where, since the functions involved are continuous, wemay now identify a+ and a~ with a. Eqn. 55 is our thirdidentity.

Using the general relation:

x[_I,(x)K0{x) + /0(x)Ki(x)] = 1

with the three identities established above gives

C = -jc

D = {E*b

G = {E*b +ja>n1alKl(Pa)Il(Pb) - K Y{pb)l

Using these expressions in eqn. 46, together with F = 0,yields the result given in the text.

476 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 7, SEPTEMBER 1988