Essays on the Formal Aspects of Electromagnetic Theory (pp. 591~24) ed. A. Lakhtakia © 1993 World Scientific Puhl. Co.

SCALAR POTENTIAL FORMULATIONS FOR MAGNETIC FIELD PROBLEMS

IOAN R. CIRIC Department of Electrical an4 Computer Engineering

The Univer1it1 of Manitoba

Winnipeg, Manitoba R3T !Nf, Canada

ABSTRACT

Applying the classical Amperian and Coulombian representations for magne­tized media, given distributions of stationary or quasistationary current are mod­eled in terms of fictitious distributions of magnetization and of equivalent magnetic charge. A scalar potential function is defined for analyzing magnetic fields on the basis of these models. The main results obtained recently by the author, relative to this new modeling method and the corresponding scalar potential formulations, are presented. A general modeling theorem and a theorem showing the necessary conditions for the new scalar potential to be a single-valued function of position have been proved. Examples of modeling volume current distributions, with new, simple formulas for their magnetic field, and of scalar potential formulations for boundary-value problems are given to illustrate their usefulness. A computational efficient finite-element formulation based on mapping unbounded regions of an ar­bitrary geometry, with anisotropic and inhomogeneous materials, onto bounded domains is also presented. The same procedures are applicable to other physical fields governed by the same equations.

a,b,c B dl,dS G H i, I J,J.

=

= = = =

List of principal symbols

conductor or coil dimensions magnetic flux density vector elements of length and surface Green's function magnetic field intensity electric currents volume and surface densities of conduction current

591

592 I. R. Ciric

Jae M,Mc ii, ii12 r,r' R r,</> t :r, y, z x,y,z v ltc µ,µo Pc1P1c

I. Introduction

= = = =

= = = =

fictitious surface current density magnetization and fictitious magnetization normal unit vectors observation and source point position vectors distance IRI = Ir - r'I circular cylindrical coordinates time Cartesian coordinates Cartesian axis unit vectors "del" operator scalar potential permeability of medium and free space volume and surface magnetic charge densities

The macroscopic electromagnetic field in the presence of motionless media is described locally by the Maxwell equations:

8B V xE=-at

8D "VxH=J+-

8t "V·D=p

"V·B=O

(1)

(2)

(3)

(4)

where the conduction current density J and the free electric charge density p

are related by the law of conservation of charge [from (2) and (3), V .J = -\7]; the electric and magnetic flux densities (or inductions) D and B are related to the respective field intensities E and H and to the (electric) polarization P and to the magnetization M by:

D = !oE+P

B = µo(H+M)

(5)

(6)

with !o and µ0 being, respectively, the permittivity and the permeability of free space. The electric and magnetic field quantities can be determined in terms of general electrodynamic vector and scalar potentials, which are completely defined by choosing certain gauge conditions in order to fix the divergence of the vector potentials.23

•24

Scalar Potential Formulation• for Magnetic Field Problem• 593

For quasistationary fields, when the displacement current density fJD/fJt can be neglected as compared to J from point of view of the magnetic field produced, Eq. (2) reduces to

V'xH=J. (7)

Now, the vector field J is solenoidal and its normal components are contin­uous across interfaces between different regions. The analysis of these fields by using vector and scalar potentials can be simplified in various ways. For instance, the magnetic field intensity in the previous equation can be decom­posed in two components: one which satisfies the inhomogeneous Eq. (7) (with simpler properties than those for H) and another one which is irrotational and can, therefore, be derived from scalar potentials. In the case of eddy-current problems, the induced current density is unknown, but the first component can be chosen to be zero in nonconducting regions, such that only the scalar potential needs to be computed there. However, the general three-dimensional eddy-current problem cannot be solved in terms of a single scalar function.3

Various formulations for quasistationary magnetic fields in terms of combined vector and scalar potentials, along with associated uniqueness theorems, are reviewed in Ref. 1.

In principle, wherever J = 0 the stationary or quasistationary magnetic field intensity is irrotational and can be derived from a magnetic scalar po­tential (from V' x H = 0, H = -V~m). The advantages of using a scalar potential consist in that it is easier to compute it from the corresponding partial differential or integral equations than the vector potential, especially in three­dimensional configurations, where the vector potential has, in general, three components. On the other hand, it is possible to visualize a scalar potential, the surfaces of constant potential being orthogonal to the field intensity. The difficulties related to the usage of this scalar potential consist in the fact that it is a multivalued function of position in the presence of current distributions. This classical scalar potential has been used for calculating the field due to line currents in a variety of magnetic devices4 and in problems of pronounced skin effect.5 Scalar potentials and the corresponding multipole expansions were also constructed for specified regions outside bounded surface or volume cur­rent distributions.2•

27 It should be remarked that the lowest nonvanishing term in the expansion of the field intensity outside a confined quasistationary cur­rent distribution corresponds to an "effective" magnetization, which is related simply to the current density. 13

594 I. R. Ciric

When the current density distribution in (7) is given, then the field inten­sity can be determined by the superposition of the solenoidal field produced by the given, localized distribution of current in a homogeneous, unbounded region and of the irrotational field due to the magnetization in the magnetic materials within the region considered. Most authors have used this type of a decomposition, with the solenoidal field, which satisfies the inhomogeneous Eq. (7), calculated by applying the Biot-Savart formula. For the irrotational component, which is derived from scalar potentials, some authors consider two different potential functions, one for subregions without currents and the other one for subregions with currents, the conditions at interfaces being imposed by using the previously computed solenoidal field component.22•17 Blewett15 chose the field component satisfying (7) to be zero outside the outer surfaces of the current coils and with a zero tangential component on these surfaces, and pre­sented a formula for this field in the case of coils of rectangular cross section; at the same time, the potentials inside and outside the coils were adjusted to be continuous within the whole region.

In this archival paper we present scalar potential formulations for magnetic field problems where the stationary or quasistationary current distributions are known, such as those regarding various electromagnetic devices, or magnetic systems in tokamak configurations, magnetohydrodynamic energy converters, particle accelerators, etc. These formulations have been recently developed by the author on the basis of a new modeling method and of an associated scalar potential, which is a single-valued function of position throughout the entire region. The modeling procedure is simple and flexible, in the sense that for a given problem one can choose a formulation which is considered to be the most appropriate in order to reduce the amount of computation required.

The magnetic field under consideration satisfies the Eqs. ( 4), (6), and (7). Across the surfaces of discontinuity of the field quantities, (7) and ( 4) are replaced, correspondingly, by

812 x (H2 - H1) = J.

812 · (B2 - B1) = 0

(8)

(9)

where n12 is the normal unit vector from side 1 to side e of each surface and J, is the surface current density eventually present. We model the given current distributions by using distributions of fictitious magnetization. It is shown that for a large class of practical boundary-value problems, this fictitious magnetization can easily be chosen such that the calculation of the field due

Scalar Potential Formu/a.tion1 for Magnetic Field Pro6/em1 595

to current distributions is reduced to that of the field determined by surface charges, with the associated scalar potential being a single-valued function of

position.

II. Modeling of Given Volume Current Distributions

The classical models of charge (or Coulombian) and of current (or Amperian) for electrically polarized and for magnetized media have been widely used to calculate the electric and magnetic fields due to such media.i 2,is These models are based on the fact that under static or stationary conditions, from the point of view of the macroscopic field produced in free space, as well as of the ponderomotive forces (forces and torques) exerted upon them in external fields, an electrically polarized or a magnetized volume element is equivalent to an elemental duplet of charges or to an elemental current loop.25 For example, in the case of a given volume distribution of magnetization M, the equivalent volume and surface fictitious charge distributions in free space are, respectively,

Pm= -µoV·M

Pam= -µoili2 · (M2 - Mi) .

(10)

(11)

These charge distributions produce in free space a field H which is identical to that due to the given magnetization, and the field B is to be determined from Eq. (6). For the same case, the Amperian model consists of the following equivalent volume and surface current distributions in free space, respectively,

lm = V xM

lam= ili2 X (M2 - Mi) .

(12)

(13)

These current distributions produce in free space a field B which is identical to that due to the given magnetization, and the field H is obtained from Eq. (6).

To illustrate our modeling method, consider a volume distribution of sta­tionary or quasistationary current of known density J. The basic idea of the method is to use both the Amperian and the Coulombian models in the fol­lowing way: first, we apply the Amperian model in a reverse sense by treating the given volume distribution of current as if it would represent an Amperian current distribution corresponding to a fictitious magnetization Mc, such that (see (12))

(14)

596 /. R. Ciric

then, according to the Coulombian model, we find the charge distribution equivalent to the distribution of Mc and determine finally the corresponding magnetic field. Since we do not impose any a priori conditions for its diver­gence, the magnetization field Mc is not uniquely determined by the Eq. (14) alone, and simple expressions for Mc can be found easily for practical current distributions. As illustrated later on in this paper, for a given current distri­bution we may construct more than one model, and the magnetization Mc may be chosen different from zero outside the regions with J 'I 0, but its curl must be zero wherever J = 0. In what follows we assume that Mc has no line or point singularities in the region considered, i.e. its closed line and closed surface integrals vanish when the path of integration shrinks to a point and the surface of integration shrinks to a line segment or to a point.

According to the Amperian model, the effect of the magnetization Mc is the same as that of the given distribution of volume current J and of a distribution of surface current over the surfaces of discontinuity of Mc, of density

(15)

Therefore, for evaluating the magnetic flux density, the given distribution of J can be replaced by the fictitious distributions of magnetization Mc and of surface current of density lac on all the surfaces of discontinuity of the tangential component of Mc. For the special case of a nonmagnetic region of permeability µ 0 we have

(16)

where BM< is due to Mc and BJ.< to lac· The field intensity HMc produced by Mc in free space is, on the other hand,

identical to that produced by the following distributions of volume charge density and of surface charge density on all the surfaces of discontinuity of the normal component of Mc (see (10) and (11)):

Pc= -µo"V ·Mc

Pac= -µon12 ·(Mel - Mei) ·

Thus

(17)

(18)

(19)

Scalar Pole11.tial Formulation. for Magnetic Fielil Problem• 59i

and from (16)

(20)

with HJ •• = BJ • ./ µo. As a result, in a region of permeability po, the field intensity due to a

distribution of volume current J can be determined as the superposition of Mc and of the field intensity due to the charge distributions Pc and P•c• and to the surface current distribution Jae.

For the simplest case of the field produced in an unbounded, homogeneous three-dimensional space by a given volume current distribution confined to a finite region, an elementary formula can be written in the form

H(r) = Mc(r) + 4~ j Jac(13

X R dS'

+ _1_ (/P•c(r')R dS' + jPc(r')R dv') (2l) 411'µ0 Ra Ra

where dS' and dv' are the elements of surface and volume, respectively, r and r' are the position vectors defining the observation point and the source point, respectively, with R = r - r', and the surface integrals are evaluated over the surfaces of discontinuity of Mc·

For regions with a linear, isotropic, and homogeneous magnetic material, the above expressions remain valid if µo is replaced by the permeability µ of the medium.

In the more general case of the magnetic systems with inhomogeneous (but isotropic) magnetic materials, of a permeability µ which may vary even within the regions occupied by the distributions of J and Mc, the models for the given volume distributions of current are constructed by keeping the local values of permeability and with the same distributions of Mc and lac satisfying the Eqs. (14) and (15). The corresponding results can be summarized in the form of the following modeling theorem7

: Let J(r) be the piecewise continuous density of a given volume distribution of stationary or quasistationary current, and

let Mc(r) be any piecewise continuous fictitious magnetization which satisfies

the equation V x Mc = J and has no point or line singularities. Let Pc = -V ·(µMc), Psc = -nn ·(µ2Mc 2 -µ1Mc1 ), and Jae= -n12 X (Mc2 -Mei) be, respectively, densities of fictitious volume charge, surface charge, and surface

current, where µ is the permeability and n12 the unit normal on the surfaces

of discontinuity of Mc and ofµ. Then, the magnetic field intensity due to J is

598 /. R. Ciric

equal to the sum of Mc and the field intensity due to the distributions Pc, Pac, and lac·

It should be noted that if Mc is chosen such that lac in (15) is zero ev­erywhere, the given volume current distribution is modeled in terms of Mc and charge distributions only; the different H - Mc can now be derived from a single-valued scalar potential and the particular expression (21) does not contain the first integral. On the other hand, if Mc is chosen such that Pc is zero everywhere, the volume current distribution is modeled in terms of Mc and only surface distributions of charge and current, and the field in (21) is expressed only in terms of surface integrals; when Mc is zero outside the region with l f; 0, then these surface integrals are taken only over the boundary of that region. It should be remarked that the field calculated by the formula (21) is identical to the field given by the classical Biot-Savart volume integral formula, as well as that by choosing various fields Mc, corresponding to the same distribution of l, one obtains various vector integral identities.

III. Modeling of Generalized Current Distributions

The modeling procedure presented for volume current distributions is also applicable to current sheets of a given surface density l 8 , with a fictitious magnetization Mc chosen to satisfy the equations V' x Mc = 0 and 812 x (Mc2 - Mc1 ) = ls instead of (14). Filamentary current loops (eventually closed at infinity) can be modeled by distributions of surface magnetization Mac over arbitrary open surfaces bounded by the loops. For a loop C carrying a current i,

Mac= iis;i (22)

with iis; being the unit vector normal to an arbitrary open surface S; bounded by C and associated with the direction of i according to the right-handed screw rule.

In the presence of volume, surface, and filamentary currents in a region D, the generalized distribution of volume current density can be written in the form20

.1(r) = l(r) + l 9 (r)6(s) + t(r)i6(c) (23)

where: l is the ordinary volume density of current, defined for r rJ. S and r rJ. C, ls is the surface density of current given on piecewise smooth surfaces S inside D, i is the intensity of current along piecewise smooth curves C inside D, t is the unit vector tangential to C, along the direction of the current i,

Sc4'ar Polential Formulation• for Magnetic Field Problem• 599

and 6(s) and 6(c) are generalized functions which satisfy the relationships21

L f(r)6(s)dv = ls /(r)dS

L f(r)6(c)dv = l f(r)dl

for any function f (r) given in D and continuous on S and C. Equation (7) can be written in a generalized form as

curl H = .J

where the generalized curl operator 20

curl H = V x H +curls H6(s) + curll H6(c)

with curls and curll being the surface curl and the line curl, respectively,

curls H :: n12 x (H2 - H1)

curll H = t Jim 1 H · dl ; c, ..... p. le,

(24)

(25)

(26)

(27)

(28)

(29)

the line curl is defined at the points Pe of a curve C which has the property that the circulation of the vector field along any closed path Ct, enclosing once the curve, is different from zero in the limit when Ct shrinks to Pe, with the unit vector t, tangential to C, associated with the direction of integration according to the right-handed screw rule.

In order to model the whole distribution of given current (23) by means of a fictitious magnetization Mc and distributions of charge only, the distribution of Mc must satisfy the generalized equation

curl Mc= .J (30)

with no restrictive conditions imposed on the divergence of Mc. As shown above the volume and surface currents in (23) can be modeled in terms of an ordinary volume distribution of magnetization, with V x Mc = J and curls Mc = J.. For filamentary currents, the volume magnetization corre­sponding to Mee in (22) is

(31)

and the surface density of (p0sitive) charge of the equivalent double layer is µi6(s;)· The line curl of the magnetization in (31) is just ti, which corresponds to the last term in (23).

600 I. R. Cine

IV. Single-Valued Scalar Potential in the Presence of Given Current Distributions

Assume that the current distribution given in a specified region is modeled in terms of a chosen fictitious magnetization Mc as shown in the previous sections. Then Eqs. (7) and (14) yield

V x (H- Mc)= 0

and hence H - Mc can be derived from a scalar potential 4'c, as

H-Mc=-V~c.

(32)

(33)

Once the potential <I>c is determined, the magnetic field intensity is given at any point by

(34)

The equation satisfied by •c for an isotropic medium, without true per· manent magnetization, for instance, can be derived from (4) and (6), with the latter equation written as B = µH, in the form

V24>c + Vµ · V~c =_Pc (35) µ µ.

where µ. is the local permeability and Pc is the volume density of fictitious magnetic charge corresponding to Mc and given by

Pc = -V ·(µMc) . (36)

In the case of a linear, isotropic, and homogeneous medium, ~c satisfies the Poisson equation

(37)

with Pc/µ= -V ·Mc. If Mc in (14) is chosen such that its volume divergence is equal to zero, then !Jc satisfies the Laplace equation within linear, isotropic, and homogeneous media,

(38)

The conditions for the tangential components and for the normal compo­nents of V~c at the surfaces of discontinuity of the quantities H, B, and Mc are derived from (8), (9), and (34), in the form

(39}

(40)

Scalar Potential Formtilationa for Magnetic Fie/ti Pro6/em1 601

where lac is given in (15) and

(41)

Equation (39) shows that wherever l. +lac = 0, the tangential component of V•c is continuous, which is equivalent to the continuity of the potential itself, i.e.

When 1'1 = µ2 = µ, Eq. (40) becomes

8ctc1 8()c3 Pac 8n12 - 8n12 = µ

with Pac/µ = -n12 · (Mc3 - Mc1 ).

(42)

{43)

Due to the flexibility of the modeling procedure presented, for a large class of practical systems the magnetic fields can be analyzed by constructing models which contain only fictitious surface charge distributions within the regions considered. By using this type of models, the calculation of the field due to general current distributions is reduced to that of the field due to surface charges. Now the potential etc is a single-valued function of position, according to the following general theorem 7: If the volume, surface, and line distributions of stationary or quasistationary current inside a region are modeled by means of a distribution of fictitious magnetization whose generalized curl is equal to the generalized volume current density, then the difference between the magnetic field intensity and this fictitious magnetization can be derived from a scalar potential which is a single-valued junction of position in that region. Indeed, applying the generalized Stokes' theorem20 to H - Mc = -V•c along any closed path r in the region considered, and using (26) and (30), yields

J (H - Mc)· dl = 1 (curl H - curl Mc)· dS = 0. (44) J'r •r

Thus, the scalar potential difference between any two points P and Po in the region is independent of the path of integration between the two points,

()cp - c)cp0

= - {p (H - Mc) • dl }po

{45)

and, therefore, c)c is a single-valued function of position, as in the case ol electrostatic fields.

602 /. R. Ciric

Magnetic field analysis and synthesis by using this scalar potential is much simpler than that based on various other methods developed previously.

V. Examples of Modeling Volume Current Distributions

The only condition for the fictitious magnetization Mc used in our modeling method is to satisfy the Eq. (14), without any other restrictions inside or outside the region occupied by the given current distribution. Therefore, for each configuration there are a few possible models to choose from in order to optimize the field calculations. For instance, in the case of practical coils consisting of straight current tubes or current tubes in the form of sections of toroid, with the current density J being constant over the cross section, one can choose Mc inside the tube as having only one component, in a direction perpendicular to J and depending in magnitude linearly on the distance along the direction of Mc x J. Such an Mc has a volume divergence which is equal to zero.

A. Long, straight conductors

In the case of a straight, theoretically infinitely long cylindrical conductor of an arbitrary cross section (see Fig. 1), carrying a current of density Jin the longitudinal z-axis direction, Mc in (14) can be chosen as

Mc(:c, y) = inside the conductor (46)

0 outside the conductor

where i: and y are the unit vectors along the positive :c and y Cartesian axes, respectively, and C1 and C2 are two arbitrary constants, C1 I C2. If the variation of J over the conductor cross section is such that V ·Mc = 0 (e.g. if J depends only on one coordinate, say :c, with C1 = 0), then, for a homogeneous medium of permeabilityµ (36), (41), and (15) become

Pc= 0

Pac = µn · Mc on Co

lac= ii X Mc on Co

(47)

Scalar Potential Formvlationa for Magnetic Field Pro6/em1 603

y

x Fig. 1. Cross section of cylindrical conductor.

with C0 being the conductor cross-sectional contour and n its outwardly ori­ented normal unit vector. Thus, the parallel-plane magnetic field intensity produced by such a conductor in an unbounded, homogeneous space can be ex­pressed in terms of contour integrals only (the corresponding two-dimensional form of (21)) as

H(r) = Mc(r) + _.!.._ J lsc(r') X R dl' + _1_ J Psc~:)R dl' . (48) 211" fco R2 211"µ !co

For a constant current density, J = const, with C1 = 0 for example, (46) becomes

I yJz

Mc(z, y) = O

and the field intensity in ( 48) is

inside the conductor

outside the conductor

H(z, y) = Mc(z, y) + :11" (fc0

z'~R di.'+ z x fc0

z'~R di.')

(49)

(50)

where n~ and n~ are the direction cosines of n in primed coordinates, with R = x(z - z') + y(y - y'). The contour integrals in (48), (50) require a

604 /. R. Ciric

y /1- D 'Sc

x

Fig. 2. Cross section of rectangular conductor.

substantially reduced amount of computation 88 compared to the integral over the conductor cross section in the Biot-Savart formula.

A rectangular crou !edion conductor carrying a current of constant density, J = iJ, can be modeled 88 shown in Fig. 2, for instance, with Mc chosen to be

y J z inside the conductor

Mc(z,y)= yJa forze(a,a'), ye(-~·~) (51)

0 elsewhere

a' being arbitrary. There are only surface charges and surface current in such a model, which are given by

±µJi: for z e (0, a), b

y=±-2 (52) Pee= b

±µJa for z e (a,a'), y=±-2

J.c = zJa for z =a', ye(-~·~) (53)

If a' = a the fictitious magnetization Mc :/; 0 only inside the conductor. When a' -+ oo, then the current sheet is removed at infinity and the model contains

Scalar Potenlial Form·llation1 for Magnetic Field Pro6lem1 605

@

J c

Fig. 3. Modeling of polygonal cross section conductor.

k-1

only surface charges. In each model the field intensity H is the same and this flexibility in constructing the model allows a much simpler formulation of re­lated boundary-value problems.8 Obviously, the same type of simple models can be constructed in the direction of negative z axis, and also with Mc cho­sen along z axis. The corresponding integrals can be performed analytically, yielding for a conductor of rectangular cross section the well known expressions obtained by double integrations in the Biot-Savart formula.19 It should be re­marked that for a - 0, but with the total current remaining finite, we have the models corresponding to an infinitely long, straight current sheet. When both a - 0 and b - 0 the models corresponds to an infinitely long, straight filamentary conductor carrying a finite current.

A general polygonal cross section conductor carrying current can be mod­eled as indicated in Fig. 3. For de solid conductors or, approximately, for

606 /. R. Ciric

k+I

L Fig. 4. Notation relative to side k of the cross-sectional polygon.

practical coil sides consisting of uniformly and closely wound wires carrying de or ac the current density is constant. The fictitious magnetization Mc can be chosen as in (49) and the model contains only surface charges and surface currents, which vary linearly along any side k of the cross-sectional polygon, according to ( 47):

lac• = Dl: X Mc = i.Mcnzl: (54)

where nz1: and n,1: are the direction cosines of the outward normal n1:. By performing the integrations in (50) corresponding to an arbitrary polygonal side k, a new, simple formula has been derived for the magnetic field produced by such a conductor at any point, in the form: 9

J [ R' ] H(z, y) = 211" ~ In R~ - ·nix R1: (55)

where the notation for each side k is shown in Fig. 4. 'Yl: = er~ - er~ is the angle under which the polygon side k is seen from the observation point and it is measured from R~ to R~, being taken to be positive in the counterclockwise direction and negative in the clockwise direction. R1: is the "altitude" vector of the observation point with respect to the side k,

Scalar Potential Formulationa /or Magnetic Fieltl Problem• 607

z

,--.:::-------/ -........

', ®J \ '- I -..........._ I

....... ______ ---~-......-

r

Fig. 5. Axial section of toroidal. conductor.

Rt = x(z - Xt) + y(y - Yt) . (56)

In terms of the Cartesian coordinates of the cross-sectional polygon vertices, we have

2 -1 (x - xi)(y - Yi:) - (x - xJ,)(y - y't)

""k = tan ' Rl:Ri + (x - xA:)(x - xi)+ (y- yt)(y-y'D

(57)

x(x1 - xlY + xl,(v: - y't)2 + (y- y'k)(x1 - xl,)(yk - Ya,) Xk =

(xi - zl:)2 + (yi- Yi:) 2 (58)

_ yl:(x1 - xJ,) 2 + y(yk - Yi:) 2 + (z - zl,)(zk - xJ,)(yk - Yi:) Yk - (xi_ z~)2 + (y~ -Yl:)2

Formula (55) is also applicable to very long, straight hollow conductors whose cross section is bounded by two arbitrary polygons. Expressions for the field from conductors whose cross-section is a regular polygon or a rectangle are obtained as special cases of this new general formula.

B. Toroidal conductors

For a toroidal coil of an arbitrary cross section, with the circular cylindrical coordinates ( r, </>, z) associated as shown in Fig. 5 and carrying a current in the

608 J. R. Cine

azimuthal direction, J = ,j,J, the fictitious magnetization Mc in ( 14) can he chosen, for example, as

Me(z,r) = -z J;

0 J(z, r)dr inside the coil

0 outside the coil (59)

where ro is an arbitrary distance from the z a.xis. If the medium is homogeneous of permeability µ and J depends only on the r coordinate, then V · Mc = 0 and models with only surface distributions of charge and current can be con­structed. With the corresponding quantities in (47), the field in (21) can he expressed in terms of integrals over the cross-sectional contour, whose inte­grands contain complete elliptic integrals.

For a toroidal coil of recta.ng-.lar cross section, with J = const, we can use the models shown in Fig. 6. Choosing

-zJ(r - a) inside the coil

0 outside the coil (60)

yields the model in Fig. 6(a), with surface distributions of charge and current,

P•e = :r-µJ(r - a) for z = ±c, r E (a,b) (61)

lsc=tbJ(b-a) forzE(-c,c), r=b.

Choosing -i.J(r - a) inside the coil

Mc= -zJ(b-a) forzE(-c,c), rE(b,oo) (62}

0 elsewhere

yields the model in Fig. 6(b). with a surface distribution of charge only,

Pac= I ::r-µJ(r - a) for z = ±c,

::r-µJ(b - a) for z = ±c,

r E (a,b)

re (b,oo} (63}

Other equivalent models for this particular volume current distribution can be obtained by cho08ing

I zJ(b - r) inside the coil Mc=

0 outside the coil {64)

Scalar Potential Form11lation1 for Magnetic Field Pro6/em1 609

z

~c µ

r

(a)

j z f;c D (_ µ.

--- ·------------- - - - - -------·--·---- -~ ~

1 I\ ®B I\ c I\ v l ~ ... v [\ l v .....

N r v [\ c LJ ~ -+++++++++++++++• + + +

f-Q -l + • ++++++ ...................

P.. b SC

(b)

Fig. 6. Toroidal coil of rectangular azimuthal section: (a) model with surface current and charge; (b) model with surface charge only.

which yields

pgc = ±µJ(b - r) for z = ±c, r E (a,b)

J.c=4'J(b-a) forzE(-c,c), r=a (65)

610 [. R. Ciric

h . 14 or by c oosmg

zJ(b- r) inside the coil

Mc= zJ(b- a) for z E (-c,c), rE(O,a) (66)

0 elsewhere

which yields only

Pac= I ±µJ(b- r) for z = ±c, r E (a,6) (67)

±µJ(b- a) for z = ±c, r E (O,a)

As in the case of a straight conductor of rectangular cross-section, math­ematically equivalent models can be obtained from the model presented in Fig. 6(b), for instance, if the distribution of fictitious magnetization Mc is ended at any r = b' > b, by placing a current sheet of density lac= t/>J(b- a) (as in(61)) at z E (-c,c), r = b'. In the next section we shall use this type of models for formulating boundary-value problems in terms of a single-valued scalar potential. For b - a -+ 0, but with the total current maintained finite, we get the models corresponding to a circular cylindrical current sheet. The particular model described by (66), (67) is now the classical model (Coulom­bian or Amperian) used for a right cylinder of finite length, which is uniformly magnetized in the axial direction. If both b - a -+ 0 and c -+ 0, then the models correspond to a circular filamentary loop carrying a finite current.

Similar models can be constructed for polygonal cross section toroidal coils.

C. General current coils

Three-dimensional magnetic fields from practical current coils can be cal­culated with a desired degree of accuracy by decomposing the coils in straight segments of rectangular cross section, whose sides are trapezoidal in general, as shown in Fig. 7(a). For a uniformly distributed current, one can choose the fictitious magnetization Mc in (14) to be different from zero only inside the coil, for each segment as shown in Fig. 7(b), perpendicular to the sides 1 and 3 (from 1 to 3), and increasing linearly with the distance from the side k = 0, where Mc = 0, to the side k = 2, where Mc = Ja. The divergence of this field is zero and, for a homogeneous medium of permeability µ, the model will have for each segment a uniform surface current of density lac = J a, flowing over the side 2 in the direction of J, and negative and positive magnetic sur­face charges on the sides 1 and 3, respectively, of a density which increases in

Scalar Potential Formulation1 Jor Magnetic Fie/ti Pro6lem1 611

magnitude linearly with the distance from k = 0, where Pac = 0, to k = 2, where IPecl = µJa. On the sides 4 and 5 there are positive or negative charges varying linearly from k = 0 to k = 2, according to (18) (with µo replaced by µ); the total surface currents given by (15) on sides 4 and 5, obtained by the superposition of the contributions from adjacent segrnents, are equal to zero.

(a)

k=2 - - T

k=O

~c {b)

Fig. 7. (a) Straight segment of current coil. (b) Cross section with fictitious magnetization and surface current and charge.

612 I. R. Ciric

x

µ

z

Fig. 8. Local coordinates for a side le of coil segment.

The contribution of a coil segment to the resultant magnetic field can be determined by calculating the field due to an arbitrary, trapezoidal side k, as shown in Fig. 8, which may have either a uniform surface current of density Jac along the positive z axis or surface charge of a density varying linearly from Pac = 0 at y = 0 to Pac = Pd at y = d. Performing the corresponding surface integrations in (21) yields the components of the field intensity at any point (a:, y, z) in the local Cartesian coordinate system:10

H(J) = Jae~ z: 41" '

H (J) - Jae II - '}', 41"

H~1 > = 0

for a side k = 2 carrying surface current of density Jae = J a, and

H(P) = ~(d + Y'l') z: 41rµd

( ) Pdd [ 1 z2 HP = - -(-y~ + z1)- -(R1 - R2)

II 41"µ d2 dll2

_ z3 - z4 (R _ R ) P12~12 P34~34] d12 3 4 + ,3 + ,3

34 12 34

H(P) _ Pdd [Ri - R2 + R3 - ~ + Q12~12 + Q34~34] a - 411'µ ll2 11. 1~2 ll..

(68)

(69)

Sca'4r Potential Form•lation1 for Magnetic Field Pro6lem1 613

for any side with surface charge, k = 1, 3,4,5, where R1, R2,Rs, and Rt are the distances from the field point to the vertices of the trapezoidal side considered, 112 and 134 are the lengths of the nonparallel sides of the trapezoid, and

P12 = -z2y +dz,

Qu = dy + z2z,

..\ _ l R1l12 - Q12 12 = n (R2 + 112)112 - Q12'

Ps4 = -(zs - z4)y + d(z - z4)

Q34 = dy + (zs - z4)(z - z4)

, _ 1 (Rs+ ls4)ls4 - Q34 1134 = n ~,34 - Q34 I

, _ 1

(R1 + z)(Rs + z - zs) z2 , zs - z4, II = n + -1112 + ---1134

(R2 + z - z2)(~ + z - z4) 112 /34

_ t -1 zQ12 - z2R~ t _1 (z - z2)(Q12 - 1~2 ) - z2~ 'Y = an zR

1d - an zR2d

-1 (z - zs)(Q34 -1~4) - (zs - z4)R~ +tan zRsd

_1 (z - Z4)Q34 - (za - Z4)R~ - tan zRtd ·

(70)

-y is just the solid angle under which the trapezoidal side surface is seen from the observation point (z, y, z) and is positive for z > 0 and negative for z < 0. The trapezoidal sides of the coil segment considered have a width d and a charge density Ptl. given by:

a fork= 1, 3

d= 6 fork= 2 (71)

a/(n2 + n2 )1/2 kMc U fork= 4,5

-µJa fork= 1

Pt1.= +µJa fork= 3 (72)

µJankM. fork= 4, 5

where nkM. and nu are the cosines of the angles made by the outwardly oriented (with respect to the segment considered) normal unit vector of side k with the fictitious magnetization Mc and the current density J, respectively. One can easily see that the contributions (69) from a side (k = 4,5) shared by adjacent segments to the fields due to those segments are equal, i.e. they

614 I. R. Ciric

are to be calculated only once for such a segment side and taken twice in the resultant field.

For almost all the current coils in electromagnetic devices and systems the density J throughout a coil has a direction which is everywhere parallel to a given plane. For such a coil, the fictitious magnetization Mc within all the segments can be chosen to have a unique direction, namely perpendicular to the above mentioned plane, for instance form the side 1 to the side 3 for all the segments, as shown in Fig. 7. In this model only three side surfaces of each coil segment contribute to the resultant field, namely k = 1, 3 with surface charge and k = 2 with surface current. The necessary amount of computation is thus correspondingly reduced, since there is no contribution form the sides 4 and 5.

The quantities in formulas (68)-(70) can be expressed in an intrinsic, vector form, independently of any reference frame. This allows the field components to be calculated directly in global coordinates, common to the entire coil, without being necessary to apply the general transformation of coordinates. The curved coil sections are replaced by an appropriate number of straight segments in terms of the desired accuracy. As illustrated for toroidal coils of rectangular cross section and for a typical tokamak field coil,10 the magnetic field computation by using the above models is substantially more efficient than that based on existing algorithms; the computation time is reduced by about an order of magnitude with respect to that when using analytical expressions containing Jacobian elliptic functions and elliptic integrals.26

The same type of surface source models, based on our modeling technique, can be constructed for coils of an arbitrary polygonal cross section, to obtain similar formulas, in terms of elementary functions only.

VI. Examples of Formulating Boundary-Value Problems

To illustrate the application of the new modeling method to the magnetic field analysis in terms of a single-valued scalar potential, we consider two ex­amples of boundary-value problems, with given volume current distributions in nonmagnetic (µ = µ0 ) bounded regions. Both relate to electromagnetic devices in which one can assume the magnetic core to be of an ideal ferromag­netic material (µ -... oo) and the electric current carried by the coils uniformly distributed over their cross sections.

Scalar Potential Formulaliona for M11.gnelic Field Pro61em11 615

A. Reciangular window

The magnetic field within the rectangular opening shown in Fig. (9) is ap­proximated to have a two-dimensional, parallel-plane istructure. The uniformly distributed currents carried by the two conductors are equal in magnitude and opposite in direction, of intensity I. Thus, the tangential component of the field intensity is zero on the ideal ferromagnetic boundary.

Fig. 9. Rectangular region with two conductors.

Each of the two current conductors can be modeled as shown in the previous section, by ending the distributions of fictitious magnetization M.;1 and Mc2

either at x = 0 or x = d, such that the model contains only surf ace charge inside the region, with the corresponding current sheets removed on the boundary. The latter have the following surface densities for the model chosen as indicated in Fig. 9:

for z = 0, YE (Yi.Yt + b1) (73)

616 1. R. Ciric

The scalar potential in (33) is a single-valued function of position inside the region x E (0, d), y E (0, h), and satisfies the Laplace equation (38) everywhere, except the points on the lines where the charge density Pac #: 0. Using (45), (33), (39), and (73), the boundary conditions can be expressed, for instance, ·as follows:

c)c1:r:=O =•c1:r:E(O,cf)=c)c1:r:=cf =0 " e (o, 111 > , = o 11 e (o, 112)

I c)cl :r: =cf = - b2 (y- Y2)

, e (112. 112 + 62)

=-1 (74)

I •c1:r:=O =-b

1(Y-Y1).

" e (111o 111 + 6i)

This is, therefore, an interior Dirichlet boundary-value problem for •c· Its solution can be obtained in terms of the corresponding Green function G(x, y; z', y'),16 as

clc(z,y) = :0 f Psc(z',y')Gdl' - f cJc(z',y');~dt' (75)

where the contour integral is over the region boundary, with the derivative 8/8n' along the outward normal. This component of clc is identical to the classical magnetic scalar potential corresponding to the field intensity which would be produced in the region by the two current sheets in (73), being independent of the position and dimensions of the given conductors along the z direction. The first term in the right-hand side of (75) is identical to the electrostatic potential which would be produced in the region by a charge distribution of density P•c• if the whole boundary were kept at zero potential. Equivalent models can be constructed with the magnetizations Mc1 and/or Mc, along the z axis. Same type of modeling and solution is valid when the conductor cross-sectional sides are not parallel to the window walls. In such a situation, the models have on the boundary not only a surface current, but also a surface charge distribution.

Scalar Potential Form·lllation11 for Magnetic Field Pro6/em1 617

B. Toroidal cavit11

Consider two toroidal coils of rectangular cross section with their axes par­allel to the axis of the magnetic core hut, in general, asymtnetrically disposed, as shown in Fig. 10. Again, the uniformly distributed currents carried by the two coils are assumed to be equal and opposite in direction, of intensity I, such that the tangential component of the field intensity is zero on the ideal ferromagnetic boundary.

z

µ.-co

µ.o

I I r

Fig. 10. Toroidal cavity with uynunetric coils.

The current distribution is modeled as illustrated in the previous section, with the distributions of fictitious magnetization Mc

1 and Mc2 ended either

at r = R1 or at r = R2. The model chosen in Fig. 10 contains surface charge inside the toroidal region and surface current only on the boundary, of densities

J1e1 =4'!/h1 forzE(z1,z1+h1), r=R1

Jee,= -4'l/h2 for i E (z2,z2 + h2), r = R2 . (76)

For such a formulation, +c in (33) is again a single-valued scalar potential inside the region r E (R1, R2), z E (0, h). It satisfies the Laplace equation

618 I. R. Ciric

except for the points on the surfaces where the surface charge density Pac :f: 0. The Dirichlet boundary conditions can be written now as:

(77)

()cl• E (•2 + h3,h) =()cl :i = h =()cl• E (z1 + h1oh) =] r=R2 re(R1,R2) r=R1

The solution has a form similar to that in (75),

CJc(r) = :0 J Pac(r')GdS' - f CJc(r') ~~dS' (78)

where G(r, r') is the Dirichlet Green function for the Laplacian, relative to the toroidal region considered, which has an analytical expression in terms of Bessel functions. 11 Thus the last integral in (78), which represents the contri­bution of the fictitious current sheets on the cavity walls, can be calculated analytically exactly. This component of the potential is axisymmetric and is independent of the position of the axes and of the radial dimensions of the two coils. As in the previous example, similar models can be constructed for even more general systems, where the coil axes are not parallel to the magnetic core axis, with the corresponding surface charge distribution present also on the boundary.

The type of boundary-value problem solution presented in these examples is much simpler and more useful for calculating local field quantities, as well as global quantities (inductances, forces, torques), than the solutions correspond­ing to vector potential or to multivalued scalar potential formulations. The single-valued scalar potential formulations presented require a substantially reduced computational effort with respect to that required by other methods developed so far. For regions of arbitrary geometry the scalar potential ~c can

Scalar Potential Formulation• for Magnetic Field Problem• 619

be computed by using appropriate boundary scalar integral equations or finite (element or difference) methods.

VII. Finite-Element Solution with Inversion Transformation for Fields in Unbounded Regions

Finite-element or finite-difference analysis of exterior-field problems is usu­ally performed by applying various techniques for the truncation of the un­bounded regions at convenient locations, with the inherent loss of accuracy. A simple and efficient method for the finite-element solution of two- and three­dimensional unbounded region field problems consists of a global mapping of the original unbounded region onto a bounded domain by applying a standard inversion transformation to the spatial coordinates.6 In this method same nu­merical values of the potential are assigned to the transformed points. The functional used for the finite-element treatment has the same structure in the transformed domain as that in the original one. This allows the implemen­tation of the standard finite-element technique in the bounded transformed domain.

Consider a three-dimensional unbounded region D, which is the exterior of a bounded domain D•, with anisotropic and inhomogeneous material proper­ties described by a symmetric tensor µ. The scalar potential c)c introduced in (33) satisfies the following equation:

'V · (µ · Vc)c) = g in D (79)

with

g :: 'V · (µ ·Mc) + µo .I( 'V · Mp (80)

in the presence of a given permanent magnetization Mp. Assume general boundary conditions in the form

c)c(r) = c)o(r),

(µ. 'Vc)c). nls, + tt(r)c)c(r) = h(r),

(81)

(82)

where c)o, u, and h are given functions of position, S1 + S2 is the boundary of D., and n denotes the outward unit normal. The finite-element solution to

620 I. R. Cirie

the above boundary-value problem can be obtained through the minimization of the functional

with trial functions which satisfy the Dirichlet boundary condition in (81). Instead of solving directly the original, unbounded region field problem, we

apply the inversion transformation to obtain a transformed functional relative to a transformed, bounded domain. Denoting the transformed Cartesian co­ordinates by z~, z~, z~, the inversion transformation of the original Cartesian coordinates z1, z2, Z3 with respect to a sphere of radius a6 and centered at (zi, z2, z3) = (61, 62, 63) is given by

i = 1, 2, 3 (84)

where

[

3 l 1/2

r6 = ~.:)zi - 6i)2 •

•=l (85)

Applying this transformation, with (6i,62,63) E D6, the original region D + S1 + S2 is mapped conformally onto the bounded domain D' + S~ + S~, and the functional in (83) is transformed in

F = L. [cv'~~> . (µ'. v'~~> + 2g' ( ~U 6

~~] dv'

+ f (0-'~:2 - 2h'~~) (a:)4 dS'. ls~ r6

(86)

V' is the del operator in primed coordinates and r~ = al/ r6 · ~~, g', o-', and h' are, respectively, ~c. g, O', and h as functions of the primed coordinates, obtained through the inverse transformation

Scalar Potential Formvlation1 for Magnetic Field Pro6/em1 621

i = 1, 2, 3 (87)

with

(88)

In a matrix form, the tensor µ' is related to µ by

µ' = (~)2 TµT- 1 (89)

where the square matrix T has the element Ti; = 8zU8z; and T-1 denotes its inverse. µ' is a symmetric tensor, as it is µ, and is interpreted as being the fictitious permeability tensor of the medium in the transformed domain. It should be remarked that the transformed functional (86) has a form which is obtained directly from that in (83) by replacing g by g'( a•/r~) 6 , and u and h by u'(a•/rD4 and h'(a•/r~)4, respectively. This functional is now minimized in the transformed domain by using trial functions which satisfy the Dirich­let boundary condition for ~~ given in (81), taking into account the point transformation in (87)-(88).

For the special case of an inhomogeneous but isotropic medium, when µ is a scalar,µ= µ(zi,z2,z3), Eq. (89) shows thatµ'= (a&/r6)2µ'(z~,z~,z~) is now also a scalar, with µ' being µ expressed as a function of the primed coordinates.

Since the transformed functional (86) is similar to the original one (83), a general finite-element computer program appropriate for the original field prob­lem can be used in a direct manner in the transformed domain. Preservation of the exact boundary conditions of the original problem in the transformed domain, including the conditions at infinity, improves significantly the accu­racy of the numerical results. The solution in the transformed domain yields, through the correspondence (87)-(88), the original field problem solution. It should be noted that global quantities related to the problem functional can be derived directly from its transformed form, without being necessary to recover the original problem potential solution through mapping. Various examples6

622 I. R. Ciric

illustrate the high efficiency of this method; even with a coarse mesh in the transformed domain, highly accurate results are produced not only for global quantities but also for the local field quantities.

VIII. Conclusions

Scalar potential formulations for magnetic field problems based on a novel method of modeling given current distributions have been presented. Using this method allows the magnetic field due to usual current distributions to be determined from that produced by corresponding fictitious surface sources. The computational effort required is substantially reduced since, for example, the magnetic field can be calculated by performing surface integrals instead of the volume integrals in the Biot-Savart formula. In addition, the modeling procedure is flexible in the sense that for a given problem one can construct the most appropriate model in order to minimize the necessary amount of com­putation. At the same time, the physical interpretation of the results is made easier. When the current distribution is modeled by employing a fictitious magnetization and only an equivalent charge distribution inside the region considered, then the associated scalar potential is a single-valued function of position throughout the entire region.

The modeling procedure is illustrated on a few examples. New, elementary formulas have been derived for the magnetic field of infinitely long, straight con­ductors of an arbitrary polygonal cross section carrying uniformly distributed currents, as well as for the contribution to the magnetic field from arbitrary coils due to their segments in the form of general straight current tubes of finite length. On the other hand, examples of magnetic field boundary-value problems demonstrate the efficiency of the models elaborated. It is also shown that by applying these models it is possible to enlarge the class of problems which admit an analytical treatment.

The formulations presented can be easily extended to magnetic fields in anisotropic or nonlinear media. A simple and straightforward solution for our scalar potential in anisotropic unbounded regions of arbitrary geometry has already been formulated by implementing standard three-dimensional finite elements in bounded domains obtained by conformal mapping through an in­version transformation. For same computational effort, this method provides

Scalar Potential Formulation• for Magnetic Field Problem• 623

a better accuracy of the numerical results than that corresponding to other methods developed so far.

References

1. 0. Biro and K. Preis, IEEE Trana. Mag. MAG-25 (1989) 3145-3159. 2. J. R. Bronzan, Am. J. Phya. 39 (1971) 1357-1359. 3. M. L. Brown, Proc. IEE Ser. A 129 (1982) 46-53. 4. H. Buchholz, Elektriache und Mognetiache Potential/elder (Springer, Berlin,

1957). 5. I. R. Ciric, Doctoral Thesis, Polytechnic Institute of Bucharest, 1969. 6. I. R. Ciric and S. H. Wong, COMPEL 5 (1986) 109-119. 7. I. R. Ciric, J. Appl. Phya. 61 (1987) 2709-2717. 8. I. R. Ciric, in Electromagnetic Fields in Electrical Engineering, eds. A. Savini and

J. Turowski (Plenum, New York, 1988). 9. I. R. Ciric, IEEE Trana. Mag. MAG-24 (1988) 3132-3134.

10. I. R. Ciric, IEEE Trana. Mag. MAG-27 (1991) 669-673. 11. J. Dougall, Proc. Edin. Math. Soc. 18 (1900) 33-83. 12. E. Durand, Electroatatique et Magnetoatatique (Mason, Paris, 1953). 13. J. D. Jackson, Clauical Electrodynamica (Wiley, New York, 1975). 14. R. W. P. King and S. Prasad, Fundamental Electromagnetic Theory and Appli­

cationa (Prentice-Hall, Englewood Cliffs, NJ, 1986). 15. M. S. Livingston and J. P. Blewett, Particle Acceleratora (McGraw-Hill, New

York, 1962). 16. P. M. Morse and H. Feshbach, Methoda of Theoretical Phyaica (McGraw-Hill, New

York, 1953). 17. S. Pissanetzky, IEEE Trana. Mag. MAG-18 (1982) 346-350. 18. R. Radulet, Bazele Teoretice ale Electrotehnicii, vol. I (Litografia lnvatamintului,

Bucharest, 1955). 19. R. Radulet, Bazele Electrotehnicii-Probleme (Editura Didactica si Pedagogica,

Bucharest, 1970). 20. R. Radulet and I. R. Ciric, Rev. Roum. Sci. Tech. Ser. Electrotech. Energ. 16

(1971) 565-591. 21. L. Schwartz, Methodea Mathematiquea pour lea Sciencea Phyaiquea (Hermann,

Paris, 1961). 22. J. Simkin and C. W. Trowbridge, Proc. IEE Ser. B 127 (1980) 368-374. 23. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1950). 24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). 25. A. Timotin, V. Hortopan, A. Ifrim and M. Preda, Lectii de Bazele Electrotehnicii

(Editura Didactica si Pedagogica, Bucharest, 1970).

624 I. R. Cine

26. L. K. Urankar, IEEE Tran.f. Mag. MAG-18 (1982) 1860-1867. 27. J. P. Wikswo, Jr. and K. R. Swinney, J. Appl. Phya. 57 (1985) 4301-4308.