SCIENCE

Unified approach to problems inelectromagnetism

J. Penman, B.Sc, Ph.D., C.Eng., M.I.E.E., and J.R. Fraser, B.Sc.(Eng.),Ph.D.

Indexing terms: Electromagnetics, Magnetic fields

Abstract: The development of a structure linking the various potentials and field vectors of the electric andmagnetic fields is presented. The approach shows that it is usually possible to find alternative formulations, fora given problem, in a relatively simple manner. These alternative formulations, when correctly chosen, poseproblems in dual or complementary form such that bounded solutions are possible. Such properties are particu-larly attractive when numerical solutions are being considered. The proposed structure is general, incorporatingboth electrostatic and magnetostatic systems as well as time varying field quantities.

1 Introduction

The inherent symmetry of structure in electric and mag-netic fields has been the subject of considerable investiga-tion since Maxwell's celebrated work [1] led to thediscovery of electromagnetic waves. The exploitation ofthis symmetry, together with the use of derived functionsthat we know as potentials, can greatly simplify the solu-tion of many problems in electromagnetism.

Recently, several authors have examined closely therelationships between the various electromagnetic fieldvectors and their associated potentials [2-4]. In Reference2, the principal effort is directed towards generalising thebasic relationships between field quantities by the use ofdifferential forms. The author also introduces the impor-tant notion of the electromagnetic flow diagram, to helpwith the visualisation of these relationships. Hammond [3]and Carpenter [4] lay great stress on the choice of theappropriate potential function for a given problem, andthe former develops a full mathematical structure aroundthe various potentials commonly used.

In the theory of elasticity and fluid mechanics, Tonti [5]also makes good use of a diagrammatical representation ofthe links between primary variables. Later, in this paper,the authors will significantly extend this technique byincorporating dual and complementary energy principlessuch that alternative formulations of a givenelectromagnetic-field problem can be chosen so as toprovide error-bounded solutions. Complementary anddual energy methods have been discussed elsewhere [6—8],but application of these techniques is restricted to rela-tively simple problems, because of the difficulties involvedin selecting suitable approximating functions when using adirect variational approach. The authors have used thesetechniques previously, in conjunction with the finite-element method, to show that realistic engineering prob-lems can be tackled [9]. Cases in which time varying fieldsare included are discussed in Reference 10, and furtherwork in this area is in preparation.

The aim of this paper is to draw together some of thepoints raised in the articles discussed above, and to showthat there is a unifying structure to electromagnetism thatlinks the notions of dual and complementary energy prin-ciples to the choice of field vector (or potential). This struc-ture is represented diagrammatically so that it may be used

Paper 2gl8A(S8), received 29th April 1983Dr. Penman is, and Dr. Fraser was formerly, with the Department of Engineering,University of Aberdeen, Marischal College, Aberdeen, Scotland. Dr. Fraser is nowwith McDermott Engineering Ltd., Union Street, Aberdeen, Scotland.

as an aid to problem formulation and as an indicator ofthe pairs of formulations required to yield error bounds.The use of such alternative formulations to provide errorbounds is of great importance, especially when this tech-nique is used in conjunction with the finite-elementmethod, for it has been shown [9, 10] that significant com-putational advantage can result. This allows larger prob-lems to be tackled, or smaller computer installations to beused. Experience has shown that with two-dimensionalstatic problems, savings of at least one order of magnitudein computer core store requirements can be expected whencalculating field-dependent parameters to an accuracy ofabout 1-2% (see Reference 9).

2 Static systems

2.1 Magnetic field due to steady currentsHere the source of the field is a current, which we canexpress in terms of a current density J. This current densityproduces a magnetic field, as described by the Maxwellequation:

\xH = J (1)

where H is the magnetic intensity. The magnetic vectorpotential may be introduced by defining

x A = B (2)

where B is the magnetic flux density. B is related to H bythe constitutive relationship B = ^H, where n is the per-meability of the medium.

Using eqns. 1 and 2 it is easy to show that

- \ x A) = J (3)

and these three relationships, together with the constitutiverelationship, may be represented diagrammatically, asshown in Fig. 1.

We see that in order to link A to J (potential to source),one may progress via B and H or go directly throughPoisson's equation (eqn. 3). Following Carpenter [4] wecan also express H in terms of a vector and a scalar func-tion so that

H= r - vn (4)

where T is often defined as an electric vector potential, andQ a magnetic scalar potential or stream function. Later,the authors suggest an alternative definition for T,reserving the term electric vector potential for anotherfunction, introduced below.

IEE PROCEEDINGS, Vol. 131, Pt. A, No. 1, JANUARY 1984 55

In order to construct a similar scheme to Fig. 1, in Tand Q, we utilise the Maxwell equation

V • B = pJ (5)

where p™ is the free magnetic pole density, and is, ofcourse, zero. We choose to include it, however, in order to

0Vx ± V x A

Vx "

o

0

Vx

Fig. 1 Basic magrietostatic relationships via A

illustrate more clearly the structure that is being devel-oped. Using eqns. 4 and 5 it is now possible to link thesources (pole distribution) to T and Q. Since

- VQ)

then

(6)

This may be represented diagrammatically as in Fig. 2

0V.

0T.vn

Fig. 2 Basic magnetostatic relationships via Q

Figs. 1 and 2 can now be linked to give the diagramfirst developed in Reference 9, and shown in Fig. 3.

Clearly, a certain symmetry exists in Fig. 3. Forexample, J and pj are sources, whilst A, Q (and T) may beregarded as potentials. However, in Fig. 1 the vector oper-ators linking A to B and H to J are V x (the curl operator)

0V x — V x A

Vx

0

0V.

Vx

0T- vn

V.p(T-VQ)Fig. 3 Combined relationships

and its adjoint, which is also V x. In Fig. 2, B is linked top™ through the operator V • (divergence operator), theadjoint of which is — V (gradient operator). Now, —V doesappear in the relationship between H, T and Q, but thebasic structure is clearly unsatisfactory. Fig. 3 also showsus that

V • (V x A) = pj (7)

Now, because V • (V x A) = 0, it would seem to be unreal-istic to even consider p"} (monopoles apart) when it mustbe identically zero.

These two anomalies in Fig. 3 need clarification. To dothis we choose to consider B as comprising a contributionfrom a scalar field and one from a vector field. We havealready done this for H. Thus

R — R A- R (K\

and

V • Bs = pj

whilst

V • B = 0

(9)

(10)

If we let Q = - VQ, H = T + Q, and we may redraw Fig. 3in the expanded form shown in Fig. 4. This structure is

Vx '

V x ! ( V x A - B s )

Vx

B ~

V. '

Vx

H= Q * T

" V.

V.jj(T-VfL)

-V

Fig. 4 Extended form of magnetostatic system

now more symmetrical mathematically, but this symmetryhas been achieved at the expense of greater complexity.However, several points leading to simplifications shouldbe noted.

If one relates A to J via the uppermost loop of Fig. 4,the expression obtained is

(11)

However, Bs results from p™, which is zero; hence Bs maybe omitted, reducing eqn. 11 to the usual form of Poisson'sequation. This also means that V x A = Bv is identical toV x A = B, as usual, and V • B = V • Bv = 0, as required.

Utilising the lower loop of Fig. 4 one can link Q to pj,as in eqn. 6, which, in the absence of current carrying con-ductors, reduces to

V • fx(T- VQ) = 0 (12)

as expected.From now on, bearing in mind the full form of Fig. 4,

we shall use the simplified style of diagram illustrated inFig. 5.

56 1EE PROCEEDINGS, Vol. 131, Pt. A, No. 1, JANUARY 1984

It should be observed that the heavier type links in Fig.5 correspond to the double links of Fig. 4. They also givethe basic relationships that must be satisfied to solve prob-lems in magnetostatics (i.e. Maxwell's equations and the

primal PDE

OVx

-V

ndual PDE

Fig. 5 Reduced form of magnetostatic system

constitutive relationship). Explicitly they are, and here weborrow terms from stress analysis to assist us:

(a) compatibility equation V • B = 0(b) constitutive equation B = \iH(c) equilibrium equation V x H — J.

In passing we see that the vector T is now firmly placedbeside H in Fig. 5. It has the same dimensions as H, and,within our purposed structures, acts like the magneticintensity vector. We therefore choose to call T the reducedmagnetic intensity vector, because it differs from H only bythe vector Q.

2.2 Alternative magnetostatic formulationsIt can be seen that there are now several choices open toone when attempting to solve a given problem and thatFig. 5 can be used to help one choose systematically. Thefirst step is to select the appropriate partial differentialequation (PDE). The usual choice in magnetostatics is theprimal form. If it is assumed that the chosen equation is tobe solved variationally, either directly, or, more usefully, inconjunction with the finite-element method, a furtherchoice must be made. We may either satisfy the compat-ibility equation and then solve for equilibrium, or satisfythe equilibrium equation and then solve for compatibility.The first choice is the commonly used A formulation, butit must be noted that compatibility is not directly satisfied;rather it is satisfied indirectly by choosing V x A = B.Essentially, with this form, known as the primal form, weconfine ourselves to the relationships of Fig. 1. If the dualform is chosen, equilibrium is satisfied indirectly by lettingH = T — VQ, and we utilise the relationships depicted inFig. 2.

The results achieved using these various forms haveinteresting properties; for example, if the primal PDE issolved for equilibrium with compatibility indirectly satis-fied, an answer is obtained that is an upper bound to thesystem energy. Solving indirectly for compatibility, withequilibrium directly satisfied, gives a lower bound.Bounded solutions can similarly be obtained from the dualPDE. For the primal problem, the required variationalstatements are in A and H to give bounds, whilst B and Oare needed for the dual variational formulations. Formula-tions in A and B are termed standard forms, whilst thosein H and Q are the complementary forms. Extending thisnomenclature, in line with Reference 9, it can be seen thatbounds are produced from the primal-standard and

IEE PROCEEDINGS, Vol. 131, Pi. A, No. 1, JANUARY 1984

primal-complementary forms, or from the dual-standardand the dual-complementary forms.

A complete formulation of the primal problem, togetherwith proof of boundedness, is given in Reference 9. Readersinterested in the application of such methods to mechani-cal systems should consult Reference 11.

2.3 Electrostatic systemsWe can proceed in an exactly similar fashion to thatadopted with magnetostatics. For electrostatic systems theprimary relationships are

V • D = p}

and

V x £ = 0

(13)

(14)

pef is the free charge density, E the electric intensity and D

the electric flux density. D is related to E through the con-stitutive relationship D — EE, where £ is the permittivity ofthe region. Also, for the conservative electric field, E can beexpressed in terms of a scalar function thus:

E= -V</> (15)

where (f) is the electric scalar potential.We can relate </>, the potential, to pe

f, the source, in asimilar way to that used in Figs. 1 and 2. This is showndiagrammatically in Fig. 6.

V. ( -0-V

0 0

and

with

and

Fig. 6 Basic electrostatic relationships via </>

Again we can expand this structure by allowing for anonconservative component in the electric field (which is,of course, zero for the static case). This is done by writing

E= F+ W (16)

K + D. (17)

- V 0 (18)

V • Ds = p} (19)

This is analogous to writing H= T — VQ and B = Bv + Bs.Note that if T is to be regarded as the electric vectorpotential, then F, to be consistent, would need to betreated as a magnetic vector potential. Some thoughtshows that F and A are closely related, but one is the timedifferential of the other, since for the nonconservative elec-tric field

E= - — - \<f> (20)

This point is expanded later, when time-varying fields areconsidered.

By following the reasoning of Section 2.1, and exami-nation of Fig. 4, Fig. 7 can be developed.

Two important new quantities are introduced by Fig. 7,

57

and they deserve some elaboration. <P is the magneticcurrent density (analogous to J), and we see that

V x £ = V x ( F + « 0 = $ (21)

In the static case, <P would be due to the steady flow of freemagnetic poles. This is obviously zero, hence eqn. 21reduces to the usual form: V x E = 0.

The vector S in Fig. 7 corresponds to A in Fig. 4, and isV . e ( F - V * )

-V

F • W = E

7 x

D M * D - D

Vx

V x j ( V x S

Vx

Fig. 7 Extended form of electrostatic system

directly related to the magnetic current in the same waythat A relates to J. Because of the relationship that A hasto J it is sometimes termed, following Maxwell, the electro-kinetic momentum. The vector S may be likewise thoughtof as the magnetokinetic momentum, or electric vectorpotential. The relationship between S and £> is given by

V x - (V x S + Ds) = (22)

which can be compared with eqn. 11.We see also that, in common with Fig. 4, Fig. 7 iden-

tifies the operators linking the scalar potential to itssources as: —V (the gradient operator) and its adjoint V-(the divergence operator), whilst the operators linking thevector potential to its sources are, V x (the curl operator)and its adjoint, which is also V x .

In passing, it is interesting to note that, whilst thecommon formulation of magnetostatic problems is via theprimal form, the usual choice for electrostatics is a dualform. This is purely because of the definition of primal anddual equation chosen by the authors, but it is justifiedbecause the usual sources of static magnetic fields aresteady currents (i.e. nonconservative sources), whilst staticcharges, which give rise to conservative fields, are thesources of the electrostatic fields.

For economy of effort later, we again reduce the form ofFig. 7 to that shown in Fig. 8 and use the terminologyindicated to identify primal and dual forms. Here we seethat we also have three principal relationships to be satis-fied. As before they are:

(a) compatibility equation V x £ = $(b) constitutive equation D = sE(c) equilibrium equation V • D = pe

f.

2.4 Alternative electrostatic formulationsIt is apparent that all that has been said concerning theavailable choices for alternative formulations of magneto-static problems will apply equally well to electrostatics. Itis unnecessary, therefore, to labour the point, but by wayof example we note that, if the primal form is chosen, wecan indirectly satisfy the compatibility conditionV x £ = 4 > , through E = —\(j), then solve for equilibriumdirectly. Alternatively, we can satisfy equilibrium directly,

58

then solve the compatibility equation indirectly. Takingaccount of the dual formulation as well, the pairs of vari-able that give bounded solutions are E and S, and (j) andD, respectively. The pair <j) and D yield the primal-

0primal PDE

-V

0

Vx I Vx

0dual PDEFig. 8 Reduced form of electrostatic system

standard and the primal-complementary forms, respec-tively, whilst the dual-standard and dual-complementaryforms are provided by the variables E and S, respectively.

3 Electromagnetic fields

3.1 Extension to time-varying fieldsWhen the sources of an electric or magnetic field undergotime variations, the two types of fields become coupled.This coupling is expressed by Faraday's law, and the fullform of Ampere's law (both in their differential form):

and

dt

(23)

(24)

respectively.If the magnetic current is included, the Faraday law

may be written

dt

which may be compared with Ampere's law.Let us define the total electric current density as

where

(25)

(26)

Jd~ dt

the displacement current, and the total magnetic currentdensity as

O, = 0 + <Pd (27)

where

" dt

The completely symmetrical forms of eqns. 23 and 24 arethus

V x E = &, = & + &d (28)

1EE PROCEEDINGS, Vol. 131, Pt. A, No. 1, JANUARY 1984

and

\ x H = Jt = J+Jd (29)

These forms appear in many standard texts on electricityand magnetism.

It is now possible to incorporate time variation in ourdiagrammatical representation. Consider the top loop onlyof Fig. 5 together with the bottom loop only of Fig. 8, andsuppose that they are placed, in planes, one behind theother to form the three-dimensional structure shown inFig. 9. To simplify matters we also suppose that no cur-

0

Fig. 9 Introduction of time variation

rents owing to free charges or poles are present. It isobvious that the operators associated with the linksjoining the planes of the two diagrams are not space vectoroperators, but time differentials.

It may be observed, in Fig. 9, that linking the electric-field plane to that of the magnetic field presupposes theoperation d/dt, whilst moving in the reverse direction indi-cates — (d/dt). The basic cubic structure of the diagram iseasily completed by noting that we have previously splitthe vectors E and H such that

E = F + W with

and

H = T + Q with T = — by analogy

E and H may be considered the sum of partial fields asshown in Fig. 10.

This diagram may be combined with Fig. 9, and byincorporating the divergence relationships of eqns. 5 and13 (which led to the completion of Figs. 5 and 8), the struc-ture shown in Fig. 11 is arrived at. It must be understoodthat the explicit relationships given by Figs. 4, 7 and 10 areimplicit here.

Fig. 11 again exhibits a striking symmetry, and its struc-

Fig. 10 Relationship between the potentials and field vectors E and H

IEE PROCEEDINGS, Vol. 131, Pt. A, No. 1, JANUARY 1984

ture confirms our use of the term electric vector potentialfor the quantity S. It should now be apparent why noattempt was made earlier to invert the diagrammaticalform of Fig. 7 to make it structurally analogous to the

primal

-V

Fig. 11 Electromagnetic system

magnetostatic counterparts. For completeness, therelationships between current and electric field, and mag-netic current and magnetic field, through the conductivitiesae and om, is also illustrated. The link between 4> and H isredundant, however, owing to the absence of free poles.

As before, the principal relationships in Fig. 11 areshown with bold lines, and again they can be typified as

(a) compatibility equations

\ • B = 0

and

V x E = 0,

which, in the absence of p™, equals — —dt

(b) constitutive equations

B = nH

and

D = eE

(c) equilibrium equations

v • D = Pyand

59

We can now use this structure to identify alternative for-mulations for electromagnetic-field problems.

3.2 Alternative formulationsThe first task is to construct the complete primal and dualforms. The primal equation is developed thus: if B isdefined a s V x / 4 , V - 5 = V - V x , 4 = 0 , so V • fl = 0 issatisfied.Also

V x £ = V x

Therefore

V x W

( dA- —

dtdA—dt

dt dt

since B = V x A.So we see that both V B = 0 and V x E = -(dB/dt)

are satisfied, i.e. the compatibility conditions are satisfied.The remaining conditions to be satisfied are b and c, whichmay be written as

["]=[» IM (30)

and

o - v ra (31)

Combining eqns. 30 and 31, and using the compatibilityconditions, gives

Vx

0

d— —

dt- V - .

1—

— e.

Vx

d. dt

0"

V UJ L-p/ (32)

We note that this equation has the form UGLU = / , whereL" and L are adjoint operators; hence it is amenable to thetreatment used in References 9 and 12 in order to con-struct standard and complementary functionals.

Eqn. 32 is also very general, for it is easy to show that ifthere are no static charges and no displacement currents, itgives the standard diffusion equation in A together withthe divergence condition on A. Furthermore, if there areno eddy currents, it reduces to the standard form of thewave equation in A, and, in the absence of time variation,it yields the primal electrostatic and magnetostatic forms.

Briefly we note that the standard-primal formulation isa direct-equilibrium method, and is in (A, 4>). It provides asolution to

Vx - —dt

0 - V(33)

The complementary primal form is in (H, D) and is anindirect compatibility method solving

Vx 0

KH-3 (34)

Hammond [13] has developed variational statements forthese variables, but uses the term dual instead of comple-mentary.

In order to construct the dual equation, we use the factthat D has been defined as D = Dv + Ds, and V x S = Dv;thus V Ds = V P.

The spatial variation of Ds can thus be obtained byintegration, as in Reference 14. Once done, this ensuresthat V • D = pe

f is satisfied.Also, since H = T + Q

V x # f - V x 5 r + V x ( Vilj

But V x 5 = D t , and therefore

dDv dD dDs

dt ~ dt dt '

Also if the time variation of Ps is such that

dt ~ ~J

then

dt

as required. Hence, by defining Dv as V x S, and ensuringthat V Ds = p}, and

dt= -J

the equilibrium conditions are satisfied. The remainingconditions to be satisfied are

e .

and

- Vd_dt1 -H

(35)

(36)

Combining eqns. 35 and 36, and writing H in terms of Sand Q, and D in terms of Ds and Dv gives

- V -

d

dt

0 '

Vx

' - A *1—e

V

[o

-V

d~

dt

X [2H»] -Elwhich has the form

Y°!W(Yv +w) = (38)

This expression is the general form of the dual equation,and is amenable to complementary variational treatment.It should be observed that posing the problem in (S, Q) is adirect compatibility method, giving the standard dualform, whilst using (B, E) is an indirect equilibriumapproach, yielding the complementary dual form.

The development of the finite-element method to solvethe eddy-current forms of eqns. 32 and 37 is the subject ofconsiderable effort at the moment, although the underlyingstructure developed here has not previously been utilised.Simkin and Trowbridge [15] have, in fact, given Galerkinformulations of both the primal and dual equations, whilstChari [16] has given a standard primal variational form.Preston and Reece [17] have presented a functional for-mulation to solve the dual equation. There remains,however, the interesting and challenging task of construc-ting a complementary primal, and a complementary dualfunctional for the general three-dimensional eddy-current

60 IEE PROCEEDINGS, Vol. 131, Pt. A, No. 1, JANUARY 1984

problems. The authors of this paper have achieved somesuccess in developing two-dimensional forms [10], andhave in preparation a paper discussing the boundedness ofsuch methods when used with the finite-element method.

4 Conclusions

A structure has been developed, in diagrammatical form,that may be used to derive alternative formulations forproblems in electromagnetism. The proposed structure notonly aids formulation, but indicates which pairs of vari-ables must be used if error-bounded solutions are of inter-est. This is a particularly useful feature, for it may beemployed with the finite-element method to greatly reducethe computational efforts required for a given accuracy ofanswer. This has been demonstrated previously by theauthors for static problems in References 9, 18 and 19, andfor a simple dynamic problem in Reference 10. The notionsof compatibility and equilibrium have been introduced sothat dynamic problems may be treated in essentially thesame manner as static ones. It also helps to reinforce thebasic symmetry that exists in the electromagnetic-fieldequations.

5 Acknowledgments

The authors would like to express their thanks to thosefriends in universities and industry who discussed the sub-stance of this work with them. Particular appreciation isaccorded to Professor E.M. Freeman of Imperial College,and to Mr. C.W. Trowbridge and his colleagues at theRutherford-Appleton Laboratories.

6 References

1 MAXWELL, J.C.: 'A treatise on electricity and magnetism' (DoverPublications Inc., 1954)

2 DESCHAMPS, G.A.: 'Electromagnetics and differential forms', Proc.IEEE, 1981, 69, pp. 676-696

3 HAMMOND, P.: 'Use of potentials in calculation of electromagneticfields', IEE Proc. A, 1982, 129, pp. 106-112

4 CARPENTER, C.J.: 'Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power fre-quencies', Proc. IEE, 1977, 124, pp. 1026-1034

5 TONTI, E.: 'On the mathematical structure of a large class of physi-cal theories', Accad. Nat. Luicei, 1972, 52, pp. 48-56

6 HAMMOND, P., and PENMAN, J.: 'Calculation of inductance andcapacitance by means of dual energy principles', Proc. IEE, 1976, 123,pp. 554-559

7 HAMMOND, P., and Penman, J.: 'Calculation of eddy-currents bydual energy methods', ibid., 1978, 125, pp. 701-708

8 ARTHURS, A.M.: 'Complementary variational principles' (OxfordUniversity Press, 1970)

9 PENMAN, J., and FRASER, J.R.: 'Complementary and dual energyfinite element principles in magnetostatics', IEEE Trans., 1982,MAG-18, pp. 319-324

10 PENMAN, J., and FRASER, J.R.: 'Complementary variational for-mulation for finite element calculation of eddy-current problems'.Symposium on Eddy-Current Problems, Rutherford-Appleton Labor-atories, 1982

11 ODEN, J.T., and REDDY, J.N.: 'Variational methods in theoreticalmechanics'(Springer-Verlag, 1976)

12 VAINBERG, M.M.: 'Variational methods and methods of monotoneoperators in the theory of non-linear equations' (J. Wiley, 1973)

13 HAMMOND, P.: 'Energy methods in electromagnetism' (PergamonPress, 1982)

14 ARMSTRONG, A., COLLIE, C.J., SIMKIN, J., and TROW-BRIDGE, C.W.: 'The solution of 3-D magnetostatic problems usingscalar potentials'. Proceedings Compumag-78, Paper 12, 1978

15 BIDDLECOMBE, C.S., HEIGHWAY, E.A., SIMKIN, J., andTROWBRIDGE, C.W.: 'Methods for eddy-current computation inthree dimensions', IEEE Trans., 1982, MAG-18, pp. 492-497

16 CHARI, M.K.V., KONRAD, A., PALMO, M.A., and D'ANGELO,J.: 'Three dimensional vector potential analysis for machine fieldproblems', ibid., 1982, MAG-18, pp. 436-446

17 PRESTON, T.W., and REECE, A.B.J.: 'Solution of 3-dimensionaleddy-current problems: the T-fi method', ibid., 1982, MAG-18, pp.486-491

18 PENMAN, J , and FRASER, J.R.: 'Efficient calculation of electro-static field problems'. Institute of Physics Conf. Publ. 66, Electro-statics 83, 1983, pp. 243-248

19 PENMAN, J.: 'A new development in finite elements'. IEE Collo-quium Digest 1983/34

7 Appendix

7.1 Some extensionsThe diagram illustrated in Fig. 11 is essentially complete,but it can be extended fairly readily. For example, if weconsider the divergence conditions given by eqns. 5 and 13,together with eqns. 26 and 27, it follows that

V Jd =

and

dt

dt

(39)

(40)

These relationships fit symmetrically into our basic struc-ture, as shown in Fig. 12. In this Figure, for the sake ofclarity, only the extensions to Fig. 11 are shown.

Fig. 12 Some extensions to electromagnetic system

The Lorentz condition

36V • A = -eono — (41)

and the equivalent expression relating the electric vectorpotential to the magnetic scalar potential

V • S=eo^o —

can also be easily incorporated if we let

-eonoX = V -A

and

= V S

(42)

IEE PROCEEDINGS, Vol. 131, Pt. A, No. 1, JANUARY 1984 61