Roth's method for the solution of boundary-valueproblems in electrical engineeringProf. P. Hammond, M.A., C.Eng., A.M.I.Mech.E., F.I.E.E.

Synopsis

Between the years 1927 and 1938 the French engineer E. Roth developed a powerful method for thesolution of boundary-value problems in electrical engineering. He applied the method to problems in heatconduction and magnetic-field problems in electrical machines and transformers, although the applicationof the method is not necessarily limited to such devices. In spite of its possibilities, the method has beenomitted from the standard treatises on electromagnetic problems, because there are certain difficultiesassociated with it. The paper gives a critical account of the method, listing both advantages and disad-vantages and relating these to the physical and mathematical basis of Roth's work.

List of symbolsA = magnetic vector potentialB — magnetic flux densityH = magnetic field strength/ = current densityV = electric scalar potential

V* = magnetic scalar potentiala, b, c, d, e,f = slot dimensions

m, n = parametersCm n = coefficient

x, y = rectangular co-ordinatesX = function of x onlyY = function of y only

Jo, J,, Yo, Y,, Z| = oscillatory Bessel functionsfM = permeabilityto = angular frequency

1 IntroductionThe solution of boundary-value problems in electrical

engineering is of great interest and importance. Whereverthere are currents surrounding, or enclosed by, iron structures,the designer is faced with the necessity of attempting to solvesuch problems. Typical examples are the air-gap and slotfields in machines and the leakage field in transformers. Inelectrostatics there are similar problems involving electriccharges in liquids and in gaseous corona in the vicinity ofinsulating and conducting surfaces, and the problems of heatconduction are solved in an identical manner.

The importance of the subject is reflected by the enormousnumber of papers and books in which it is discussed.Analytical, graphical, numerical and experimental methodsvie for the reader's attention, and an adequate survey isalmost impossible. However, as often happens, the firstimpression of an abundance of solutions is contradicted bythe designer's experience when he tries to find the solutionto his particular problem. Then it seems as jf all the availablemethods have been applied to the same problems and thatthese are of little relevance to the problem in hand.

In such a situation it is important to discover whetherthe well known methods are necessarily limited to the examplesdiscussed in the literature and whether there may not be othermore suitable methods which have been overlooked.

The paper deals with the method developed by E. Roth(Appendix 9.2) in a series of remarkable papers from 1927 to1938. It is a method which is ignored by most writers,although Hague1 has a lengthy and enthusiastic section onit, in which he says that 'no student of these problems canafford to neglect' Roth's work. Unfortunately, Hague's dis-cussion is not free from error and contains some claims forthe method which cannot be sustained. The work is alsomentioned by Bewley2 and by Billig,3 who praises its soundtheoretical basis and suggests that, in this respect, it issuperior to other methods. There is an extended account in

Paper 5429 J, first received 1st May and in revised form 26th July 1967Prof. Hammond is with the Department of Electrical Engineering, TheUniversity, Southampton, Hants., EnglandPROC. IEE, Vol. 114, No. 12, DECEMBER 1967

a book by Binns and Lawrenson,4 but they also repeatHague's misleading conclusions. (Sections 5.1 and 5.2.)

Hague, in 1929," called these papers 'a complete treatise',and his praise would have been even more justified if he hadhad access to Roth's later work. Nevertheless, the wordtreatise is misleading. Roth's papers deal with particularproblems which he needed to solve in the course of his profes-sional work. The applications are dominant, and he does notattempt a critical account of the scope and limitations of hismethod. The marks of his work are courage, determinationand industry rather than critical appraisal. The algebra isoften exceedingly intricate and involves the manipulation ofinfinite series containing circular, hyperbolic and Besselfunctions. Summation of the series is carried out by hand, and7-figure logarithms have occasionally to be employed. Billigcomments on the 'unreasonable amount of labour' involvedand suggests some useful simplifications. Nevertheless, Rothproduced a large number of solutions, and his papers containmany valuable and interesting flux maps.

The scope of Roth's work is considerable, and his methoddeserves to be widely known. There are, however, pitfallsto be avoided if it is to be applied with safety. It is likely thatmany workers have avoided the method because they wiselydoubted some of the claims made for it. It is hoped that theshort critical account in this paper will enable designers tosee both the scope and the limitations of Roth's methodwithout having to undertake the labour of investigating theoriginal sources. Once the scope is understood it is easy todecide when to use the method and when to avoid it.

2 Short account of Roth's papersRoth's first paper5 deals with the flow of heat in

electrical machines. The work is based on Fourier's analysis,and Roth pays particular attention to 2-dimensional rect-angular co-ordinates. Practical results are given, and there isan extensive bibliography.

The second paper6 discusses the thermal and magneticfields of various current-carrying conductors in a rectangularslot (Fig. 1). The magnetic boundary conditions are those ofinfinite permeability along AB, AA' and A'B', while BB' is aflux line. The centre line of the slot is a line of symmetry.

Roth next7 directs his attention to the leakage field intransformers. He considers several different arrangements oftransformer windings. The discussion is limited to the2-dimensional field in a rectangular transformer 'window'.The cross-sections of the windings are rectangular, but theirposition is arbitrary. The permeability of the iron boundariesis assumed to be infinite, and the magnetising current isneglected. The paper contains various flux plots and photo-graphs of a 3-dimensional model of the magnetic field.

In a paper8 written with his colleague G. Kouskoff,published a month after the paper on transformers, Rothdeals with some mathematical difficulties of his method. Hissolutions are in the form of doubly infinite series of productsof circular functions. The convergence of the series is poor,and it is desired to discover finite representations of the

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series over the appropriate interval. This aim is achieved-bychoosing a representation in hyperbolic functions whichreduces the double series to series in a single circular functionmultipled by a hyperbolic function of the same order.

Fig. 1

Symmetrical arrangement of conductors in a slot

Armed with the results of this mathematical investigation,Roth9 returned to the consideration of the leakage field intransformers in a long paper published in two sections. Heconsidered the arrangement of Fig. 2. The boundaries AB,

windingscore iron

Fig. 2Transformer window and core

BC and DA represent the yoke of the transformer and haveinfinite permeability. The boundary CD lies along the centreline of the core and is, by symmetry, a flux line. The new stepin this paper is the inclusion of the shaded region ECDFrepresenting the core. This region has a constant finitepermeability.

Four years later, in 1932, Roth10 generalised his study ofthe field in the slot of a rotating machine. The arrange-ment considered is illustrated in Fig. 3. The windings andboundaries are still rectangular in section, but the conditionsspecified in the 1927 paper have been relaxed. The windingsare no longer symmetrically disposed about the slot centreline. The permeability on the boundary AA' is still infinite,but along AB and A'B' the tangential magnetic field is finiteand of arbitrary strength. The line BB' is also no longer aflux line.

The problem of cylindrical co-ordinates is considered ina massive paper published in three sections in 1936." Solu-tions are obtained in terms of series of Bessel and circularfunctions, and the latter are represented by hyperbolic func-tions as before. The treatment is extended to transformershaving more than two windings. Infinite permeability is

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assumed for all iron boundaries, but, in spite of this simplifica-tion, the labours involved are prodigious.

A further paper in 193812 deals with the magnetic field of

Fig. 3

Arbitrary arrangement of conductors in a slot

a system of rectangular parallel conductors. Roth first con-sidered a region by flux lines and then relaxed the conditionto allow the discussion to apply to an unbounded region.

Outline of Roth's method3.1 Poisson's equation in rectangular co-ordinates

Roth's method applies to boundary-value problemswhich can be described in terms of Poisson's equation. Thisrelates a potential function within a region to a distributionof sources throughout the region. In electrostatics, forexample, the electric field strength E can be derived from ascalar potential V by the relationship E = — grad V. Theelectric flux density D is related to the sources of the fieldby div D = p, where p is the volume charge density andD — eE, where e is the permittivity. Hence div gradV— —pic This is written as V2K— •— p/e, which isknown as Poisson's equation.

With the exception of one paper," Roth confined hisattention to 2-dimensional rectangular co-ordinates. In thissystem

*x2 VA similar equation arises in magnetostatics, whereH = — grad V*, and V* is the scalar magnetic potential.

If the magnetic field is due to steady electric currentsrather than magnetic poles or dipoles, a vector potential Ais introduced, defined by curl A = B and div A = 0. Since

curl curl A = grad div A — V2<4 = — V2A

and curl B = /x curl H = \xJ

we have V2/4 = — fxJ where J is the current density perunit area. The vector equation V2A = — /x7 describes threeequations of the Poisson type. In the simple, but important,case in which J x — Jv = 0, we have the single equation

+ (2)

where A and J are both in the ^direction.

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

Poisson's equation appears also in problems of heat andfluid flow and in many other situations, but Roth confinedhis discussion to electrostatics, magnetostatics and steady heatconduction. The method of solution for the equation is tofind a function for the potential which satisfies the equationsubject to certain boundary conditions. The usual method isto find the complementary function and particular integralof the equation, but Roth has developed a technique forfinding a single function which satisfies the entire equation.His method is best explained by considering a simple example.

3.2 Magnetic field of a conductor in a slot

Fig. 4 shows a conductor in a slot. The boundaryconsists of regions of infinite permeability on three sides and

For the special case when m = 0,

_ 2aJ sin nf — sin ne

Fig. 4

Single conductor surrounded by three boundaries of infinite per-meability and one boundary which is a flux line

a line of magnetic flux on the fourth side, which forms themouth of the slot. Steady, uniformly distributed current flowsin the conductor in a direction perpendicular to the plane ofthe diagram. The origin of co-ordinates is taken as the bottomleft-hand corner of the slot. Eqn. 2 applies and is subject tothe boundary conditions

^— — 0 at x = 0 and x = aox

— = o at y = 07)y

and A = constant at y = b. Without loss of generality, thisconstant can be set equal to zero.

Roth postulates the solution

A = 2 £ Cmn cos mx cos ny (3)m n

which satisfies the boundary conditions at x — 0, y = 0. Theremaining boundary conditions are satisfied if sin ma = 0,and cos nb — 0, i.e. if m = h-n\a, where h = 0, 1, 2, 3, . . . ,and n = (2k + \)7r/2b where k = 0, 1, 2, 3, . . . Thus mais an even multiple of 77/2 and includes the value zero, whilenb is an odd multiple of TT/2. The equation itself is satisfied if

£ £ (w2 + n2)Cm „ cos mx cos ny = \xJ . . (4)m n

The coefficients Cmn can be obtained by Fourier's method:

r" r6

(w2 + n2)Cmn I cos2 mjrr/x: cos2 nydy0 0

= /xJ cos mxdx cos «y</y

_ 4ju.y 1 sin md — sin we sin nf — sin «e*•" ~ ~aT m2 + n2 m n " ( 5 )

PROC. 1EE, Vol. 114, No. 12, DECEMBER 1967

• • ( 6 )

The coefficients can now be computed, and the problem issolved. The magnetic field can be obtained from trie relation-

7)A ~bAships Bx — ^— and Bv = — ^—. The vector B is therefore

x ^y y ^xperpendicular to the vector F — — grad A, which has thecomponents

^ A AT? ^ A

Fx = — ^— and F = — x—x ^x y oy

This implies that flux lines are also lines of constant A, and aplot of A is therefore a flux plot. This enhances the value ofthe solution in terms of the vector potential.

4 Advantages of Roth's methodThe simplicity of Roth's approach can best be appre-

ciated by comparing it with the alternative and more usualmethod of the separation of variables. This very generalmethod was applied by Rogowski to problems similar tothose considered by Roth, who was familiar with Rogowski'swork.

Consider again the problem discussed in Section 3.2. Inorder to solve it by the method of separating the variablesthe region of the slot must be subdivided as indicated inFig. 5. This is necessary because the current distribution is

Fig. 5

Single conductor in a slot showing the three regions required for thesolution by separation of variables

limited to a part of the slot. The regions are ACC'A',CDD'C and DBB'D', numbered 1, 2 and 3, respectively. \nregions 1 and 3, the magnetic field is described by Laplace'sequation

1?A+ ^ 2 -

(7)

In region 2 the field is described by Poisson's equation(eqn. 2). Consider solutions of the form A = XY, where Xis a function of x only and Y is a function of y only.Laplace's equation gives

\(8)

Since both terms are dependent on only one variablethey must be equal to constants of equal magnitude andopposite sign. Thus

I ~d2X , J I ^ Y ,- ;—y = - ml and — ^ - y = + mL

X ox2 Y Oy2

The variables have separated and solutions are of the form

X = sin mx or cos mx

Y = sinh my or cosh my

1971

(9)

For the special case m = 0,

X = a or fixY = y or 8y

(10)

To obtain A, products XY are formed which will satisfy theboundary conditions. The choice of circular functions for thexdirection implies a Fourier representation in terms of x ofthe current distribution in region 2.

J — 2 Jm c o s mx> where m = hrr/a .

m

Therefore

2 f' , 2J sin md — sin meJm

= - \ J c o s mxax =a l am

and Jn =J(d - c)

(11)

(12)

(13)

Hence, in region 2

t>2/4 ^ 4 _ _ yj

c)x2 <)y2 a— sin m

m)cos mx

(14)

The particular integral of this equation is of the form

A = *LKm cos mx + K0y2 (15)

2iiJ (sin md — sin me)where Km = — ,

a mi

and Kn = —- c)

2a

(16)

(17)

The complementary function has to be added, and thecomplete solution for region 2 is

A 2 = ]£ {(/>m cosh my + Em sinh my) cos mx} + a2 + j82j>

J^//j N 2 A

m3 cos

where Dm, Em, cc2 and j32 are constants.

In region 1, making use of -̂— = 0 at y — 0,

(18)

A{ = cosh m^ cos mx) (19)

and in region 3

A3 = 2 {(Fm cosh my + Gm sinh m^) cos mx} + a3 + fi3ym

. . . . (20)

Region 3 is bounded by a flux line. Thus A — constant aty = b. Jf, as in Section 3.2, we put A = 0 at y = b, and usethe fact that on the boundaries between the regions, A

and ^— are continuous, (these relationships imply that r—oy ox

is also continuous) we obtain equations from which the'coefficients Cm, Dm, Em, Fm, Gm, Km, a,, a2, a3, ^2 and ^3

can be found.After considerable reduction,

sin md — sin mc\

(21)

cosh m(b — e) — cosh m(b — f)cos mx ; ; cosh my

cosh mb

sin md — sin me

. , sinh me sinh m(6 —cos mx-< I

cosh mb

cosh my cosh m(b — / )cosh

- C) {2b(f - e ) - p - ley) (22)

sin md — sin mea m3

sinh m/ — sinh me .cos mx cosh mb

sinh m(6 —

(23)

Jt is clear that this method is far more cumbersome than theone suggested by Roth. His solution gives a single expressionfor the entire region of the slot, while the method of theseparation of variables involves three different expressionsand requires the use of four boundary equations between theregions, in addition to the boundary conditions at the peri-meter of the slot.

Limitations of Roth's method

5.1 Periodicity of the solutionIn Roth's method the functions representing the

potential have to be orthogonal and therefore oscillatory inevery direction. Thus in the example in Section 3.2 thex variation is cos mx and the y variation is cos ny.

Similarly, the current distribution has to be expressed inoscillatory functions. These functions are orthogonal over asuitable interval, and this makes it possible to obtain theindividual coefficients Cmn by Fourier's method.

The periodicity restricts solutions to problems in whichthe source distribution is also periodic. Such a distribution isshown in Fig. 6, in which the direction of current is indicated

1972

Fig. 6

Doubly infinite array of conductors

by the plus or minus sign. Such infinite distributions of currentare, however, only of academic interest. What the engineerneeds is a solution of the field in a finite region, such asthat indicated by the boundaries in Fig. 6. The questiontherefore arises as to whether there are any physicalboundaries which give the effect of periodicity and representthe outside sources correctly both in position and magnitude.From the theory of images, we know that there are twopossible boundaries: those of infinite permeability and thoseof zero permeability. The first type represents an approxima-tion to a physical situation, because permeabilities are nevertruly infinite. Nevertheless, the field in air outside an ironboundary very closely resemoles the field outside a boundaryof infinite permeability. Zero permeability can be postulatedalong a flux line, and so does not represent an approximation.By analogy the boundaries in an electrostatic problem must

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

have infinite or zero permittivity. Roth's method applies aslong as the boundaries are either scalar equipotentials

(=— = 0) or flux lines (A = constant).

It follows that Roth's method cannot be applied to regionsbounded by material of finite permeability or permittivity.Nor can it be applied to any physical problem in which thefield has both normal and tangential components at aboundary.

This restriction should be compared with the conditionsapplicable to a solution by the method of separation ofvariables. In this method the use of oscillatory functions forall the variables is not allowed for static problems. At leastone function must be nonoscillatory. Thus, in Section 4 thesolution was obtained in terms of cos mx and sinh my orcosh my. Alternatively, it could have been obtained by usingcircular functions of y and hyperbolic functions of x.

Only oscillatory functions impose,the boundary conditionsof zero and infinite permeability. Thus, the method of-separation of variables can be applied to regions havingmixed boundary conditions in the direction of the hyperbolicfunctions. The magnetic field in a slot, for instance, can beobtained if the bottom of the slot has finite permeability,and the top of the slot need not be bounded by a flux line.Even the two sides of the slot need- not be of infinite per-meability, because solutions of the type e±Jmxe±r"y can beadded to solutions of the type e±mxe±jmy. Thus there is noneed for periodicity in either x or y. The scope of the methodof separation of variables is far more extensive than thescope of Roth's method. All problems that can be solvedby Roth's method can also be solved by the separation ofvariables, but the converse is not true. This conclusion is atvariance with the conclusions reached in References 1 and 4.

5.2 Superposition of solutions

The simplicity of Roth's solution depends on the factthat a single expression-satisfies the entire Poisson equationwith its boundary conditions. Eqn. 3 in Section 3.2 waswritten as a double Fourier series. It should be noted explicitlythat the individual terms of the series do not satisfy Poisson'sequation, nor can the individual terms be identified with thesource distribution of current or with particular parts of theboundaries. Only the complete expression has physicalsignificance. Every problem amenable to Roth's method hasits own complete solution. Superposition of solutions isrestricted to the superposition of sources within the sameboundaries. In particular it is not possible to superpose theeffect of magnetised surfaces and currents in the mannersuggested by Hague.1 .

This is in sharp contrast with the method of separation ofvariables. In Section 4 it was shown how that solution wasbuilt up from a complementary function and a particularintegral. The latter deals with the current distribution butignores the top and bottom of the slot. The solution is thencompleted by adding terms which account for these boundariesand also for the limited extent of the current in the ^direction.Every individual term in the resulting series solution is alinearly independent solution of Laplace's equation, and,with a little care, these individual terms can be identified withthe different parts of the boundary. These boundaries aretreated by the solution as additional sources. Thus the seriessolution is a true summation of the effect of individualsources. This accounts for the great flexibility of* the methodand also for the complexity of the algebra. Unlike Roth'smethod, the method of the separation of variables allowsthe superposition of solutions with different boundary con-ditions. Complicated conditions can be built up by combiningsolutions of simpler cases. Moreover, the method of separa-tion of variables can deal with regions where the sources areconfined to the boundaries. Such Laplacian problems cannotbe solved by Roth's method.

5.3 Time-varying problems5:3.1 Eddy-current problems

In his papers Roth confined his attention to staticfields. He did, however, mention the fact that the knowledgeof the magnetic field calculated by his method could be usedPROC. IEE, Vol. 114, No. 12, DECEMBER 1967

to determine the eddy-current loss in the conductors in a slot.Presumably he had in mind the low-frequency approximation,in which the magnetic field of the eddy currents is neglectedin comparison with the magnetic field of the applied current.

It is interesting to investigate whether this restriction isnecessary or whether the method can be applied moregenerally.

The electromagnetic equation governing eddy-current phe-nomena is known as the diffusion equation. In terms of themagnetic vector potential it is written

r Tit r

where a is the conductivity and / ' is the applied currentdensity. A solution in Roth's form demands that the con-ductivity and permeability must be constant throughout theregion. It thus excludes most problems of practical interest.Solutions will be of the type

A =m n to

S Cm>,,it0 cos mx cos ny cos cut (24)

and the boundary conditions will again have to be of infiniteor zero permeability. These conditions are more difficult tomeet than in the static case, because currents may be inducedin the boundaries, and the effective permeability will be acomplex function of the frequency.

The method of the separation of variables does not sufferfrom the restrictions imposed by the use of circular functionsin both x and y. This method makes it possible to examinethe interaction of applied and induced currents and to solvecomplicated problems by dividing the region to be consideredinto a number of simpler subregions linked to one anotherby suitable boundary conditions.

5.3.2 Microwave problems

The relevant equation is

Roth's method is again a possible one, but the boundaryconditions are extremely restrictive. The interaction of electricand magnetic fields can no longer be neglected, as in staticand quasistatic problems. The electric boundary conditionsare infinite or zero permittivity, and the magnetic conditionsare infinite or zero permeability. Every boundary mustpossess a pair of these properties. However, there are notreally four independent conditions. Consideration of thecontinuity of electricity shows that infinite permittivity islinked to zero permeability and that a similar linkage holdsbetween zero permittivity and infinite permeability. Theproblem is even more restricted because, in practice, highpermeabilities are almost unobtainable at high frequencies.This leaves one boundary condition, namely infinite per-mittivity and zero permeability, or in other words normalelectric field and tangential magnetic field. In theory, theboundary would therefore have to be superconducting, butan ordinary conductor will give a reasonable approximation.Roth's method is therefore restricted to regions enclosed byhighly conducting boundaries.

Again the method of separation of variables is preferablebecause it is not restricted to enclosed regions but can dealwith the field of radiating sources in free space. Even in wave-guides, Roth's method does not allow the separate considera-tion of regions free from conduction current and is thereforenot easily applied.

5.4 Slow convergence of the seriesIt is a feature of Roth's method that his solution is of

the form

A = ,n c o s mx c o s ny • • • (3)

Thus, for a problem in two dimensions, the solution involvesthe summation of a doubly infinite series. The indices m andn are independent of each other. The comparable solution byseparation of variables is of the form

A = c o s mx c o s n

1973

Instead of a double series in m and n, there is a single seriesin m only. This use of a single index, or separation constant,is made possible by choosing every term of the series to be asolution of Laplace's equation. In other words, the physicalconditions of a vector field free from sources are included inthe solution at the beginning, whereas in Roth's methodthese conditions are not applied to the individual terms butto the whole series. This accounts for the fact that Roth'smethod has slow convergence in comparison with the methodof separation of variables.

6 Extensions of Roth's methodRoth was aware of the difficulties which could be

encountered in the use of his method and he laboured toremove them, although unhappily he did not investigate thephysical reasons for the difficulties. His papers give theimpression that he believed that the method could be extendedindefinitely by an improvement in the algebraic formulation.Writers like Hague1 seem to have accepted this point of view,and the excessive claims made for Roth's method may haveled to its disuse.

6.1 Magnetic field in a space containing iron andelectric current

Roth considered this problem in a long paper dealingwith the leakage fields of a transformer. Fig. 2 is taken fromthis paper. The transformer windings are located in the left-hand side of the figure, and the iron core is shown on theright. The line CD is the centre line of the core and is, bysymmetry, a flux line. The yoke is represented by the surfacesAB, BC and DA, and it is assumed that the permeability ofthe yoke is finite. The boundaries are therefore all of thekind demanded by Roth's method, and the method can beapplied to the solution of the problem. The difficulty of theproblem stems from the presence of the core iron in the spaceECDF, since this iron will modify the magnetic field of thecurrents in the space ABEF and there are thus two sets ofmagnetic sources. The current sources are assigned and areknown, but the effect of the iron is unknown.

Roth overcame the difficulty by replacing the iron by anadditional current distribution on the surface EF. This currentdistribution is written as an arbitrary function of theco-ordinates, and a formal solution of the vector potentialis obtained. Roth then used this solution to obtain the dis-continuity in the tangential magnetic field strength at thesurface. Finally he wrote the step in tangential magnetic fieldin terms of the permeability of the iron, and thus obtainedan equation for the equivalent surface current in terms ofthe permeability. All the current sources are then known,and the solution proceeds as before.

This is an ingenious method and extends Roth's approachto all problems in which iron as well as current is included inthe bounded space. However, some words of warning needto be said. Although he was aware that the surface currentrepresents an additional source in Poisson's equation, Rothdescribed the extra term due to the surface current as a'Laplacian field'. He stated that he was superimposing aLaplacian on a Poissonian distribution of field. This is anunfortunate nomenclature, because Roth's method is inprinciple incapable of dealing with a Laplacian field, andthis is one of its severe limitations. This point seems to havebeen overlooked by subsequent writers.

A further conceptual difficulty arises from Roth's apparentassumption that the equivalent current on the iron is confinedto the surface EF in Fig. 2. This is misleading. In general,there will be surface layers of current on all the surfaces ofthe iron. It is only where these surfaces lie along theboundaries of the space considered that it is possible toignore such further surface currents, because in this specialcase the effect of the surface current is embodied in the valueof the potential at the boundary. Roth's single layer ofcurrent does not satisfy the condition of continuity. This iseasily seen if, instead of surface current, we use surface poles.Roth's explanation would then imply the presence of onekind of polarity more than of another which is not physicallypossible. Similarly, there can be no net induced flow of directcurrent in iron. It is curious that Roth seems unaware of the

1974

difficulty, and that in discussing a problem in heat conductionhe introduces surface distributions of heat which do not addup to zero. They modify the energy input and do not satisfythe principle of conservation of energy and can thus easilylead to erroneous conclusions.

6.2 Improvements in the convergence of seriessolutions

It must be remembered that Roth did not have theaid of computers. The many valuable flux plots which heobtained required great labour and, with the aid of G.Kouskoff,8 he devised a means of improving the convergenceof the double Fourier series encountered.

The method used was to find a finite sum for the Fourierseries, and Roth and Kouskoff investigated the use of hyper-bolic functions. Typical results are given in Appendix 9.1.When the coefficients Cm „ have been found, the independentparameters m and n in the double Fourier series are reducedto the single parameter m, and the series of circular functionsin n are replaced by a single hyperbolic function in m. As aresult there is a marked improvement in the convergence. Itis, however, found that it is no longer possible to use a singlerepresentation for the entire bounded space. Differenthyperbolic functions need to be used in different regions.

Applying Roth's and Kouskoff's results to the problem ofthe slot in Section 3.2 and Fig. 4, we find that the regionneeds to be subdivided into three: region 1 for 0 < y < e,region 2 for e < y < / a n d region 3 f o r / < y < b.

Remembering

A = 2 2 CTO „ cos mx cos rt.y (3)m n

we have in region 1 by the use of eqns. 31 and 33 (Section 9.1)

cosh m(b

sin md — sin

e) ~ cosh m(b — / ) .> cos mx cosh mycosh mb

(25)

which is identical with eqn. 21. Similarly, A2 and AT, can bederived and will be found to be identical with eqn. 22 andeqn. 23. It is extraordinary that neither Roth, Hague norBinns and Lawrenson seem to have realised the significanceof this identity. Inevitably the substitution of hyperbolic forcircular functions has led to the form of solution obtained bythe separation of variables.

Roth's substitutions merely provide an alternative way ofobtaining this form of solution. Either one can subdivide thebounded space into different regions and use the boundaryconditions between regions to obtain the solution, or one canwrite the solution first in terms of circular functions and thenuse Roth's substitutions. If a computer is readily available,the direct method is more straightforward, but Roth's sub-stitution may be simpler where the problem is worked byhand. In either case the ultimate solution is the same. Roth'ssubstitutions provide a stepping stone to the separating ofvariables.

6.3 Magnetic field in a region bounded by iron offinite permeabilityIn a paper published in 1932,l0 Roth endeavoured to

extend his method to problems involving boundaries whichare neither of infinite nor of zero permeability. Fig. 3 istaken from this paper. The boundary AA' has infinitepermeability, but the teeth are saturated, and there is anassigned and varying tangential magnetic field strength H{along AB, and H2 along A'B'. Finally, there is an arbitraryvarying vector potential along BB', which is therefore nolonger a flux line.

It should be noted that there is a great difference betweenthis problem and the one discussed in Section 6.1, in whichthe iron was enclosed within the boundaries. Indeed, if our

PROC. IEE, Vol. 114, No. 12, DECEMBER 1967

previous analysis of Roth's method is correct, the method isnot by itself applicable to the problem of Fig. 3. This is borneout by a study of Roth's paper.

He first derived a vector potential for the field of thecurrents when AB, A'B' and AA' have infinite permeabilityand BB' is a flux line. He then added, by the separation ofvariables, a vector potential which has no sources in the slotand has no derivative at the three iron boundaries. Thispotential was chosen to have the correct value on BB'. Lastlyhe added a further vector potential, obtained by separationof variables, which satisfied the tangential magnetic field onAB and A'B'. Thus the problem was solved by a mixture ofRoth's own method and that of the separation of variables.

This example shows that it is possible to extend the applica-tion of Roth's method by the addition of solutions obtainedby the separation of variables. In Section 6.2 it was pointedout that any solution by Roth's method has an equivalentsolution by the separation of variables. Since it is possibleto superpose solutions by separation of variables, it is alsopossible to superpose such solutions on a solution by Roth'smethod. The converse is not true however. The method ofthe separation of variables cannot be extended to newproblems by adding solutions of Roth's type. All suchsolutions are only particular cases of the method of separationof variables.

Although, in principle, Roth's method is therefore unneces-sary, it may in practice be very convenient in examples wherethe method of separation of variables demands the solutionto be split into many subregions. In the example of Fig. 3 themixture of the two methods is appropriate and convenient.

6.4 Magnetic field in an unbounded regionIn a paper published in 193812 Roth investigated the

magnetic field of a set of parallel rectangular conductors inair. He first obtained this field in a region bounded by fluxlines and used his own method of double Fourier series. Hethen made use of the substitution mentioned in Section 6.2and converted the solution into the form obtained by separa-tion of variables. The boundaries were then moved to infinityby expressing the hyperbolic functions in terms of exponentialsand the circular functions in terms of a Fourier integral.

Except for the initial step, no use is made of Roth's method,and it is clear that he himself preferred to use the separationof variables in solving this problem.

6.5 Roth's method in cylindrical co-ordinates»

In Section 5.1 it was shown that the functions used inRoth's solution must all be orthogonal over the interval con-sidered. Many functions possess this property, and in a mas-sive paper in 1936" Roth.investigated the field of cylindricaltransformer windings by means of his method using Besselfunctions of the first and second kinds and of order unity.

Fig. 7 shows half of a transformer window. The core ADand the yoke AB and DC are assumed to have infinite per-

\\

D 0 s

\ \ \ \

Fig. 7Succession of slots and teethPROC. IEE, Vol. 114, No. 12, DECEMBER 1967

meability. The cylindrical surface CB carries no tangentialflux because the net m.m.f. enclosed by ABCD is zero. Theonly component of current is the circumferential / e , and thereis no 6 variation of the field. As a result, the vector potentialhas only the component A$, and Poisson's equation becomes

~o2A 1 ~bA A ~b2A+ _ _ + = = _ / x y . . . . ( 2 6 )

or r or r oz

subject to the boundary conditions:

7)AB= - ^ - = 0 at z = 0 and z = b

oz

<^ A nB7 — -^— + — = 0 at r = a* and r = aor r

Choosing a circular function cos nz for the z variation, wehave sin nb — 0 or nb = kir where k is an integer or zero.Choosing Bessel functions 3\(mr) and Y,(mr), the solutionis of the form

Z,(mr) = .

\J0(ma)

subject to the condition

Y0(ma0)\ J0(ma) Y0(ma)

J,(/nr) Y,(mr)

_

(27)

(28)

The roots of eqn. 28 give the values of m and can be obtainedfrom tables.13

The solution is of the form

A = Cm/)Zi(mr)cos«z

which has to obey Poisson's equation, so that

S 2 Cm,n{m2 + «2)Z,(mr) cos nz = fxJ

(29)

• (30)

whence the values of Cmn can be obtained by making useof the orthogonality of Zt(mr) and of cos nz. The solution isvery simple in form and, like the solution in circular functions,can easily be remembered.

There is, however, an additional difficulty to be overcomein finding the values of the coefficients Cm „. Integrals ofBessel functions arise in the calculation, and these integralsare not themselves Bessel functions and are not tabulated.They would, therefore, have to be computed separately.

As before the convergence of the series is poor, and Rothmade use of his usual substitution of hyperbolic functionsfor circular functions. This reduces the double series in mand n to a single series in m. The solution is one which couldhave been obtained by separation of variables, but it is notthe best solution. The method of separation of variablesadmits the possibility of another solution in circular functionsof z and modified (nonoscillatory) Bessel function of r. Insuch a solution integrals of Bessel functions do not arise,and this solution is much to be preferred to Roth's solution.It appears, therefore, that although an extension of Roth'smethod to cylindrical co-ordinates is possible, the methodcannot be recommended.

It is, of course, true that Roth could have derived a sub?stitution of oscillatory for nonoscillatory Bessel functions andthat any solution by separation of variables can be derivedfrom Roth's solution by appropriate substitutions. There issomething rather artificial in such a procedure, and Roth'ssolution is only really useful where it can be applied in itsoriginal form. Nevertheless, if computer time is expensive ordifficult to obtain it may be worthwhile to proceed in thismanner.

Roth did not investigate other co-ordinate systems. Thiscan be done in the way shown in this Section. It is, however,very unlikely that the conclusions would differ from thosethat have just been stated, namely that Roth's method isonly convenient in rectangular co-ordinates.

7 ConclusionsBy means of Roth's method, a single solution for the

magnetic field in a region containing electric current can be

1975

written by inspection, whereas the region has to be subdividedif the method of separation of variables is used.

The convergence of the series used in Roth's method isslow but it can be greatly improved by making substitutionsof the type described in Appendix 9.1. By this procedure it ispossible to obtain the solutions of the method of the separationof variables without having to solve a large number of simul-taneous equations. This can be a considerable help whereproblems have to be worked by hand. Even when a computeris available, the cost of using it can be reduced by workingthe first part of the problem by hand in Roth's manner andusing the computer to sum the series obtained after thesubstitutions have been made.

Roth's method is best applied to systems in rectangularco-ordinates having boundaries which are flux lines or areof infinite permeability. It is possible to extend the method toother co-ordinate systems and to more complicated boundaryconditions, but the simplicity and benefit of the method tendto be lost.

8 References1 HAGUE, B. : 'Electromagnetic problems in electrical engineering'

(Oxford University Press, 1929), pp. 313-3252 BEWLEY, L. v.: 'Two-dimensional fields in electrical engineering'

(Dover, 1963), pp. 81-833 BILLIG, E. : The calculation of the magnetic field of rectangular

conductors in a closed slot and its application to the reactance oftransfer windings', Proc. IEE, 1951, 98, Pt. 4, pp. 55-64

4 BINNS, K. J., and LAWRENSON, P. J. : 'Analysis and computation ofelectric and magnetic field problems' (Pergamon, 1963), pp. 109-117

5 ROTH, E.: 'Introduction a l'etude analytique de Pechauffement desmachines electriques', Bull. Soc. Franc. Beet., 1927, 7, pp. 840-954

6 ROTH, E. : '£tude analytique du champ propre d'une encoche',Rev. Gen. ilect., 1927, 22, pp. 417-424

7 ROTH, E. : '£tude analytique du champ de fuites des transformateurset des efforts mecaniques exerces sur les enroulements', ibid., 1928,23, pp. 773-787

8 ROTH, E., and KOUSKOFF, G.: 'Sur une mdthode de sommation decertaines series de Fourier', ibid., 1928, 23, pp. 1061-1073

9 ROTH, E. : '£tude analytique des champs thermique et magnetiquelorsque la conductibilitie thermique ou la permeabilitie n'est pasla meme dans toute l'etendue du domaine considere', ibid., 1928,24, pp. 137-148 and 179-188

10 ROTH, E.: '£tude analytique du champ resultant d'une encoche demachine electrique', ibid., 1932, 32, pp. 761-768

11 ROTH, E. : 'Inductance due aux fuites magnetiques dans les trans-formateurs a bobines cylindriques et efforts exerces sur lesenroulements', ibid., 1936, 40, pp. 259-268, 291-303 and 323-336

12 ROTH, E. : 'Champ magnetique et inductance d'un systeme debarres rectangulaires paralleles', ibid., 1938, 44, pp. 275-283

13 JAHNKE, E., and EMDE, F. : 'Tables of functions' (Dover, 1945),pp. 204-206

14 BETHENOD, i.: 'Edouard Roth', Rev. Gen. Elect., 1939, 46, pp. 131-134

9 Appendixes9.1 Summation of certain Fourier series

In Reference 8 various formulas are derived by meansof which double Fourier series can be converted into singleseries. The following formulas are relevant to Section 6.2 ofthis paper. If nb is an odd multiple of TT/2, then for 0 < y < b

sin nb' cos ny

if

and

if

also

if

and

if

n{m2 + n2)

y<b',

sin nb' cos ny

b cosh mb — cosh m(b2

b') cosh my

*t n{m2 + n2)

y>b\

sin nb' cos ny

sin nb' cos ny bb'

y> b'

m2 cosh mb

b sinh mb' sinh m(b — y)2 m2 cosh mb

(3D

(32)

(33)

(34)

9.2 Edouard Roth

Born on 17th January 1878 at Mulhouse, E. Rothstudied at the Technische Hochschule at Zurich. At the ageof 22 he joined the firm Societe Alsacienne de ConstructionsMecaniques at Belfort. He made notable contributions tothe design of induction motors and turboalternators. Hebecame chief alternating-current engineer of his companyand, on the formation of the company Alsthom, he becamechief engineer (technical services). He remained with thecompany till his death on 28th April 1939. He publishedthree books and 29 papers, which show a remarkable blendof mathematical skill and physical insight. A short account ofhis life and a complete bibliography are given in Reference 14.

1976 PROC. IEE, Vol. 114, No. 12, DECEMBER 1967