Exact solution of electrostatic problem for a system of

Exact solution of electrostatic problem for a system of parallel cylindrical conductors V. Alessandrini, H. Fanchiotti, C. A. García Canal, and H. Vucetich Citation: Journal of Applied Physics 45, 3649 (1974); doi: 10.1063/1.1663832 View online: http://dx.doi.org/10.1063/1.1663832 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/45/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Parallel iterative solution of large‐scale acoustic scattering problems using exact non reflecting conditions on distributed memory computer systems J. Acoust. Soc. Am. 113, 2252 (2003); 10.1121/1.4780416 Cylindrical liquid bridges squeezed between parallel plates: Exact Stokes flow solutions and hydrodynamic forces Phys. Fluids A 4, 1105 (1992); 10.1063/1.858229 Electrostatics of systems of parallel plates Am. J. Phys. 50, 899 (1982); 10.1119/1.12968 Exact solution of Kondo problem Phys. Today 34, 21 (1981); 10.1063/1.2914564 Proximity Effect in Systems of Parallel Conductors J. Appl. Phys. 43, 2196 (1972); 10.1063/1.1661474

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Exact solution of electrostatic problem for a system of parallel cylindrical conductors v. Alessandrini*, H. Fanchiotti t , C. A. Garcia Canal*, and H. Vucetich Departamento de Fisica, Universidad Nacional de La Plata, La Plata, Argentina (Received 31 October 1973)

The problem of the calculation of the electric field distribution of a system of infinite parallel cylindrical conductors in the presence of a grounded plane is reexamined. The exact solution of a class of two-dimensional harmonic problems in multiply connected domains proposed by Burnside in 1893 is reviewed. This is shown to agree with recent generalizations of the method of images. An improved version of the exact solution which is more convergent than the conventional methods when the conductors are very close to one another is presented. The potential usefulness of the theoretical ideas and the mathematical apparatus used here for technological applications is emphasized.

I. INTRODUCTION

The problem of determining the electric field distribution of a system of infinitely long parallel cylindrical conductors placed either above a grounded plane or inside a grounded cylinder arises in a variety of engineering problems. The knowledge of the electric field in the neighborhood of the conductors is necessary for example in the design of extra-high-voltage ac or dc overhead transmission lines in order to calculate the corona effect and the associated power loss and radio interference. Mathematically, this problem boils down to the solution of the Laplace equation in the exterior of the conductors, this is to say, to the solution of a harmonic problem in a multiply connected domain. By a multiply connected domain we mean, of course, a two-dimensional domain with holes, the boundaries of the holes being, in the example under consideration, the boundaries of the cylindrical conductors. In addition to the problem of the cylindrical conductors, a wide variety of other technological problems can be reduced to the determination of harmonic functions in a multiply connected domain_ As another example we might mention the determination of the resistivity of thin films. Thin films are usually obtained by a metallic deposit in the interior of a spherical balloon. But this metallic deposit does not fill all the surface of the sphere, because there are circular holes left out: two of them are the boundaries of the electrodes that establish a current in the film, the other ones are holes used to introduce the metallic vapor. The resistance R of the film is obtained from Ohm's law, and the resistivity a is given by R = aG, G being a geometrical factor. The calculation of this geometrical factor G can be rephrased again in terms of the solution of a harmonic problem in a multiply connected two-dimensional domain.

even in the case in which the holes are not circular but have any shape-in terms of the so-called "automorphic functions"5 which are a generalization for surfaces with many holes of the elliptic functions, solutions of the harmonic problems in surfaces with only one hole. These exact analytic solutions are given in terms of series called "Poincare theta series"6 and the main purpose of this paper is to show that these exact solutions are easier to handle in practical technological applications than the approximate solutions presented in recent years in engineering literature. The 'general results that we shall present here are contained in a beautiful mathematical paper published by Burnside in 1893. 7 We shall see that the solution to the problem under consideration can· be reinterpreted in terms of the electric field generated by an infinite set of image charges located in the interior of the cylindrical conductors. Indeed, we shall find that the original version of the Poincarl! theta series proposed by Burnside coincides with the generalization of the method of images for the present case proposed by Sarma and Janischewskyj.l However, the general theoretical ideas on automorphic functions underlying Burnside's solution allows one to recast the original exact series solution in a way that is more convenient for practical applications. To the best of our knowledge, these improved series are only known in the field of high-energy nuclear physics where very recently they have become useful in the solution of harmonic problems in multiply connected Riemann surfaces related to theoretical models of elementary particles. 8,9

The determination of the electric field distribution of the system of conductors we are dealing with requires solution of the Dirichlet problem: the potential is given on the boundaries. This problem has received considerable attention in engineering literature in recent years where various approximate methods of varying degrees of accuracy have been devised in order to solve it. 1-3 However, the construction of harmonic functions in multiply connected domains is well known in general and extensive mathematical literature exists on the subject. 4 This problem can be exactly solved-

This paper is organized as follows: In Sec. II we collect a series of definitions and properties of projective transformations of the complex plane that are necessary for the understanding of the ideas developed in the rest of the paper. In Sec. III we start by recalling some general facts concerning the solution of harmonic problems in multiply connected domains and proceed next with the construction of the explicit solution for the problem under consideration in terms of Burnside automorphic functions. We give the exact formulae for the potential distribution, and compare them with the method of images of Sarma and Janischewskyj. We also give exact formulae for the Maxwell coefficients (generalized capacities) of the system of conductors. 10 Finally, we conclude that section by carrying out an instructive exercise that provides a great deal of in-

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Copyright © 1974 American Institute of Physics

Journal of Applied Physics, Vol. 45, No.8, August 1974

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Alessandrini et al.: Electrostatic problems for parallel cylindrical conductors

sight into the general problem: we discuss in detail how the general solution yields the well-known result for the case of only one cylindrical conductor in the presence of a grounded plane. The analysis and the example of Sec. III show that the exact solution of the problem can be regarded as a method of successive approximations in which at each step one cancels a given set of image charges inside the conductors and replaces it by a new set of image charges at well-defined positions. This coincides with the method proposed in Ref. 1. We call this mechanism the phenomenon of infinite cancellations of image charges. Clearly the process must converge to some well-defined distribution of image charges. In Sec. IV we show how this infinite cancellati,on can be explicitely eliminated in general and how the exact solution can be written in terms of an improved series where no infinite cancellation occurs. This series is expected on general grounds to be more rapidly convergent than the Sarma and Janischewskyj method, especially in pathological cases where the cylindrical conductors are very close to each other or to the grounded plane. This is substantiated with a numerical analysis of some examples, to show the power of the method. Finally, in Sec. V, a discussion of the results is given. II. PROJECTIVE TRANSFORMATIONS

The exact solution of two-dimensional potential problems in multiply connected domains rests heavily on the systematic use of projective transformations. We shall therefore present here, without proof, most of the elementary properties of these transformations that are used in the rest of the paper. A projective transformation is a conformal transformation of the form z' =T(z)=(az+b)/(ez+d).

(1)

These are the only conformal transformations that map the whole of the complex plane onto itself. They also map circles or straight lines into circles or stright lines-a stright line is considered here as the particular case of a circle with the center at infinity. The parameters a, b, e, d are in general complex. Notice however that the transformation (1) depends only on three of them because, being a homogeneous transformation, it does not change if we multiply all parameters by the same complex constant. We can therefore always choose one of them to be equal to 1 or alternatively, as we shall do in what follows, we can impose the normalization condition ad - bc = 1. Another important point to be noticed is that, since we have three parameters at our disposal, we can always find a projective transformation that maps three given pOints into three other given points in the complex plane.

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plex numbers). Then it follows that (1) can be written in matrix notation as (3)

USing this representation it can now be easily checked that if we perform two successive projective transformations z"=T"(z)=T'(T(z»=

a"z+b" , e"z+d"

(4)

with T(z) given by (1) and T'(z) given by T'(z)=a'z+b'; e'z+d'

7"= (

a' e'

b'd' ) ,

dd' - b'e' + 1; (5)

then the parameters of the resulting projective transformation T" (z) are given by a 2 x 2 matrix T" equal to the product 7'" = 1"' 1".

(6)

Notice that 7'" is automatically normalized because det7'" = det7" det1" = 1. We conclude therefore that the operation of performing successive projective transformations is entirely equivalent to the matrix multiplication of the associated matrices. One therefore usually refers to the projective transformation T" (z) = T'(T(z)) as the product of the transformations TT' . We shall need very often the concepts of "invariant points" and "multiplier" of a given projective transformation T(z). The equation T(z) = z

(7)

has in general two dist~nct complex roots that we shall denote by ~, 11 and which are called the invariant points of T. It may happen that ~ = 11 in which case the transformation T(z) is called "parabolic", but we shall never need to consider parabolic transformations in what follows, so from now on we shall assume that ~ *11. The invariant points are easily obtained from the representation (3) of T(z) in terms of homogeneous coordinates. Notice that the couples (x, y) and (Ax, AY) give the same complex number z=x/y. Let us now compute the eigenvalues and the eigenvectors of 1": (8)

We shall find in general two complex eigenvalues AI' A2 and associate complex eigenvectors (xo' Yo) and (Xl' YI)' respectively. Now (8) is just (7) in homogeneous coordinates, so we conclude that the eigenvectors of 1" are the homogeneous coordinate representation of the invariant pOints. Then we find (9)

We finally define the multiplier K as the ratio of the two eigenvalues of 1":

It is very convenient to associate to the projective transformation T(z) given by (1) a 2 x 2 matrix (2)

with det1" = ad - be = 1. This matrix 1" arises very naturally if one introduces homogeneous coordinates in (1) and writes z = x/y; z' = x' /y' (x, y; x', y' are com-

(10)

so the eigenvectors and eigenvalues of 1" determine the invariant points and the multiplier of T(z), respectively. One can now use the three complex parameters ~, 11, and K to characterize the projective transformation

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T(z). In terms of these parameters T(z) can be written T

as T(z) - ~ T(z)-11

_K z- ~. z-11

(11)

The parameter K is called multiplier because one can always perform a projective transformation z' = 5(z) that maps (~,11) to (0,00) as follows: z' =5(z) =(z - ~)/(z -11).

(12)

In terms of this transformed variable, (11) reads T(z')=Kz'

(z)- az+b _ -ad/e+(a/\el)exp(i and the projective transformation T(z,) reduces to a scale transformation in the z' complex plane.

(22)

An important point to be emphasized here is that one can always ehoose the ratio of the eigenvalues AJA 2 in such a way that IKI < 1, for (11) shows that changing K into K- 1 simply amounts to interchanging ~ and 11. So we define ~ as the invariant point determined from the eigenvector of T associated with the smallest eigenvalue (in magnitude). This choice of what we call ~ ensures that IK I < 1. Finally we mention that the relation between (~, 11, K) and (a, b, e, d) can also be obtained by computing 1'(z) from (11) and comparing the result with (1 ).

Another useful concept that we need in what follows is that of "isometric circle" of a transformation T(z) and of its inverse transformation r-1(Z). Given T(z) by (1), its inverse transformation has an associated matrix 7'"1, and is !~iven by T-l(Z)= (dz - b)/(- ez + a);

da - be= 1.

(14)

The isometric circle of T(z)-denoted by I(T)-is the circle defined by (15)

lez+dl =1

so its center J and radius R are given by (16)

J=-d/e.

Likewise the isometric circle of T-I(z)-denoted by I(T-I)-is defined by

1- ez + a = 1

(17)

1

so we succeeded in checking that, if z EO J( T), T(z) EO J(T-I). In the same way one can check that if

z EO J(T- 1 ), T-I(z) EOJ(T). Since a circle is uniquely determined by three pOints in the complex plane, and projective transformations transform circles into circles and are also uniquely determined by stating how three given pOints in the complex plane transform into three other given pOints, it is clear that one can always find a projective transformation T that maps a given circle C into another given circle C'. However, if C and C' have the same radius the problem of finding T is particularly simple, because C will be the isometric circle of T, and C' the isometric circle of T- 1 • The formulae written above allow us to find the parameters of T in terms of the radius R and centers of the circles. We shall come back to this point in the next section. Finally, we want to conclude this section with a few remarks that will play an important role in our future developments. We shall need very often to compute a power of a projective transformation, T"(z), defined as the successive application of T(z) n times. Clearly its associated matrix is Tn, and the eigenvectors of Tn are the same as those of T. Therefore, the invariant pOints of T"(z) are the same as those of T(z). However, the eigenvalues of T" are Af, A~, where AI' A2 are the eigenvalues of T, and then it follows that the multiplier K" of T"(z) is given by (23)

I

so the radius R and the center J- are given by (18)

These concepts are very important because in the next section the radii and centers of the isometric circles of some projective transformations will be identified with the radii and centers of the cylindrical conductors. Moreover, the fundamental result that we shall need in what follows is that T(z) maps its isometric circle I(T) into I(T- I ) and the interior (or exterior) of I(T) into the exterior (or interior) of I(T- I ). Likewise, T- 1(z) maps I(T- 1 ) into I(T) and the interior (or exterior) of I(T- I ) into the exterior (or interior) of I(T). We present here a proof of some of these statements because they play a crucial role in our future developments. Indeed, if z is on I(T) it can be written as z =J + R

exp(icp) =.:... d/e + (1/ Ie I) exp(icp);

then

K being the multiplier of T(z). This makes it Simple

therefore to compute the powers of a projective transformation T(z). If the invariant points of T(z) are mapped to (0, 00) by the transformation z' =5(z) of (12), then Tn(z,) = Knz' .

(24)

Going back to (11), we can write T"(z) as

(25) Since IKI < 1, when n- 00 K"-O and we see immediately from Eq. (25) that

(19)

lim T"(z) ==~. n-

~

(26)



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o 3

o \

o 2

o N

,

o

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The harmonic measures can be calculated once the Green's function of the system is known, but we shall obtain them in a different way. Once the harmonic measures wj(z) are known, the solution of the Dirichlet problem in which I/>(z) takes the value 1/>1 on the ith boundary is given by (28)

Grounded Plane

FIG. 1. N cylindrical conductors placed above a grounded plane.

This result means that after a sufficiently large number of transformations T any point in the complex plane is mapped into a point arbitrarily close to the invariant point ~ of T. Later on we shall rephrase this result by saying that the successive "images" of a point Z cluster at the invariant point ~, where by image of Z we understand T(z). Moreover, one can easily show that T-1(z) has the same multiplier K as T(z), but the invariant points are interchanged: ~'=7) and 7)' =~. Then we can also conclude that lim. T-"(z) =7). n~

+

as it can easily be checked by inspection. However, not all N + 1 harmonic measures can be independent of each other. This can be seen considering the solution of the Dirichlet problem in which all I/> 1 = 1; that is to say, the potential is 1 in all boundaries. Since a harmonic function cannot have local maxima or minima, the solution is a constant equal to 1. Then N+l

I/>(Z)=6Wj(Z)=1.

(29)

iet

We conclude then from (29) that there are only N independent harmonic measures that we shall take as being the ones associated with the N cylindrical conductors. Using (29) to eliminate W N+l(z)-the harmonic measure of the grounded plane-in (28), we find (30)

(27)

(x, y) takes a constant value 1/>1 over the ith boundary. In this case, it is very convenient to introduce a set of N + 1 harmonic functions W I(X, y) (i= 1, "', N + 1) called "harmonic measures" that are solutions of the Laplace equation with the boundary condition that the ith harmonic measure WI takes the value 1 on the ith boundary and zero on the others. Each conductor and the grounded plane have their associated harmonic measure. The physical interpretation is clear: they are solutions of the electrostatic problem in which one of the conductors is set at a potential V = 1 while the others are kept at V = O.

so we see that the only relevant physical quantities of interest for the determination of the electric field distribution are the potnetials of the conductors measured with respect to the grounded plane potential, as it was to be expected. Without loss of generality we shall set I/> N+l =0 in what follows.

B. Explicit solution We shall now proceed to construct the desired solution using Burnside's results. Clearly what one has to do is to obtain an exact analytic solution for the harmonic measures. However, Burnside's paper contains enough information for constructing explicit formulae also for the Green and Neumann functions if needed. The first step is to construct the images of the circular holes representing the cylindrical conductors with

o

Grounded Cylinder

N

Q FIG. 2. N cylindrical conductors inside a grounded cylinder.




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30

1

0

C;OA ;

0

d

(34) is pure imaginary. We shall take the plus sign. Using this result in (32) we obtain aj=df· G.p.

,. l'

0

0

c:o ;'

3' 0

o N'

FIG. 3. N cylindrical conductors and tbeir images with respect to the grounded plane. C j and Cj are the isometric circles of T j and Tjl, respective ly, These transformations map A into A' and vice versa.

respect to the grounded plane, as in Fig. 3, and consider an electrostatic problem in the whole of the complex plane with 2N holes. We shall be using here a particular case of Burnside's results-what he calls the "symmetrical case" in which there are N pairs of circles and each of the pairs are the inverses of each other with respect to a circle Co which in this case is taken to be the real axis. The second step is to construct a set of N basic projective transformations T j (i = 1, "', N) defined very simply as follows: If we call C j the original circles representing the cylindrical conductors and C/j their images with respect to the grounded plane (real axis), then T j is the projective transformation that maps C j into q. The mapping is to be defined in such a way that the point A on C j is mapped to the point A' on C/j and that, when Z moves on C j in a counterclockwise direction z' = T j(z) moves on Ci in a clockwise direction (see Fig. 3). In other words, if we were to fold the complex plane along the real axis, every point Z on C j should fit on top of its image T j(z) on C/j' The parameters of the projective transformation T j are entirely determined in terms of the center and radius of the circle C i' We can use here formulae (16)(18) of Sec. II because C i and C'i have the same radii and then C i coincides with the isometric circles of T j and Cj with the isometric circle of Tj1. If we write T j(z) = (ajz

+ b j)/(cjz + d j ),

(31)

witha jd j -b j c j =l, thenR j =lc j l-1 • Moreover, the centers J j and Jlj are the complex conjugate of each other, so, from (16)-(1~) - dT/cf=a/c j •

(32)

The requirement that the point A is mapped to the point A' means that, when the phase cJ> of (z -J j ) is zero (z on Cj)' the phase of (T(z) - J~) should also be zero. From (22) we determine the phase a of C j because it follows that 11 + 2a = 0

(modulus 211)

and then

(33)

(35)

Finally, d j is obtained from (16): d j = - cjJj = - iRjlJ i •

A'

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(36)

To sum up, given c j and d j by (34) and (36) in terms of the radius and center of c p a j is computed from (35) and b j from the normalization condition ajd j - bjc j = 1. The projective transformation T j depends only on three real parameters. The N projective transformations T j (i = 1, "', N) that map each circular boundary into its image with respect to the grounded plane are then completely determined by the geometrical configuration of the N cylindrical conductors. Now that we have at our disposal this basic set of N projective transformations T j and their inverses Tjl (i = 1, "', N), the third step is to construct an infinite set of projective transformations to be denoted by {T ",} by taking the identity and aU possible products of any number of transforms of the basic set T j (which we shall call "generators" from now on) and of their inverses. Let us give an example in order to clarify these ideas. Suppose that we have only one conductor in the presence of the grounded plane. Then we have only one basic projective transformation T 1 (one generator), and the infinite set of projective transformations consists simply of the identity I, and all possible positive and negative powers of T 1: T~, Tin (n>O). An important point to note is that this infinite set of transformations forms a group. Indeed, (a) given an arbitrary transformation T~ of the set there exists its inverse Tin; (b) the product of any two transformations of the set gives another transformation of the set (37)

Let us now consider the case of two cylindrical conductors. There are then two generators, T 1 and T 2' and the elements of the infinite set can be arranged as follows: (i) The identity I. (ii) Elements containing one generator: T 1, T 2 , Til, T;l. (iii) Elements which are product of two generators (but not the identity) TIT l' T 1 T 2 • TIT;'\ T 2 T 2 • T 2 T 1 , T2Til, and their inverses Ti1Ti\ T;lTil, T;lT;l, ''', and then we follow with elements which contain three generators like T 1T 2 T 1 • and so on. The calculation of the elements is easily done because it Simply involves the multiplication of 2 x 2 matrices. The labeling of all of them in the more general case of N generators may look rather complicated, but this labeling as well as the calculation of the T", is easily done with a computer program. We want to emphasize once again that one special property that we found in the example of only one generator 1s always true: the set of all T", form a group, This group {T",} is called the group of automorphism of the surface with N holes under consideration and is completely determined by the geometry of the problem. 6 It is therefore an intrinsic property of the two-dimensional domain under consideration. The basic set of

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transformations Ti (i = 1, "', N) is referred to as the generator of the group of automorphism. This group of projective transformations, which is essentially an infinite set of unimodular 2 x 2 matrices is a discontinuous group in the sense that not any unimodular 2 x 2 matrix belongs to the set but only a subset of them, and that there are no continuous parameters involved here that allow one to go smoothly from one element to another. In particular, there is no element of the group arbitrarily close to the identity, as it is the case with continuous (Lie) groups. The elements of the group are labeled by discrete indices. Finally, a remark about the notation that we use: whenever we refer to a generator of the group we use a Latin index (T i ), and whenever we refer to an arbitrary element of the group we use a Greek index (T ",).

=6

.p (c",z +1 deY T ",(z)1 - a '

(38)

where the projective transformation T ,,(z) has parameters

In (T,,(Z)-J/),

"

(40)

T,,(zo)-J j

form a basic set of N analytic functions in terms of which the harmonic measures can be constructed. In (40), Zo stands for an arbitrary point of the complex plane where ¢;(z) is normalized to zero. The N analytic - functions ¢/(z) themselves have a nice physical interpretation which follows from three basic properties shown by Burnside: (i) The functions ¢;(z) are analytic functions of z for any value of z in the region outside the 2N holes. Indeed, if T", is not the identity I, we can write In[T (Z)_J,l=ln(a"z+b" -J,\ "

Now, after the construction of the group of automorphism of the surface under consideration, we can introduce a set of analytic functions that are the building blocks of the harmonic measures. The basic object is the Poincarl! theta series defined as e(z, a) =

3654

,

c"z+d",

'l

= l/a,,[z - (d"J; - b,,)/(- c",J; \ c,,(z + d,,/c) = In

[z - T';1(J i) ] + const z-J"

+ a,,)l)

'

where J" is the center of the isometric circle of T ,,' Now it is a simple exercise to compute the center of the isometric circle of the transformation TiT" -which we shall indicate by J j ", and show that ( 42)

(39)

and the summation in (38) extends to all the elements of the group of automorphism including the identity I. The proof of the convergence of the Poincar~ theta series is one of the basic results in Burnside's paper. We shall discuss in the next section in more detail the relevant geometrical parameters that control the convergence of the series. For the time being we only mention that the key convergence factor is (c"z + d,,)-2, that occurs in every term of the series. This factor is always smaller in modulus than 1 in the region of interest because it is 1 on the isometric circle of T ", and bigger (or smaller) than 1 inside (or outside) that isometric circle. Now a fundamental result concerning the group of automorphism states that the isometric circle of T" is always inside the isometric circle of one of the generators, i. e., inside some C j or q (i = 1, "', N), so we are sure that in the region of interest (outside the 2N circles C i and C'j) this convergence factor is smaller than 1. There are all sort of results concerning the distribution of isometric circles of the elements of the group {T" P; it is possible to know where I(T ,,) is simply by knowing how T", is built in terms of the generators T j of the group. But it is always true that the centers J" of the isometric circles I(T ,,) are inside the 2N circles C p Ct. As we shall see later on, their radii tend to zero.

(41)

so we can write ¢j(z) in the form ¢j(z) = In(z - J i ) + 6 1n "H

(Zz-J" - J;,,) + const.

(43)

This means that Re¢i(z} is the potential generated by an infinite set of positive and negative point charges located at the centers of isometric circles. Since we have already mentioned that the centers of the isometric circles of any element of the group are always inside some of the 2N circles C j or Ci, ¢iz) is analytic in the region outside the cylindrical conductors. (ii) If we write the two harmonic functions associated with ¢j(z) (44)

then the functions uj(x, y) = Re¢;(z} are constant on the 2N circular boundaries and on the real axis. This means that they are solutions of N Dirichlet problems in which the potentials take some (yet to be determined) constant values on the boundaries and on the real axis. (iii) The last basic property of the functions ¢I(Z) is that uj(x, y) are single valued in the domain under consideration, but the conjugate functions vj(x, y) are not. This follows from the fact that

In terms of this Poincar~ theta series, Burnside introduces N functions e(z, J j ) (i = 1, "', N), where J j is the center of the circle C / (the isometric circle of T;). The integrals of these functions,

( 45) ' P , rj

d z = - 2' -d¢j 1TIO", dz tJ

where rj (or

If) are

(46)

closed paths that enclose the



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circles C j (or Ci). Burnside established this result from (43). He showed that in the series that gives cp/(z), there are inside C j (j i) as many centers of isometric circles associated with positive charges as centers of isometric circles associated with negative charges. So when we turn around C j the phases of the logarithms (±27ri) cancel by pairs and CPI(Z) is single valued. On the other hand, inside C I there is an odd number of centers of isometric circles and there is always one phase (+ 27ri) which is not cancelled, The same thing happens inside C'I where there is a phase (- 27ri) that is not canceled. This is the content of (45) and (46).

'*

We have already remarked that the functions u/(x, y) are the solutions of Dirichlet problems in which the circular boundaries and the real axis are at constant potentials. Let us consider the electric field distribution generated by the potentialu/(x, y). We can use Gauss's theorem to compute the total charge per unit length Qj inside the jth cylindrical conductor. Using the CauchyRiemann relations, we find

.::::i ds=i av/(x,y) yr.; oUI(x,y) an yr.} as

QJ

ds::::27r5

j

),

(47)

so we conclude that the harmonic functions ul(x, y) give the potential distribution corresponding to a physical situation where the conductor C j has a net charge per unit length Q I :::: 27r, its image C~ a net charge Q'I =- 27r, and all the other conductors have zero net charge per unit length. Let us go back for a moment to (43) and examine more closely the structure of the function cp/(z), whose real part ul(x, y), as we have just seen, represents an electrostatic potential distribution generated by the system of N cylindrical conductors kept at some constant (but yet unknown) potentials but with a net charge per unit length equal to 1 inside the ith conductor and zero on the others. By looking at (43) one could be tempted at this point to say that the centers of the isometric circles are the places where we have set an infinite number of image charges. This statement is not wrong, but it must be qualified. The point is that it is true that we have arranged an infinite set of image charges at the centers of the isometric Circles, but there are infinite cancellations among the charges so arranged. Indeed, the starting point of the series (43) is a charge + 1 at J j) the center of the ith conductor-the term In(z - J 1). This term alone would satisfy the charge condition, but of course it does not satisfy the boundary condition that its real part is a constant on the real axis and on the remaining circular boundaries. This charge + 1 at J I would also be the starting point of the method of images developed by Sarma and Janischewskyj. The remaining terms in (43), that is to say, the sum over elements of the group {T "'} different from the identity, corrects the deficiency and forces cf>j(z) to satisfy the boundary condition. As we said, the sum over O! extends to all elements of the group (except for the identity). We have then a contribution, for example, from the element T", = T j which gives a charge + 1 at Jii=J(T~)-the center of the isometric circle of T~-and a charge - 1 at J; that cancels the one we use to start with. When we consider

the contribution of T a :::: T~ to the sum we find a new charge + 1 at J(T1) and a charge - 1 at J(T~) that again cancels the positive charge introduced in the previous step, and so on. This is a general fact. Every term in the series (43) gives rise to a positive and a negative charge that sooner or later are canceled by negative or positive charges generated by other terms in the series. Image charges also appear, of course, inside the neutral conductors; but the net charge is always zero. In the last part of this section we shall work out the example of only one conductor to clarify the picture. However a careful analysiS of the general case shows that the series (43) can be arranged in such a way as to exactly coincide with the method of Sarma and Janischewskyj: At each step of the calculation one replaces a given set of image charges by a new set obtained by inverting all the old charges with respect to all the boundaries of the cylinders external to the charge in consideration. This process clearly converges to a well-defined infinite set of image charges. In other words, we would like to get rid of these infinite cancellations of image charges. We shall give the solution to this problem for one particular example in Sec. mc, and for a general case in Sec. IV. The final result will be that cp/(z) can be interpreted as the potential generated by an infinite set of image charges located at the invariant pOints of the transformations T", of the group rather than at the centers of the isometric circles. That it should be so is guaranteed by the general theory of the CPj(z) functions, called "first abelian integrals" in the mathematical literature. We know that their only Singular points are the invariant pOints of the group elements. Remember that the invariant pOints of T '" and T~! are also inSide the isometric circles of T a and T-!

'" .

Finally we come to the pOint of determining the boundary values of the set of potentials ul(x, y) (i= 1, "', N). First note that since they are defined up to an arbitrary constant, we can always choose this constant so as to make them vanish on the real axis. Going back to (40), this is accomplished by chOOSing
(z) -1>(zo)' Using T-n(J)=J ••+1 (n ~2), we find that the negative powers of T contribute 1 z-J' z-Ja +In Z -JI ( 1nZJ 1I) +In ~J' Z-2 3

=

1

l'

/_rr; In(z - J~)

1 = ~i_n.! In[z - y-n(J)]

1

and the invariant points and multiplier are easily computed with the result

n,

00

(73)

= In(z -1]) ; so the final result is 1>(z) = log[(z - O/(z -1]) 1+ canst.

(74)

If we want 1>(z) to vanish on the real axis we write (zo real)

1>(Z)=ljz-~ ZO-1]).

"\z -1]

Zo - ~

(75)

We can compute now the period 'matrix A (there is only one matrix element in this Simple case) using (74)

Let us now compute the function 1>(z), which in this J. Appl. Phys .• Vol. 45. No.8, August 1974


3658


ments belonging to a certain class 0 j ' This class 0 i is defined as containing the identity I and the group elements T", such that when written in terms of generators, it does not end up with T~ or Tin (n> 0). In other words, group elements of the form TBT7 or TBTt are not to be included. This result is indeed very reasonable because these are precisely the group elements that have been summed up in order to obtain (SO). Indeed, the set of image charges TaT'I(Jj) converges to TB(~I)' and the set TBTj"(J/) converges to T B(7) I)'

2

Q

D

G.p I

Q

Q

3658

1,30

FIG. 5. Two cylindrical conductors above a grounded plane. R is the radius of the conductors, D is the distance between their centers, and h is the height above the grounded plane.

=In(T(Z)-~ T(z) -

T/

1,28

Z-T/)

(76)

Z - ~

and using (ll),

1,27 ".

(77)

A=lnK,

a constant depending only on the geometrical parameters, as expected. The harmonic measure is then W(X,y)= __2_ln(z-~ InK Z- T/

Zo-~),

1,26

(7S)

Zo - T/

-r----~------r-----~----~----_.----_.~&

which is the well-known solution to this problem. Moreover, the capacity per unit length of the cylindrical conductor is, as expected, €

=-

41T InK

=-

r. (Y -

(y2 - R2)1 /2\J 41T ~n y + (y2 _R2)1/2}

-1

.

(79)

This trivial example we have worked out in detail shows explicitly that the series (43) and (65) for the calculation of the abelian integrals and period matrices are not very convenient tools for practical calculations. Our example exhibited the infinite cancellation of image charges at the centers of the isometric circles. The exact solution can be interpreted as the potential generated by a positive and a negative charge at the invariant pOints ~ and 7), respectively. We shall carry out a similar simplification for the general case in Sec. IV.

o

60

120

180

240

300

360

(a)

.....

0,29

2

.

-'-'-'-'-.

~~'.". 7;-";:';~~+"~2~;: =c-c=ccc=c~ -,-.-.- -'-

0,26

0,27

IV. ANALYSIS OF SOME NUMERICAL APPLICATIONS

The infinite cancellation of image charges can be worked out for the functions cfJ/(z) in the general case in the same way as in the previous example. The result is very simple, and can be written as 9

-+----~----~-----.~--~~--~~--~~~~

o

60

120

180

240

300

360

(h)

(i=l, .. ·,N)

(SO)

where the sum does not extend now over the complete group of automorphism, but only to those group ele-

FIG. 6. Potential distribution (in arbitrary units) on the surface of conductors of Fig. 5, in terms of the angle measured with respect to the horizontal, for D/R~4 and h~2. Different curves correspond to taking into account the contribution of all elements of {Ta} containing up to 1, 2, 3, and 4 generators, respectively. (a) Conductor 1, carrying net charge 1 per unit length. (h) Conductor 2, with no net charge.



3659

3659


physical grounds. What we have obtained is a new method of images where the starting points are charges ± 1 located at the invariant points ~ i inside C i and 7) i inside Ci. We note the fact that the number of images is much less here than in the previous method. The potential (81) already satisfies the boundary condition on the real axis and on the ith boundary, so in order to correct the first approximation (81) one starts by inverting the charges with respect to another boundary j different from i. This is the origin of the restriction in the summation in (80). Note moreover that T6Ti(~i) =T6(~i)' because ~i is an invariant point of Ti' so we do not really get new image charges by including the terms we have left out in (80).

0,

A similar improved formula can be derived for the period matrix Aii' The result is 9 0,4

(82) -+------~-----~,------~----~----~------,~~.~

o

60

120

160

240

300

360

(a)

where the summation is' again restricted to a class of group elements 0 ii' The class 0 iJ is defined as the set of group elements that do not have Tj on the right and T'[' on the left (n, m;O 0). We also exclude the identity I from 0 ii; the contribution of the identity is explicitly indicated in the first term of the right-hand side of (82), Ki being the multiplier of T j ' The result (82) has some obvious advantages with respect to the previous formula (65): In addition to the fact that it is more rapidly convergent, it no longer exhibits the spurious dependence on a variable z, making it clear that Aii only depends on the geometry of the problem. At this point we should like to comment on a possible criterion for the convergence of the series involved in (80) and (82). For this purpose it is useful to introduce the concept of anharmonic ratio. We understand by anharmonic ratio of four pOints Zl' Z2' Z3' Z4' the expression

0,12

(Zl'

(Zl -

Z2)(ZS - Z4)!(Zl - Z3)(Z2 - Z4)

(83)

that has the property of being invariant under any projective transformation of the complex plane onto itself. Namely,

0,10

o

Z2' Z3' Z4) =

i

60

• (}

360

(h)

FIG. 7. Potential distribution (in arbitrary units) on the surface of conductors of Fig. 5, in terms of the angle measured with respect to the horizontal, for D/R=2.4 and h=1.2. Different curves correspond to taking into account the contribution of all elements of {T,,} containing up to 1, 2, 4, and 6 generators, respectively. (a) Conductor 1, carrying net charge 1 per unit length. (h) Conductor 2, with no net charge.

(Zl'

Z2' Z3' z4)=(T(Zl)' T(Z2)' T(Z3)' T(Z4»'

(84)

Now for any term of the series (80), the argument of the logarithm is certainly an anharmonic ratio. So we can perform a projective transformation of the type we called S",(z)-given by (12)-that maps the invariant points of T",(z) (~",71,,) to (0, 00) and in this way, that term will take the form (85)

Note that the first term in the series (80) is the potential cl>i(z)=ln[(z- ~i)/(Z-71i)]'

while retaining its value. (81)

The result given by (80) can be easily understood on

In Eq. (85) we see that K", is the driving parameter for the convergence of the series. Indeed, having defined IK", I < 1, the parentheses in (85) can be expanded



3660

3660


The main purpose in treating that sample of problems is to show that pathological configurations-as we call the situation when conductors and grounded plane are very near to one another-can be easily attacked with our method, even if it implies the necessity of taking into account some more terms of the series in order to have greater accuracy in the results.

6Gc, (degrees)

......,. -1

2

---f..:: .....'--~/::......,-_·--i3r-..·=-...,·-F-":...-=-=-=:...-=-~.-~.---'-~'::::-,",,',....;.,~-=:;3;..;00~-...,4-_ _ ._.. G-

o

60

4

120

160

240

_._._. : 3 0

-0,5

The cases under consideration (Fig. 5) are studied when the conductor labeled 1 carries net charge l-in arbitrary units-per unit length, while conductor 2 has no net charge in it. In Figs. 6(a) and 6(b), we show the potential distribution-in arbitrary units-on the surface of the conductor 1 and 2, respectively, forD/R=::4 and h=::2.

-1

"&]:, (degree.)

(a) M~C2

(degrees)

6 ····1

4 _

o

4

- " - " - " - ' - " - ' - ' - _._ .. :::"::-... 60 180 120

300

i /\\:

: I

4 120 .,;::~ __ ,i 240 :. __ - _.-' ~0-F~~~6~0~\~-~3~-~/~.~~.:=~·~·~~I~eo~~~:\~-~._-·-~·~.~~~oo~--~~~o-----+~

_

.;-..... ...... ", : 2 .:

~2

\...... 1.:

: ._.... : :

-4

,"

...

-8

:'

(a)

-4

.....

(b)

FIG. 8. Departures from the normal direction of the electric field at the surface of the conductors of Fig. 5, for the same conditions as those for Fig. 6. (a) Conductor 1. (b) Conductor 2.

in a power series of Ka, as 1 + K",(~j - 'I1j) (l/z~ - liz') + O(K!).

(86)

This shows that the rate of convergence of our series is controlled by the smallness of K",. Clearly, the above discussion can also be applied to the series in (82) for the elements of the period matrix. As stressed in Sec. I, the method we have just discussed can be applied to a wide variety of electrostatic problems. However, we shall present only a couple of examples to show how the general method works. We have studied numerically the case of two cylindrical conductors above a grounded plane, Fig. 5, for two different values of D/R. D is the distance between the centers of the conductors and R is their radius. The examples also differ on the height above ground that we call h.

(b)

FIG. 9. Departures from the normal direction of the electric field at the surface of the conductors of Fig. 5, for the same conditions as those for Fig. 7. (a) Conductor 1. (b) Conductor 2.



3661


3661

The different curves in those figures correspond to taking into account the contributions of all elements of {T a} containing up to 1, 2, 3, and 4 generators, respectively. On the other hand, Figs. 7(a) and 7(b) show the potential distribution when D /R = 2. 4 and h = 1. 2. In this case, elements that contain up to 6 generators were considered. The rate of convergence of the method is clearly appreciated from the figures.

such as groups of automorphisms. Nevertheless, the mathematical structures underlying this exact solution of harmoniC problems on multiply connected domains are very powerful indeed for practical pruposes. For example, had we not used the properties of projective transformations and groups of automorphisms, we would have never been able to derive the improved solution for the harmonic measures and Maxwell coefficients.

In Figs. 8 and 9 we plot the departures from the normal direction of the electric field at the surface of the conductors for all cases considered. Here again the convergence of the proposed method becomes clear.

We have used here only a small fraction of the interesting information available in the mathematical literature. In our opinion, the most promising methods for practical applications are the ones based in the socalled "domain variational theory". 4 The domain variational theory provides us with accurate expansions of the harmonic measures, period matrices, and Green and Neumann functions of a given multiply connected domain when the boundaries-conductors-are distorted, when their radii shrink to zero, or when they are almost touching each other. We hope to come back to the application of these methods to problems of practical interest in future publications.

In connection with the previous discussion on the criterion of convergence of the series involved in our calculations, we should like to point out that the maximum value of K" for the case with D/R =4 is Km=O. 07 while for the more pathological configuration with D /R = 2.4 it is Km =O. 42. For that reason, in the latter case it was necessary to consider up to 6 generators in the elements of {T a } in order to get similar accuracy in the results when compared with the ones obtained for the previous case. V. CONCLUSIONS

The exact analytic solution of the problem of the electric field distribution of a system of infinite parallel cylindrical conductors discussed in this paper proved to be as handy as the more heuristic method for performing practical calculations. The original version of the exact solution is equivalent to the generalized method of images of Sarma and Janischewskyj. The improved version we also discuss is even more convergent, and this improvement is fully appreCiated in cases in which conductors are close to one another or to the grounded plane and a lot of image charges are needed to satisfy the boundary conditions within a given error. Needless to say, all the exact solutions of simpler conductor configurations quoted in the literature are particular cases of the general solution discussed above, whIm appropriate changes of variables are made. There is one major point that we want to strongly emphasize here. It is granted that one can solve the problems discussed in this paper with the simple method of images without worrying about unfamiliar concepts

*Member of the Consejo Nacional de Investigaciones Cient(ficas y Tecnicas of Argentina. tAlso Departamento de Electrotecnia, Universidad Nacional de La Plata, La Plata, Argentina. tM. P. Sarma and W. Janischewskyj, IEEE Trans. Power Appar. Syst. PAS-SS, 1069 (1969). 2M. S. Abou-Seada and E. Nasser, IEEE Trans. Power Appar. Syst. PAS-SS, 1802 (1969). 3M. S. Abou-Seada and E. Nasser, IEEE Trans. Power Appar. Syst. PAS-90, 1822 (1971). 4See , for example, M. Schiffer and D. Spencer, Functio1Ulls of Finite Riemann Surfaces (Princeton U. P., Princeton, N. J., 1954), and references contained therein. A nice introduction to this subject can be found in G. Springer, Introduction to Rieman Surface (Addison-Wesley, Reading, Mass., 1957). 5L. Ford, Automorphic Functions, 2nd ed. (Chelsea, New York, 1951J. 6J. Lehner, Discontinuous Groups and Automorphic Functions (American Mathematical SOCiety, Providence, R.I., 1964). See also Ref. 5. 7W. Burnside, Proc. Lond. Math. Soc. 23, 49 (1893). BV. Alessandrini, Nuovo Cimento A 2, 321 (1971). 9E. Cremmer and J. Scherk, Nucl. Phys. B 48, 29 (1972>tOE. Clarke, Circuit A1Ullysis of A-C Power Systems (Wiley, New York, 1965), Vol. I, pp. 434-489.



Exact solution of electrostatic problem for a system of

Exact solution of electrostatic problem for a system of

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