IEEE Transacfions on Dielectrics and Electrical Insulation W. 4 No. 4, Augusf 1997 439

Electric Field of Insulated Wire at the Interface of Two Dielectric Media

Yu. P. Emets Department of Electrophysics, Institute of Electrodynamics, National Academy of Sciences, Kiev, Ukraine

ABSTRACT The electric field of a straight charged wire is calculated when the wire is coated with a cylindri- cal insulation and is placed on the plane boundary between two dielectric media. The exact so- lution of the problem is derived in form of a rapidly convergent series. The technique used for the solution is based on the effective methods of the theory of functions of complex variables. The peculiarities of the electric field configuration are discussed. Some remarks are included concerning possible generalization of the solution to a system with multiple wires.

1. INTRODUCTION the principle of analytic continuation are employed which demand only purely algebraic manipulations. The problem can be solved also by an alternative method that is based on the method of images; this approach is discussed in Section 5.

HE purpose of this study is to investigate an electric field in sys- tems that consists of an infinitely long charged wire, of negligible T

cross section, coated with a cylindrical dielectric insulation and placed on the plane boundary between two different media, Figure 1. The for- mulation of the problem is concerned with some applications to calcu- lations of nonuniform electric fields in HV installations, in cables and insulated wires. 2.1. BASIC RELATIONS

2. BOUNDARY CONDITIONS

The problem under consideration here is to calculate the electric field of an infinitely long charged wire coated with a cylindrical insulation of radius T and permittivity ~ 2 . The insulated wire is placed on the plane boundary between two dielectric media of permittivities ~1 and EQ as shown in Figure 1. When there are no distributed charges in the isotrop- ic dielectric media, the electric field is described by

l3 = E l 3 Figure 1. An insulated wire at the interface of unlike media.

In the case where a charged straight filament, without insulated coat- ing, is placed near an infinite plane boundary, the electric field is calcu- lated easily by the method of images and has a simple expression [l].

In the system under study, the solution of the problem is complicated because the electric field must be found in three regions: in the dielectric cylinder around the wire and in the two adjoining halves of the space oc- cupied by different materials. Nevertheless, the formulated problem ad- mits an explicit solution in an analytic form under the general assump- tion that the permittivities of the materials can take arbitrary values. In the final form, the solution is presented as a rapidly convergent series of line poles for which coefficients and coordinates are obtained from the solution of the corresponding problem.

In a plane normal to the cross section of the wire, the electric field is two-dimensional and, hence, one can introduce the complex functions of an electric field E ( z ) and an electric displacement D ( z )

E ( z ) = E , - iEy D ( z ) = D, - iD, (4

z = z + i y

According to Equation (l), these functions satisfy the Cauchy-Riemann equations, ie., they are analytical functions of the complex variable z .

The techtuque used here is based on the methods of the theory of functions of a complex variable. The conformal transformations and

At the interface between unlike dielectric media, the usual boundary conditions are performed under which the tangential components of the

1070-9878/97/$3.00 @ 1997 IEEE

I

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A A1 A2

A,k 13'

E ( z ) = E, - iEv

h

IC L1 Ll

Emets: Electric Field at the Interface of Two Dielectric Media

Table 1. Table of symbols.

constant determined in Equation (5) constant determined in Equation (42) constant determined by Equation (54) coefficients of line poles electric displacement vector

electric displacement function electric field vector electric field inside the cylinder electric field outside the cylinder electric field in the right half of the space two parts of the field E2

complex function of electric field

sectional analytic function three parts of the function f ( < ) two parts of the function f 1 (<) distance between the cylinder axis and the plane boundary infinite set of natural numbers: 1,2, . . . boundary contour, the circle boundary contour, the straight line

unit normal line charge

electric field vector and the normal components of the electric displace- ment vector are continuous. In terms of the function E ( z ) , the bound- ary conditions are written as

!I? {n ( t ) E I E I ( t ) } = {n ( t ) ~ v E v ( t ) } 9 {n ( t ) El ( t ) } = 9 {n ( t ) Ev ( t ) ) v = 1 + p t E L p p = 1 , 2

(3)

where n ( t ) is the unit normal to the boundary contours L1 and L2

radius of the cylinder region outside the circle L1 region inside the circle L1 right half of the z plane bilinear transformation determined by Equation (39) point on boundary contours L1 and L2 points of symmetry complex variable coordinates of poles in the region S2 coordinates of poles in the region Ss non-dimensional parameter deter- mined by Equation (12) relative permittivities determined by Equation (10) permittivity permittivity of free space permittivity of the cylinder permittivity of the medium surround- ing the cylinder permittivity of the right half of the space complex variable boundary contours, straight lines in the < plane point on the boundary contours hl,h2 analytic functions

I O

(4) Figure 2. The conformal transformation of a circle tangent to a straight line on two parallel straight lines. t€L2,0<0<27C

The contours L1 and L2 divide the z-plane into the three distinct regions S,, = 1,2, 3, occupied by different dielectric media with permittivities E,, respectively, see Figure 2(a).

The charged wire can be taken into account by means of the singular point at the origin, the line pole of the first order of the function E2 ( 2 )

E2 ( 2 ) = Ea (2) + E ; (2) = ; + E2 ( 2 ) A ii

4 (5) A=--

2ZE2

Here q is the line charge of the wire, E: ( z ) is the analytic function everywhere within the region S2. At the point at infinity the electric field of the charged wire tends to zero Equations (5) and (6) serve as additional conditions to the problem.

They must be considered simultaneously with the boundary conditions (3). El (CO) = E3 (CO) = 0

IEEE Transactions on Dielectrics and Electrical Insulafion W. 4 No. 4, August 1997 441

2.2. THE GENERALIZED RIEMANN PROBLEM

In expanded form, the conditions (3) are expressed by

where the overline indicates complex conjugation.

After eliminating the function El ( t ) from each pair of Equations (7), the boundary conditions are reduced to

This equation can be rewritten as

where the relative permittivities

E1 - Ev a,, = ~ (-1 < a,, 6 1) v = 2 , 3 E1 + Ev

are introduced.

Thus, finding of the electric field in the examined system is reduced to solving the boundary-value problem (9) with the additional conditions (5) and (6). It is the generalized Riemann problem with homogeneous conditions, known also as the two-element Riemann problem. The com- plete solution of this problem is given in Appendix 1.

3. ELECTRIC FIELD

3.1. EXACT SOLUTION OF THE

CHARACTER ISTICS

PROBLEM

The electric field in the system under study is expressed by

Here A is the non-dimensional parameter, the generalized permit- tivity

A = A12. A13 - l < A < l (12) where the parameters A12 and A13 are determined by Equation (10). The origin of the coordinate system is taken to be the charged wire, coin- cided with the center of the circle in the z-plane as shown in Figure 2(a).

A direct check shows that Equations (11) really satisfy the boundary relations (9) and the additional conditions (5) and (6). As a matter of fact, Equations (11) are derived directly from the solution of the boundary- value problem developed in detail in Appendix 1.

It is seen from Equation (11) that the solution is obtained in the form of rapidly convergent series of line poles. One set of these poles with the coordinates along the x-axis

z 2 k = r ( k - 1) / k k = 1 ,2 , ... (13) is located inside the cylinder (region S2) and the other set with the co- ordinates

is placed in the region S3, see Figure 3. With increasing k, the poles are concentrated near the point of tangency of the circle L1 with the straight line L2; the higher the k , the nearer they are to the contact point.

The electric field in the region S1 is given by the sum of the field of all the line poles. In the region 5'2, the field E2 ( z ) is the resultant field due to the one internal pole in the point ~ 2 1 and the externally placed poles in the points ~ 3 k . In the region S3, the field E3 ( 2 ) is determined only by the fields of the poles in the points z2k.

It is important to remember that only the pole at the origin, having the coordinate z21, at the center of the circle, is actual and corresponds to the given charged wire; the others are fictitious poles.

442

Figure 3. Therelative position of poles.

E +

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 Figure 4. The distribution of modulus of electric field along the z -

axis; solid curves for €1 = 1, ~2 = 5 and ~3 4 00 (Al2 = -0.667, A13 = -l), dotted lines for E I = ~2 = 1 and E3 --f 0O (Aiz = = -1).

-2 0 -1.5 -1.0 -0.5 0 0.5 1.0 Figure 5. The distribution of modulus of electric field along the z-axis

forEl = 1 , ~ 2 = 15 andE3 4 00 (A12 = -0.875,&3 = -1).

3.2. ANALYSIS OF THE SOLUTION

Emets: Electric Field a t the Interface of Two Dielectric Media

E +

-2.0 -1.5 -1.0 -0.5 0 0.5 1 .U Figure 6. The distribution of modulus of electric field along the z-axis

for ~1 = 1, €2 = 100 and ~3 4 00 (A12 = -0.98, A13 = -1).

of Equation (11) for the relative values

E, = E / E o E, = q / 2 n ~ l r

,U= 1 , 2 , 3

where E, is the permittivity of free space. Further, the asterisks are omit- ted. The distribution of the electric field is calculated for different per- mittivities of the insulation ~ 2 , for the case when the insulated wire is immersed in air (EI = 1) and is placed on a plane metal surface ( ~ 3 + 00). Hence, the parameter A13 given by A13 = -1. The dotted curve in Figure 4 corresponds to a charged wire without insulat- ed coating (e2 = 1, which gives A12 = 0).

C Y

Figure 7. The distribution of modulus of electric field along the plane boundary in the neighborhood of its contact with dielectric cylinder; solid curves for €1 = 1, € 2 = 5 and ~3 t 00 (A12 = -0.667, ai3 = -I), dotted lines for €1 = € 2 = 1 and ~3 + 00 (A12 = 0, A13 = -1).

It is seen from Figures 4,5 and 6 that the electric field in the insulation of a wire has a minimum from the side of the plane boundary It is held for all cases when ~2 # 1. The minimum of the function E2 (x) lie5 between the center of the circle x = 0, where the charged wire is, and

The features of the electric field in the examined system are illustrat- ed in Figures 4,5 and 6 in which the curves show a distribution of the electric field along the x-axis. The curves are constructed with the aid

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 4 No. 4, August 1997 443

Figure 8. The distribution of modulus of electric field along the plane boundary in the neighborhood of its contact with dielectric cylinder for c1 = 1, EZ = 15 and ~3 t 00 (AIZ = -0.875, A13 = -1).

the point of contact of the circle with the straight line z = 1. From physical point of view such behavior of the electric field is explained by a concentration of the field near the contact of the dielectric cylinder with the metal surface. The concentration of the electric field increases with increasing permittivity of the insulation, ~ 2 . It is demonstrated in Figures 7 and 8 where the curves show a distribution of the electric field at a boundary between air and metal in the vicinity of the contact point. The electric field is calculated with application of the first expression (11) for relative values. The dotted curve in Figure 7 corresponds to the case of a noninsulated wire ( ~ 2 = 1, i.e., A 1 2 = 0).

FUNCTION 3.3. FIELD NONUNIFORMITY

The influence of a plane boundary on the distribution of the electric field in a insulation of the wire can be estimated by the function E; ( z ) which is

E; ( z ) = E 2 ( z ) - E; ( z ) (16)

where E; ( 2 ) is the electric field of a charged wire in a uniform medium of permittivity ~ 2 , see Equation (5),

4 1 2XE2 z

E; ( z ) = --

In accordance with (ll), the function E; ( z ) is given by the following expression

4 E t (2) = ---A13 (1 - A 1 2 ) (1 - A) x 2 n E 2

It becomes zero when there is no a plane boundary and, hence, the in- sulated wire is immersed in a uniform dielectric: ~1 = ~3 ( A 1 3 = 0). For other cases, the function E; ( z ) shows the influence of a plane boundary on the electric field in the insulation.

Graphically, expression (18) is represented in Figure 9 where some numerical results are given for the system when the insulated wire is placed in air on a metal surface: ~1 = 1, ~3 -+ CO ( A 1 3 = -1). The

-1.0 - 0.5 0 0.5 1 .o Figure 9. The function showing the nonuniformity of electric field in the

cylinder; curve 1 for E I = ~3 (A13 = 0), curve 2 for EI = ~2 = 1 and ~3 + 00 (A12 = 0, A13 = -l), curve 3 for EI = 1, E Z = 5 and ~3 t 00 (A12 = -0.667, A13 = -l), curve 4 for EI = 1, E Z = 15 and ~3 4 00 (A12 = -0.875, A13 = -1), curve 5 for ~1 = 1, E Z = 100 a n d ~ 3 + 03 (A12 = -0.98, A13 = -1).

function E; ( z ) is plotted along the real axis in the insulation of a wire for ~2 =1,5,15 and 100 using the relative values (15).

It can be seen from Figure 9 that the function E; ( z ) varies between the limits 0 and 1, taking the value 1 at the point ~ 7 : = 1 for all cases. The nonuniformity of the function E; ( z ) grows with increasing ~ 2 . In general, the behavior of the function E; ( z ) reflects the main features of the electric field in the insulation of the wire. For the analyzed sys- tem, the field becomes more asymmetric and more nonuniform as the permittivity ~2 increases.

3.4. FIELD STRENGTH AT THE CONTACT POINT

In the neighborhood of the point of tangency of the circle L 1 and the straight line L 2 (Figure 2(a)), the electric field may be substantially nonuniform when the permittivities of materials in the system are con- siderably differed from each other. It can be evaluated, in particular, from the expression of electric field strength at the contact point, z = T .

At this point, Equation (11) after summing of the infinite series has the following simple form

Equation (19) determines different limiting values of the electric field strength at the contact point from different sides of the boundaries. There are only the normal components of the electric field on each side of the boundary contours L1 and L 2 . The electric field strength at the point z = T , and hence in the immediate vicinity to it in each region S, of different permittivities, depends on the relationship between the permittivities E,,.

It is seen that the electric field between a dielectric cylinder and a plane boundary increases with growth of the permittivity ~2 of the cylinder, when ~1 is small and ~3 is very large. The electric field in the cylinder near the contact point, on the contrary, decreases as the

444 Emets: Elecfric Field a f fhe Interface of Two Dielecfric Media

permittivity E Z increases. The field E2 ( T ) has its largest value, that is limited, as the perniittivities ~1 and EZ are very small and the per- mittivity ~3 is very large, so if E L = ~2 = E, and ~3 t 00 (AL2 = 0 and A13 = -1) then E2 ( r ) = q / 2 m o r .

3.5. PARTICULAR SOLUTIONS

Some simple well-known solutions can be derived from the gener- al solution (11) by variation of the permittivities of the materials in the systcm under study,

in the whole z-plane.

3.5.2. CHARGED WIRE WITH AN I N SU LATl ON COATI N G

When

F1 = E3 ( a 1 3 = 0) (Figure lo), then Equation (11) becomes

3.5.3. CHARGED WIRE PARALLELED TO A PLANE BOUNDARY BETWEEN

UNLIKE MEDIA

If (Figure 11)

Equation (11) turns to Figure 10. An insulated wire in an infinite medium.

El (2) = E2 ( 2 ) = -

(25) z - 2r

Rz > r 4 l - A l 3 E3(z ) = ~~

2xs3 z

3.5.4. CHARGED WIRE ON A PLANE BOUNDARY BETWEEN UNLIKE

MEDIA In the case of Figure 12,

T = 0 €1 = &2 (A12 = 0) (26) Figure 1 1. A charged wire near the interface of unlike media. and Equation (11) reduces to

%z < 0

8Zz 3 0

l+a13 El (z) = ~~

(27) 2XE1 Z

4 1-a13 E3 ( 2 ) = G y

taking into account that

(28) 1

1 - A

00

= p k - l = ~

k = l

It is to be noted that the solution (25) is usually obtained by the method of images [l].

Figure 12. A charged wire on the plane boundary between different me- dia

3.5.1. CHARGED WIRE IN A HOMOGENEOUS MEDIUM.

In this case

~1 = ~2 = ~3 = E (A12 = A13 = 0 ) (20) Then, it follows from (11) thdt

El ( 2 ) = E2 ( z ) = E3 ( z ) = E (2)

Figure 13. An insulated wire near the interface of unlike media. 4 1 (21) E ( x ) = --

2ze Z

IEEE Transactions on Dielectrics and Electrical Insulation bl. 4 No. 4, August 1997 445

(4 Figure 14. The conformal transformation of a circle near a straight line

on concentric circles.

4. INSULATED WIRE NEAR BOUNDARY BETWEEN TWO

MEDIA

The problem discussed in the preceding Sections can be generalized to the case when an insulated wire is placed at an arbitrary distance from a plane boundary between different materials, see Figure 13.

Let, as in the previous system, a straight wire of a small cross-section be coated with a dielectric cylinder of a radius T and a permittivity E Z .

The insulated wire is parallel to an infinite plane boundary between two media of permittivities and c3. It is immersed in a medium of per- mittivity ~1 and located, in contrast to the previous case, at a distance h 3 T from the plane boundary as shown in Figures 13 and 14(a).

A solution of this problem can be obtained in just the same way as for the above problem using the method of the theory of functions of complex variable. All the differences between the two solutions follow only from the distinct conformal transformations. In the first case, the system maps onto parallel straight lines while the present system trans- forms onto concentric circles, compare Figures 2 and 14.

the regions S2 and S,; they are defined by

21 - 22y z2k =

Z Q ~ = 2h - 2 2 1 ~

2k

1 - y2k

where XI and 2 2 are

and y is the following parameter

(32)

Here, 21 and 2 2 are the points of symmetry with respect to the circle L1 and the line L2, see Figure 14(a). The other symbols in (29) and (30) have the same meaning that they have in Equation (11).

=21

2 2 2

' 2 3

z24

z 3 1

'32

Figure 15. The diagram of calculation of the pole coordinates.

Finally after the solution of the corresponding boundary-value prob- lem, the electric field in the system in which we are interested is ex- pressed by

In the present case, the solution is also represented by a series of two sets of line poles: one set of poles with coordinates zzk is located inside the circle L1 (region SZ), the other set of poles with coordinates %3k

is placed in the region 273, symmetrically to the first set, as shown in Figure 15. One can note that Equation (29) superficially is similar to the previous solution (11). Equation (29) differs from Equation (11) only in the coordinates of the poles.

It is apparent from Equation (29) that the properties of the electric field in the system are in outline analogous to the previous case if the insulated wire is situated closely to the plane boundary As the distance between the insulated wire and the plane boundary increases, the solu- tion (29) tends to the solution of a single charged wire with an insulated coating (23).

4 E1(z) = --(1- A) x 2Z&1

z E s1

+ L A n ( 1 - AlZ)(l - A) x Ez(z ) = i z z 2nez (29)

z E s2 5. SOLUTION BY A CONSTRUCTIVE METHOD Ak-1

E3(z) &(I - A13)(1 - A) (--> z - z2lc The solution (11) of the initial boundary-value problem is obtained

strictly by the methods of the theory of functions of complex variable. This problem can be solved also by using the constructive method that seems to be simpler and easy to grasp. This method is discussed below,

k = l

z E s,

Here zzk and z 3 k are the coordinates of poles located respectively in

446

5.1. MAIN PROPOSITIONS

Emefs: Electric Field a f the Inferface of Two Dielecfric Media

same way as above give (see Figure 16) 221 = 0 231 = 2 h - 221 = 2 h The electric field in the each region S, can be expressed ab iiu'fiio form

as an infinite sum of line poles 0 0 "

where A C l k and z,k are the coefficients and coordinates of the line poles, respectively.

The problem under consideration is now stated as follows: it is re- quired to define the unknown parameters of line poles, A p k and z,k, in Equation (33) satisfying the boundary relations (9) and the additional conditions (5) and (6).

In Equation (33), only the pole situated at the origin is actual: it corre- sponds to the charged wire. All the other poles are fictitious; their dispo- sition can be determined by using the reflection principle in the follow- ing manner. The actual pole located at the point 221, as shown in Fig- ure 3, defines the position of the first image pole at its mirro-symmetric point 231, with respect to the straight line Lz, The inversion transfor- mation of the point 231 with respect to the circle L1 give the position of the second image pole at the point 222, see Figure 3. And so, continuing this procedure makes possible to define step-by-step the location of all the fictitious poles in the system, Z2k and z 3 k .

The sequence of transformations is shown schematically in Figure 15. The direct calculations give the set of the pole coordinates

221 = 0

222 = r 1231 = 7-12 223 = r /232 = 2 r / 3

231 = 21- - 221 = 2r

232 = 21- - 222 = 3 r / 2

233 = 21- - 223 = 4 r / 3 (34)

2

2

and so forth. The general terms of the sequence (34) are 221 = 0

22k + Z 3 k = 27-

ZZ(k+l ) z3k = r2 (35)

k = 1 , 2 , . . .

Comparison of the relations (35) with (13) and (14) shows that they give the same results although they have a different method of compu- tation.

, , , I

Figure 16. The relative position of poles.

In a similar manner to the above, it is possible to determine the dis- position of pole coordinates in the system of the insulated wire placed near a boundary between different materials. Calculations in just the

and so forth. Here, the general terms of the sequence (36) are 221 = 0

Z2k i Zgk = 2 h 2

22 (k+l )23k = 7'

k = 1,2,

(37)

Some remarks can be made concerning the relations (37). They are, in fact, the recurrence formulas for finding of the disposition of pole co- ordinates in the system when the dielectric cylinder does not touch the plane border. The calculation of the coordinates follows, as in a previ- ous case, to the diagram represented in Figure 15. The relations (37) are, obviously, a generalization of the above relations (35). Indeed, in the limit h + r, the dielectric cylinder is tangent to the plane border and (37) turns to (35). Earlier, in Section 4, the pole coordinates in the system under consideration were defined by the explicit expressions (30) to (32) which differ from those given here by the manner of calculation.

After the calculation of the coordinates of the poles in Equation (33), the next step is to determine the pole coefficients A , k . For this purpose, the solutions of the following additional problems can be used:

1. a line charge q in a medium of permittivity ~1 near a plain bound- ary behind which is a medium of permittivity ~2 (problem 3 in Section 3.4);

2. a line charge q in a medium of permittivity ~1 near a circular cylinder of permittivity E ~ ;

3. a line charge q inside a circular cylinder of permittivity ~2 out- side of which is a medium of permittivity ~ 1 .

The solutions of these simple problems are well known [l]. They are used to calculate the coefficients A,k. Finally, after some noncomplex algebraic manipulations, omitted here for brevity, the calculations give the expressions of A p k , which coincide with the corresponding expres- sions in Equations (11) and (29).

It is quite apparent from the previous consideration that the con- structed solution differ from the explicit solution only in computational details and satisfies the boundary conditions of the initial problem.

5.2. COMMENTS

In this and preceding Sections, it is shown that there are two distinct methods of solution of the formulated boundary-value problem (oth- er methods can be, evidently, suggested). While the techruques used in these methods are essentially different the forms of solutions are sim- ilar and the only difference is in the manner of calculation of the pole coordinates.

In the first method, direct, based on the complex variable theory, the positions of the poles are determined by the explicit general terms: the

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 4 No. 4, Augus f 1997 447

location of any pole is calculated individually, independently from the others. In contrast, the positions of the poles in the second constructive method are specified by sequential procedure, one by one, that is, the location of the pole, say, with index k can be determined only if the co- ordinate of the previous ( k - 1)th pole is known.

Hence, for the calculations in which the value of k is sufficiently large, the application of the explicit expressions are more suitable, saving in time and efforts. But the use of the sequential procedure is to be pre- ferred if it is required to determine the positions of the first n poles of moderate number. When the calculations is to be carried out manually either by the first or second approach, it is only the length of computa- tions that determines which should be selected. This is, however, not an important matter when computing facilities are available.

The general consideration of the two discussed methods gives the possibility to make the following notes. The advantage of the direct method is that it has logically explicit rules of calculations and its fi- nal results are presented in an exact form, convenient for applications. But it must be recognized that this method is more complicated than the constructive method and it involves lengthy manipulation which is not obvious at first sight. The constructive method has the virtue that the solution can be determined by a combination of the basic solutions al- ready found. This method is simpler and calculations at any stage of solution have a clear physical interpretation. The main disadvantage of the constructive method, in general, is that there is no routine approach of developing solution. Very often, the simple constructive method can be suggested a posferiory after an exact solution obtained by any ana- lytical method, as it did in this paper.

6. CONCLUSIONS ERTAIN general points should be emphasized. In the formulated

C P roblem, the electric field depends on many parameters. For this reason, the general analytical solution of the problem has an evident ad- vantage over numerical methods of solution, as it gives an overall pic- ture of the different dependencies. The solution is obtained in the form of a series of line poles, convenient for analysis and computations. For many practical problems, the series is rapidly convergent. This gives the possibility to achieve acceptable accuracy of calculations taking a finite number of terms; often only the first N 3 terms are needed.

The electric force acting in the system under consideration has been determined earlier [2].

And finally, it should be noted that the offered constructive method can be used also for a solution of more general problems, for instance, it can be applied for a calculation of the electric field in the dielectric cylinder that contains a few charged wires. The exact solution in this case will be complex but the approximate solutions can be obtained with satisfactory accuracy.

7. APPENDIX 7.1. SOLUTION OF THE

BOU N DARY-VALUE PROBLEM

The explicit analytical solution to the boundary-value problem (9) with additional conditions (5) and (6) is given by using the methods of the theory of functions of a complex variable.

The first step is to transform conformally the system with given boundary configuration, a circle tangent to a straight line, into the sys- tem with more simple boundary: parallel straight lines. This can be done by the bilinear function

< = t + i q

Equation (38) transforms the circle L1 and the line L2 in the z-plane into the corresponding straight lines hl and h2 in the <-plane; conse- quently, the regions S, (p = 1 , 2 , 3 ) are mapped into the regions Q,, (see Figure 2). The positive directions of tracing along the boundary are indicated by the arrows. The origin and the point at infinity in the z- plane are transformed respectively into the points < = -1 and < = 1 in the <-plane.

The inverse transformation to (38) is also the bilinear function

< + I z = T (<) = r- < - I

(39)

In the <-plane, the sectionally analytic function f (<) is introduced

f (<I = E (T (<)I (40)

for which the boundary conditions are identical to the condition (9)

According to the condition (5) , the function f 2 (<) has the following representation in the region Q2

where the function f . (<) is analytic at every point in the region Q 2 .

The former condition at infinity (6) is written now in the <-plane as

f l (1) = f 3 (1) = 0 (43)

The boundary relations (41) together with additional conditions (42) and (43) represent the conditions of the generalized Riemann problem for finding the sectionally analytic function f (<) over the whole <- plane with the pole of the first order at the point < = -1.

The function f1 (<), defined in the infinite strip Ql, can be present- ed, due to the Laurent theorem [3], as the sum of the following two func- tions

fl (<I = fi' (<I + f l (<) = f : (0 R< < 1 (44)

= f,- (<I K 3 1

448 Emefs: Electric Field at the Interface of Two Dielectric Media

Then, relationship (41) can be rewritten in the form

mined afier finding thefunctions f: (<) and fc (<). These functions

From these equations and in accordance with an analytic continua- tion, the two analytic functions, @ (<) and KP (<), can be introduced as

Here the relationship

T __ < - 1

m-m (47)

is taken into account. The procedure of reflection transformations with respect to the lines 11 and 1 2 is shown in Figure 2 (b) by the dotted line.

By Equations (42) and (43), the functions f1 (r), f z (<) and f3 (c) are defined at specific points. Using this fact, (46) can be expressed, on the basis of the Liouville theorem, in the following form

Q(<) = o Therefore, Equation (46) is reduced to

c - 1

(1 + A 1 3 ) f T ( < ) + a l 3 f 3 (2 -t) = 0 %< < 1

From Equations (44) and (49), it is seen that the sectionally analytic function f (r) can be represented as

f l (0 = fi' (r) + fi- (<I 0 < % < 6 1 W 6 0

f 3 ( 0 = + A13)fc (<) %< 3 1 f z ( < ) = (1 + aiz)fT(T) + A i E

(50)

It is apparent from Equation (50) that the function f (<) will be deter-

in turn can be obtained from Equation (49). The set of Equation (49) con- tains, in fact, the four unknown functions, but this set can be reduced to one equation with a single unknown function, f $ (<) or fc (<). The procedure of manipulations is as follows.

An application of the reflect transformations, with respect to the lines 11 and 1 2 , to Equations (49) gives

f2c-Q

(1 + A 1 2 ) f l (r) (51)

f 3 ( < ) - (1 + a 1 3 ) f c (0 = 0 R< 3 1

Elimination now of the function f2 (-t) from the first pair of Equa- tion (51) and the function f 3 (<) from the second pair yields

1 - a,, < - 1 1 + a 1 2 <+ 1

Ai-- %< 3 0

a 1 3 f F ( < ) + f : ( 2 - r ) = o %<>

From these equations, it is possible to eliminate one more function, for example, f; (0. Then, the results is

where

On making the reflect transformation with respect to the line A2,

Equation (53) may be rearranged as

This is the functional equation for the unknown function fF<). Equation (55) can be solved by the iterative method. The recurrence for- mula of an iterative process is

r - 3

(56) f,'(< - 4) = Az-

< - 7 and so forth.

IEEE Transactions on Dieleetries and Electrical Insulation Vol. 4 No. 4, August 1997 449

The sequential substitution in Equation (55) gives, after n transfor- mations,

fl" = A2(< - x n

[Ak--'(< - 2k + l)-'(< - 2k ~ 1)-l] k = l

In the limit n + CO, the last term on the right-hand side of Equation (57) tends to zero, giving

00

f t ( < ) = A2(< - 1)2 [A"-'(< - 2k + 1)-l x k = l (58)

The final step is to return to the physical plane, the z-plane, with the help of Equation (38) to (40). It gives the desired expressions of the elec- tric field in the system under study. While making corresponding trans- formations, it is necessary to resolve the expressions in the square brack- ets in the right-hand side of Equations (58) and (59) into partial fractions. This calculations are very convenient to perform after returning into the z-plane. Simple algebra leads to

00

9 f, '(z) = -A13(1 - A) k = l

2ne1

k = l

with final results that are given by Equation (11).

REFERENCES Now, the function f; (<) can be found from the second Equation [l] L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media,

Pergamon Press, New York, 1968. [2] Yu. l? Emets, N. V. Barabanova, Yu. l? Onofrichuk, and L. Suboch, "Force on

Insulated Wire at the Interface of Two Dielectric Media", IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 1, pp. 1201-1204, 1994.

(52), after some algebra,

00

f c (<) = -A2A13(< - 1)' [A"'(< + 2k - x

[3] R. V. Churchill, Complex variables and applications, McGraw-Hill, New k = l

x ( < + 2 k - 3 ) - 1 ] R<> 1 (59) York, 1960.

Thus, the two parts of the function f l (c ) are defined completely and, hence, in accordance with Equation (50), the whole sectionally analytic function f (<) can be determined.

Manuscript Was receivedon 9MaY 199%