c Qf t

F. Olyslager I .V. Li nd el I

Indexing terms: Capacitance relations, Conductor configurations, Conformal mapping

Abstract: A relation, discovered during the 1950s, between the two-dimensional capacitances of four almost touching shells with an arbitrary shape is revisited. A short proof of this relation is presented based on Schwarz-Christoffel conformal mapping. The relation is then applied to find closed form expressions for the capacitances in configurations of more than four almost touching conductors. The results are important for making precise capacitors, capacitor benches and as benchmark results for testing capacitance software.

1 Introduction

In 1956 a remarkable relation between elements of the capacitance matrix of four, almost touching, conduct- ing shells (Fig. 1) was discovered [l, 21. It was shown that the elements CI3 and C,, of the capacitance matrix, due to the fields inside the shell of conductors, satisfy the relation:

regardless of the particular shape of the conductors. Throughout the paper C denotes capacitance per unit length in two-dimensional structures. The same relation holds for the capacitances due to the fields outside the shells. If, from symmetry, it is known that CI3 = C,, this relation gives a fixed value for the capacitances (CI3 = E& In 112).

In [2] the relation eqn. 1 was proven by means of a conformal mapping of the structure of Fig. 1 on a cir- cle. It was also mentioned in [2] that one could prove the relation (eqn. 1) with a Schwarz-Christoffel map- ping. This approach allows a very short and effective proof as we will illustrate here. From that proof we will also obtain some properties of the other capacitances in the capacitance matrix.

In [3] and [4] the idea by [2] was generalised to con- figurations with five and six, almost touching, conduc- 0 IEE, 1996 IEE Proceedings online no. 19960281 Paper first received 18th September 1995 and in revised form 15th Decem- ber 1995 F. Olyslager is with the Department of Information Technology, Univer- sity of Ghent, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium I.V. Lindell is with the Electromagnetics Laboratory, Helsinki University of Technology, Otakaari 5A, SF-02150 Espoo, Finland

tors. In this contribution we will apply eqn. 1 to study symetric configurations with an arbitrary number of conductors. It will be shown that the capacitances between these conductors have fixed values and follow from the solution of a coupled set of polynomial equa- tions. We will propose a general solution technique to solve this complicated system and give the actual solu- tions for up to 20, almost touching, conductors, most of them in closed form.

p12

Fig. 1

p34

4 ConJiguration with four almost touching conducting shells

The remarkable result that, in a symmetric configura- tion, the capacitances have fixed values makes these configurations of conductors very suited to build pre- cise capacitors as was mentioned in [l] for the case of four conductors. The idea to use five or six conductors as suggested in [3, 41 has the advantage that the system of equations, relating the different capacitances, is overdetermined and that these extra relations between the capacitances can be used to check the accuracy of the measurment set-up. Another advantage of using more than four conductors in a standard is that the values of the fixed capacitances become somewhat larger when the number of conductors increases.

The use of a large number of capacitances allows the construction of a bench of standard capacitors, perhaps with a switch, to find a capacitor of known value and preferred magnitude.

Another, and more up-to-date application of these capacitor configurations is that they can be used as benchmarks for numerical techniques. Since the bound- aries of the conductors can be shaped arbitrarily, it is posssible to test different aspects of the software. In numerical techniques, such as surface integral tech- niques [5] , the charge density on the conductors is approximated by some basis functions with unknown coefficients and the surface of the conductors are

302 IEE Proc.-Sci. Meas. Technol., Vol. 143, No. 5. September 1996

approximated by a polygon or by some second-order functions. The benchmark results allow one to deter- minje the accuracy of the different approximations in the code such as the modelling of curved boundaries and the modelling of the charge density singularities [6] at edges. This application as a benchmark is illustrated in the last Section.

In [4, 71 the five conductor capacitor standard was usedl to have an absolute determination of the ohm. By using precise bridges and standards for time, it is possi- ble to transform the standards for capacitance to standards for ohm. More recently, in [9], Theorem (eqni. 1) was used to build a more direct resistance standard out of superconducting materials. Since DC resistor problems are governed by potential problems, the results of this paper are also applicable to construct precise resistors.

It is also possible to use the Thompson-Lamipard the- orern (eqn. 1) to build standard inductors, because two-dimensional inductance problems are also described by a scalar potential problem as will be dis- cussed at the end of the paper. An alternativle way to construct standard inductors is presented in [8] where, instead of touching conductors, discrete wires are used. In [13] a system of four, five and eight wires is used.

1 -a 2

2 Relation between the capacitances of four almost touching conductors

Consider the two-dimensional structure of Fig. 1, con- sisting of four metal conducting shells whiclh almost touch. One could see the structure as a cylindrical hol- low tube with four narrow slots which isolate the four parts of the tube. The conductors are numbered from 1 to 4. We now prove that the special relation (eqn. 1) between the capacitances CI3 and C24? due to the fields inside the shell of conductors, is valid independent of the specific geometry of the conductors.

First, approximate the curve c of the four conductors by a polygon with N sides. The number N can be taken as high as necessary to improve the approxiniation of the original boundary. Eventually, one can see from the fortlhcoming reasoning that, without restriction, one can take the limit N .+ +W.

z-plane

3 b 4

PP

’23 I p3fA 5

Fig.2 rl = (1 axis

Complex z = 5 + iq plane with conductors of Fig. 1’ mapped to

Secondly, we use a Schwarz-Christoffel conformal mapping [lo] to map the internal region of the curve c on the upper half of the complex z(= < + iq)-plane. The conductors are mapped on the real axis of the z- plane. In the transformation, one has the freedom to choose the mapping of two points. Let us map the point P41 on 241 = and the point P23 on 223 = 0. The other two points are then defined by the Schwarz- Christoffel mapping. Let us indicate the mapping of P12 by 212 = -a and the mapping of PT4 by z34 = b, where a and b are positive numbers which we do not know, and which depend on the particular geometry of the curve c. The final geometry in the z-plane is shown in Fig. 2.

IEE ProL -Sa Meas Technol, Vol 143, No 5, September 1996

Thirdly, we determine the capacitances CI3 and CZ4 in the transformed geometry. Because we used a con- formal mapping, these capacitances are the same as in the original geometry. Let us start with Cl3. To find this capacitance, we have to put conductor 1 on a potential V, and the other conductors on zero poten- tial. The complex potential w(z) is then given by

(2) a d W

W ( Z ) = $(z ) + Z$(Z) = -Vo In ~

71 z + a where @(z) is the real potential and ~ ( z ) is the flux function. To determine the required capacitance, we need the charge Q3 induced on the third conductor due to this potential. It is easy to show that this charge is given by

( 3 ) E O V O a Q3 = Q,[$(x = b) - $(x = O ) ] = ~ In -

7r b + a This means that the capacitance CI3 is given by

(4) which has a negative value. The capacitance C24 fol- lows immediately from C,, by interchanging a and b

€0 C24 = - In ~

7r b + a ( 5 )

Since a and b are positive, we have proved that eqn. 1 is valid whatever the values of a and b may be.

It is also interesting to look at the capacitances CI1 and C12. Both the self-capacitance CI1 and neighbour capacitance C12 go to infinity when the distance between the conductors becomes smaller. Now let us determine how these capacitances go to infinity.

For the capacitance CI2 we need the charge on con- ductor 2. Let us assume that there is a gap ranging from < = -a to < = -a + til2 between conductor 1 and conductor 2 in the transformed plane. The charge Q2 is then given by

-U + d12)] = -In EOVO - 612 (6) n u

This means that CI2 goes to infinity as

(7 )

where I is some typical length of the conductor cross- section, for example, the length of the curve c. This length I is introduced to make the argument of the log- arithm dimensionless. Since locally a conformal map- ping can be seen as a linear mapping which just stretches lengths, the previous expression remains valid with 6 the distance between the conductors in the orig- inal configuration of Fig. 1. Since Q4, the charge on conductor 4, will have the same behaviour as Q2 and since the total charge on the four conductors should be zero, it follows that the self-capacitance CI1 behaves as

C11 z --In €0 - 641612

7r 12

where 841 is the distance between conductor 4 and 1. An interesting special case emerges when the geome-

try of Fig. 1 contains a symmetry line as in Fig. 3. In this case Cl3 = C24 and thus from eqn. 1 it follows that

c1 = -CIS = -cZ4 = 3 1n2 (9) 7r

where we introduced the notation C1 for later usage. The same reasoning as above can be repeated for the

capacitances corresponding to the external fields. These capacitances will also satisfy eqn. 1. However, by map-

303

ping the internal and the external region, the points P12 and P34 will not necessarily be mapped on the same points of the real axis in the z-plane. This means that the values of a and b need not be the same for the internal and the external mapping or that CI3 due to the internal fields will differ from C,, due to the exter- nal fields. So, in general, we cannot write down a sim- ple relation for the total capacitances Ctp,l and Cy1 of the structure of Fig. 1. In symmetric situations, such as Fig. 3, however, both the values of C:.,t and are equal to 2e01~ In 112.

I _ _ _ _ _ _ - _ - - -

Fig. 3 Symmetrical configuration with four conductors

Fig. 4 Configuration with four conductors with finite thiclcness

3

4 Fig. 5 try lines

Four conductors with noncoinciding external and internal symme-

Now consider the structure of Fig. 4 of four touch- ing conductors with finite thickness. The results for the internal capacitances and external capacitances remain valid. Sometimes we know that the CI3(C&) due to the internal fields is equal to C1,(C2,) due to the external fields. This will be the case when there exists a confor- mal mapping which maps the external region on the internal region, such as the mapping z -+ l/z. In this case the total capacitances satisfy

304

Another interesting example is shown in Fig. 5 where both the internal and the external region each have a symmetry axis, which do not coincide. In this case still c t o t - 13 - Cy; = ~ E ~ / J C In 1/2. In the next Sections we will concentrate on the internal capacitances. For symmet- ric configurations the external capacitance can be taken into account by a factor of 2 in the capacitance expres- sions provided that the external boundary of the con- ductors is also symmetric.

By applying shielding conductors, one can construct measurement setups where only the internal or the external capacitances come into play. This was done in the standard capacitors of [I] where circular cylinders were surrounded by a square shield.

Finally we want to remark that it is allowed that parts of the conductors are located at infinity as shown in Fig. 6. The Schwarz-Christoffel transformation also handles these situations.

4 Fig. 6 Configuration with four conductors with parts at infinity

4 Fig. 7 Configuration with five conductors

3 Configuration with five conductors

Consider the five almost touching conductors shown on Fig. 7. If we now apply eqn. 1 and connect the conductors 3 and 4 we find

where, from now on, xii denotes 225 + 213514 = 1 (11)

By connecting other neighbouring conductors also, the following relations hold:

2 1 4 + 2 2 5 x 3 5 1 (13) 2 1 3 + 2'25x24 zz 1 (14)

IEE Pvoc -Sei. Meas. Technol., Vol. 143, No. 5, September 1996

2 2 4 + 2 1 3 2 3 5 = 1

2 3 5 + 2 1 4 2 2 4 = 1 (15)

(16) These are not independent equations which uniquely define the five different values xii. Our needs to specify two values of xij to find the other three. This means that, out of three of these equations, the other two can be constructed. The generalisation of eqn. 1 to five almost touching conductors takes then the following form:

Now assume that the structure is symmetrical as in Fig. 8. Let C1 denote the capacitance between a con- ductor and its second neighbour as we did in eqn. 9 i.e. CI3 == CI4 = C,, = C2, = C,, = -C1 and let x1 denote

x1= exp (- 3) (20)

Fig. 81 Symmetrical configuration with Jive conductors

From eqn. 11, it then follows that x1 satisfies

or that 2 1 + x: = 1

A-1

(21)

(22) 21 = ~

2 and ifinally

CO 4 5 - 1 Cl = --ln- T 2

-

Fig. 9 Configuration with Jive conductors with symmetry axis

This result coincides with the result in [4]. In the struc- ture of Fig. 9 there is only one line of symmetry (i.e.

IEE Proc -Sei. Meas. Teclinol., Vol. 143, No. 5, September 1996

xi3 = x35 and ~ 2 5 = xI4). The set of eqns. 11-16 now reduces to

and

Fig. 10 Symmetrical configuration with six conductors

4

Now consider the symmetric structure of Fig. 10 with six conductors. In addition to the second-neighbour capacitances C1, there is also the third-neighbour capacitance C2 which remains finite when the gaps between the conductors become smaller. To find these capacitances we can again apply eqn. 1. If we first con- nect the conductors 3, 4 and 5 and apply eqn. 1, we find

Symmetric configuration with six conductors

21 + 2:z, = 1 (26) A second equation is obtained by connecting the con- ductors 1 and 6 and also the conductors 3 and 4. This results in

This set of two equations is easily solved, it requires only the solution of linear equations, the result is

2 2 1 = -

3

3 2 2 = -

4

2 2 + 242; = 1 (27)

(28)

(29)

and

5 Symmetric configuration with seven conductors

For a symmetric configuration with seven almost touching conductors, there are again two finite capaci- tances C1 and C,. The equations for the corresponding variables x1 and x2 are

and 21 + 21.; = 1

2 2 + 2$x$ = 1

(30)

(31)

305

The solution now reduces to the solution of a third degree polynomial equation:

2: + 32: - 421 + 1 = 0 (32) and x2 follows then from eqn. 30. The result is:

(33) 8 r + 2 a rc t an (3A)

2 1 = - 1 + 2 -cos 6

(34) 2 2 f i r - a r c t a n ( 3 4 ) 3 3 3

2 2 = -- + -cos

6 Symmetric configuration with eight conductors

For eight symmetric conductors, there are three differ- ent finite capacitances C1, C2 and C,. The equations for the corresponding xl, x2 and x3 are given by

(35)

2 2 +.:&E; = 1 (36)

2 3 + 59x;2; = I (37)

2 2 2 1 + 2 1 x 2 5 3 z= 1

and

Solution of this set of equations is simplified considera- bly by connecting pairs of adjacent conductors and applying the result (eqn. 9) corresponding to four sym- metric conductors. This gives the relation

(38) €0 1 2C2 + 2C3 = -- In - 7 r 2

or ~2x32 = 112. The values of xl, x2 and x3 are now easily found by solving just some quadratic equations. The result is

1

= z 2 2 = 2 J z - 2

2 + &

and

2 3 = ~

4

7 Symmetric configuration with nine conductors

Now the three equations for xl, x2 and x3 are

2 1 +2:22;2; = 1

2 2 + 2:x;x; = 1

2 3 + .:x;5; = 1

and

It is still possible to express the solution in closed form as

(45) 7r 27r

~1 = -2 - COS - + Gcos - 9 9

1 + 2 c o s ~ 2 2 = 3 (46)

and r

5 3 = -1 + 2cos - 9 (47)

8

It is possible to write down the system of polynomial equations for a symmetric configuration with M > 3 conductors in compact form. If A4 is an even number,

Symmetric configuration with M conductors

306

there are N = (M-2)/2 unknowns xi (i = 1, ..., N). The set of equations for these unknowns can be written as

N

2=1

where i = 1, ..., N and T~ is: i 2 j # N : rij = 2 j i < j # N : rij = 2i j = N : rt3 = i

If A4 is an odd number, there are N = (M-3)/2 unknowns x (i = 1, ..., N). The set of eqn. 48 remains valid; however, the definitions for zii change:

(49)

(50) i > j : r i 3 = 2 j i < j : rij = 2i

Due to the special structure of the -G = [-cy] matrix it is possible to formulate an elimination procedure for the system of equations. Let us start with splitting the product in eqn. 48 in two parts (j = 1, ..., i-1 and j = i, ..., N) and rearranging that equation

i-1

(51) 1 - xi x3ii = ~

n, Xj..l

N j = S

3 = 2

If this expression is inserted in eqn. 48 for i-1, using the structure of Z, one can express xiPl as a function of xi with j z i alone:

N-1

with (5 = 1 when A4 is even and (5 = 2 when A4 is odd. Applying this expression recursively allows us to express each xi(i = 1, ..., iV) as a rational function of xN only. A polynomial expression for xN is then found by inserting all the rational expressions for the xi in, for example, eqn. 48 for i = 2.

Let us assume that M = 2L and that L is even. Now we will show that it is easy to express the solutions xi for the A4-conductor case as a function of a = xk with k = (L-2)/2 for the L-conductor case. When M = 2L, we can connect the conductors two by two and use the result of the L-conductor case:

( 5 3 ) 2:2&-1 = cv A second equation relating xN and x ~ - ~ follows from eqn. 52 for i = N:

2 X N - 1 (54) 2 N - 1 = ~

XN The solution of both these equations is readily obtained:

?--.

and

(55)

Now, by using the recursion formula (eqn. 52), we can find all the other xi, i = 1, ..., N-2. In the case A4 = 2L with L odd, things are somewhat more complicated. By connecting the conductors in pairs we now have

with a the solution a = xk, with k = (L-3)/2 of the L- conductor case. Two other equations follow from

(57) 2

X N X N - ~ ~ N - ~ =

IEE Proc.-Sei. Meas. Technol., Vol. 143, No. 5, September 1996

eqn. '52 for i = N and i = N-1: Table 1

M XN

13 0.94188363485210405210

14 I7+2d7cos[(n-arctan(3I'3))/311/12

15 0.95629520146761 127586

16 I2+d/(2+d2/2)1/4

17 0.9659461 9936780355656 1 8 19 0.97272260680544474721

20 [4+I'2./(5+./5)1/8

I 1 +cos( d9) 112

and

(59)

The solution of this system of equations is given by

and

When M is odd, it is in general not possible to write the solution for A4 conductors in terms of the solution of another number of conductors in closed form.

To conclude this Section, a few specific cases, other than the ones handled in the previous sections, are included to illustrate the above results. When M = 10, we can use the five conductor result to obtain

and

2 23 = z

5 + 6 2 4 = -

8

(64)

(65)

The 11-conductor case is the first one which (does not allow a closed form solution. After elimination, one finds that x4 satisfies the following quintic equation:

x i + 4x: + 22: - 52: - 2x4 - 1 = o (67) for which no solution in elementary functions was found. The numerical solution for the A4 = 11 case, however, is given by:

2 1 = 0.728445870661 17882056 (68)

2 2 = 0.86103089809392755117 (69)

x3 = 0.90407261232771053712 (70) ~4 = 0.91898594722899477978 (71)

2 1 = &- 1 (72)

For the 12-conductor case, we can use the result of six conductors and obtain:

- d3

2 2 = - 2 (73)

2 4 = 2 4 2 - h) (75)

(76) 2 + v 5

2 5 = ~

4 Finally, Table 1 gives xN for some other values of M :

IEE Proc.-Sei. Meas. Technol., Vu1 143, No. 5, September 1996

. .

Fig. 11 Possible configuration of capacitor bench

9 resistors

Precise capacitor benches, inductances and

In [l] a precise capacitor was constructed using four circular cylinders. By considering a configuration with a large number of almost touching cylinders sur- rounded by a cylindrical shield (Fig. l l ) , one could construct a capacitor bench allowing a wide range of fixed capacitances. This can be achieved by connecting the cylinders with a switching network allowing differ- ent interconnection configurations. One interesting pos- sibility would be the usage of 2n+2 (n 2 0) cylinders. The value of x1 in this case is then given by

1 (77)

IC1 = T5 and the value of xN by

2 + J 2 + Jm X N 1 4 (78)

where there are n square roots. The value of X N defines the smallest possible capacitance in the bench. Another possibility would be a configuration with n conductors where n has a large number of different divisors. This also allows a wide range of possible capacitances.

Since, in two dimensions, inductances per unit length also follow from a two-dimensional potential problem the presented results can also be used to construct pre- cise inductors. In the equations one just has to replace eo by l/po and C, by l/L,. This means, for example, that the definition for xi now becomes

xi = exp (-?) (79)

307

In electrostatics, the current flow in conductive materi- als is also governed by a two-dimensional Laplace problem. This means that the presented results are also applicable for making precise resistors. For example, as shown in the configuration of Fig. 12 where there is a conductive region (conductivity 0) surrounded by a number of almost touching electrodes. In the equations one now has to replace E~ by the conductivity o and C, by the conductance G, per unit length.

6

’3 Fig. 12 Precise resistor with six electrodes

C,pF/rn 1.961 .

1953 , , , I ” I ’ 1 ‘ I /

number of divisions per cylinder 20 LO 60 ao

Fig. 13 Benchmark test of two-dimensional capacitance software puck- age

10 Application as a benchmark

In this Section we illustrate the application of the standard capacitance values as benchmarks for numeri- cal techniques. We consider a configuration of four cir- cular cylinders symmetrically arranged in a square. The radius of each cylinder is 1.9, and each two adjacent cylinders are separated by a distance equal to 0.2. As a numerical technique we use the integral equation tech-

nique described in [5]. The integral equation is discre- tised numerically with the method of moments. The surface of each cylinder is divided into a number of small segments and within each segment the surface charge density is assumed to be constant. Fig. 13 shows a graph of the capacitance between two opposite cylinders as a function of the number of divisions on each cylinder. The theoretical value eqn. 9 is 1.953549044pFim for co = 8.85418782pFim. Since the software was written in single precision, an accuracy of more than 5 digits is not to be expected. The software shows that, even for a relatively large gap between the cylinders, the accuracy is rather good.

11 Conclusion

A two-dimensional capacitance problem of a system of conductors studied in the 1950s was revisited and gen- eralised, and a study was made of the fixed values of partial capacitances corresponding to symmetric con- ductor configurations. Explicit results, most of them in closed form, were presented for up to 20 conductors.

11 Acknowledgment

Frank Olyslager is a postdoctoral researcher of the Bel- gian National Fund for Scientific Research. This work was done during his visit at the Electromagnetics Labo- ratory of the Helsinki University of Technology spon- sored by a travel grant from the Belgian National Fund for Scientific Research. The present problem was initi- ated by Professor F. Stenman from the Department of Physics, University of Helsinki. The authors also wish to thank the anonymous reviewers for their useful sug- gestions.

13 References

THOMPSON, A.M., and LAMPARD, D.G.: ‘A new theorem in electrostatics and its application to calculable standards of capac- itance’, Nature, 1956, 177, pp. 888 LAMPARD, D.G.: ‘A new theorem in electrostatics with applica- tions to calculable standards of capacitance’, Proc. IEEE, 1957, 216C, pp. 271-280 FIEBIGER, A., and FLEISCHHAUER, K.: ‘Ein Kreuzkonden- sator nach Thompson und Lampard mit sechs kreiszylindrischen Elektroden’, PTB Mitteilungen, 1975, 85, (4), pp. 271-274 ELNEKAVE, N.: ‘An absolute determination of the ohm based on calculable standard capacitors’, IEE Conf Pub., 1977, 152, pp. 53-55 OLYSLAGER, F., FACHE, N., and DE ZUTTER, D.: ‘New fast and accurate line parameter calculation of multiconductor transmission lines in multilayered media’, IEEE Trans., 1991,

VAN BLADEL, J.: ‘Singular electromagnetic fields and sources’ (Clarendon Press, Oxford, 1991) DELAHAYE, F., FAU, A., DOMINGUEZ, D., and BEL- LON, M.: ‘Absolute determination of the farad and ohm and measurement of the quantized hall resistance RH(2) at LCIE’, IEEE Trans., 1987, IM-36, (2), pp. 205-207 PAGE, C.H.: ‘A new type of computable inductor’, J. Res. Nut. Bureau Standavds B. Math. Math. Phys., 1963,67B, (l), pp. 31-39 FRENKEL, R.B.: ‘A superconductor analogue of the Thompson- Lampard theorem of electrostatics and its possible application to a new SI standard of DC resistance’, Metrologiu, 1993, 30, pp.

MTT-39, (6), pp. 901-909

117-113 _ _ I _”I

10 DURAND, E.: ‘Electrostatique et Magnetostatique’ (Masson & Cie, Paris, 1953), pp. 328-332

308 IEE Proc.-Sei. Meas. Technol., Vol. 143. No. 5, September 1996