- Email: [email protected]
On capacitance corrections for unguarded disc electrodes
J o u r n a l of
ELECTROSTATICS ELSEVIER
Journal of Electrostatics 33 (1994) 371 378
On capacitance corrections for unguarded disc electrodes K.Y. Kung a, H.J. Wintle b, A. Blaszczyk c, P.L. Levin a'* a Computational Fields Laboratory, Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA b Department of Physics, Queen's University, Kingston. Canada K7L 3N6 c Asea Brown Boveri Corporate Research Center, Speyerer Strasse 4, 69115 Heidelberg, German)'
Received 30 July 1993; accepted after revision 23 March 1994
Abstract
The edge capacitance corrections for thin unguarded disc electrodes placed on opposite faces of a dielectric sheet or film were computed by Wintle and G o a d [1]. In their paper, they mention another calculation done by Sato and Zaengl using a surface charge simulation method that seemed to show some discontinuity between e2 = 1 and e2 = 2, where e2 is the dielectric constant of the insulator. In this paper we have tried to reproduce the result with a different surface charge simulation method, as well as a region-oriented charge simulation approach and a finite element program in order to take a closer look at the discrepancy.
1. Introduction The dielectric constant of solid material is best measured using a guarded electrode system by placing a sheet or thin film of the material between two electrodes. However, it is common to employ the simpler unguarded electrode system, which is shown in Fig. 1. The capacitance of an unguarded disc capacitor with a dielectric sheet placed between electrodes is approximately rl; a 2 go gr
Cge°m =
h
(1)
This calculation ignores the edge effect of the electric field and some correction is normally required. Indeed, much work has been done on this problem dating back to empirical formulae established by Scott and Curtis [2]. To date none of these provide a satisfactory degree of precision, and as the accuracy of measuring equipment has increased, these correction formulae are unacceptable for certain applications. More
* Corresponding author. Tel.: 508 831 5231. Fax: 508 831 5491. 0304-3886/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 3 8 8 6 ( 9 4 ) 0 0 0 0 4 - G
K.Y. Kung el al./Journal o[ Electrostatics 33 (1994) 371 3L~
372
iihlml £1
II Ra
i_
a
t Ra
(a)
(b)
Fig. 1. An unguarded disc capacitor with infinitesimally thin electrodes and an unguarded disc capacitor with electrode of finite thickness. R is the ratio of dielectric disc radius to electrode radius, a is the radius of the electrodes, h is the thickness of the dielectric between the electrodes, and t is the thickness of the electrode. Our purpose is to calculate numerically the capacitance as t tends to zero.
recently, Wintle and G o a d [1] reexamined an analytical formulation to address this problem, in which they claimed that the charge on the two electrodes maintained at potential of -I- Vao~,d2 is well defined by
Vappl ~ ( 1 - g ~ I~: ~ j o ( p p \ eoe2 / d o
) 1 - ehP 1 - - K e hpdp
(p G a)
(2)
and
f ~ A(p)Jo(pp)Op = 0 (p > a ) ,
(3)
0
where K = (~ - ez)/(~l + e.2), p is a parameter varying from zero to infinity, h and p are the thickness and radius of the dielectric disc, and the expression inside the second integral is the well-known Fourier-Bessel expansion of the charge density function [3], which is expressed in terms of the distribution function A(p). These integrals can be solved numerically and the results are far more accurate than any preceding analytical or numerical approach. In their article, Wintle and G o a d refer to a numerical study by Sato and Zaengl which attempted to deal with this problem using a surface charge simulation method. Their results for ~2 = 1 agree with the capacitance values of Wintle and G o a d to four or five significant figures. However, their work shows an unexpected j u m p between ~2 = 1 and ~2 = 2, and their values for E2 > 1 disagree with the analytical result. These discrepancies are characteristic of certain charge simulation formulations. Since this problem is of practical concern, we have taken a closer look.
2. Computing free charge based on the Gauss' law boundary condition We begin our discussion with a description of the difficulties involved in using a surface charge simulation p r o g r a m to calculate the capacitance of a system with
K.Y. Kung et al./Journal o f Electrostatics 33 (1994) 371- 378
373
infinitely thin electrodes. Later, we will compare this technique to two other methods to corroborate the earlier results of Wintle and Goad. Neither of the alternatives rely on approximating the discontinuity of displacement field near the electrodes, which we demonstrate here is the largest source of error. The basis of the surface charge simulation method is to establish a system of linear equations by introducing equivalent surface charges distributed in a vacuum. Polarization charges which appear on dielectric boundaries are solved for explicitly by imposing the continuity of flux condition, and the electric field in the region of interest can be evaluated directly as a superposition sum of the approximate surface charge distribution. The surface charge simulation program used here employs a Galerkin boundary element formulation [4] to simulate the electric fields in the structure shown in Fig. 1. We specified electrodes of finite thickness because it is extremely difficult to obtain numerically an accurate value of the charge density for an infinitesimally thin electrode, except for the special case when g2 = el' To see this, begin by considering the Gauss' law jump condition governing the electric field on either side of a material interface [5]: O'SCSM :
O'free -~- a b o u n d e d = gO
(4)
E1 -- go E2,
where a indicates a surface charge density and ascsM is the simulated charge resulting from the surface charge simulation method. The normal components of the electric field on either side of the boundary are represented by E, and eo is the permittivity of free space. The values of aSCSMare readily found by the numerical technique; however, one is typically only interested in the value of afree. The bound charge on a dielectric
P2),
(5)
which for linear materials is a simple function of the electric field abounded =
--
(eoZ1E, -
80~(2E2)
(6)
implying O'free -~- Co(1 + z t ) E , -- Co(1 +
~(2)E2
(7)
where (1 + Zl) and (1 + ~2) a r e the relative permittivities of the two media. Note that if Xl = X2 or either E1 or E2 is zero, then afree = (1 + Z) aSCSM.In these two cases, we do not have to measure El and E 2. However, if Xl # Z2, then we may have to calculate the values of both El and E2 arbitrarily close to the boundary, which can dramatically compromise the accuracy of the simulation. Clearly, any formulation that can sensibly determine the abrupt discontinuity of field directly on an infinitesimally thin boundary will avoid these kinds of errors. The test model used in this paper is shown in Fig. 2 and the number of strips we employed for the surface charge simulation method is shown in Table 1. The structure is simulated with axisymmetric strips using a Galerkin formulation of the BEM and
374
K.Y. Kung et al./Journal o[' Electrostatics 33 (1994,) 371
378
axis of rotation
I i
m ] ....................... n i~ 2
o
plane of ground symmetry Fig. 2. This graph shows the geometry used to model an unguarded disc capacitor. The letters I, rn, n, o represent the n u m b e r of strips on each boundary. In order to obtain strips of approximately equal length, we let n = (R - 1 ) / a n d o = hl/2a, where R, a, h are defined in Fig. 1.
Table 1 Data of the test model C,~om/ne, oe~a = a/h
R
I
m
n
o
0.100'* 0.100" 0.100 0.333 1.000 3.333
10 10 5 5 5 5
20 20 20 20 20 20
2 2 2 2 2
180 180 80 80 80 80
100 100 100 30 10 4
R is the ratio of the electrode radius to the dielectric sheet radius. The actual number of strips used in each simulation for each boundary is D times the number shown in the table, where D is the discretization ratio. Note that only models with discretization ratios of 1, 2, 4 are used for extrapolation; 8 is discarded due to the unexpected b u m p shown in Fig. 3. The quantities I, m, n, and o are described in Fig. 2.
the results obtained are the capacitance of different disc capacitor configurations with electrodes of finite thickness. These data are extrapolated to zero thickness electrodes using standard Richardson extrapolation [6], which assumes that the capacitance value obtained from the simulation is the actual capacitance value C plus an error related to the thickness of electrode t. The error is represented by a Taylor series expansion, and the extrapolation recursively eliminates higher-order terms based on simulation results for different values of t. We thus obtained the capacitance of the disc capacitor for an electrode of zero thickness, and then repeated the process assuming that the data has an error function related to the reciprocal of the discretization ratio and extrapolated the data to
K.Y. Kung et al./Journal of Electrostatics 33 (1994) 371-378
375
Table 2 Comparison of semi-analytical values against the surface charge simulation result
Cgeom/~oe2a = a/h
e2
Cex/~eoe2a,Re~ [1]
Extrapolated CSP
Percentage error
0.1**
0.5 1 2 5 lO 20
1.91320 1.25920 0.92982 0.73096 0.66440 0.63106
1.89242 1.25902 0.90453 0.70616 0.63990 0.60671
1.08 0.01 2.71 3.39 3.68 3.85
0.1*
0.5 1 2 5 10 20
1.91320 1.25920 0.92982 0.73096 0.66440 0.63106
1.92844 1.25940 0.92122 0.71698 0.64822 0.61361
0.79 0.01 0.92 1.91 2.43 2.76
0.1
0.5 l 2 5 10 20
1.91320 1.25920 0.92982 0.73096 0.66440 0.63106
1.94591 1.25940 0.90862 0.68851 0.60072 0.56871
1.70 0.01 2.27 5.80 9.58 9.87
0.333
0.5 1 2 5 10 20
1.94755 1.25581 0.90213 0.68582 0.61277 0.57603
1.97822 1.25607 0.89120 0.66936 0.59455 0.55688
1.57 0.02 1.21 2.39 2.97 3.32
1.000
0.5 1 2 5 10 20
2.09180 1.31830 0.91253 0.65896 0.57210 0.52813
2.10702 1.31873 0.91046 0.66727 0.58109 0.53719
0.72 0.03 0.23 1.26 1.57 1.71
3.333
0.5 1 2 5 10 20
2.43850 1.50443 1.00121 0.68043 0.56919 0.51258
2.38675 1.50541 1.02772 0.70882 0.59237 0.53155
2.12 0.06 264 4.17 4.07 3.70
The program is called CSP. Error increases as the dielectric permittivity. The accuracy improved dramatically when we doubled the radius of the dielectric disc for a/h = 0.1", which verified that an important source of error in our simulation is the truncation of the dielectric radius. 0.1 ** is obtained by directly using infinitesimally thin electrodes; these results are less satisfactory than the extrapolated values.
zero strip length I-7]. In our case, we used 4 different values of t and 3 different values of 1/D. The extrapolated values are then used to calculate the normalized excess capacitance values that are shown in Table 2.
376
K.Y. Kung et al./Journal of Electrostatics 33 (1994) 371 378
percentage error
10 9
E, = 20
8
£, = 10
7
£,
6
5
5 4 3
£ , = 0.5
£,=2
2 1
O.r 0
,
0.I
~.
0.2
0.3
,
0.4
?
0.5
.
0.6
.
.
0.7
.
£,=I
0.8
0.9
1
l/D, where D is the discretization ratio Fig. 3. Graph of percentage error of extrapolated excess capacitance Ce~/~eoe2a as a function of discretization ratio for the case a/h = 1. This graph demonstrates the influence of using a dielectric disc of finite radius. Notice that there is an unexpected rise in the extrapolated data when D is increased from 5 to 8. The increase in error with 82 is due to the difficulty of imposing continuity of flux through dielectric interfaces.
The semi-analytical results of [1] assume that the radius of the dielectric plate approaches infinity, which is not possible to model numerically. Approximating the dielectric with a finite radius (Ra in Fig. l) also introduces considerable error, particularly when a/h is small and the permittivity of the dielectric material is high. The sensitivity of the results to the radius of the dielectric is clearly seen by the second and third series of numerical experiments, and it appears that the truncated dielectric is a significant source of numerical error. Indeed, the accuracy for e2 - - 2 0 was c o m p r o m i s e d by m o r e than a factor of three. The close agreement between the extrapolated and semianalytical values for the case ~32 = 1 lends confidence to our extrapolation. Notice in Fig. 3 that the accuracy drops dramatically when e 2 :~ 1, and increases as e2 increases. Finally, the worsening accuracy for e2 = 1 as the mesh becomes very refined points to a further problem. We surmise that the matrix becomes steadily more ill-conditioned as its size increases. We presume that the effect is more acute for e2 # 81, thus giving rise to the b u m p s seen in Fig. 3, and we suggest that this apparently inconsistent behavior is spurious.
K.Y. Kung et al./Journal o f Electrostatics 33 (1994) 3 7 1 - 3 7 8
377
3. Results and conclusions Calculating the free surface charge on an infinitely thin conductor that separates regions of different permittivity is a numerically daunting task. Employing the Gauss' law jump condition is theoretically straightforward; however, approximating this condition numerically can introduce significant error, as demonstrated by comparing the values of a/h = 0.1"* and 0.1" in Table 2. The results in Tables 2 and 3 are displayed in terms of the excess capacitance, which is defined by C,o,,, = Csoom+ C,x.
(8)
The values are obtained by numerically determining Ctotal, substituting for Cgeomfrom Eq. (1), and normalizing by a factor of rt~oe2a. Notice that the abrupt discontinuity with respect to 82 found in an earlier numerical simulation based on the charge simulation method is not apparent here. Table 3 shows a comparison of the previously reported results of Wintle and Goad and three codes based on different methods. Two of the programs employ integral formulations, and one solves Laplace's equation directly using finite elements. The surface charge simulation code minimizes the residual function in a Galerkin sense over every material interface. Richardson extrapolation was employed to evaluate the capacitance as t --* 0, where the independent variable for extrapolating the strip length to zero is taken here as 1/D. The capacitance of infinitely thin elements calculated by this process is better than the result obtained by evaluating E at points arbitrarily close to the boundary, but not nearly as accurate as the two other methods that avoid this restriction altogether. For example, the region,oriented charge simulation i[8] implements Eq. (7) exactly on each side of the conductor boundaries, and can measure the discontinuity of flux
Table 3 Comparison of all three numerical techniques to the semianalytical result for a/h = 3.333 Normalized extra capacitance for a/h = 3.333 ~2
Ref. I-1]
CSP (% error)
EMS (% error)
ELFI (% error)
1 2 5 10 20
1.50443 1.00121 0.68043 0.56919 0.51258
1.50541 1.02772 0.70882 0.59237 0.53155
1.51041 1.00576 0.68418 0.57864 0.51615
1.5028 (0.11) 1.0014 (0.02) 0.6815 (0.16) 0.5705 (0.23) 0.5142 (0.32)
(0.06) (2.64) (2.84) (4.07) (3.70)
(0.39) (0.45) (0.55) (1.66) (0.69)
The best result is from ELFI; the program employed 500 region-oriented smulation charges. The EMS results is based on a second-order finite element scheme that provides very fast solutions and runs on 486-class personal computer.
378
K.Y. Kung et al./Journal of Electrostatics 33 (1994) 371-378
without resorting to extrapolation. Moreover, it explicitly forces the net flux through all dielectric interfaces to be zero, which dramatically improves accuracy when e2 >> 1. The boundary conditions are imposed in a collocation sense; however, this apparantly does not mitigate the quality of the result for integrated quantities like net charge. The finite element approach is based on a proprietary formulation that automatically discretizes the problem using an adaptive solver option. The solutions were created on a 486-class personal computer in less than 30 s; less than 3 s were required for the final iteration in all cases which included as many as 1500 second-order nodes.
Acknowledgements We wish to thank Dr. S. Sato for some helpful discussions. H.J.W. wishes to thank NSERC (Canada) for continuing financial support. The Computational Fields Laboratory is underwritten in part by Richard Bergener GmbH. The authors gratefully acknowledge the use of ElectroMagnetic Solutions' "2D Toolset", which was provided by Dr. Arthur Butler.
References [1] H.J. Wintle and D.G.W. Goad, Capacitance corrections for disc electrodes on sheet dielectrics, J. Phys. D: Appl. Phys., 22 (1989) 1620-1626. [2] A.H. Scott and H.L. Curtis, J. Res. NBS, (1939) 747-757. [3] E. Durand, Electrostatique, Vol. I, Masson, Paris, 1964, pp. 259-260. I-4] P.L. Levin, A.J. Hansen, D. Beatovic, H. Gan and J.H. Petrangelo, A unified boundary-element finite-element package, IEEE Trans. Electrical Insulation, 28 (1993) 161-t67. 1-5] H.A. Haus and J.R. Melcher, Electromagnetic Fields and Energy, Prentice-Hall, Englewood Cliffs, N J, 1989, p. 21 I. [6] J.H. Ferziger, Numerical Methods for Engineering Application, Wiley-lnterscience, New York, 1981, pp. 30-31. I-7] H.J. Wintle, The capacitance of the regular tetrahedron and equilateral triangle, J. Electrostatics, 26 (1991) 115-120. 1-81 A. Blaszczyk and H. Steinbigler, Region-oriented charge simulation, Conference Proceedings, COMPUMAG, Miami (1993) 154-155 (extended version under review by IEEE Trans Magnetics).
ELECTROSTATICS ELSEVIER
Journal of Electrostatics 33 (1994) 371 378
On capacitance corrections for unguarded disc electrodes K.Y. Kung a, H.J. Wintle b, A. Blaszczyk c, P.L. Levin a'* a Computational Fields Laboratory, Department of Electrical and Computer Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA b Department of Physics, Queen's University, Kingston. Canada K7L 3N6 c Asea Brown Boveri Corporate Research Center, Speyerer Strasse 4, 69115 Heidelberg, German)'
Received 30 July 1993; accepted after revision 23 March 1994
Abstract
The edge capacitance corrections for thin unguarded disc electrodes placed on opposite faces of a dielectric sheet or film were computed by Wintle and G o a d [1]. In their paper, they mention another calculation done by Sato and Zaengl using a surface charge simulation method that seemed to show some discontinuity between e2 = 1 and e2 = 2, where e2 is the dielectric constant of the insulator. In this paper we have tried to reproduce the result with a different surface charge simulation method, as well as a region-oriented charge simulation approach and a finite element program in order to take a closer look at the discrepancy.
1. Introduction The dielectric constant of solid material is best measured using a guarded electrode system by placing a sheet or thin film of the material between two electrodes. However, it is common to employ the simpler unguarded electrode system, which is shown in Fig. 1. The capacitance of an unguarded disc capacitor with a dielectric sheet placed between electrodes is approximately rl; a 2 go gr
Cge°m =
h
(1)
This calculation ignores the edge effect of the electric field and some correction is normally required. Indeed, much work has been done on this problem dating back to empirical formulae established by Scott and Curtis [2]. To date none of these provide a satisfactory degree of precision, and as the accuracy of measuring equipment has increased, these correction formulae are unacceptable for certain applications. More
* Corresponding author. Tel.: 508 831 5231. Fax: 508 831 5491. 0304-3886/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 3 8 8 6 ( 9 4 ) 0 0 0 0 4 - G
K.Y. Kung el al./Journal o[ Electrostatics 33 (1994) 371 3L~
372
iihlml £1
II Ra
i_
a
t Ra
(a)
(b)
Fig. 1. An unguarded disc capacitor with infinitesimally thin electrodes and an unguarded disc capacitor with electrode of finite thickness. R is the ratio of dielectric disc radius to electrode radius, a is the radius of the electrodes, h is the thickness of the dielectric between the electrodes, and t is the thickness of the electrode. Our purpose is to calculate numerically the capacitance as t tends to zero.
recently, Wintle and G o a d [1] reexamined an analytical formulation to address this problem, in which they claimed that the charge on the two electrodes maintained at potential of -I- Vao~,d2 is well defined by
Vappl ~ ( 1 - g ~ I~: ~ j o ( p p \ eoe2 / d o
) 1 - ehP 1 - - K e hpdp
(p G a)
(2)
and
f ~ A(p)Jo(pp)Op = 0 (p > a ) ,
(3)
0
where K = (~ - ez)/(~l + e.2), p is a parameter varying from zero to infinity, h and p are the thickness and radius of the dielectric disc, and the expression inside the second integral is the well-known Fourier-Bessel expansion of the charge density function [3], which is expressed in terms of the distribution function A(p). These integrals can be solved numerically and the results are far more accurate than any preceding analytical or numerical approach. In their article, Wintle and G o a d refer to a numerical study by Sato and Zaengl which attempted to deal with this problem using a surface charge simulation method. Their results for ~2 = 1 agree with the capacitance values of Wintle and G o a d to four or five significant figures. However, their work shows an unexpected j u m p between ~2 = 1 and ~2 = 2, and their values for E2 > 1 disagree with the analytical result. These discrepancies are characteristic of certain charge simulation formulations. Since this problem is of practical concern, we have taken a closer look.
2. Computing free charge based on the Gauss' law boundary condition We begin our discussion with a description of the difficulties involved in using a surface charge simulation p r o g r a m to calculate the capacitance of a system with
K.Y. Kung et al./Journal o f Electrostatics 33 (1994) 371- 378
373
infinitely thin electrodes. Later, we will compare this technique to two other methods to corroborate the earlier results of Wintle and Goad. Neither of the alternatives rely on approximating the discontinuity of displacement field near the electrodes, which we demonstrate here is the largest source of error. The basis of the surface charge simulation method is to establish a system of linear equations by introducing equivalent surface charges distributed in a vacuum. Polarization charges which appear on dielectric boundaries are solved for explicitly by imposing the continuity of flux condition, and the electric field in the region of interest can be evaluated directly as a superposition sum of the approximate surface charge distribution. The surface charge simulation program used here employs a Galerkin boundary element formulation [4] to simulate the electric fields in the structure shown in Fig. 1. We specified electrodes of finite thickness because it is extremely difficult to obtain numerically an accurate value of the charge density for an infinitesimally thin electrode, except for the special case when g2 = el' To see this, begin by considering the Gauss' law jump condition governing the electric field on either side of a material interface [5]: O'SCSM :
O'free -~- a b o u n d e d = gO
(4)
E1 -- go E2,
where a indicates a surface charge density and ascsM is the simulated charge resulting from the surface charge simulation method. The normal components of the electric field on either side of the boundary are represented by E, and eo is the permittivity of free space. The values of aSCSMare readily found by the numerical technique; however, one is typically only interested in the value of afree. The bound charge on a dielectric
P2),
(5)
which for linear materials is a simple function of the electric field abounded =
--
(eoZ1E, -
80~(2E2)
(6)
implying O'free -~- Co(1 + z t ) E , -- Co(1 +
~(2)E2
(7)
where (1 + Zl) and (1 + ~2) a r e the relative permittivities of the two media. Note that if Xl = X2 or either E1 or E2 is zero, then afree = (1 + Z) aSCSM.In these two cases, we do not have to measure El and E 2. However, if Xl # Z2, then we may have to calculate the values of both El and E2 arbitrarily close to the boundary, which can dramatically compromise the accuracy of the simulation. Clearly, any formulation that can sensibly determine the abrupt discontinuity of field directly on an infinitesimally thin boundary will avoid these kinds of errors. The test model used in this paper is shown in Fig. 2 and the number of strips we employed for the surface charge simulation method is shown in Table 1. The structure is simulated with axisymmetric strips using a Galerkin formulation of the BEM and
374
K.Y. Kung et al./Journal o[' Electrostatics 33 (1994,) 371
378
axis of rotation
I i
m ] ....................... n i~ 2
o
plane of ground symmetry Fig. 2. This graph shows the geometry used to model an unguarded disc capacitor. The letters I, rn, n, o represent the n u m b e r of strips on each boundary. In order to obtain strips of approximately equal length, we let n = (R - 1 ) / a n d o = hl/2a, where R, a, h are defined in Fig. 1.
Table 1 Data of the test model C,~om/ne, oe~a = a/h
R
I
m
n
o
0.100'* 0.100" 0.100 0.333 1.000 3.333
10 10 5 5 5 5
20 20 20 20 20 20
2 2 2 2 2
180 180 80 80 80 80
100 100 100 30 10 4
R is the ratio of the electrode radius to the dielectric sheet radius. The actual number of strips used in each simulation for each boundary is D times the number shown in the table, where D is the discretization ratio. Note that only models with discretization ratios of 1, 2, 4 are used for extrapolation; 8 is discarded due to the unexpected b u m p shown in Fig. 3. The quantities I, m, n, and o are described in Fig. 2.
the results obtained are the capacitance of different disc capacitor configurations with electrodes of finite thickness. These data are extrapolated to zero thickness electrodes using standard Richardson extrapolation [6], which assumes that the capacitance value obtained from the simulation is the actual capacitance value C plus an error related to the thickness of electrode t. The error is represented by a Taylor series expansion, and the extrapolation recursively eliminates higher-order terms based on simulation results for different values of t. We thus obtained the capacitance of the disc capacitor for an electrode of zero thickness, and then repeated the process assuming that the data has an error function related to the reciprocal of the discretization ratio and extrapolated the data to
K.Y. Kung et al./Journal of Electrostatics 33 (1994) 371-378
375
Table 2 Comparison of semi-analytical values against the surface charge simulation result
Cgeom/~oe2a = a/h
e2
Cex/~eoe2a,Re~ [1]
Extrapolated CSP
Percentage error
0.1**
0.5 1 2 5 lO 20
1.91320 1.25920 0.92982 0.73096 0.66440 0.63106
1.89242 1.25902 0.90453 0.70616 0.63990 0.60671
1.08 0.01 2.71 3.39 3.68 3.85
0.1*
0.5 1 2 5 10 20
1.91320 1.25920 0.92982 0.73096 0.66440 0.63106
1.92844 1.25940 0.92122 0.71698 0.64822 0.61361
0.79 0.01 0.92 1.91 2.43 2.76
0.1
0.5 l 2 5 10 20
1.91320 1.25920 0.92982 0.73096 0.66440 0.63106
1.94591 1.25940 0.90862 0.68851 0.60072 0.56871
1.70 0.01 2.27 5.80 9.58 9.87
0.333
0.5 1 2 5 10 20
1.94755 1.25581 0.90213 0.68582 0.61277 0.57603
1.97822 1.25607 0.89120 0.66936 0.59455 0.55688
1.57 0.02 1.21 2.39 2.97 3.32
1.000
0.5 1 2 5 10 20
2.09180 1.31830 0.91253 0.65896 0.57210 0.52813
2.10702 1.31873 0.91046 0.66727 0.58109 0.53719
0.72 0.03 0.23 1.26 1.57 1.71
3.333
0.5 1 2 5 10 20
2.43850 1.50443 1.00121 0.68043 0.56919 0.51258
2.38675 1.50541 1.02772 0.70882 0.59237 0.53155
2.12 0.06 264 4.17 4.07 3.70
The program is called CSP. Error increases as the dielectric permittivity. The accuracy improved dramatically when we doubled the radius of the dielectric disc for a/h = 0.1", which verified that an important source of error in our simulation is the truncation of the dielectric radius. 0.1 ** is obtained by directly using infinitesimally thin electrodes; these results are less satisfactory than the extrapolated values.
zero strip length I-7]. In our case, we used 4 different values of t and 3 different values of 1/D. The extrapolated values are then used to calculate the normalized excess capacitance values that are shown in Table 2.
376
K.Y. Kung et al./Journal of Electrostatics 33 (1994) 371 378
percentage error
10 9
E, = 20
8
£, = 10
7
£,
6
5
5 4 3
£ , = 0.5
£,=2
2 1
O.r 0
,
0.I
~.
0.2
0.3
,
0.4
?
0.5
.
0.6
.
.
0.7
.
£,=I
0.8
0.9
1
l/D, where D is the discretization ratio Fig. 3. Graph of percentage error of extrapolated excess capacitance Ce~/~eoe2a as a function of discretization ratio for the case a/h = 1. This graph demonstrates the influence of using a dielectric disc of finite radius. Notice that there is an unexpected rise in the extrapolated data when D is increased from 5 to 8. The increase in error with 82 is due to the difficulty of imposing continuity of flux through dielectric interfaces.
The semi-analytical results of [1] assume that the radius of the dielectric plate approaches infinity, which is not possible to model numerically. Approximating the dielectric with a finite radius (Ra in Fig. l) also introduces considerable error, particularly when a/h is small and the permittivity of the dielectric material is high. The sensitivity of the results to the radius of the dielectric is clearly seen by the second and third series of numerical experiments, and it appears that the truncated dielectric is a significant source of numerical error. Indeed, the accuracy for e2 - - 2 0 was c o m p r o m i s e d by m o r e than a factor of three. The close agreement between the extrapolated and semianalytical values for the case ~32 = 1 lends confidence to our extrapolation. Notice in Fig. 3 that the accuracy drops dramatically when e 2 :~ 1, and increases as e2 increases. Finally, the worsening accuracy for e2 = 1 as the mesh becomes very refined points to a further problem. We surmise that the matrix becomes steadily more ill-conditioned as its size increases. We presume that the effect is more acute for e2 # 81, thus giving rise to the b u m p s seen in Fig. 3, and we suggest that this apparently inconsistent behavior is spurious.
K.Y. Kung et al./Journal o f Electrostatics 33 (1994) 3 7 1 - 3 7 8
377
3. Results and conclusions Calculating the free surface charge on an infinitely thin conductor that separates regions of different permittivity is a numerically daunting task. Employing the Gauss' law jump condition is theoretically straightforward; however, approximating this condition numerically can introduce significant error, as demonstrated by comparing the values of a/h = 0.1"* and 0.1" in Table 2. The results in Tables 2 and 3 are displayed in terms of the excess capacitance, which is defined by C,o,,, = Csoom+ C,x.
(8)
The values are obtained by numerically determining Ctotal, substituting for Cgeomfrom Eq. (1), and normalizing by a factor of rt~oe2a. Notice that the abrupt discontinuity with respect to 82 found in an earlier numerical simulation based on the charge simulation method is not apparent here. Table 3 shows a comparison of the previously reported results of Wintle and Goad and three codes based on different methods. Two of the programs employ integral formulations, and one solves Laplace's equation directly using finite elements. The surface charge simulation code minimizes the residual function in a Galerkin sense over every material interface. Richardson extrapolation was employed to evaluate the capacitance as t --* 0, where the independent variable for extrapolating the strip length to zero is taken here as 1/D. The capacitance of infinitely thin elements calculated by this process is better than the result obtained by evaluating E at points arbitrarily close to the boundary, but not nearly as accurate as the two other methods that avoid this restriction altogether. For example, the region,oriented charge simulation i[8] implements Eq. (7) exactly on each side of the conductor boundaries, and can measure the discontinuity of flux
Table 3 Comparison of all three numerical techniques to the semianalytical result for a/h = 3.333 Normalized extra capacitance for a/h = 3.333 ~2
Ref. I-1]
CSP (% error)
EMS (% error)
ELFI (% error)
1 2 5 10 20
1.50443 1.00121 0.68043 0.56919 0.51258
1.50541 1.02772 0.70882 0.59237 0.53155
1.51041 1.00576 0.68418 0.57864 0.51615
1.5028 (0.11) 1.0014 (0.02) 0.6815 (0.16) 0.5705 (0.23) 0.5142 (0.32)
(0.06) (2.64) (2.84) (4.07) (3.70)
(0.39) (0.45) (0.55) (1.66) (0.69)
The best result is from ELFI; the program employed 500 region-oriented smulation charges. The EMS results is based on a second-order finite element scheme that provides very fast solutions and runs on 486-class personal computer.
378
K.Y. Kung et al./Journal of Electrostatics 33 (1994) 371-378
without resorting to extrapolation. Moreover, it explicitly forces the net flux through all dielectric interfaces to be zero, which dramatically improves accuracy when e2 >> 1. The boundary conditions are imposed in a collocation sense; however, this apparantly does not mitigate the quality of the result for integrated quantities like net charge. The finite element approach is based on a proprietary formulation that automatically discretizes the problem using an adaptive solver option. The solutions were created on a 486-class personal computer in less than 30 s; less than 3 s were required for the final iteration in all cases which included as many as 1500 second-order nodes.
Acknowledgements We wish to thank Dr. S. Sato for some helpful discussions. H.J.W. wishes to thank NSERC (Canada) for continuing financial support. The Computational Fields Laboratory is underwritten in part by Richard Bergener GmbH. The authors gratefully acknowledge the use of ElectroMagnetic Solutions' "2D Toolset", which was provided by Dr. Arthur Butler.
References [1] H.J. Wintle and D.G.W. Goad, Capacitance corrections for disc electrodes on sheet dielectrics, J. Phys. D: Appl. Phys., 22 (1989) 1620-1626. [2] A.H. Scott and H.L. Curtis, J. Res. NBS, (1939) 747-757. [3] E. Durand, Electrostatique, Vol. I, Masson, Paris, 1964, pp. 259-260. I-4] P.L. Levin, A.J. Hansen, D. Beatovic, H. Gan and J.H. Petrangelo, A unified boundary-element finite-element package, IEEE Trans. Electrical Insulation, 28 (1993) 161-t67. 1-5] H.A. Haus and J.R. Melcher, Electromagnetic Fields and Energy, Prentice-Hall, Englewood Cliffs, N J, 1989, p. 21 I. [6] J.H. Ferziger, Numerical Methods for Engineering Application, Wiley-lnterscience, New York, 1981, pp. 30-31. I-7] H.J. Wintle, The capacitance of the regular tetrahedron and equilateral triangle, J. Electrostatics, 26 (1991) 115-120. 1-81 A. Blaszczyk and H. Steinbigler, Region-oriented charge simulation, Conference Proceedings, COMPUMAG, Miami (1993) 154-155 (extended version under review by IEEE Trans Magnetics).