—

SENSOR AND’SIMULATION NOTE

NOTE 220

February 1976

Isolated Capacitance and Equivalent Radiusof the

Rectangular Parallelopiped

by

T. H. ShumpertAuburn University

Auburn, Alabama 36830.

and

D. J. GallowayDynetics, Inc.

Huntsville, Alabama

ABSTRACT

In order to predict the increase in the capacitance of a space

system when inserted in a system-generated electromagnetic pulse

(SGEMP) simulator, it is useful to have information concerning the isolated

capacitance of the space system. This note attempts to provide pertinent

data which may be used in obtaining preliminary values for the isolated

capacitance of the often irregularly shaped space system. The isolated

capacitances and equivalent radii of perfectly conducting parallele-

pipeds is calculated using the method of moments, and this data is

presented and compared graphically with the capacitances of certain

bounding geometries to indicate their validity. “It was related by

Kirchhoff, that shortly before his death Dirichlet solved the problem

of the distribution of electricity on a rectangular parallelopiped.

If so, the solution has been lost.’’[l]

In discussing the simu’

INTRODUCTION

ation of system generated EMP in exoatmospherc

conditions, one important aspect is the electrical effect of the vacuum

chamber utilized as a necessary part of the simulator. The capacitance

to the chamber walls, cavity resonances, and reflection of higher fre-

quencies from the cavity wall are some of the electromagnetic interactions

of the space system with the test chamber which must be considered [z].

This note addresses itself to the problem of the system capacitance.

In free space far from any other objects the space system has a certain

capacitance to infinity. The simulator chamber inherently increases

this capacitance thereby changing the low frequency characteristics

of the space system. The first step towards obtaining a value for this

capacitance between the space system and the test chamber is to have a

knowledge of the capacitance of the space system to infinity. In this

onote, the isolated free space capacitance of perfectly conducting rectangular

parallelepipeds is determined and appropriate equivalent radii are presented

in graphical form. It is believed that this data would provide useful

estimates of the isolated capacitance of the actual space systems to be

tested.

Rectangular Plate: Formulation and Numerical Results

A charged rectangular conducting plate is shown in Figure 1. A

cartesian coordinate system is shown centered on the plate, which has

dimensions of 2a x 2b. Let ~(x’,y’) represent the surface charge density

on the plate, assumed to have zero thickness. Following the same

general procedures as did tlarringtori[3]

square, the isolated capacitance of this

2

for the charged conducting

rectangular plate is

z

4-----2” -/

Figure 1. Charged Rectangular Conducting Plate.

3

where

r .—. —r= (X-X’)* + (y-y’! . (2)

Through the method cf nments, divide the rectangular plate into N squire

subsections, Asn; represent the charge density by

P(x’,y’)= ~ CtJn ,11=1

where the an’s are the unk~own coefficients arid

(1 on ASfn = n

O elsewhere )

(3)

(4)

and satisfy equation (1) at the midpoints of the subsections, (xm, yin).

These operations result in

V= fitmnanIn =1,2, .... Nn=l

Q=!! anAsn~n=l

with

!/J{

1=mn Axn Ayn 4ncorm dx’ dy’

4

(5)

(6)

(7)

(8)

kmn is the potential at the center of Asm due to a uniform charge density

of unit amplitude over Asn. Thus,

(9)

In order to evaluate J!llln,treat the non-diacjoml terms as those which

would arise from point charges and analytically evaluate the self terms

[2], such that

i’ J. 1.7628 abnn-7reo --K

2~ ab

(-)

1mn NITEO rmn ‘

where

(10)

(11)

(12)

Reitan and Higgins [4J applied numerical techniques to the charged

plate many years acjo. Figure 2 illustrates the excellent agreement

between this moment method formulation and their crude “sub-areas” approach.

Rectangular Parallelopiped: Formulation

Consider the rectangular parallelopiped, or box, composed of six

individual plates as shown in Figure 3. Let P(s’) be the surface charge

density on the box, such that the electrostatic potential at any point

in space is

5

. .

----.

m

square plate_

Moment Method● ● ● Reitan 8 Higgins

El T3

1--’-’-1

r infinitsiy thin necdlo

:t

o 0.2 0.4 0.6 0.8 1.02a/2b

Figure 2. Normalized Capacitance of Rectangular Conducting Plate.

2b

/ x

Figure 3. Charged Rectangular Conducting Parallelopiped.

7

fl(x,Y,z)=J

ok’) ~~, ,4ncOR

s’(13)

where

R=

After placing

condition can

v =

+

+

where superscripts refer to one of the sixparticular plates comprising

the box and

[1~/

R(l) = (X-M2+(y-y’)*+ (Z-z’f ‘ 2

[ 1~

R(z) = (X-x’)z+ (y-y’)z+ (z-2b)2

[1~

R(3) = (x-x’)*+ (y-2a)2+ (Z-Z’)2

[ 1~

~(4) = X2 + (y-y’ )*-+ ;,-,’)*

[ 1

~~(5) = (X-X’)*+ (y-y’)z +22

[ 1&

R(6) = (X+’)* +yz + (2-,’)2

8

(15)

C=Q ~!l~~”(j)(’’(j))~’’(j)—=v

—...——v .

Define basis functions JS

+j) =n

(16)

1 on all Asn(j)

o(17)

d!sewhere

where n represents the subsection and j the particular plate. The charge

density is expanded as

~(j) ()~t(j) =~(j)~ Jj)f (j) j

=1, 2, 3, ... 6 (18)n=l n n

(“)where N J is the number of square zones on rectangular plate j,Substitute

this into equation (?4) and satisfy the equation at the midpoints of each

zone, The matrix representation can be put into the form of

T=[1

J/l) ; or ~

“ ! J!w) Jd pn = 1,2,...,6

q=l ‘n(19)

where

9

\ [:

+1)

+2)

.●.

+6)

=an

-.

(20)

(1)an

(2)an

●

..

~n(6)

.

● .

. . .

,

J16ymn

J66).ml

—

(21)

(22)

(Note that ~m~q} is the potential at the center of Asm on plate p due

to a uniform charge density of unit amplitude over ASn on plate q. As

before, subsections were chosen to be squares, Barrington’s approximations

(in the evaluation of fim~q)were used, and full advantage was taken of thc

resulting symnetry of the equations. It was found that the isolated

capacitance of the rectangular parallelopiped could be expressed simply

as

[1C = 8W2 f ~ ~b?] ‘1n=l rri=l ‘n mn

(23)

10

70

fJ

50

/“

/’/

//

—— .

—————————

/---

I

I

Moment Method

—— Reiton S Higgins

450 50 100

Number of 5 ones on Cube

Figure 4. Capacitance of a Conducting Cube

With

(24)

[1 L&21)+R(24) ~(22)+L(25’j

.1

~(23)+g(26)Q(Pq) = mn mn mn mn mn mnmn

J31 )+A(34) ~(32)+%(35) ~(33)+$36)mn mn mn mn mn mn

and 2W is the length of a subsection side. With the total number of

subsections on the box equal to N, this matrix is N x N, as opposed to

one ii which synrnetrywas not utilized, which would be 2N x 2N.

Rectangular Parallelopiped - Numerical Results

The only numerical results available for comparison are those of

Reitan and Higgins [1] and then only for the cube. Figure 4 illustrates

the convergence of data for this special case of the cube as the number

of subareas increases. Both solutions approach a value of approximately

73pf, but the moment method formulation gives much better results for a

small number of subsections.

In a previous note [51, Shumpert defined an equivalent radius aS

‘c%q’=~ “ -

(25)

Also, the note advanced the supposition that the isolated capacitances OF

the inscribed and circumscribed ellipsoids could be considered as

approximate bounds on the capacitance of more complex shaped bodies.

12

Figure’5 presents these bounds, the mcment method results, as \!cllas a

surface area approximation established by Polya and Szego [6] , !;hichis

c ()s+= 4TE0G- 9 (26)

where S is the total surface area of the solid. The surface area approxi-

mation gives reasonable results for a box whose di~-.i~sionsare comparable’

to a cube, but as the parallelopiped approaches di;.:nsionscomparable to

a wire, this approximation breaks down. The inscribed ellipsoid yields

a better approximation to the moment method solution than an average of

the two ellipsoidal curves would give.

Figures 6 and 7 depict the normalized equivalent radius of the

rectangular parallelopiped versus its dimensions. NQte that the degenerate

cases--infinitely thin needle, rectangular plate, and the cube--appear

as boundary lines or points on these general plots.

13

4 ●1

2,!

2.

1.

1.

0.

0,

— .-——— ——

●

✎

✎

✎

✎

.—

---—.

Inscribed Ellipsoid (Prolate Speroid)

C[rcurnscribed Ellipsoid (Prolate Speroid)

Surfaco Area Approximation -Polyo a Szego

5 10

x

t5 20

Figure 5. Equivalent Radius of a Rectangular Conducting Parallel opipedwith Unit Cross-Sectional Area and Arbitrary Length.

“ w

I .0

0.8

req. 0.6z

0.2

0

cube

\ .

/

/

infinitely thin needle

1 I I I

o 0.2 0.4 0.6 0.8 I .02b/2c

r<m,,we G Nmwmal-iTO~FnllivaientRadius of Rectangular Conducting[ Iyulc u. I!vl a!..-. ---- -T–

Parallelopiped of Arbitrary Dimensions.

I .0

0.8

006

0.4

0.2

0

plate (20= O)

‘/i- nfinitely thin needle

o 0.2 0.4 0.6 0.8 1.0

2a12c

Figure 7. Normalized Equivalent Radius of Rectangular Conductin9Parallelopiped of Arbitrary Dimensions.

20

2cA2b&2a

o

Conclusions

The isolated capacitance of complex objects is a problem of interest.

Approximations of and bounds for these isolated capacitances can be

approached from many fronts. Inscribed and circumscribed ellipsoids

yield bounds for many complex shaped objects. Perhaps, the rectangular

parallelopiped could be used as a “building block” in order to arrive at

estimations for the isolated capacitance of more complex bodies.

17

REFERENCES

[1] 11~:e~::: and J. Higgins, “Calculation of the Electrical Capacitance,“ Jour. Appl. Phys., Vol. 22, No. 2, pp. 223-226, February,

1951.

[2] C. E. Baum, “A Technique for Simulating the System Generated Electro-magnetic Pulse Resulting from an Exoatmospheric Nuclear WeaponRadiation Environment,” Sensor and Simulation Note 156, 18 September1972.

[3] R. F. Barrington, Field Computation’s by Moment Methods, MacMillanco., New York, N.Y., 1968.

[4] D. Reitan and J. Higgins, “Accurate Determination of the Capacitanceof a Thin Conducting Rectangular Plate,” AIEE Trans., Vol. 75,Part I, pp. 761-766, J;ntiarv. ;’?5?.

[51 T. H. Shwnpert, “Capacitance Calculations for Satellites”, Part I.Isolated Capacitances of Ellipsoidal Shapes with Comparisons toSome Other Simple Bodies, Sensor and Simulation Note 157, September 19?7

o[6] G. Polya and G. Szego, “Isoperimetric Inequalities in Mathematical

Physics,” Annals of Mathematical Studies, Number 27, Princeton )University Press, Princeton, N.J., 1966.