Journal of Electrostatics, 27 ( 1992 ) 211-222 Elsevier

211

Analysis of electrostatic field in the system: charged dielectric-discharging electrode

Leszek Ptasifiski and Lestaw Turkiewicz Faculty of Electrical Engineering, Automatics and Electronics, Academy of Mining and Metallurgy, aL Mickiewicza 30, 30-059 Krakdw, Poland

(Received January 26, 1990; accepted in revised form September 18, 1991 )

S u m m a r y

Investigations were performed for the configuration: dielectric object-grounded spherical elec- trode. An axissymetrical system was considered and electric field distribution along the axis of this system was found, in particular at the electrode surface. A model of the electrostatic field in the system under investigation was proposed in order to determine the field intensity. Analysis of this model together with calculation of electric field for given dielectric surface and space charge distribution and for a parametrically variable geometry of the system is the objective of the paper.

1. Introduction

Insight into electrostatic discharges, especially in the configuration: charged dielectric material-discharging electrode have been described by several au- thors. The investigations concerned in particular discharge parameters [ 1,2 ], effect of geometrical dimensions of the system: dielectric-electrode [3-5 ], ef- fect of the velocity at which the electrode is moving towards the dielectric and of its resistivity [6 ] and of the capability to ignite flammable gases and vapours mixed with air by these discharges [3-5 ]. Attempts have been made to relate the results of these investigations to the quantities that characterise the state of the dielectric charging, such as surface charge density [4 ] or the dielectric surface potential [5 ].

This paper presents a calculation method of electric field intensity in the system of the investigation, assuming the stationary distribution of source charge in a homogenous space. In contrast with the references mentioned above, an analytic method to determine the electrostatic field intensity along the sym- metry axis was elaborated. In particular, the explicit formulae were derived for two cases of electric charges distribution: (i) surface charges on opposite sides of a dielectric disk; (ii) space charge in the dielectric disk volume.

0304-3886/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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2. Model description

Accordingly to the assumptions mentioned above we analyse a system con- sisting of a charged rotational solid and a grounded spherical electrode situated in the axis of the solid as shown in Fig. 1 (a). The subject of analysis is the electrostatic field intensity E (x) along the symmetry axis of the system and particularly on the electrode surface (point M) assuming constant surface charge density qs or space charge density qv and the predetermined system geometry. The crucial question in the considerations is how the presence of a grounded conducting sphere influences the field intensity distribution along the axis. It is well known that in presence of a conducting object the field can be treated as a superposition of fields caused by source charges and charges induced at the conductor surface. The latter are usually replaced by pertinent simulating charges. The effect of simulating charges together with the source charges should ensure equipotentiality of the surface of the conducting envi- ronment, after its removing.

In the problem under consideration, the charge induced at the sphere surface exhibits axissymmetry, owing to which there is

/~(x) =EQ(x)+Ei(x)--f~(EQ(x) +Ei(x) ) (1)

for points lying on the system axis. Intensity of the field produced by charges of the solid (referred to as the

source field in the sequel) will be denoted by the symbol EQ(x)= f:,EQ(x), while the field intensity from charges induced on the sphere surface will be denoted as ~i(x)= f~Ei (x).

Fig. 1 (a). Analized system: dielectric-electrode.

Fig. I (b). Substituted system.

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The effect of the induced charge (along the system axis) can be simulated by a field originating from an equivalent point charge whose value Q~ and po- sition on the axis (at a distance h from the sphere centre determines the E ~ (x) component.

The equivalent system (a solid and a point charge) and thu trace of the sphere surface are shown in Fig. 1 (b).

In the interval of interest xe [0,d], field intensity along the axis can be ex- pressed as

E ( x ) _ E Q ( x ) + E i ( x ) _ E Q ( x ) _ Qi 4~ZEo ( d.i_ h _ x ) 2 (2)

while x >I d + 2r to the right of the sphere.

Qi E ( x ) f E Q ( x ) -} 4~ .o(d+h_x) 2 (3)

In fact, the vectors of both fields have the same directions for xe [0,d] and the opposite ones for xe [d + 2r, ao ] as the induced charge must be of opposite po- larity, to that of the solid. Thus, determination of the Ei(x) component for a known source field E Q (x) and for the assumed geometric parameters d,r can be reduced to calculation of the quantities: Qi and h. These quantities result from the following conditions concerning potential differences (voltages) which must be satisfied both in the original system and in the equivalent one. (i) All points on the electrode have the same potential hence, in particular,

potential difference between the points M and N must be equal to zero,

~gM --~N = UMN ~ 0 (4)

(ii) Grounding the electrode means, from the theoretical point of view, equal- ization of the sphere potential and the potential at infinity,

-~® =UN® •0 (5)

We shall write the conditions (4) and (5) in the form of definite integrals, and it must be taken into account that the potentials of points M and L, as shown in Fig. 1 (b), the equivalent point charge are equal:

which implies a corresponding modification of the lower limit of integration of the Ei(x) function.

Considering (6) the conditions (4) and (5) are d + 2r d-i- 2r

UMN UQMN'~" i f Z Q ff -- ~-~ ULNm~ (x)dx'~" Ei (x )dx 0 (7) d d + 2 h

214

oo

v oo = + = d + 2 r

GO

EQ(x)dx+ f d + 2 r

E (x)dx=O (8)

where in the equivalent system, function E i (x) is expressed by the relationship (3) in the interval of [ d + h, oo ).

Following the integration of the function E i (x), the equation system (7) and (8) is solved and after elementary operations general formulae describing the position h and Qi of the equivalent point charge are obtained:

h = 2r vl Ve+172 (9)

V] 172 (10) Qi = - 47teo h V~ = -87teorv, + V2

~ o Go

d d + 2 r

Interpretation of the V, and V2 is both interesting and obvious. These are simply potentials of the points M and N in the source field if we assume the conventional method of potential normalization, ~o~ •0.

In the sequel, they will be referred to as potential constants. It should be stressed that analytical determination of the Vt and V2 requires calculating the following limit

lim E~(x)dx

Moreover, it can be seen, that the positive direction of the source field (EQ(x) > 0) implies, as expected, a negative sign of the Qi charge (and con- versely), because V~ > 0 and V~ > 0.

Substitution of (9) and (10) into (2) yields the sought intensity distribution along the axis of the original system

E(x) =Eq(x) + 2rV~ 112 xE [0,d] (12)

(V,+V2) d4 V +V2 x

The maximum value of the intensity that occurs in the point M of the electrode (x = d) is particularly interesting

E(d) =EM =E~ + V~ ( V~ + V2)=E ~ + 2rV2 h

(13)

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Application of results of the described procedure will be presented in the fol- lowing section. Two simple and important cases will be considered: investiga- tion of a field produced by surface and spatial charges of a plane-parallel disk.

3. Resul ts of ea leula t ions

3.1. Surface distribution of charge Let us consider a disk with a radius R and thickness g (Fig. 2) whose bases

are surface charged with densities q~-qs and qsb = kq~ where k is an arbitrary constant. The formula describing field intensity along the x axis (whose origin, O, lies at the centre of the right hand side base with the q~ charge) can be easily derived by integrating the field intensity caused by the charged linear ring, and then adding the contributions of the q~ and kq~ charges,

,: k(x+g) ] EQ(x)----2~ ° l+k-x /R2+x 2 - ~/R2+ (x+g)2 j (14)

The evaluation procedure requires the function (14) be integrated and then the limit of the integral for x--, oo be found

lax EQ(x)d .x :=~o ( l+k)x-x/R2+x2-kx/RZ+(x+g) 2 = (x) (15)

The limit of the I (x) function (of the oo-ootype ) can be written in the form of a sum of two limits:

X - b O O X - ~ O O X ---~ OO

The first one is equal to zero, while the second one, after appropriate modifi- cations, yields the result: kg.

From (11 ) the potential constants can be calculated

' .N- . . . . . .

Fig. 2. Disk with surface charge.

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V 2 m qs [x/R2+ (d+2r) 2 +kx/R2+ (d+g+2r) 2 2Co L

- (l +k) (d+ 2r)-kg 1

(18)

Then, using (13), the field intensity EM-- EMS at the point M of the electrode is determined. To make the relationships more compact and possibly universal we shall express the geometric parameters, potential constants and the EMS intensity in the dimensionless forms: d/r=d; g/r=g; R/r= R_;

V1,2 - 2Co V,.2 q~ r

(19)

2~o eMS -- EMS (20)

qs

It can be easily verified (14) that the dimensionless quantity eMS, determines the ratio of intensity EMS and the source field intensity at the centre 0 of the disk charged at one side, putting x--0 and k--0.

Having introduced the modifications mentioned above, a formula describing eMS can be derived on the basis of (13), (14), (17) and (18). Furthermore we shall use the following notation for simplicity reasons:

P, -~=. 2+d2; P2-- R.~(d+g. )2 (21)

p,

obtaining the potential constants in the form

v, =p, -t-kp2 - (1 +k)d-kg B

v2-p3"bkp4- (l"bk) (d+2)-kg m

The final formula reads:

(22)

d k(d+g)]+vl(vl+v2) eMS ffi l + k - - - - " - (23)

P, P2 2v2

The first term (in square brackets) concerns the effect of the source charges and the second one the charge induced on the sphere.

3.2. Space distribution of charge As the field source we consider a charge with constant volume density qv

deposited in the disk volume as shown in Fig. 3. A formula describing the field

,, / 9 , ' I I ' t "

R I*, , 1,4" 0

I | • l i T

I 1 +

• Z

dz

Fig. 3. Disk with spatial charge.

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along the x axis can be derived by integrating the field intensity produced by the charge of disk with the infinitesimal thickness, dz,

g

E Q ( x ) - E ~ l x ) = ~ c ° 1 - x/R2 + (x+z) 2

qv

=2Co (x+g)

(24)

Although the integral of eqn. (24) is an elementary function [ 7 ]"

qv x R2 l n ( x + ~ / R ~ x 2 ~Ev(x)dx-~o[gX+ 2 x/R2+x2 + ~" _ + )

x+g . /R2+ (x+g) ~ ln (x+g+ x/R 2 - 2 ~ --2- +(x+g)2 (25)

ffi2 ol(x)

The evaluation of its limit is complicated and requires quite specific transformations.

Because of space limitations of the paper, only the final result is presented. It proved to be surprisingly simple

lim I(x) = - ½ g2 (26)

Dimensionless quantities were introduced, as earlier, defined in such a way, so that it would be possible to compare certain quantitative results concerning surface charges. Moreover, the auxiliary notation (21) was used. We define the modified potential constants vl, v2 as:

4~__oo V1,2 vl,2- qv r 2 (27)

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and we determine them according to ( 17 ) and (18) by using the dimensionless overall quantities d, g, R:

d+g+p2 (28) Vl=(d+g-)p2-dpl-2dg--g-2+R21n d+pl m

d+g+p4 + 2 v 2 = ( d + g _ + 2 ) p 4 - ( d + 2 ) (2g_+p3-g2+R 21n d + p 3 + 2

If the dimensionless field intensity at the point M is determined

(29)

eM ---- eMv -- 2Co EMv (30) rq~

then, according to (27), formula (13) becomes

eMv__eQMv. } VI(Vl +V2) (31) 4v2

Consideration of (29) and (21) yields an abbreviated, final formula, being an equivalent to formula (23)

eMv = (Pl --P2 +g) + Vl (Vl +V2) (32) - 4v2

4. Discussion of results

On the basis of formulae (23) and (32) a series of calculations was per- formed of the dimensionless field intensity eM as a function of geometrical parameters, expressed in relative units (d, g, R). Figure 4 shows the variation in field intensity as a function of the non-ain~ensional distance d for various

m

ratios k. Figure 5 presents exemplary characteristics for surface distribution (eMS)

and spatial distribution (eMv) when the global charge is equal in both cases. The results of calculations enable us to draw following conclusions.

In the case of surface charge eM as well as its derivative [ (deMs/d_d) [ de- crease with increasing d and depend strongly on value of k. The highest values occur for k= 1, i.e. when the equal charges are situated at both surfaces of the disk.

Increase in disk diameter (R) causes eM increase and this effect becomes more pronounced with increasing distance d.

From the results presented it follows that the form of the EM=f(d) char- acteristic can be effectively influenced by suitable choice of the geometrical dimensions of the system: dielectric (disk)-discharging electrode (R, r, g).

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il8 & k.O ! 4 k=4s" g k-4,O m ,,.4s o k= .1,0 i

4B

to!

0 2 8 4~ _d

Fig. 4. Electrostatic field intensiW (dimensionless) eMs as a function of distance (dimensionless) d forg=O,2.

e~S 10 ~IV' A e,~ i B=2° ;k,O & eHv ; R. ,2O 0 e~s; 8--~o/k-o m eMvi .R ,~o

4 o

O l ~ , ~ ~ • ~ , ~ . . . . . . . - .

d

Fig. 5. Electrostatic field intensity (dimensionless) eMs and eMv as a function of distance (dimen- sionless) d for R = var and g = 1.

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The absolute values of electric field intensity were compared on the basis of formula

EMS g eMS

EMv -- 1 + k eMv - - , k ¢ - 1 (33)

and p

E~s ~,~ E " MS e ~IS

(34)

where E~s , e~s are values of EMS and eMS for k=0, and E~s , e~s for k= - 1. The formula (33) is valid for the same global values of charge.

Results of the calculations are illustrated by data given in Tables 1 and 2. It follows from Table 1 that the values EMS and EMv are nearby equal for

unipolar charges at both surfaces of a dielectric plate, while they can differ in the case of bipolar charges.

TABLE 1

Comparison of values of electric field intensity for surface and spatial charge distribution and the same global values of charge. Maximum values EMs/EMv (for d=0)

No. R g k EMs/EMv . B - -

1 2O 1,0 0 1,03 2 20 1,0 0,5 1,01 3 20 1,0 1,0 1,00 4 20 1,0 -0,5 1,08 ,5 20 1,0 -0,9 1,48

6 10 1,0 0 1,05 7 10 1,0 0,5 1,02 8 10 1,0 1,0 1,00 9 10 1,0 -0,5 1,15

10 10 1,0 -0,9 1,97

11 20 0,5 0 1,01 12 20 0,5 0,5 1,00 13 20 0,5 1,0 1,00 14 20 0,5 -0,5 1,04 15 20 0,5 -0,9 1,24

16 20 0,1 0 1,00 17 20 0,1 0,5 1,00 18 20 0,1 1,0 1,00 19 20 0,1 -0,5 1,01 20 20 0,1 -0,9 1,05

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TABLE 2

Comparison of values of electric field intensity for surface charge distribution for k = 0 (E hs ) and k= - 1 (E~s) . Minimum values Ehs/E~s (for d = 0 )

No. R g E'Ms/E'~s

1 20 1 20 2 20 0,5 40 3 20 0,2 100 4 20 0,1 200 5 20 0,01 1990 6 10 1 10 7 10 0,5 20 8 10 0,2 50 9 10 0,1 100

10 10 0,01 990

Independently of the polarity of surface charges, the distinction of the EMS

and EMv values decreases with increasing R values and decreasing g values. In the case of suitably chosen dimensions of tl~e dielectric disk (R~) and diameter of the discharging electrode (r) it is possible to obtain EMS ~ EMv. This is rel- atively simple in the case of an unipolar surface charge (k >i 0) and becomes more difficult for bipolar surface charges (k < 0 ). Equality of the EMs and EMv quantities means that, from the viewpoint of electric field intensity, in the system: charged dielectric-discharging electrode, the volume charge of the di- electric can be replaced by surface charges with the same global value. In par- ticular, these charges can be distributed at only one surface of the dielectric plate (k ffi 0).

From data given in Table 2 it follows that the electric field intensity, in the case of unilateral surface charge distribution, Ehs (kffi0), significantly ex- ceeds the value of field intensity in the case of bilateral surface charge distri- bution (equal in respect to absolute values and opposite sign E~s (kffi - 1 ) ). Relatively close values can be obtained only in the case of large thicknesses (g) and small radius (R) of the dielectric disk.

The presented formulae and analysis are the preliminary results of investi- gation aiming at determining the conditions of initiating and spreading out of electrostatic discharges. In particular it has been possible tc estimate the effect of system geometry on maximum electric field intensity in chosen cases of dielectric charge distribution. Results concerning the problem of discharges will be addressed in a separate study.

Acknowledgement

This work was sponsored by the Institute of Electrical Engineering in War- saw under the Department Programme of Fundamental Research 02.7.

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References

1 S. Bach, A. Kasprzyk and L. Ptasi~ski, Investigations of spark discharges from surfaces of charged dielectrics, J. Electrostatics, I0 (1981) 153-160.

2 E.U. Landers, Distribution of charge and field strength due to discharge from insulating surfaces, J. Electrostatics, 17 (1985) 59-60.

3 K.G. L~vstrand, The ignition power of brush discharges - on the critical charge density, J. Electrostatics, 10 (1981) 161-168.

4 K. Gibson and F.C. Lloyd, Incendivity of discharges from electrostatically charged plastics, Brit. J. Appl. Phys., 16 (1965) 1619-1631.

5 Y. Tabata and S. Masuda, Minimum potential of charged insulator to cause incendiary dis- charges, IEEE Trans., IA-20 (1984) 1206-1211.

6 I. Berta and N. Gastanek, The energy of electrostatic discharges, Inst. Phys. Conf. Ser. No. 48 (1979) 67-72.

7 H.B. Dwight, Tables of Integrals and Other Mathematical Data, Mac-Millan, New York, 1961.