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Unformatted text preview: H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 2) 3) 4) 5) 6) 7) hydrocarbon/air mixtures [1], the chargetransfer chargecollection efﬁciency associated with a typical test setup (disc
with same area as a 150 mm square sheet) is 0.81, 0.75 or 0.69
with 10 mm, 12.5 mm and 15 mm radius electrodes respectively. Thus the typical degree of underestimation due to
inducedcharge errors is modest but not negligible.
The lowest chargecollection efﬁciency occurs with the largest
sphere and the smallest possible disc. The largest sphere we
considered was 15 mm radius and for this, with our best estimate of the postdischarge charge distribution on the disc
surface, the minimum chargecollection efﬁciency we found
for a 60 nC chargetransfer was 0.56. With worstcase
assumptions about the breakdown voltage and the distribution
of charge on the disc surface after the discharge, this reduced to
an overall worstcase chargecollection efﬁciency of 0.40.
However the combination of conditions associated with this
low level of efﬁciency is thought unlikely to occur in practice.
It is possible to allow for a reduced chargetransfer chargecollection efﬁciency by applying a correction factor to the
measured chargetransfers. The results in Conclusion 2) indicate that, for results relating to a 60 nC chargetransfer
threshold and with the expected postdischarge charge distribution, a correction factor of Â2 would ensure that hazardous
equipment was not wrongly classiﬁed as safe. A correction
factor of Â2.5 would deal with our worstcase charge distribution. These corrections would, of course, only be needed for
fast, unshielded probe measurements and would not be
required for measurements made with slow and/or shielded
probes as these have a high chargetransfer chargecollection
efﬁciency.
Lower chargetransfer thresholds have been suggested [1] for
more sensitive materials (30 nC for Gas Group IIb and 10 nC for
Gas Group IIc). There is an inverse relationship between
chargetransfer and chargecollection efﬁciency so efﬁciencies
are lower for these smaller chargetransfers. In addition, for
smaller chargetransfers, the efﬁciencies vary more steeply
with the geometric parameters. Because of the low chargecollection efﬁciency and increased sensitivity to parameter
changes it seems best to avoid the use of fast, unshielded
probes for measuring chargetransfers around the 30 nC and 10
nC thresholds.
Our results suggest an overall lower limit to the chargetransfer
chargecollection efﬁciency of around 0.2 although this
conclusion depends on the relationship we have used between
discharge patch size and discsphere separation, which is
unveriﬁed. If this relationship, or an equivalent, is ultimately
conﬁrmed, it may be acceptable to interpret data for more
sensitive materials with the lowerlimit efﬁciency and hence
get results that will err on the safe side. With such low levels of
chargecollection efﬁciency, the circuit chargetransfers corresponding to the 10 nC and 30 nC thresholds would only be 2
nC and 6 nC respectively and it is debatable whether there
would be any beneﬁt in measuring and interpreting such small
circuit chargetransfers rather than simply requiring that no
discharges should be detected by a measurement circuit of
adequate sensitivity.
Although our results were obtained with charged discs, they
are expected to apply approximately to ﬂat surfaces of other
shapes and equivalent area.
The calculations in this paper cover discharges from isolated
charged discs to earthed spheres. We expect measurement
efﬁciencies to be higher for the following cases:
a. Discharges from discs located near earthed planes (because
the presence of the earth plane will reduce the charge
coupling between disc and sphere). 17 b. Discharges from convex charged objects (because the
convex shape will reduce coupling to the sphere).
c. Discharges with higher chargetransfers.
If chargetransfer results for cases like these are interpreted
using the measurement efﬁciencies estimated in this paper, the
resulting hazard assessments should err on the safe side. Thus
hazardous items should not be wrongly classiﬁed as safe although
some (marginally) safe items may be wrongly classiﬁed as
hazardous.
8) Charges on a concave plastic surface that, to some extent,
wraps around a test sphere would have more coupling to the
sphere than the charges on a plane disc and hence we would
expect a reduced chargecollection efﬁciency. We therefore do
not recommend the use of fast, unshielded chargetransfer
probes for measuring discharges from concave surfaces
without further investigation.
Acknowledgements
The author would like to thank J. Chubb, J. Smallwood, U. von
Pidoll and P. Holdstock for valuable discussions during the preparation of this work.
Appendix A. Analytical equations for analysing breakdown
and chargetransfer of an earthedsphere/chargedinsulatingdisc gap
A.1. Basic equations
Equations for the voltages and onaxis axial electric ﬁeld
components in the discsphere system have been given by, for
example, Heidelberg [14]. In order to highlight key parameters we
present them in dimensionless form using the characteristic
quantities L*, E*, F*, s* and Q* which are deﬁned in the Glossary. We
found it useful to link the characteristic electric ﬁeld to the breakdown criterion in order to explicitly include the role of breakdown
phenomena.
With the above characteristic quantities, the dimensionless
forms of Heidelberg’s equations for the voltage and axial electric
ﬁeld on axis can be written: !
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
À
À
Á
Á
F0 ðz0 Þ
¼
D0 À z0 2 þR02 À D0 À z0
s0
À
E0 ðz0 Þ s0 1
z02 !
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
À
Á2 À
Á2 À
Á
D0 z0 À 1 þ R0 z0 À D0 z0 À 1 (A.1) À0
Á
D À z0
¼ À1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
À
Á2
D0 À z0 þR02
9
8hÀ
i
>
> D0 z0 À 1ÁÀD0 z0 À 2Á þ R02 z02
<
Á=
À
1
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
À D0 z0 À 2
À 03
Á2
À
>
>
z:
;
D0 z0 À 1 þR02 z02 (A.2) and the charge induced on the sphere by the charge on the disc is
0
Qs s0 ¼ À2 i
hpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
D02 þ R02 À D0 (A.3) where the primes indicate dimensionless quantities and the
symbols are as deﬁned in the Glossary. 18 H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 charge density of Às ad ¼ Às0 s*ad. The discharge chargetransfer,
0
DQd , is given by: Useful special cases of equations (A.1) and (A.2) are:
a) The voltage on axis at the disc surface (z0 ¼ D0 ): ÀÁ
F0 D0 s0 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2 À
!
Á2
1
D02 À 1 þ R0 D0 À D02 À 1
¼ R0 À 02
D 0
0
02
DQd ¼ Àad rd2 s0 ¼ reff s0 (A.4) a) The axial electric ﬁeld component on axis (in the air) at the disc
surface (z0 ¼ D0 ): (A.8a)
0
DQc The circuit chargetransfer is
¼
þ
where DQi0 is
the change inqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
induced charge on the sphere. From equation (A.3),
0
DQi0 ¼ 2s0 ad ½ D02 þ rd2 À D0 and hence: & 0
DQc ¼ s0 ad 2 The 0
DQd DQi0 !
'
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
02
02
D02 þ rd À D0 À rd chargecollection efﬁciency (A.9)
can be deﬁned as 0
0
3m ¼ DQc =DQd and hence is given by: E0 À 0Á
D s0 8h
9
i
>
02
02
02 02
=
>
1 < D À1 D À2 þR D
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
¼ À1 À 03
À D02 À 2
2
>
D>
:
;
D02 À 1 þR02 D02 (A.5) 3m ¼ 1 À 2ad h
02
reff qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
i
02
D02 þ reff =ad À D0 If the discharge patch radius is roughly dependent on the
0
separation of disc and sphere centres we can set reff ¼ KD0 and
thus write:
0
DQd ¼ ÀK 2 D02 s0 a) The axial electric ﬁeld component at the sphere surface (z0 ¼ 1): 3m ¼ 1 À
E 0 ð1Þ s0 8hÀ
9
i
> D0 À 1ÁÀD0 À 3Á þ R02
>
<
Á=
À0
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
¼À
À D À3
À
Á2
>
>
:
;
D0 À 1 þR02 (A.6) (A.10a) 2ad
K 2 D0 !
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ K 2 =ad À 1 (A.8b)
(A.10b) The various forms of equations (A.8) and (A.10) give the results
we require for the chargetransfer and chargecollection efﬁciency
in terms of the disc charge density, discharge geometry and the
characteristics of the discharge patch (mean radius and charge loss
fraction). A.2. Breakdown conditions Appendix B. Critical breakdown ﬁelds and voltages Because we have chosen the characteristic electric ﬁeld to be the
critical breakdown ﬁeld at the sphere surface, the breakdown
criterion is E0 (1) ¼ 1. From this, equation (A.6) leads directly to the
dimensionless disc charge density required for breakdown at the
distance, D0 , (we ignore polarity effects): The most accurate breakdown criterion (see, for example, the
review in [16]) is generally thought to be that of Pedersen [15],
which states that breakdown occurs when: 8hÀ
9À1
i
> D0 À 1ÁðD0 À 3Þ þ R02
>
<
=
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
s0 ¼
À ðD0 À 3Þ
À
Á2
>
>
:
;
D0 À 1 þR02 Zg ½MaxfðEðzÞ À pBÞ; 0g2 dz ¼ pC 2 (B.1) 0 (A.7) By deﬁnition, the characteristic charge density, s*, is determined
by the critical breakdown ﬁeld, Eb, at the sphere surface. Thus the
real disc charge density, s ¼ s0 s*, required to cause breakdown with
a given discsphere separation (or gap) can be determined by
combining equation (A.7) with a calculation of Eb (see Appendix B).
Alternatively, if the charge density on the disc is given, equation
(A.7) can be solved for D0 to obtain the equivalent breakdown gap
(g0 ¼ D0 À 1).
A.3. Discharge chargetransfers and chargetransfer collection
efﬁciency where E(z) is the axial electric ﬁeld in the gap, g is the gap length, z
is the distance from the surface of the highly stressed electrode
(which is the earthed sphere in the cases of interest to us), p is the
pressure and B and C are constants. The electric ﬁeld can be written
as E(z) ¼ E0 (z)E0 where, in the case of a gap between a charged disc
and an earthed sphere, E0 is the electric ﬁeld at the surface of the
sphere. The dimensionless ﬁeld, E0 (z), depends only on the geometry and it is known, equation (B.1) can be solved to ﬁnd Eb, which is
the value of E0 that produces breakdown. Solving equation (B.1)
with lengths in mm, p in bar, B ¼ 2.42 kV/mm bar and C ¼ 2.08 kV/
(mm bar)0.5 gives Eb in kV/mm. The breakdown voltage, Fb, is then: Fb ¼ Eb Zg E0 ðzÞdz (B.2) 0 We assume that, when a discharge occurs, it affects a patch of
radius rd on the disc and that within this patch the charge density
changes by a ﬁxed multiple, Àad of the initial charge density, s. We
discuss how to obtain the equivalent top hat radius and mean
charge loss factor in Appendix C.
With the above simpliﬁed representation of postdischarge
conditions, the postdischarge voltages, electric ﬁelds and charges
are found by adding the contributions from two coaligned charged
discs, the original disc and a second of radius rd carrying a uniform This gives voltages in kV when used with the constants above
and lengths in mm.
The Pedersen method is relatively complex because it requires
the solution of equation (B.1). Heidelberg [14] quotes a suggestion
by Bouwers [18], that the critical ﬁeld can be obtained from the
solution of: ða=lÞðEb l=2vÞexpð À v=Eb lÞð1 À Eb l=2vÞ ¼ 1 (B.3) H.L. Walmsley / Journal of Electrostatics 68 (2010) 5–20 where l is the mean free path of an electron (taken as 3.7 Â 10À7 m)
and v is the ionization potential of O2 (taken as 12.5 V).
Landers [17] uses an expression by Steinbigler [19], which may
be written as (B.4) where E00 ¼ 2.28 Â 106 V/m and acrit ¼ 10À2 m.
In the present work we solved equation (B.1) for Eb in the discsphere geometry using equations (A.2) and (A.6) to deﬁne E0 (z) and
compared the results to the values obtained from the simpler
expressions in equations (B.3) and (B.4) (see Figs. 2 and 3 of the
main text). When testing the incendivity of discharges from plastics
without earthed backings, we are usually concerned with electrode
radii between 10 mm and 15 mm, and gaps of a few tens of mm. The
discussion in the main text shows that equation (B.4) is accurate
enough for our purpose under these conditions and its simplicity
matches that of the analytical solutions. We have therefore adopted
this breakdown criterion for our calculations. It leads to a characteristic charge density of: s* ¼ 23a 30 Eb ¼ 23a 30 E00 1 þ ðacrit =aÞ1=3
01=3
¼ 23a 30 E00 1 þ acrit
(B.5) where a0
crit ¼ acrit =a. For sphere radii between 10 mm and 15 mm
0 1 =3
we have 2:0 < ð1 þ acrit Þ < 2:145. Hence, to a fair approximation,
we can simplify equation (B.5) to Eb w 2.07E00 ¼ 4.72 MV/m, which
leads to an approximate ﬁxed characteristic charge density of
s* ¼ 43a30E00 ¼ 84 mC/m2.
Once s* is established it leads, via equation (A.7), to the disc
charge density required for breakdown.
The integral in equation (B.2) is already available as equation
(A.4), which can be written: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
&
2 À
!'
Á2
1
0
0
Fb ¼ Eb as R À 02
D02 À 1 þ R0 D0 À D02 À 1
D
(B.6)
This expression was used to generate Fig. 4 of the main text. Appendix C. Representing the discharged patch as a charge
disturbance with a ‘‘tophat’’ distribution
The distribution of removed charge in a symmetrical discharge
patch can be expressed as: a(r) ¼ (s(r) À s)/s , where r is the distance
from the centre of the patch, s(r) is the charge density in the patch
after discharge and s is the initial (uniform) charge density. Our
simple model deals only with uniform charge densities so we need to
approximate this distribution of charge by an equivalent tophat
charge distribution. To do this we use the tophat distribution that has
the same mean radius and total charge deﬁcit as the real distribution.
The mean radius is r¼ ZN ar2 dr= 0 ZN ardr (C.1) 0 For a uniform tophat distribution of radius r1, equation (C.1)
gives a mean radius of (2/3)r1. Thus the radius, r1, of the tophat
distribution with the same mean radius as a is 3
r1 ¼
2 ZN
0 ar2 dr= ZN
0 ardr and the appropriate uniform a value within the top hat (i.e. for r <
r1) is: a¼2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Eb ¼ E00 1 þ 3 acrit =a (C.2) 19 ZN 2
ardr=r1 0 13 0 N
12
0N
Z
Z
8@
2A
a r dr A = @ a r dr
¼
9
0 (C.3) 0 The actual distribution may be represented by a power law such as aðrÞ ¼ a0 1 À ðr=r2 Þn ; r
À Á r2 : aðr Þ ¼ 0; r > r2 (C.4) Substituting this in equations (C.2) and (C.3) gives: r1 ¼ r2 nþ2
nþ3 (C.5) and ðn þ 3Þ2 a ¼ a0 n (C.6) ðn þ 2Þ3 The power law distribution approaches a top hat as n tends to
inﬁnity and, consistent with this, in this limit (C.5) reduces to r1 ¼ r2
and (C.6) to a ¼ a0 .
Alternatively, the charge distribution with an earthed backing
has been ﬁtted [23] to an expression of the form: aðrÞ ¼ a0 1 À ðr=r2 Þ2 2 ;r r2 : aðr Þ ¼ 0; r > r2 (C.7) With this function equations (C.2) and (C.3) evaluate to: r1 ¼ 24
r ¼ 0:686r2
35 2 (C.8) and a¼ 1225
a ¼ 0:709a0
1728 0 (C.9) Expressions (C.5–C.9) illustrate how to convert more detailed
representations of the charge distribution in a discharged patch
to approximate tophat distributions for use in our simple
model. References
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incendiary discharges, IEEE Trans. IA20 (1984) 1206–1211. Glossary8
a: Sphere radius, m
acrit: Characteristic radius used in breakdown criterion (¼10À2 m)
g: Discharge gap (g ¼ D À a or g0 ¼ D0 À 1), m
gq: The quenching distance, m
p: Pressure (used in breakdown criterion), bar
rd: Radius of patch affected by discharge, m
2
2
reff: Effective patch radius ðreff ¼ ad rd Þ, m
z: Distance along the axis of symmetry from the sphere centre, m
B: Constant in Pedersen breakdown criterion, (¼2.42 kV/mm bar)
C: Constant in Pedersen breakdown criterion (¼2.08 kV/(mm bar)0.5)
D: Separation of disc and sphere centres, m
E: Electric ﬁeld component along the axis of symmetry, V/m
E0: Actual electric ﬁeld at the surface of the sphere, V/m
E00: Characteristic electric ﬁeld in Steinbigler breakdown expression (2.2106 V/m)
Eb: Breakdown ﬁeld at the surface of the sphere, V/m
Ebs: Breakdown ﬁeld at the surface of a plastic sheet, V/m
E*: Characteristic ﬁeld (taken as Eb), V/m
K: The ratio of reff to D
L*: Characteristic length (taken as sphere radius, a), m
Qi: Induced charge on the sphere, C
Q*: Characteristic charge (pa2s*), C
R: Disc radius, m
ad: Fraction of initial charge removed from patch affected by discharge
30: Permittivity of free space, F/m
3a: Relative permittivity of air (w1)
3m: the chargetransfer collection efﬁciency
l: Mean free path of an electron in air, m
n: The ionization potential of O2, V
s: Charge density on disc (uniform), C/m2
s*: Characteristic charge density (23a30E*), C/m2
smax: The maximum charge density that can exist on a plastic surface in air without
causing breakdown, C/m2
DQg: The charge transfer across the gap, C
DQi: The change in induced charge on the sphere during the discharge, C
F: Potential, V
Fb: Breakdown potential
F*: Characteristic voltage (¼E*L*), V 8
Dimensionless quantities normalized by the characteristic values are indicated by
primes. ...
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