Download Zahn, M., Charge Injection and Transport in a Lossy Capacitor Stressed by a Marx Generator, IEEE Transactions on Electrical Insulation EI-19, 179-181, June 1984

IEEE Transactions on Electrical
Tnsulation
Vol.
EI-19 Nos.3 June 1984
179
CHARGE INJECTION AND TRANSPORT IN A LossY
CAPACITOR STRESSED BY A MARX GENERATOR
M. Zahn
Department of Electrical Engineering
and Computer Science
High Voltage Research Laboratory
Massachusetts Institute of Technology
Cambridge, MA
ABSTRACT
A theoretical solution is given
electric field and charge density
capacitor with a charge injecting
being charged by a Marx capacitor
Conservation of Charge: VJ+
INTRODUCTION
Injected charge of density q and mobility , increases
where q
the effective ohmic conductivity a to
is time and space dependent. Then, open circuit decay
curves at moderate voltages can show a dielectric relaxation time that decreases with increasing voltage
while at very high voltages the open circuit decay
Such anomalous behavcurves can be non-exponential.
ior has been measured in highly purified water [1] and
would be present in any material with enough net charge
to distort the electric field. A drift dominated conduction model is used to solve for the electric field
and charge density distributions and the terminal voltage-current behavior of a lossy capacitor where one
electrode injects charge. The capacitor is connected
to a Marx capacitor bank so that the injected charge
decreases as the Marx voltage decays, the decay rate
in turn being in part determined by the injected
charge.
a+lqp|
DRIFT DOMINATED CONDUCTION MODEL
Gauss's Law: V.(EE)=q
-X
=
q/c
Constitutive
--
E)
St
=
.T=0a E+q-iE
Law:
(4)
,
x=x/Z, v-v/Vo,
E=EZ/VO,
J=JZ3/(CiVG2)
t=-pVOt/Z2,
q=qZ2/(CV0)
(5)
[=
-
pVO/Z2
Then (l)-(4) reduce to
, =()
a+
=-a
+k aU
at
(6)
%
Dt
~X
T
where :tY) is the total terminal current per unit
electrode area.
=
TDX
T
equation for the charge
(7)
q
+
at
an
SX
2
X-
tJ
I
I, A.
C.:--
X2
t
(2)
-.
a-(J+
Cx
It is convenient to nondimensionalize these equations by normalizing all variables to the electrode
spacing Z, the initial voltage V and nominal injected charge transit time Zl /(P@)
d+IL%+
/ D' C:.
If
B-T"67,.-'94
=0
while integrating (6) between electrodes relates the
current and voltage
(1)
Edx=zv
Conduction
q
c
St
(3)
Differentiating (6) gives
density
We consider parallel plate electrodes located at
x=O and x=l, where the x=O electrode is assumed to be
a source of positive charge with constant mobility ip.
and intrinThe dielectric has constant permittivity
sic ohmic conductivity a. Neglecting edge effects,
the electric field and conduction current are x directed and all quantities only depend on x. The governing equations are then
Irrotational Electric Field: VxE=O-
for the
of a
electrode
bank.
v !.
1 79V
-+.
.
t .
..
T '-.
EE
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1EEE Transactions
InSO
rF-l}i
1.0 -/
0.0
on
2.0
q
E
-
Electrical Insulation
/
-
0.2
A
{,t-;
El-19l
June
No.3,
1984
TO i the time that the charge is injected at
For 0=O, there is a charge front which propagates towards the opposite electrode and reaches it
at time
there
x=O.
=
0.4
Vol.
(14)
Tin['/('-1)]
X
The injected chaVe density of (12) is found by
evaluating (6) at x=O
dko Wo
dko
(15)
{,+ {, + qoko = ,! + -, + Wko2 = °
T
d
T
dt
ko
-A
_.I
u.u
_
_
1.0
O.b
_
_
_
2.0
1.6
2.6
3.0
3.5
4.0
with solution
Fig. 1: Charge transport trajectories efor an ini
aZZy charged but open circuited Zossy capacitor
where the Zower x=O eZectrode injects charge pr.
portionaZ to the instantaneous ZocaZ eZectric f
The demarcation curves emanating from the origi:
various vaZues of dieZectrzic reZaxati on
shown,for
time
separate the initiaZ vaZue probZem with
S=O)=O, 2(k, O)=1 from the subsequent char(ge
V(x,
injection probZem.
EoN1
^v
1~~~~1
=
I
Ir {
(16)
(2+l/- J exp [t/-c ] -2 I
The solutions for field and charge density
are
then
T,
0
flu
X
E(x, tJ=
]T(1-exp[-t/T])<X1
(exp [-
(17)
rv-j
t
l+x
Method of Characteristics
c{
(2+1/lbexpj/T ]-A%}
The partial differential equations (6) and (7) c
be converted to ordinary differential equations by
jumping into the frame of reference of the moving
charges.
a 9S+ d:x'
dt 't dt
d3
ax
dt
_aq,d! 9q=_(u2 +/l(j"
d't 9't
{(/l'xp htl ZO<x<T
(9)
The
V(t)
open
q(t)
where
=
qO {t0)
"U
Ili
(b rk,
ni
'IV
lb
(1+q 0 (t 0 )T) exp[ (t-t 0 )/Tl -q, (.,) T'
qo(TO)
(1-exp[-W/r"])
(18)
circuit terminal voltage is then
=0p1/tTb]_ -[/- ]
d
2
=
This last equation ~~~~~to
is directly
O
integrated
%
,%,
T(1-expT
0
(10)
q/.
dt ax
~~~Osxs5 T (l-exp [-t/T]l
r
1+1/2
T{
(Af+1/T )exp
(19)
[ t/T ] -A
}
(1 1)
is the injected charge density at time
.
The problem is not completely specified until we
define the charge injection law. We expect the injected charge density to increase with electrode
electric field. Although numerical analysis can
treat any injection law, we consider a simple linear
law which allows analytic solution
q(x--O,t)=AE(x=O,t)-*
q%o=;rEo ;
A=Azlc
(12)
OPEN CIRCUIT DECAY
Fig. 2:
Consider an initially charged capacitor which is
then open circuited for t>O (J t)=O). The solutions
to (9) are
.(T)=.O(To) exp[-(Mo)/T1
x
(.) =ko (.o ) "Ti
-
;
exp F(
various
Open circuit voZtage decay of (19) for
v Jues
point if T 2>1.
of A and
T.
(13)
.-To )/,Tl
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There is
an
infZection
Zahn: Charge Injection and Transport in a Lossy Capacitor
MARX CAPACITOR CIRCUIT
Another typical configuration is to have a Marx
capacitor bank with total capacitance Cm connected
directly across the test dielectric of area S. We
neglect the Marx shunt resistance and assume the test
dielectric is lossless (u->-). Then the terminal
current is also given by
6mSZ/
m
3m
m
(20)
To allow closed form solutions e also assume space
charge limited injection where A- so that 20=0.
Then with 2(x=1,)=-i, (8)-(11) reduce to
y=
m~
av
m
d'
diX + _, rE12
(lim
)
(22)
2-T
=
=
(1+Cm )
InitiaZ Injection Problem
Approaching the x0=O, T=0 origin from along the
axis has (x=0, 0-)=-, while approaching the origin
from the t axis has E(x=0, t=0)=O for space charge
limited conditions. This disgontinuity results in a
family of injected charge at t=0 all with charge density q=l/?. As the charge propagates, Coulombic repulsion causes the charge to separate maintaining a
uniform charge densitq and thu, a linear electric
field. Using (6) at x=1 with T= and (21) yields
for t>tl a single differential equation in the variable TE1
Pt
_
(ItE1)2
m
m
t
m i; t
(24)
et
rb
-0, . (. ),;,Zo
Zo,0=0-,iE(
0)=O
10>0,X(10)=O
0
0
1/r\.,
=
i/(T_ TO)
(30)
1~
t[l- rnl
In
xt/t
m
dx = E
X
q
+0
2(1+ )
(23)
+ C v = constant
m
dt
m
(29)
Ct
2 - m
:(tE,
=
(~i-1)
-l
- ~
mis
El2=
dky
et
d
(21)
2(1+2 m
lei
(25)
is found from (26). This solution
where ESE1
is valid unttj time t2 wJen the last of the charge
injected at t=0 reaches x=1. For later times it is
not possible to obtain analytic solutions, and numerical integration is necessary.
The charge transport trajectories obtained by solv-
ing (24) break the solution into three spatial regions.
Initial VaZue Problem
Ahead of the charge front, =0, so that the electric
field is uniform. The field
at x=1 is found using
(6) wi th (21) for q=O and T==
dT
-S
-
2(+m)
;
1=0
-*
21
(26)
=
1
t-0
t
-
2 (1l +C)
This solution is valid until the time t1 when the
initially injected charge first reaches the opposite
electrode
1,
2(1+f )
=
,
[ l-exp [-3v'(2 (1+3m ))] }
(27)
m~~~~~
0.0
The voltage and current for
dkl
~~m
2(1+C6) [1 -
Ot.tl
(28)
m
]2
0.5
1.0
1.5
2.0
3.1
2.5
are
Fig. 3: The terminqZ current Y(t) of (21) for
various values of Cm. AnaZytiz oZutions zre
given in the time intervaZs 0:t1T61 and T1-gtW2.
NumericaZ inteqgation is required to find the
soZu ion for ttt2. The dash d curves show
the solution
and 12 for each case. When
is that of a step voZtage from rest.
%o,
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Tj
IEEE Transactions on Electrical Insulation Vol. EI-19
182
No.3,
0.8
8
0.6
\\\
0.\\
\\\:\4
[1]
\
2
\
_ t05
0.4
CM=
0
~
\
N
M. Zahn, D. B. Fenneman, S.
Voldman,
Fig. 4:
of Cm
0.5
1.0
1.5
2.0
2.5
and
T. Takada, "Charge Injection and Transport in
High Voltage Water/Glycol Capacitors", J. Appl.
Phys. 54, pp. 315-325 (1983).
This paper was presented at the 1st International
Conference on Conduction and Breakdown in SoZid
Dielectrics, 4-8 JuZy 1983, Toulouse, France.
Manuscript was received 21 January 1984.
0.0
1984
REFERENCES
1.0
Q.0
June
3.0
TerminaZ voZtage v t) for various values
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