Download Zahn, M., S.C. Pao, and C.F. Tsang, Effects of Excitation Risetime and Charge Injection Conditions On the Transient Field and Charge Behavior for Unipolar Ion Conduction, Journal of Electrostatics 2, 59-78, 1976.

Journal of Electrostatics, 2 (1976) 59--78
59
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
EFFECTS OF EXCITATION RISE-TIME AND CHARGE INJECTION
CONDITIONS ON THE TRANSIENT FIELD AND CHARGE BEHAVIOR
FOR UNIPOLAR CONDUCTION
MARKUS ZAHN, SHING-CHONG PAO and CHEUNG FUNG TSANG
Department of Electrical Engineering, University of Florida, Gainesville, Florida 32611
(USA)
(Received January 1, 1975; in revised form August 26, 1975)
Summary
Recent analysis using a unipolar ion mobility conduction model is continued and extended by taking into account the experimental realities of excitation rise-time and charge
injection threshold conditions. The transient behavior of the bulk electric field and space
charge distributions as a function of time and position as well as the resulting time dependence of the terminal voltage and current are found for an imposed voltage or current
excitation with non-zero rise-time, contrasting the behavior between field or charge
specification at the emitter electrode with charge injection onset once the emitter electric
field exceeds a threshold value. Representative plots of field and charge distributions as
well as voltage and current time dependence for various excitations, rise-times, and boundary conditions are presented. Important time constants which can be correlated to experiments under these various conditions are tabulated and where possible, results are presented in closed form.
1. I n t r o d u c t i o n
Recent analysis has used a unipolar ion mobility conduction law to model
electrical conduction mechanisms in electrical insulators [1,2]. The analysis,
also appropriate for solid state conduction [ 3], neglects diffusion which is
allowed if applied voltages are much greater than the thermal voltage
(v >> kT/q) and if the charge density gradients are not very large [4].
Under these assumptions, this recent work solved for the transient time
and spatial responses of the electric field and space charge distributions as
well as the terminal voltage--current characteristics for any type of applied
excitation and any physically allowed initial and b o u n d a r y conditions. Examples presented in those works [1,2] include step changes in excitation of applied current or voltage to an initially unexcited or pre-stressed system with
specification of the electric field at the emitter electrode as a b o u n d a r y condition. Space charge limited conditions are treated when the emitter electric
field is zero, and as a model for a system where charge injection occurs at
some threshold field strength, the emitter electric field was also allowed to be
60
clamped at some non-zero value. Specific applications include correlation to
Kerr electro-optic measurements where precision electric field measurements
are made optically for those dielectrics (typically nitrobenzene) which become
significantly birefringent when stressed b y high voltages [5,6].
We disagree with other related work which has argued that specification of
the electric field at the injecting electrode as a b o u n d a r y condition is n o t
valid, b u t rather that the charge density must be specified [7,8]. The mathematics requires kn.owing the emitter electric field either because it is specified
or calculated from a model relating the emitter charge density to the electric
field. In earlier work, the specific case where the emitter charge density is
proportional to the emitter electric field was treated, which would be appropriate for conduction b y particulate impurities charged u p o n contact with
the electrodes. We do agree that specification of the emitter charge density is
often more convenient as solutions can be obtained for all values of charge
density (0 ~< q < o~) which however only corresponds to a finite range for the
emitter electric field (0 <~Eo <~ V/d). For specification of E0 outside these
bounds no physically realizable solution is possible. Thus, the b o u n d a r y condition to be used depends on what is known about the physical nature of the
emitting electrode or, for analytical purposes, on convenience. As we will
show here, for many cases specification of the emitter electric field allows for
closed form solutions over wide intervals, while with the emitter charge density specified, numerical techniques must often be used.
The work presented here is an extension and continuation of this recent
analysis. We continue our practice of considering voltage or current excitations where the analytically simpler case of current specification is usually
appropriate for semiconductor applications, while for dielectric liquids, the
voltage is usually imposed. For completeness, we repeat the cases of step
voltage and current excitations, now specifying the charge density at the
emitter electrode rather than the electric field. We also examine the effects
of excitations which have non-zero rise-times and tabulate important time
constants for various rise-times and b o u n d a r y conditions. These time constants are important to related experimental measurements of ionic mobility
by time-of-flight techniques [3]. Where possible, analysis will be presented in
closed form.
In our earlier examples [1,2] with a step voltage excitation and specification of the emitter electric field t w o characteristic curves emanate from the
origin. These t w o curves were due to the field discontinuity at x = 0, t = 0,
where approaching the origin along the line t = 0, the field was given b y the
voltage divided b y the electrode spacing, while approaching the origin along
x = 0, the electric field was clamped at some lower value. Here the electric
field is continuous at the origin for all cases, so only one characteristic curve
emanates from there.
61
2. Review
of
past analysis
We consider again a parallel plate geometry, where the electrode at x = 0
is a source of positive ions with mobility p in the dielectric. All physical
parameters are assumed constant, and with the neglect of fringing so that the
electric field and current are in the x-direction and only depend on the
coordinate x, the field equations reduce to:
~E
e
0x
~Jc
(1)
=q
aq
4---=0
~x
~t
(2)
Jc = q u E
(3)
z
f Edx = v
(4)
v
0
where e is the fluid permittivity, E is the electric field in the x-direction, q is
the charge density, Jc is the c o n d u c t i o n current, and v is the voltage. Substituting eqns. (1) and (3) into eqn. (2), we obtain:
aE
--
at
+ pE
~E
8x
= J(t)/e
(5)
where J ( t ) is the current per unit electrode area flowing in the wires to the
electrodes due to b o t h conduction current in the fluid bulk given by eqn. (3)
as well as displacement current due to the time rate of change of the surface
charge on the electrode. By integrating eqn. (5) and using eqn. (4) we obtain
the voltage--current relation
dv
-
-
dt
+ (p/2)[E:(l,t)
- - E2(0,t)] = J ( t ) l / e
(6)
Note that the first term on the left yields the familiar capacitive current, while
the second term is the modification due to the presence of bulk charge with
mobility p.
Equation (5) falls into the class of quasi-linear partial differential equations
of first order which is solved in the standard way of Lagrange and the m e t h o d
of characteristics using the subsidiary equations [ 9]
dt
--
1
dx
=
ttE
edE
-
J(t)
(7)
U p o n integration, we obtain the solutions:
E = Q(t)/e + e,
(8)
62
q = qo/[1 + gqo(t -- t0)/e]
(9)
t
x = gtc, + (g/e) ; Q(t')dt' + c2
(10)
o
where Q(t) is the total charge per unit electrode area flowing through the
electrodes
t
t
Q(t) = f J(t')dt' = (e/l){v + (g/2) ; [E2(l,t ') -- E2(O,t')]dt ' }
o
(11)
o
Equations (8) and (9) give the electric field and charge density on the family
of curves in the x--t planes with the trajectories given by eqn. (10). The
parameter q0 is the charge density on a particular trajectory at any time t = to.
Usually to is picked as the time where a particular characteristic trajectory
begins at a boundary, usually t = 0 or x = 0. The parameters cl and c2 are
constants of integration to be determined from initial and boundary conditions.
These constants are invariant on any characteristic curve but generally differ
between curves.
As discussed in our earlier work, if the parameter Q(t) is known, it is a
simple straightforward procedure to solve for E(x, t) and q(x,t) from eqns.
(8)--(10). From eqn. (11), Q(t) is determined by a simple integration if J(t)
is known. However, if the voltage is imposed, then the last equality of eqn.
(11) must be used. Generally, Q(t) cannot be found in closed form this way
because the electric field at the electrodes is n o t k n o w n until the total solution is obtained, so finite difference schemes must be used.
In the following section, we will use the results of eqns. (8)--(11) to consider excitations of step voltages or currents with the boundary condition at
x = 0 being the specification of the charge density. Then we will generalize
past analysis by including voltage and current excitations with non-zero rise
times and specification of the electric field at the emitter electrode.
3. Specification of the charge density at the emitting electrode
3.A Step current excitation
We assume an input
J(t)
= t0
J0
t< 0
(12)
t>0
Then from eqn. (11)
Q(t) = Jot
t> 0
We introduce the normalized quantities
(13)
63
= x/l,
~ = [eMJoll'~E,
~" = [ g J o / e l ] v ' t ,
Y = (eu/Jol3)'av,
= [pl/eJoll~q
(14)
The initial and b o u n d a r y conditions of interest are
E(.~,~" =
0) = 0,
~(E = 0,~') = qo
(15)
All t h a t remains to be done is to find the parameters cl and c2 in eqns. (8) and
(10). Those characteristic curves which emanate from the ~ = 0 boundary, as
in Fig. 1, have zero values for E(E,~ = 0) and Q(t = 0), so that cl = 0. Similarly
I.O
0.8
/
0.6
0.4
0.2
0
0
0.5
1.0
[5
20
i
Fig. 1. T r a j e c t o r i e s o f c h a r a c t e r i s t i c curves in n o r m a l i z e d x - - t space for a s t e p c u r r e n t
e x c i t a t i o n w i t h t h e n o r m a l i z e d charge d e n s i t y specified at t h e e m i t t i n g e l e c t r o d e as q'0 -- 1.O
q0 = 0, so from eqn. (9) the charge density remains zero along those curves.
Using eqn. (13) in eqns. (8) and (10) yields the solutions in this region as:
E - - E 0 =t'2/2,
E=~,
~ =0
(16)
where c2 = x0 is the position where a characteristic curve begins at ~ = 0. This
region extends over the time interval of 0 < ~ < 2 ~ which is obtained by
solving for the time the trajectory in eqn. (16) which starts at x0 = 0 and
reaches the other electrode at E = 1. The solutions of eqn. (16) are valid to
the left of this demarcation curve, exemplified by the heavy curve in Fig. 1.
To the right of this demarcation curve, characteristic curves begin at the
= 0 b o u n d a r y with the charge density specified as ~(E = 0,~) = q0. From
eqn. (9), we immediately know the charge density along these curves as
= 5o/[1 + qo(}'-- }~o)]
(17)
where To is the time where the characteristic curve begins at E = 0. However,
we cannot obtain the equations of the characteristic curves given by eqn. (10)
until we find the parameter c,. This parameter can be found from eqn. (8)
64
when the solution for the emitter electric field is known. FindingE(~ = 0,~)
is easy when the current is constant, as eqn. (5) at ~ = 0 in normalized form:
ct~(~ = 0,~)
d{
+ ~oE(~ = 0,~) = 1
(18)
has solution
E(~ = 0,~) = [1 -- exp(--~ot)]/qo
(19)
Then using eqn. (19) we find cl from eqn. (8), which when substituted into
eqn. (10) yields the trajectories and electric field as
= (t'-- t'o)[(~-- to)/2 + (1 -- exp(--qot'o))/qo]
(20)
= t - - t'o + (1 -- exp(--~oto))/~o
(21)
Note that as }'o > > 1/~o, the steady state solutions are
tim
/~(~) = [2~ + 1 1 ~ ] '/2
~-~
~(~) = iI[2~ * II~] '~
(22)
Typical results for q0 = 1 are plotted in Fig. 1 for the characteristic trajectories, and for the time and spatial dependences of the electric field and
charge density in Fig. 2. The heavy line emanating from the origin separates
those curves emanating from the ~= 0 boundary and from the ~ = 0 boundary. The curves to the left of this demarcation curve obey eqn. (16) while to
the right they obey eqn. (20). Note that the demarcation curve reaches the
boundary ~ = 1 at T = t~ = 2 '/2,independent of the boundary condition at
= 0. Figures 2(a) and (b) show the electric field and space charge distributions as a function of ~ for various times. For ~ < t'l, at a fixed time, the
electric field distribution has curvature due to the space charge injected over
the limited spatial region below the demarcation curve. The charge is zero
above the demarcation curve, resulting in the zero slopes shown in this region
in Fig. 2(a). At }'= ~,, space charge fills the volume, and as time increases
further, the electric field increases as in eqn. (21), finally reaching the steady
state of eqn. (22). Figures 2(c) and (d) show the electric field and space charge
distributions at various positions as a function of time. Here to the left of the
demarcation curve, the electric field increases linearly with time with no space
charge as given by eqn. (16). To the right of the demarcation curve the field
and charge distributions are given by eqns. (17) and (21).
Once the electric field distribution is known, the terminal voltage can be
found from eqn. (4). Figure 3 plots this voltage versus time for various values
of q0- Note that for ~ < ~,, we can calculate the voltage, in closed form using
eqn. (6), to be
~(~) = ~-- t3/6 + ( 1 / 4 ~ ) [ 2 ~ o ~ - - exp(--2~ot) + 4exp(--~o}') -- 3]
0 ~< ~
For ~ > ~,, numerical techniques are necessary.
t~
(23)
65
1.5
1.5
io
I0
05
o
|
o
025
05
075
ID
(c)
(o)
1.0
'~'00 '~
J
fQ'~
075
075
!"
"'-2o
~
BUr2 : o , , ~
Oglo
0.5
O2'5
on
0
0.2,5
1"
05
0
075
0
1.0
1.0
2.0
3.0
40
(d)
(b)
Fig. 2. Normalized electric field and space charge distributions as functions of normalized
t i m e and position for a step current e x c i t a t i o n with q'0 = 1.0.
1.5
~°,a5~7
~
1.0
io.
5 1.0 ~
0.5
0
0
I
I
I
IO
20
3.0
4D
Fig. 3. Normalized terminal voltage for a step current e x c i t a t i o n as a f u n c t i o n o f normalized
t i m e for various values of normalized e m i t t e r charge densities, q0.
66
3 . B S t e p voltage e x c i t a t i o n
For a step current excitation, the analysis was straightforward because the
parameter Q ( t ) was known. For a step voltage excitation, the parameter Q ( t )
is not easily obtainable because it depends on the electric fields at the electrodes
as given by eqn. (11). Despite this difficulty, with the b o u n d a r y condition of
a constant emitter electric field, large parts of the solution were obtained in
closed form. Here, with specification of the charge density at the emitter at
x = 0, we have not been able to obtain any part of the solution in closed form
so numerical techniques are used throughout.
With a step voltage applied:
v(t) =
i
Vo
t>0
0
t<O
(24)
initially the field is uniform with no space charge
E ( x , t = O) = Vo/l,
(25)
q ( x , t = O) = 0
We also maintain the constant charge density boundary condition at the
emitter electrode
(26)
q ( x = O,t) = qo
We now normalize all variables to the voltage as
= x/l,
E = El/Vo,
t = u V o t / 2 l 2,
J = 2Jl3/epV~,
5 = ql:/eVo
(27)
All characteristic curves emanating from the ~= 0 boundary have 5 = 0,
while from eqns. (9) and (26) those emanating from the ~ = 0 boundary at a
time t0 have
5 = 50/[1 + 2~0(F-- F0)]
(28)
However, we cannot calculate the characteristic trajectories directly until the
parameter cl is evaluated, which as previously stated cannot be obtained
until E(~ = 0,F) and Q ( t ) are known. Unlike the development for a step current in eqn. (18) the equation for J~(~ = 0,T) obtained from eqns. (5) and (6):
dE(~ = 0,~)
dF
+ 250E(£ = 0,~) = [E~(~ = 1,t) --/~2(~ = 0,~)]
(29)
cannot be directly solved until the electric field at ~ = I is known, although
finite difference methods can be easily applied to obtain solutions. Analogously
to Fig. 1 a demarcation curve emanates from the origin. To the left of this
curve the charge density is zero while to the right the charge density is given
by eqn. (28). Figure 4 shows typical plots of the time and spatial dependences
of the electric field and space charge distributions for q0 = 10. An important
time constant of the system is the time T1 for the demarcation curve to reach
= 1. For space charge limited conditions (q0 -* 0% ~(~ = 0,F) -* 0), past work
has shown that T1 = 1 - - exp(--1A) ~ 0.3935. As q0 -+ 0 so that E(~ = 0,~) -~ 1,
67
]
qo ~ I 0 . 0
.,~
7, - j
15
1.5
£,i .o
0.9
08
I .25
1.25
O?
06
ID
7_-
--
-
I0
-
05
0 4 - 0.3
075
).75
0 . 2 - -
).5
0.5
Ol
- -
I
.05
0.25
-
-
L25
O0
,
OJ
[ll
0
0.2
0'.4
OJ6
o's
fo
o:,,
0
I
0.'8
20
1.6
i"
(o)
(c)
ro
;'=0,0 j
e1
D04-
6
.01
0
2
~
4
J
2¸
.............
OI
0
I,
0.25
I ,
Q5
I ,
0.75
T'..._
I.O
.
.3-
0
0
04
Q8
[2
16
T
(b)
(d)
Fig. 4. N o r m a l i z e d electric field a n d s p a c e charge d i s t r i b u t i o n s as f u n c t i o n s o f t i m e a n d
p o s i t i o n f o r a s t e p voltage e x c i t a t i o n w i t h q'0 = 10.
we have that ~1 -~ 0.5. For charge densities in between these values, ~, is bounded b y these t w o values as listed in Table 1 for various values o f q0.
Figure 5 shows the current per unit electrode area as calculated from eqn.
(6) with the time derivative of the voltage being zero for ~r> 0, b u t an impulse at ~-- 0 due to the instantaneous capacitive charging. Note that for
finite emitter charge densities, besides the impulse current, the current density
approaches zero at T~ 0, while for space charge limited conditions, q0 = oo,
the normalized current density jumps to unity.
68
TABLE 1
Characteristic time constant ~ for a step voltage excitation for various values of emitter
charge densities
r,
0.0
0.01
0.1
0.25
0.5
0.7
1.0
1.5
2.5
10.0
oo
0.5000
0.4992
0.4923
0.4826
0.4700
0.4621
0.4527
0.4415
0.4277
0.4034
0.3935
2.5
2o
25
15
~5
J
lOI.O
o 7 - -
0,5
o25
0
0
Q25
0.5
0:75
IO
125
Fig. 5. Normalized terminal current for a step voltage excitation as a function of normalized
time for various emitter charge density values.
4. C u r r e n t e x c i t a t i o n w i t h n o n - z e r o r i s e - t i m e
We w i s h t o c o n s i d e r h e r e a c u r r e n t e x c i t a t i o n o f t h e f o r m
J ( t ) = J0(1 - - e x p ( - - t / r ) )
t > 0
(30)
N o t e t h a t i n t h e l i m i t as r -~ 0, o u r r e s u l t s s h o u l d r e d u c e t o t h o s e p r e v i o u s l y
reported for a step current [1]. From eqn. (11):
Q(t) = Jot + Jor(exp(--t/~)
- - 1)
(31)
69
and since E(x, t = 0) = 0, t h o s e curves e m a n a t i n g f r o m t h e t = 0 b o u n d a r y
have t h e p a r a m e t e r c, d e f i n e d in eqn. (8) as zero. Thus, t h e solution in this
region f r o m eqns. (8)--(11) is:
(32)
-- Xo = t2/2 -- ~r" + F2( 1 -- exp(--~/~'))
F, = ~+ ~'(exp(--~/r') -- 1)
(33)
~=0
(34)
where we use the normalizations d e f i n e d in eqn. (14). Again, we have the
d e m a r c a t i o n curve which starts at the origin with E = 0.
To the right of this curve, we m u s t specify a b o u n d a r y c o n d i t i o n at E = 0.
We t a k e t h e p o i n t of view here t h a t for t h e e m i t t e r electric field less t h a n
some critical value Ec, no charge emission occurs, b u t once this value is
reached t h e e m i t t e r electric field is c l a m p e d at this value. Thus for ~" < Ec the
s y s t e m acts as a capacitor w i t h no charge injection, so t h e p e r t i n e n t e q u a t i o n
in normalized f o r m is, f r o m eqn. (5), [~/~x] [E(x = 0, T)] = 0:
dE(E
=
0,~)
= 1 -- exp(--~/F),
d~
w i t h solution:
=
=
l
E(E = 0,£) ~< Ec
~+ F(exp(--~/F) -- 1)
0 ~< ~
(35)
~/
(36)
where T/is d e f i n e d as t h e t i m e t h a t E(E = 0,Y) just reaches Ec. Thus, as s h o w n
in Fig. 6 for F = 0.1 and Ec = 0.5, we have t w o d e m a r c a t i o n curves, one
f
0.8
06
04
0.2,
.?
ti
?t
TI T2 2
3
4
Fig. 6. C h a r a c t e r i s t i c t r a j e c t o r i e s f o r a n imposed current w i t h n o r m a l i z e d r i s e - t i m e ~" = 0 . 1
and critical emitter electric field E c = 0.5.
70
emanating from the origin and one emanating from ~j. To the left o f the latter
curve, the space charge density remains zero so the electric field in this region
is still given by eqn. (33) but the trajectory o f the curves is n o w given by the
relation
= (}-2 _ }-o2)/2 _ ~(~ _ t0) + ~2(exp(--~0/~) -- exp(--~/~'))
(37)
For those curves starting at times ~ > ~/the electric field, space charge and
trajectories are given by the relations
= (~'-- t'0)(/~c -- ~'0 -- 7exp(--~'o/~) + (~2 _ T~)/2 + 72(exp(--~0/~') --
-
exp(--F/~))
= Ec + ( ~ - - ~o) + ~'(exp(--}'/~') -- exp(--to/7))
"~ = ll[.Ec + (1"--
(38)
(39)
~'o)]
(40)
Figure 7 shows the time and spatial dependences for the electric field and
space charge distributions for ~'c = 0.5 and F = 0.1. The pertinent time constants
are ~',, where the curve emanating from the origin reaches the other electrode
at ~ = 1, ~., the time when charge injection begins, and }'2, the time w h e n the
15
~c:05
:01
~
c =05
T :01
15
T,t
/
QQ
j
I0
~
T,
05
9~
E"°/
~2
~o
9~
Q9
0.5
o2
0
&
oi~
oi.
o18
i
I0
}i r
0
t
i
i
3
~l 2
T
(c)
(o)
~'9
2.0
~c=o5
"f" =Ol
I.fi
15
~i
t.o
O.4
0.5
,T,
02
06
O4
(b)
Q8
I0
~2
(d)
Fig. 7. Time and space distributions of the electric field and space charge density for
current excitation with ?'= 0.1 and/~c = 0.5.
71
TABLE2
Characteristic t i m e constants ~ , ~ and ~ for c u r r e n t e x c i t a t i o n w i t h non-zero rise-time ~"
for various values of steady state e m i t t e r electric field E ¢
0.0
0.05
0.5
1.0
2.5
5.0
t2
0
1.4142
0
1.4633
0
1.8277
0
2.1142
0
2.6784
0
3.2730
t2
0.1000
1.4177
0.1474
1.4675
0.3534
1.8570
0.4832
2.1277
0.7421
2.7027
1.0345
3.3128
t2
0.2000
1.4283
0.2497
1.4782
0.5250
1.8557
0.7068
2.1539
1.0714
2.7482
1.4841
3.3858
t2
0.3000
1.4457
0.3500
1.4957
0.6687
1.8816
0.8889
2.1892
1.3335
2.8074
1.8381
3.4784
t2
0.4000
1.4697
0.4500
1.5197
0.7988
1.9139
1.0501
2.2318
1.5611
2.8769
2.1427
3.5849
t~
t2
0.5000
1.5000
0.5500
1.5500
0.9207
1.9518
1.1983
2.2807
1.7669
2.9543
2.4159
3.7011
t2
0.6000
1.5362
0.6500
1.5862
1.0372
1.9950
1.3375
2.3351
1.9573
3.0380
2.6669
3.8243
t~
t=
0.7000
1.5780
0.7500
1.6280
1.1499
2.0429
1.4701
2.3942
2.1363
3.1266
2.9012
3.9526
0.8000
1.6248
0.8500
1.6748
1.2598
2.0952
1.5976
2.4573
2.3062
3.2192
3.1222
4.0844
0.9000
1.6763
0.9500
1.7263
1.3676
2.1514
1.7211
2.5240
2.4687
3.3149
3.3325
4.2187
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t~
1.0
0.8
0.6
0.4
0.2
I
2
3
4
5
Fig. 8. Terminal voltage v e r s u s t i m e with current e x c i t a t i o n for various values of rise-time
with Ec ffi 0 and E c = 0 . 5 .
72
curve beginning at ~ reaches ~ = 1. For space charge limited conditions ~ = 0,
and ~1 = ~2. The time ~1 can be found from eqn. (32) with x0 = 0 and ~ = 1
while }) can be found from eqn. (36) with E = Ec, and Y2 can be found from
eqn. (37) with ~ = 1 and T0 --- ~/. The transcendental nature of these equations
requires numerical techniques for solution. Table 2 lists these various time
constants for different values of Ec and ~. Figure 8 plots the terminal voltage
v e r s u s time for various rise-times ~ with Ec = 0 and Ec = 0.5.
5. V o l t a g e e x c i t a t i o n w i t h n o n - z e r o rise-time
In practice, the voltage imposed across a dielectric or insulating liquid has
a non-zero rise-time because the capacitive charging current is limited to a
finite value. If this rise-time is small compared to the transit time for an injected charge to reach the other electrode, the step voltage solutions obtained
in previous work and here in Section 3.B are approximately correct. Normalizing all variables to the voltage as in eqn. (27), the rise-time ~ must obey the
inequality ~"< < 0.5 as seen from Table 1. For ~ comparable or greater than
0.5, the problem must be reconsidered using eqns. (6) and (8)--(11) where we
choose the voltage to be given as
v(t) =
V0(1 -- exp(--t/r))
t/> 0
(41)
As found in earlier examples, a characteristic trajectory emanates from the
origin in the x - - t plane which separates those characteristics arising from
initial conditions at the Y= 0 b o u n d a r y from those due to the charge emitter
boundary condition at ~ = 0. We again assume that charge injection begins
only after the electric field at the emitter electrode reaches a threshold value,
Ec and that for higher voltages the emitter electric field remains clamped at
this value. For lesser field values, no charge emission occurs. Thus for the
excitation of eqn. (41), the electric field remains uniform given by the instantaneous voltage divided by the electrode spacing with no charge injection
until the normalized field strength reaches ~'c. The normalized time this threshold field strength is reached is defined as ~/. Two other pertinent time constants
and their relative values to the rise-time ~ describe the system. We define tl as
the time when the curve emanating from the origin, (~ = 0, ~= 0), reaches the
other electrode at ~ = 1. The time the characteristic trajectory which begins at
time ~j reaches the other electrode at ~ = 1 is defined as t2. These system time
constants, shown in Fig. 9 and very important in correlation between analysis
and experiments, are tabulated for various rise-times and charge emission
threshold conditions in Table 3.
At time t2, space charge is distributed over the entire region between the
electrodes. For ~ < < ~2, the electric field and space charge distributions are
very close to the steady state distributions at Y2 as can be seen in Figs. 10 and
12 for Y = 0.1 with Ec = 0 and Ec = 0.5. If ~ is comparable or greater than t2,
the field and charge distributions gradually approach the steady state distributions for times greater than t2 as demonstrated in Figs. 11 and 13 for
73
,.o,o,- o,.-[./f,
o' ////
No charge
injection
until Tj
: / 2/ / /// //
t
Onset of
charge injection
Electric field
cloml~d at I~.c
for T I "fj
Fig. 9. Typical characteristic trajectories for an applied voltage excitation with normalized
rise-time ~. Charge injection begins at time ~ when the emitter electric field reaches value
/~c- Note that ~ can be greater or less than t~.
TABLE 3
Characteristic time constants ~, ~/, and ~ for voltage excitation with non-zero rise-time ~"
for various values of steady state emitter electric field Ec
/•c•
0
~
=~
0.0
0.1
0.5
1.0
1.5
2.5
0
0.3935
0
0.4886
0
0.8115
0
1.0722
0
1.2753
0
1.5996
0.25
~/
0.0
0.3976
0.0288
0.5072
0.5096
0.1438
0.8384
0.8546
0.2877
1.1169
1.1575
0.4315
1.3388
1.4095
0.7192
1.7003
1.8455
0.50
~
0.0
0.4118
0.0693
0.5307
0.5443
0.3466
0.8845
0.9755
0.6932
1.1591
1.3978
1.0397
1.4115
1.8082
1.7329
1.7669
2.5610
0.75
~
tL
~2
0.0
0.4396
0.1386
0.5662
0.6117
0.6931
0.9190
1.2535
1.3863
1.2000
1.9831
2.0794
1.4200
2.6937
3.4657
1.7700
4.0976
0.90
t~
t~
0.0
0.4701
0.2303
0.5904
0.7190
1.1513
0.9300
1.6747
2.3026
1.2000
2.8388
3.4539
1.4200
3.9919
5.7565
1.7700
6.3073
= 1.0 with Ec = 0 and Ec = 0.5. Note that for space charge limited conditions
= o),
=
Figure 14 plots the time dependence of the resulting terminal current for
Ec = 0 and Ec = 0.5. Note that for ~ small, the current is very large near
-- 0 due to the large capacitive charging current necessary. If ~ -- 0, this
charging current is infinite (an impulse) at ~= 0, but immediately drops to
finite values for ~ > 0. For non-zero but small ~, the initial current is large
but still finite, quickly dropping down with some small oscillations as it approaches the steady state. For large ~, the initial charging current is small and
74
1-751
I¢o.o
j
,./.-/'"
,.~5~
75
I
o,_1
~
0.75
0.75
025
025
/
1
i
0.25
05.
,;'°1
0.75
r~.o
o.,
oz
~
'
~
:
i
1.0
0.5
o
.
o
5
i
--
i
1.0
1.5
2.0
T
(al
(c)
6.0
4.0
2.0
l
"? :o.i
50
I~¢=0
"C=OI
40
005
\L
5.0
20
~a3
"-..
~ -
0.2
O.4
o.~Qt,.o -
10
I0
0.25
0.5
(b)
075
I0
05
I0
15
20
(d)
Fig. 10. Electric field and space charge distributions for an applied voltage with Ec = 0 and
~'= 0 . I .
there are no oscillations as the current approaches the steady state. These
curves are important in correlation to time-of-flight measurements, which
determine the ionic mobility # through measurement of the time for the
current to reach its peak (~= ~1) when a step voltage is applied. In an experi-
75
15
125
~¢= 0
1.5
//J]
"~o,o
//
~',=°0
;~J/I
//
1.0
075
0.75
05
0.5
0 25
025
015
0 8 "~----
i
0.~5
,0
I
I
I
I
2
3
/~;"
~o.o
I
2.5 I~,-o
15
1.5
I0
I0
°s
I
025
I
'I
08
05
y
(b)
4
r
{c)
(Q)
z5
~''°~
2o
1.0
0.15
/
I
125
075
IO
005/
0z~
~ o f - -
o5
I
2
3
4
T
(d)
Fig, 11. Electric field and space charge distributions for an applied voltage with Ec = 0 and
F'= 1.0.
ment, if the voltage rise-time is small, t h e current peak at ~ = tl is so m u c h
smaller than t h e initial charging current that it can be easily o v e r s h a d o w e d
and difficult t o measure. If F is large there is n o current peak at ~ = tl so t h a t
the ionic m o b i l i t y c a n n o t be measured. These problems make terminal measurements o f this t y p e difficult. Further r e f i n e m e n t o f bulk optical measurem e n t s using Schlieren p h o t o g r a p h y and the Kerr electro-optic effect will
allow greater precision in measurements w h e n used in c o n j u n c t i o n w i t h terminal measurements.
76
1.5
Ec=0'5
"~=01
T,-
1.25
'~~o~
°4-
1.5
c=05
I. 25
=
//~
"
f
I0
~
08
O6
,o
///
o.
1.0
///
~
O4
0.2
02
075
i'=0.1
0 1 - -
0.5
~~
t=lj=00693
T:005
0.5
0.25
i:oo3
025
' ~ = 0 - -
~{:ooi
I
0.25
I
~l'=0
05
I
1.0
075
~j
I
i
i
O4
O l8
I 2
(c)
(o)
20
-
1.6
f
~c :0.5
20
00
E¢:0.5
'T :01
010
1.5
1.5
025 - "70I
7=01
-..
ID
"~
05
0.5
025
i
05
(b)
05--
T2.
~:iO-
0.5
Q5
i;" ~il
-
i
i
I
0.75
ID
Q4
08
12
16
T
(d)
Fig. 12. Electric field and space charge distributions for an applied voltage w i t h Ec = 0.5
and ~'= 0.1.
6. Concluding remarks
The analysis presented here is of great value in correlation to experimental
measurements of the conduction and breakdown mechanisms in dielectric
liquids as well as semiconductors for single carrier charge injection. By fitting
77
I .25
~c=O..~
.~-'"
f" : ~.o
J
./ /
I0
0 75
/.'J"
12 5
Ec =0.5
;: ~o / "
i", i.o
2.0
oo.B.6~
1.0
/~
04~
02~
0.75
1,0-
/
. / ~
T=OB
O0
05
0.5
--05
- -
0 25
- -
0.25
0 3
OI
I
i
0 5
i
0.5
0.75
iI
1,0
i
i
2
:5
4
R
(o)
(c)
20 ~ ~c=0.5
2.0 l I~c=O.5
~:l.O
I
~,oo ~
~
~I
02 ~
1.0
~
O. 5
°
1 ~
iO
o 4 ~
'
0.5
Tz
I_
0.25
05
0,75
I0
ijt I $[, T~
2
3
4
T
(b)
(d)
Fig. 13. Electric field and space charge distributions for an applied voltage with Ee ffi 0.5
and 7 = 1.0.
experimental measurements with the analysis, values of ionic mobilities and
other physical parameters can be calculated. However, as emphasized in this
work, the pertinent time constants measured and derived from analysis are
strongly related to the emitter boundary conditions as well as the rise-time of
the applied excitation. The results presented here, both for the bulk electric
field and space charge distributions as well as the time dependence of the
terminal voltage and current, allow for greater precision in the calculation of
physical parameters by taking into account these experimental realities of excitation rise-time and charge injection thresholds.
78
[ ~, Vo,
i ,~
1 2 /
I0
]
~
--
i
.¢
/
,,"
/
".....
f"
"
. t5
05~.25
"~
~o
j
./
7
2
3
Fig. 14. Terminal current for an applied voltage excitation for various values of ~ w i t h
= 0 (solid lines) and E c = 0.5 (dotted lines).
Ec
Acknowledgements
Ms. Marjorie A. Niblack and the photography staff of the Office of Instructional Resources at the University of Florida are gratefully acknowledged for
their help in the preparation of the plots presented in this work.
This work was supported by the National Science Foundation under Grant
Nos. GK--37594 and ENG72--04214 A01.
References
1
2
3
4
5
6
M. Zahn, C.F. Tsang and S.C. Pao, J. Appl. Phys., 45 (1974) 2432.
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R.B. Schilling and H. Schachter, J. Appl. Phys., 38 (1967) 841.
E.C. Ca~idy and H.N. Cones, J. Res. Natl. Bur. Stand. (U.S.), 73C (1969) 5.
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EI--9 (1974) 43.
7 P. Atten and J.P. Gosse, J. Chem. Phys., 51 (1969) 2804.
8 P. Atten, C. R. Acad. Sci., 266 B (1969) 1188.
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