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Operation
2-33
Because of the charging of C
IN
, the electrometer follows the
exponential curve shown in Figure 2-26B. At the end of one
time constant (R
S
C
IN
), the voltage will reach approximately
63% of its final value. At the end of two time constants
(2R
S
C
IN
), the voltage will reach about 86% of its final value,
and so on. Generally, at least five time constants should be al-
lowed for better than 1% accuracy.
The amount of time that must be allowed will, of course, de-
pend on the relative values of R
S
and C
IN
. For example,
when measuring a voltage with a source resistance of 10G
Ω
with an input capacitance of 100pF, a time constant of one
second results. Thus, at least five seconds must be allowed to
achieve a better than 1% accuracy figure. Table 2-9 summa-
rizes voltage values and percentage error values for ten dif-
ferent time constants (
τ
= R
S
C
IN
).
*
τ
= R
S
C
IN
The most obvious method to minimize the slowing effects of
input capacitance is to minimize the amount of capacitance
in the circuit. Using low-capacitance cable and keeping the
cable as short as possible are two ways to do so. However,
there is a limit to the amount of capacitance reduction that
can be achieved. In those cases, especially where high im-
pedance levels are involved, guarded operation (see para-
graph 2.7.4) may be necessary.
While input capacitance does increase rise-time, it can help
to filter out some noise present at the input by effectively re-
ducing electrometer bandwidth. If we assume that all input
capacitance is lumped into a single element, the half-power
(-3dB) point of the circuit in Figure 2-26A will be:
Thus, if R
S
has a value of 10M
Ω
, and C
IN
has a value of
100pF, the half-power point will be 159Hz.
2.13.7 Source resistance
As shown in Table 2-10, a minimum value of source resis-
tance is recommended for each AMPS range. The reason for
this limitation can be understood by examining Figure 2-27.
C
S
and C
F
do not affect low-frequency noise and drift and
can be ignored for the purposes of this discussion.
Figure 2-27
Simplified model for source resistance and source capaci-
tance
Input amplifier noise (E
NOISE
) and offset (E
OS
) appearing at
the output can be calculated as follows:
Thus, it is clear that, as long as R
S
>> R
F
, Output E
NOISE
= In-
put E
NOISE
. However, as R
S
decreases in value relative to R
F
,
Output E
NOISE
increases. When R
F
= R
S
, Output E
NOISE
= 2
×
Input E
NOISE
, and the same relationship applies for E
OS
.
Table 2-9
Voltage and percent error for various time constants
Time
constant*
V
M
%Error
τ
2τ
3τ
4τ
5τ
6τ
7τ
0.632E
S
0.86E
S
0.95E
S
0.982E
S
0.993E
S
0.9975E
S
0.999E
S
37%
14%
5%
1.8%
0.674%
0.25%
0.09%
f
3dB
–
1
2
π
R
S
C
IN
--------------------------
=
Table 2-10
Minimum source resistance
Range
Minimum source
resistance
All pA
All nA
All µA
All mA
100G
Ω
100M
Ω
100k
Ω
100
Ω
TO RANGING
AMPLIFIER
ENOISE
EOS
ES
CS
CF
RF
OutputE
NOISE
InputE
NOISE
1
R
F
R
S
-------
+
×
=
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