Measurement of Small Signals – Measurement System Model and Physical Limitations
61
Current noise on the same resistor under the same conditions is 0.41 fA. To place this number in perspective, a
10 mV signal across this same resistor generates a current of 100 fA, or again an error of up to 0.4%. Again,
reducing the bandwidth helps. At a noise bandwidth of 1 mHz, the current noise falls to 0.013 fA.
With E
s
at 10 mV, an EIS system that measures 10
11
Ω at 1 Hz is about 2½ decades away from the Johnson
noise limits. At 10 Hz, the system is close enough to the Johnson noise limits to make accurate measurements
impossible. Between these limits, readings get progressively less accurate as the frequency increases.
In practice, EIS measurements usually cannot be made at high-enough frequencies that Johnson noise is the
dominant noise source. If Johnson noise is a problem, averaging reduces the noise bandwidth, thereby
reducing the noise at a cost of lengthening the experiment.
Finite Input Capacitance
C
in
in Figure 9-1 represents unavoidable capacitances that always arise in real circuits. C
in
shunts R
m
, draining
off higher-frequency signals and limiting the bandwidth that can be achieved for a given value of R
m
. This
calculation shows at which frequencies the effect becomes significant. The frequency limit of a current
measurement (defined by the frequency where the phase-error hits 45°) can be calculated from:
f
RC
= 1/(2πfR
m
C
in
)
Decreasing R
m
increases this frequency. However, large R
m
values are desirable to minimize the effects of
voltage drift and voltage noise in the I/E-converter’s amplifiers.
A reasonable value for C
in
in a practical, computer-controllable, low-current measurement circuit is 20 pF. For a
6 nA full-scale current range, a practical estimate for R
m
is 10
7
Ω.
f
RC
= 1/6.28 (1 × 10
7
)(2 × 10
−12
) 8000 Hz
In general, try to stay two decades below f
RC
to keep phase-shift below one degree. The uncorrected upper
frequency limit on a 6 nA range is therefore around 80 Hz.
One can measure higher frequencies using the higher current ranges (i.e., lower impedance ranges) but this
reduces the total available signal below the resolution limits of the “voltmeter.” This then forms one basis of
statement that high-frequency and high-impedance measurements are mutually exclusive.
Software correction of the measured response can also be used to improve the useable bandwidth, but not by
more than an order of magnitude in frequency.
Leakage Currents and Input Impedance
In Figure 9-1, both R
in
and I
in
affect the accuracy of current measurements. The magnitude error caused by R
in
is
calculated by:
Error = 1− R
in
/(R
m
+ R
in
)
For an R
m
of 10
7
Ω, an error < 1% demands that R
in
must be greater than 10
9
Ω. PC board leakage, relay
leakage, and measurement device characteristics lower R
in
below the desired value of infinity.
A similar problem is the finite input leakage current I
in
into the voltage-measuring circuit. It can be leakage
directly into the input of the voltage meter, or leakage from a voltage source (such as a power supply) through
an insulation resistance into the input. If an insulator connected to the input has a 10
12
Ω resistance between
+15 V and the input, the leakage current is 15 pA. Fortunately, most sources of leakage current are DC and
can be tuned out in impedance measurements. As a rule of thumb, the DC leakage should not exceed the
measured AC signal by more than a factor of 10.
The Interface 1010 uses an input amplifier with an input current of around 1 pA. Other circuit components
may also contribute leakage currents. You therefore cannot make absolute current measurements of very low
pA currents with the Interface 1010 n practice, the input current is approximately constant, so current
differences or AC current levels of less than one pA can often be measured.
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