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Chapter 8: Measurement of Small Signals--Measurement System Model and Physical Limitations
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
at 10 mV, an EIS system that measures 10
s
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 Figure 8-1 represents unavoidable capacitances that always arise in real circuits. C
in
draining off higher-frequency signals and limiting the bandwidth that can be achieved for a given value of
R
. This calculation shows at which frequencies the effect becomes significant. The frequency limit of a
m
current measurement (defined by the frequency where the phase-error hits 45
f
= 1/(2πfR
RC
Decreasing R
increases this frequency. However, large R
m
voltage drift and voltage noise in the I/E-converter's amplifiers.
A reasonable value for C
For a 6 nA full-scale current range, a practical estimate for R
f
= 1/6.28 (1 × 10
RC
In general, try to stay two decades below f
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 8-1, both R
and I
in
R
is calculated by:
in
Error = 1− R
/(R
in
7
For an R
of 10
Ω, an error < 1% demands that R
m
leakage, and measurement device characteristics lower R
A similar problem is the finite input leakage current I
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
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 1000 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 1000 n practice, the input current is approximately constant, so current
differences or AC current levels of less than one pA can often be measured.
C
)
m
in
in a practical, computer-controllable, low-current measurement circuit is 20 pF.
in
7
−12
)(2 × 10
to keep phase-shift below one degree. The uncorrected upper
RC
affect the accuracy of current measurements. The magnitude error caused by
in
+ R
)
m
in
11
Ω at 1 Hz is about 2½ decades away from the Johnson
values are desirable to minimize the effects of
m
7
is 10
Ω.
m
) 8000 Hz
must be greater than 10
in
below the desired value of infinity.
in
into the voltage-measuring circuit. It can be leakage
in
8 - 3
shunts R
,
in
m
o
) can be calculated from:
9
Ω. PC board leakage, relay
12
Ω
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