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Chapter 5 -- Measurement of Small Signals -- Measurement System Model and Physical Limitations
For purposes of approximation, the Noise bandwidth, δF, is equal to the measurement frequency. Assume a
10
11
ohm resistor as Z
cell
rms. The peak to peak noise is about 5 times the rms noise. Under these conditions, you can make a voltage
measurement of ± 10 mV across Z
reduce the bandwidth by integrating the measured value at the expense of additional measurement time. With
a noise bandwidth of 1 mHz, the voltage noise falls to about 1.3 µV rms.
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 will generate a current of ± 100 fA, or again an error of up to ± 2%.
Again, reducing the bandwidth helps. At a noise bandwidth of 1 mHz, the current noise falls to 0.012 fA.
With E
at 10 mV, an EIS system that measures 10
s
noise limits. At 0.1 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 5-1 represents unavoidable capacitances that always arise in real circuits. C
in
off higher frequency signals, limiting the bandwidth that can be achieved for a given value of R
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
f
= 1/ ( 2 ω R
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
3 nA full scale current range, a practical estimate for R
f
= 1/ 6.28 (1x10
RC
In general, one should stay two decades below f
upper frequency limit on a 30 nA range is therefore around 80 Hz.
One can measure higher frequencies using the higher current ranges (i.e. lower impedance ranges) but this
would reduce 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 5-1, both R
and I
in
calculated by:
Error = 1- R
/(R
in
. At 300
o
K and a measurement frequency of 1 Hz this gives a voltage noise of 41 µV
with an error of about ± 2%. Fortunately, an AC measurement can
cell
C
)
m
in
in a practical, computer controllable low current measurement circuit is 20 pF. For a
in
7)
(2x10
-12
) ≈ 8000 Hz
RC
affect the accuracy of current measurements. The magnitude error due to R
in
+R
)
m
in
11
ohms at 1mHz is about 3 decades away from the Johnson
o
) can be calculated from:
values are desirable to minimize the effects of
m
7
is 10
ohms.
m
to keep phase shift below one degree. The uncorrected
5-3
shunts R
, draining
in
m
. This
m
is
in
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