Gamry Reference 600+ - Johnson Noise in Zcell

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measurement can 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 V 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  0.4%. 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 8-1 represents unavoidable capacitances that always arise in real circuits. C
in
shunts R
m
,
draining off higher-frequency signals, 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/(2f R
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, 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.
You 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 exercise forms one
basis to the statement that high-frequency and high-impedance measurements are mutually exclusive.
You can use software correction of the measured response 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
in
and I
in
affect the accuracy of current measurements. The magnitude error caused by
R
in
is calculated via:
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 resistance of
10
12
Ω between +15 V and the input, the leakage current is 15 pA. Fortunately, most sources of leakage