Measurement of Small-current Signals – Measurement System Model and Physical Limitations
75
where:
k
= Boltzmann’s constant 1.38 × 10
–23
J/K
T
= temperature in K
F
= noise bandwidth in Hz
R
= resistance in Ω.
For purposes of approximation, the noise bandwidth,
F
, is equal to the measurement frequency. Assume a
10
11
Ω resistor as
Z
cell
. At 300 K and a measurement frequency of 1 Hz this gives a voltage noise of 41
V rms.
The peak-to-peak noise is about five times the rms noise. Under these conditions, you can make a voltage
measurement of
10 mV across
Z
cell
with an error of about
0.4%. Fortunately, an AC 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 10-1 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 10-1, 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
)
