Johnson Noise In Z - Gamry Instruments Interface 1000 Operator's Manual

Chapter 8: Measurement of Small Signals--Measurement System Model and Physical Limitations
Unfortunately, technology limits high-impedance measurements because:
Current measurement circuits always have non-zero input capacitance, i.e., C

Infinite R cannot be achieved with real circuits and materials.

in
Amplifiers used in the meter have input currents, i.e., I

The cell and the potentiostat create both a non-zero C

Additionally, basic physics limits high-impedance measurements via Johnson noise, which is the inherent
noise in a resistance.

Johnson Noise in Z

cell
Johnson noise across a resistor represents a fundamental physical limitation. Resistors, regardless of
composition, demonstrate a minimum noise for both current and voltage, per the following equations:
E = (4kTRF)
I = (4kTF/R)
where
k = Boltzman's constant, 1.38 × 10
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 . At 300 K and a measurement frequency of 1 Hz, this gives a voltage noise of 41 µV
cell
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
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 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 generates a current of 100 fA, or again an error of
Figure 8-1
Equivalent Measurement Circuit
R shunt
C shunt
Icell
R in
1/2
1/2
−23
with an error of about 0.4%. Fortunately, an AC
cell
8 - 2
Rm
C in
> 0.
in
and a finite R
shunt shunt
J/K
> 0.
in
.