Sub-Harmonic Sampling
for Electrochemical
Impedance Spectroscopy
|
||
|
Gamry
Instruments has developed a unique approach to making Electrochemical
Impedance Spectroscopy (EIS) measurements.
The Sub-Harmonic Sampling Technique uses a dedicated sine-wave
generator and the signal measurement
hardware in the potentiostat to make the measurement and digital signal
processing techniques to transform the data into the frequency domain.
This approach allows EIS measurements to be made to frequencies as
high as 1 MHz with no external device such a Frequency Response Analyzer. The
benefits of this novel method are a simplified instrument configuration
with a small footprint at greatly reduced cost. The
voltage waveform is applied to the sample using Direct Digital Synthesis
(DDS). The DDS electronic
circuitry is built into every Gamry Potentiostat.
DDS employs digital signal processing techniques to generate a true sine wave voltage waveform at the desired
frequency (1 MHz – 1 Hz) and synchronize it very precisely to the
Potentiostat’s data acquisition timebase.
Frequencies below 1 Hz are generated using the Potentiostat’s
Digital/Analog Converter. Current
and voltage waveforms recorded at excitation frequencies exceeding 8 Hz
are transformed to lower frequencies prior to analysis.
Sub-Harmonic Sampling makes this transformation by using the
Analog/Digital Converter on the Potentiostat Board to sample the current
and voltage. The A/D data
acquisition frequency is chosen to take samples at different points on
different cycles of the AC waveform (see Figure).
At 100 kHz, for example, the system takes data on every 100th cycle
of the current and voltage waveforms.
Note that the phase is incremented between data points.
This phase increment can be precisely controlled because the DDS
and data acquisition clocks are both derived from the same crystal
oscillator. The
result of this process is a sine wave with the same amplitude as the
original current or voltage waveform.
The relative phase shift between voltage and current is also
preserved in these lower frequency curves. After
the current and voltage curves have been sampled they are each transformed
into the frequency domain using a discrete Fourier Transform (DFT). For the fundamental frequency this is given by:
where
m = 1 for the fundamental frequency and N = the number of samples per
cycle. For Gamry EIS300 EIS software operating a Series G Potentiostat, N =
128. The DC
component is calculated similarly by:
The
remaining power in the signal after subtracting DC and fundamental
frequency components is a measure of the noise in the signal.
Several cycles of data may be averaged to reduce the noise to an
acceptable level. A
similar calculation is made for
Z is a
complex quantity which is plotted as Zmag, Zph vs. F
in a Bode plot or as Zim vs. Zre in a Nyquist plot. There
are also some correction factors applied to Z.
First, there is a correction for sample time delays between the V
and I signals. Also,
there are corrections for the errors due to roll-off in the electrometer
and I/E amplifiers, for mismatch between the I and V channel filters, and
for roll-off in the I and V channel range amps.
All of these errors are reproducible and have been calibrated
either in post-production test or by the AC calibration routine.
|
||
|
Home |
Products | App
Notes | Sales |
Contact | News
| Support |
Search
|
||
|
Gamry Instruments © 1997-2008
|
||
.
