Transcript Cell Current Noise

Cell Current Noise Analysis – a Worked Example
Regarding
Examination of Statistical and Spectral Characteristics
of
Noise present in Fuel Cell Current Waveforms
Caused by
Bubble Formation and Release
Craig E. Nelson - Consultant Engineer
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Purpose of Cell Current Example Noise Analysis:
1.
Demonstrate how to analyze the statistics and spectral energy
distribution from “noise” current waveforms from fuel cell devices.
2.
Demonstrate how to use time series analysis results to aid
understanding of physical processes in fuel cell electrode structures
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Summary of Results for Cell Noise Current Analysis:
1.
Noise patterns have well defined morphology that relate to bubble accumulation and
release.
2.
Noise patterns are loosely determined by average cell current and operation time.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Nature of the Measurement:
1.
Current is measured by a potentiostat
2.
The measured data is entered into an Excel spread sheet
Nature of the Analysis :
1.
The “raw” time series is detrended to produce a zero mean time series with the
very long term variations removed.
2.
The detrended time series is divided up into three “chunks” so that variation of
extracted noise parameters during the “beginning”, “middle” and “end” of the test
run can be compared. These chunks consist of 512 ( 512 = 2^9 ) data points ( a
power of 2 - is required by fast Fourier analysis algorithms) and represent 8.5
minutes of data collection.
3.
Descriptive statistics (probability distribution function histogram – moments Power Spectral Density (PSD) and Autocorrelation Function are then calculated
and plotted on “Summary Report” sheets in an Excel workbook.
4.
Examination of the Summary Reports allows information about gas formation in
the electrode structures and fluid manifolds to be inferred.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Current (milliamps)
Raw Data
70
60
50
40
30
20
10
0
0
500
1000
1500
2000
Time (Seconds)
Detrended Data
Current (milliamps)
10
5
0
-5
-10
0
200
400
600
800
1000
1200
1400
1600
1800
T ime (Seconds)
Typical Cell Current Waveform Before and After “Detrending”
by Fitting a Fourth Order Polynomial to the “Raw” Data Time Series
(The time series is split into three “chunks” to allow variations in parameter values during the course of an experiment)
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Probability Density Function - "Histogram"
( All Data )
200
Count
150
All Data
First Third
Middle Third
Last Third
100
50
0
-15
-10
-5
0
5
10
15
"Noise" Current (Milliamps)
Histogram Approximation to the Probability Distribution Function
Notice that the PDF is clearly not “Normal – Gaussian” – it is more full in the middle and does not have long “tails”.
Also notice that there is a fair amount of “skew”.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
The First Through the Fourth Moments of a Probability Distribution Function
First Moment = the Mean



=

x F( x) d x
“Center of Gravity”
2
“Radius of Gyration”

Second Moment = the Variance
=












x  F( x) d x

Third Moment = the Skew
=
3
“Measure of Asymmetry”
4
“Measure of Central Tendency”
x  F( x) d x

Fourth Moment = the Kurtosis
=
x  F( x) d x

These four parameters quantitatively describe the shape, spread and location of a probability distribution
function. Each parameter is the integrated result of all the data in a particular time series and thus may be used to
compare the histograms from similar but different fuel cell noise current waveforms. Use of these parameters
represents the classical statistical analysis approach to knowledge inference from time series data consisting of
information submerged in random data.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
The First Through the Fourth Moments of a Probability Distribution Function
Kurtosis - “Measure of skinniness”
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Detailed Descriptive Statistics:
Mean
Standard Error
Median
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(1)
Smallest(1)
Confidence Level(95.0%)
All Data
First Third
9.3E-06
0.05
-0.10
2.20
4.85
-0.45
0.25
11.77
-5.32
6.45
0.02
1792.00
6.45
-5.32
0.10
0.109
0.109
-0.063
2.465
6.076
-0.640
0.210
11.769
-5.323
6.447
56.076
513.000
6.447
-5.323
0.214
Middle Third Last Third
-0.155
0.094
-0.215
2.125
4.515
-0.326
0.302
10.662
-4.794
5.869
-79.361
512.000
5.869
-4.794
0.184
0.0265
0.0851
-0.0852
1.9246
3.7041
-0.5075
0.2446
10.1077
-4.7200
5.3877
13.5587
512.0000
5.3877
-4.7200
0.1671
Descriptive Statistics
As expected by examination of the histogram, the Kurtosis and Skewness are non-zero and significant
(the kurtosis < 0 means “flat topped” Kurtosis > 0 means “pointy topped”. The kurtosis = 0 for the
normal-Gaussian distribution)
Note: The “Mode” is not given because, due to very high data acquisition resolution
and subsequent numerical manipulation, no two data points had the exact same value
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
The Power Spectral Density Function
Fourier Transform = Magnitude and Phase Spectrum
=
F(s) =




f( x)  e
 i 2  s
dx

Inverse Fourier Transform = Real or Complex Time Series
=
f(x) =




F( s )  e
 i 2  
ds

Power Spectral Density = PSD
=






F( s )

2
ds
or





F( s )  F( s ) d s


where F( s ) is the complex conjugate of F(s) and s is the complex frequency ( j* )
f(x) is the time series to be analyzed and F(s) is the complex (mag and phase) Fourier Transform of the time series
The Power Spectral Density Function tells us at which frequencies there is energy within the time series
that we are analyzing. A plot of amplitude, power or energy vs. frequency is called a “Spectrogram”
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
The Power Spectral Density Function for a Noised Sine Wave
This Half is Usually
Not Plotted
Clean Sinewave Time Series
Several Noisy Spectrums
Average of Several Noisy Spectrograms
Noise + Sinewave Time Series
Sinewave is “buried in the noise”
Sinewave
Frequency
Nelson Research, Inc.
Spectrum
Line of
Symmetry
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
What to Look for When Using the Power Spectral Density Function
Spectral Energy Peaks with a Harmonic
Relationship (f2 = 2 * f1 etc.)
Roll-off
Slope
f
1
Broadband
noise in a
frequency
range
f
2
f
0
f
3
0
Frequency (Hz)
Sub-Harmonic
or long period
feature
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
PSD
Power Spectral Density
( All Data )
Linear – Full Range
2.0E+05
1.8E+05
1.6E+05
1.4E+05
1.2E+05
1.0E+05
8.0E+04
6.0E+04
4.0E+04
2.0E+04
0.0E+00
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
Frequency ( Hz )
PSD
Power Spectral Density
( All Data )
Linear – Expanded Range
1.0E+04
9.0E+03
8.0E+03
7.0E+03
6.0E+03
5.0E+03
4.0E+03
3.0E+03
2.0E+03
1.0E+03
0.0E+00
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
Frequency ( Hz )
Power Spectral Density
( All Data )
Log10 – Full Range
1.0E+06
PSD
1.0E+05
1.0E+04
1.0E+03
1.0E+02
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
Frequency ( Hz )
Power Spectral Density
“Noise Source with Roughly 20 dB per
Decade Power Density “Roll-off”
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High Frequency Wideband
“Noise” Source
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
PSD
Power Spectral Density
( All Data )
2.0E+05
1.8E+05
1.6E+05
1.4E+05
1.2E+05
1.0E+05
8.0E+04
6.0E+04
4.0E+04
2.0E+04
0.0E+00
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.100
Frequency ( Hz )
.005 Hz
(200 sec)
.016 Hz
(62.5 sec)
.033 Hz
(30.3 sec)
.049 Hz
(20.4 sec)
Fundamental and first Two Harmonics
of the Primary “Sawtooth” Noise waveform
Long Time Constant “Sub harmonic”
Power Spectral Density – Magnified Plot of Low Frequency Range
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
The Autocorrelation Function
Auto-Correlation Function = ACF()

=




o r
F   x  F  d
or




PSD e
 i 2   
ds


where F  is the complex conjugate of F() ,  is the relative correlation time delay
and s is the complex frequency ( j* )
The Autocorrelation Function measures how similar a time series is to itself when compared at
different relative time delays. Because the Autocorrelation Function is the inverse Fourier
transform of the Power Spectral Density Function, it represents the same information … but …
in a different way.
The PSD relates the time series and its energy at different frequencies. The ACF relates the time
series to a time delayed copy of itself. Because each is the Fourier transform of the other, a
feature in the time series that repeats itself at a fairly regular time intervals will be represented
by a peak in the Autocorrelation function at a time delay equal to the repetition interval. The
same feature will appear in the Power Spectral Density plot as a “peak” at a frequency equal to
the inverse of the time delay ( freq = 1 / time ).
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
The Autocorrelation Function
1
-1

Nelson Research, Inc.
= 0
Magnified and explained on
the next page
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
What to Look for When Using the Autocorrelation Function
1
1
Maximum
correlation = +1
at zero time
delay
0
3
-1
Roll-off
Slope
Nelson Research, Inc.
Positive
Correlation
Peaks
2


= Relative Time Delay
Negative
Correlation
Peak
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Autocorrelation Function
( All Data )
Autocorrellation Function
1.00
0.50
0.00
-0.50
-1.00
0
50
100
150
200
250
Time Delay ( Seconds )
30 Sec
(.033 Hz)
80 Sec
(.0125 Hz)
60 sec
(.0167 Hz)
170 Sec
(.0059 Hz)
225 Sec
(.0044 Hz)
120 sec
(.0083 Hz)
Autocorrelation Function
The positive and negative peaks (autocorrelation maxima for “in phase” and “out of phase” waveform
features) are at time delays that are nominal inverses of the frequency peaks in the Power Spectral
Density Function. At zero time delay, the autocorrelation function is always 1.0.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Autocorrellation Function
Autocorrelation Function
( All Data )
20 dB per Decade “roll-off”
0.80
0.40
0.00
-0.40
1
10
100
1000
Time Delay ( Seconds )
4.9 Sec
(.204 Hz)
2.25 sec
8 sec
(.44 Hz)
(.125 Hz)
60 Sec
103 Sec
(.017 Hz) (.0097 Hz)
29 Sec
87 Sec
(.345 Hz) (.0115 Hz)
Autocorrelation Function – Log10 Time Delay
The positive and negative peaks (autocorrelation maxima for “in phase” and “out of phase” waveform
features) are at time delays that are nominal inverses of the frequency peaks in the Power Spectral
Density Function. At zero time delay, the autocorrelation function is always 1.0.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com
Summary:
1.
These powerful statistical measures of cell current “noise” analysis may be
helpful in determining various bubble production, release and extraction
mechanisms.
2.
These tools may be helpful in making comparisons of surfactants.
3.
These tools can give a quick check on whether a fuel cell test run has
performed in a reasonably correct manner.
Additional Comments:
1.
Once the process is automated, it is very easy to run a battery of statistical tests
on all fuel cell run data. Since we have a large library of data already in hand, we
can quickly learn a lot about how bubble induced cell current noise behaves
under a wide range of experimental conditions.
2.
The methods presented in this tutorial are all well established and commonly
used by statisticians and reliability engineers.
3.
The combination of moments, spectra and autocorrelation functions are powerful
tools that can help us resolve difficult questions.
Nelson Research, Inc.
2142 – N. 88th St. Seattle, WA. 98103 USA 206-498-9447 Craigmail @ aol.com