Wavelet analysis for Ar double arc plasma electrical signal
Before we analyze Argon double arc plasma electrical signal, we
also need to extract data from all of them. So we determine to extract one data
from 1000 data, after that, we obtain 1000 data. Then we use the same method to
draw wavelet time-frequency graph with using Matlab as shown in Figure 1, 2.
Figure 1: Wavelet
time-frequency graph for U1
Figure 2: Wavelet time-frequency graph for U2
From above two figures, we can discover that there is no band
around high frequency, which means that there is no high frequency components
for electrical signal. This result is agreement with that of Fast Fourier
Transform. For observing the characteristics of low frequency, we need to zoom
them in as shown in Figure 3, 4.
Figure 3: Zoom in wavelet time-frequency for U1
Figure 4: Zoom in
wavelet time-frequency for U2
We can observe that
main frequency is located at 150 Hz for U1 and U2, This is the power source
frequency, and its corresponding period is 6.67 ms. For finding the changing of
intensity of frequency related to the power supply, we need to zoom in Figure 3
and the input electrical signal U1 as shown in Figure 5, 6.
Figure 5: Detailed time-frequency from
0.03 to 0.032 s
Figure 6: Detailed input signal U1 from 0.03 to 0.032 s
The undulation of input signal U1 is not large
enough, the peak to peak value is under 1 V, the intensity of frequency changes
regularly. So the fluctuation of input signal U1 cannot affect the
intensity of frequency. This is different from Ar-N2 double plasma electrical
signal. However, the non-stationary of power supply is still main reason
to produce fluctuation of plasma jet.
February, 22nd - Session 4
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| Figure 1. Time-frequency of plasma electrical signal. |
In this
laboratory session we have been working on graphical representation of
sinusoids are building an electrical signal of the first sample. This type of
information could not have been presented by Origin Pro, so we decided to use
Mathcad for drawing this graph. The output is presented in the figure 1, the
colors are showing the intensity of the particular frequency. From the graph we
can observe definitely the most intense colours are oscillating in the low
frequency region - below 100Hz. It is the base frequency. Second most intensive
frequency we can see is 4000 Hz. It is the high frequency component of the
signal. We can also notice that intensity of the signal is decreasing
approximately in the middle of the graph and is much higher at the beginning
and in the end.
February, 15th - Session 3
Figure 1. New X-axis - number of samples replaced with time (in seconds).
During the last week sessions all graphs were shown as voltage against number of samples graph. Origin Pro refused to use time as a domain. After few attempts, we have finally changed it.
February, 8th - Session 2
 |
| Figure 1. On the left: magnification of the signal; on the right: Fast Fourier Transform |
Then we attempted to perform a Fast Fourier Transform to the reduced set of samples. We observed that there is one fundamental frequency in the low region between 0 and 100 Herz and another one at approximately 4kHz. As the amount of data was reduced we weren't able to extract all the information from the FFT and it was necessary to repeat the analysis with all the samples.
FFT for Ar double arc plasma
For ensuring the accuracy of the results and preventing the lose of data, we use all of data to do Fast Fourier Transform for Ar plasma double arc as shown in Figure 1, 2.
Figure 1: FFT for electrical signal U1
Figure 2: FFT for electrical signal U2
In the two figures, the electrical signals do not have high frequency components (1-15 kHz). This is the main characteristics which is different from Fast Fourier Transform for Ar-N2 double arc plasma jet. However, the high frequency component is caused by the arc root motion on the anode wall or the large-scale shunting of the plasma jet. Therefore, the graphs which do not have high frequency components indicate this special design torch can reduce the high frequency fluctuations for argon plasma successfully. If we want to observe the characteristics of low frequency of pure argon plasma, the two figures need to be zoomed in as shown Figure 3, 4.
Figure 3: Zoom in the low frequency region for U1
Figure 4: Zoom in the low frequency region for U2
As shown in the last figure, the prominent peak is located at 150 Hz, so the corresponding period is 6.67 ms. And its harmonics are around 300 and 600 Hz. 150 Hz is the power source frequency, so the frequencies only depend on fluctuation of rectified power supply, it is not related to the current intensity and gas flow rate. In other words, undulation of input voltages causes the non-stationary of output plasma jet.
In fact, the Y-axis does not represent the real value of amplitude, it is relative amplitude.
In the two figures, the electrical signals do not have high frequency components (1-15 kHz). This is the main characteristics which is different from Fast Fourier Transform for Ar-N2 double arc plasma jet. However, the high frequency component is caused by the arc root motion on the anode wall or the large-scale shunting of the plasma jet. Therefore, the graphs which do not have high frequency components indicate this special design torch can reduce the high frequency fluctuations for argon plasma successfully. If we want to observe the characteristics of low frequency of pure argon plasma, the two figures need to be zoomed in as shown Figure 3, 4.
Figure 4: Zoom in the low frequency region for U2
As shown in the last figure, the prominent peak is located at 150 Hz, so the corresponding period is 6.67 ms. And its harmonics are around 300 and 600 Hz. 150 Hz is the power source frequency, so the frequencies only depend on fluctuation of rectified power supply, it is not related to the current intensity and gas flow rate. In other words, undulation of input voltages causes the non-stationary of output plasma jet.
In fact, the Y-axis does not represent the real value of amplitude, it is relative amplitude.
Wavelet analysis for Ar-N2 double arc Plasma electrical signal
We can convert the electrical signals in time domain to time-frequency domain with using wavelet as following steps:
Figure 2: Wavelet time-frequency for U2
1. Extract the data from original 1,000,000 data to reduce the number of data. We select one data from every 1,000 data. So that 1000 data can be obtained for each variable.
2. Export the data sheet from ORIGIN to the text.
3. Import the data sheet to the content of the Matlab.
After that, we can draw the time-frequency using Matlab code as the following algorithm:
1. Import the data sheet to a matrix A.
2. Extract the first column data as variable Time.
3. Extract the second and third column data as input voltage U1 and U2 respectively.
4. Set the sampling frequency is 10,000 and the scale is 128.
5. Calculate the central frequency of wavelets.
6. Change the scale into the frequency.
7. Calculate the continuous wavelet coefficients.
8. Draw the wavelet time-frequency graphs of electrical signals such as U1 and U2.
We obtained the wavelet time-frequency graphs for U1 and U2 as shown in the following Figure:
Figure 1: Wavelet time-frequency for U1
Figure 2: Wavelet time-frequency for U2
From the above two figures, the X-axis represents the time domain and the Y-axis means that the frequency domain, therefore, the signals in time domain are converted into time-frequency domain with using wavelet analysis. The different color in figure represents intensity of the frequency.
As shown in two figures, there are two brilliant bands appearing at around 150 Hz and 4.1 kHz. This results are in agreement with the Fast Fourier Transform. Therefore, the power source period is 6.67 ms and the breakdown period is 0.25 ms respectively.
We will zoom in the time-frequency of U1 and find some characteristic in detail. First, we need to analyze the low frequency at 150 Hz as shown in Figure 3, 4:
Figure 3: Detailed wavelet time-frequency graph at 150 Hz from 0 to 2 ms
Figure 4: Detailed graph of input electrical signal U1
We can compare the last two graphs. When the input signal is increasing, the color around 150 Hz will change from dark blue to light blue, which means that the intensity of the frequency becomes larger. However, when the input voltage is decreasing, the color will be from light blue to dark blue so that the intensity of frequency is smaller. From the above phenomenon, the input voltage affects the intensity of frequency. So the fluctuation of plasma jet is influenced by the undulation of rectified power supply because non-stationary of input signal will affect the intensity of frequency so that output jet is not stationary.
Second, we need to analyze high frequency (4000 Hz) to find another characteristic related to the frequency band as shown in Figure 5, 6.
Figure 5: Detailed wavelet time-frequency graph at 150 Hz from 0.05 to 0.053 s
Figure 6: The corresponding input signal U1
As shown in Figure 5, the frequency band is different from others at 0.05 to 0.053 s. There exists a very narrow frequency band and then band will become wider. So we want to investigate the reason caused it. Why a normal frequency band becomes narrow?
We extract the input electrical signal U1 from 0.05 to 0.053 s to compare with the time-frequency figure. We can observe that the frequency band will be large if the input peak to peak voltage is large enough. In practice, the normal peak to peak voltage is more than 10 V, however, in the last figure, some peak to peak voltages is nearly 5 V. So the narrow frequency band is caused by this kind of voltage whose peak to peak voltage is nearly 5 V. And the large band is due to voltage whose peak to peak value is large.
We need to provide another example to prove this kind of phenomenon as shown in Figure 7, 8.
Figure 7: Detailed wavelet time-frequency graph at 150 Hz from 0.055 to 0.062 s
Figure 8: The corresponding input signal U1
As shown in Figure 5, the frequency band is different from others at 0.05 to 0.053 s. There exists a very narrow frequency band and then band will become wider. So we want to investigate the reason caused it. Why a normal frequency band becomes narrow?
We extract the input electrical signal U1 from 0.05 to 0.053 s to compare with the time-frequency figure. We can observe that the frequency band will be large if the input peak to peak voltage is large enough. In practice, the normal peak to peak voltage is more than 10 V, however, in the last figure, some peak to peak voltages is nearly 5 V. So the narrow frequency band is caused by this kind of voltage whose peak to peak voltage is nearly 5 V. And the large band is due to voltage whose peak to peak value is large.
We need to provide another example to prove this kind of phenomenon as shown in Figure 7, 8.
As shown in Figure 7, there still exists narrow frequency bands from 0.057 to 0.062 s. And then we extract the electrical signal with using the same methods. And we can observe the same phenomenon when we compare the time-frequency and input signal. Also, if the voltage whose peak to peak value is less than 10 V is input, the output narrow band will be appearing. Wider band exists because the peak to peak voltage is more than 10 V.
Therefore, input voltage affects the undulation of the frequency so that it also causes the fluctuation of the Argon-nitrogen double arc plasma jet.
Therefore, input voltage affects the undulation of the frequency so that it also causes the fluctuation of the Argon-nitrogen double arc plasma jet.
FFT for Ar-N2 Plasma double arc electrical signal
We need to do fast Fourier transform for Ar-N2 plasma double arc electrical signal. There are three kinds of data such as time, U1 and U2, however, there are 1000,000 values so that we cannot obtain FFT graph with using ORIGIN because there are too many values. Therefore, we need to extract one value from every ten values so that 100,000 values can be obtained for time, U1 and U2 respectively.
The following graph shows the FFT for electrical signal U1 and U2:
As shown in the above figures, the shapes of FFT for U1 and U2 is nearly same. The prominent peak is located at 150 Hz (period is 6.67 ms) representing the frequency of power source and its harmonics is at 300 and 600 Hz. The fluctuation of frequency is caused by the undulation of the rectified power supply, not depend on the current and gas flow rate.
Another peak is around 4100 Hz and its corresponding arc breakdown period is 0.244 ms. Its first harmonic is located at 8.2 Hz. The frequency is caused by the arc root motion on the anode surface. Also, the gas flow rates affects the position of this frequency. Recent investigation has proven that the current mainly affects the fluctuation of the plasma arc in the restrike mode.
The y-axis represents the relative amplitude of the electrical signals not real amplitude. For example, 150 Hz peak occupies a major proportion compared to the first one. And the amplitude of U1 is nearly twice amplitude of U2 at 150 Hz.
The following graph shows the FFT for electrical signal U1 and U2:
Figure 1: Fast Fourier Transform for U1
Figure 2: Fast Fourier Transform for U2
As shown in the above figures, the shapes of FFT for U1 and U2 is nearly same. The prominent peak is located at 150 Hz (period is 6.67 ms) representing the frequency of power source and its harmonics is at 300 and 600 Hz. The fluctuation of frequency is caused by the undulation of the rectified power supply, not depend on the current and gas flow rate.
Another peak is around 4100 Hz and its corresponding arc breakdown period is 0.244 ms. Its first harmonic is located at 8.2 Hz. The frequency is caused by the arc root motion on the anode surface. Also, the gas flow rates affects the position of this frequency. Recent investigation has proven that the current mainly affects the fluctuation of the plasma arc in the restrike mode.
The y-axis represents the relative amplitude of the electrical signals not real amplitude. For example, 150 Hz peak occupies a major proportion compared to the first one. And the amplitude of U1 is nearly twice amplitude of U2 at 150 Hz.
Difference between FFT and Wavelet transform
Before we comparing the wavelet analysis and FFT, we need to know the essential concept related to these two kinds of transforms first (Table 1).
Table 2: The main difference, same point and application between FFT and Wavelet transform
FFT (Fast Fourier transform)
|
This is a method to calculate the
discrete Fourier transform and its inverse. It breaks down a signal into
sinusoids of different frequencies transforms from time domain to frequency
domain.
|
Wavelet
transform
|
It is one of the methods of the
time-frequency-transformations. It decomposes the signals into different
frequency ranges and allows extraction of features relating to quality.
|
Table 1: The basic concepts of FFT and Wavelet transform
Next, we will show the main difference between FFT (Fast Fourier Transform) and wavelet transform in detail (Table 2). FFT and wavelet transform have different characteristic so that they are used to deal with different kinds of signals.
FFT
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Fourier
Transform
|
|
Difference
|
In terms of trigonometric polynomial.
Convert time domain to the frequency domain.
Extract only frequency information, losing time information.
In whole time axis, cannot analyze at instant particular frequency
rises.
|
In terms of translations and dilation of mother wavelet.
Convert time domain to the time-frequency domain.
Extract both time evolution and frequency composition of a signal.
Provide more accurately localized temporal and frequency
information.
Has multiresolution capabilities.
|
Same
point
|
Deal with
expansion of functions in terms of a set of basic functions.
|
|
Application
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Ø Long-lived, stationary signals.
Ø Suitable for time invariant signal.
|
Ø Transient, intermittent behavior.
Ø Suitable for time-varying phenomena.
Ø Can be used to analyze non-stationary signal.
|
What is wavelet analysis?
Mother Wavelet
Consider a complex-valued function ѱ which satisfies the following relationships:
Continuous Wavelet Transform
If ѱ satisfies the above condition, the continuous wavelet transform can be defined as:
where ѱ' represents the complex conjugate of ѱ and the parameter "a" means the scale of the analyzing wavelet while parameter "b" is the time shift. Therefore, the function s(t) in time domain can be mapped into the other domain that described by parameter "a" and "b".
In actual, wavelet can be much more accurately localized in temporal and frequency domain because wavelet transform can be regarded as a microscope to visualize the signal s(t). And parameter "a" called scale parameter represents the magnification and "b" chooses the position to be observed.
In actual, wavelet can be much more accurately localized in temporal and frequency domain because wavelet transform can be regarded as a microscope to visualize the signal s(t). And parameter "a" called scale parameter represents the magnification and "b" chooses the position to be observed.
February, 1st - Session 1
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