Online Diagnostic Method and System Design of Arc Plasma Jet Characteristics Based on Wavelet Transform and Linear Regression Theory

For monitoring the online jet characteristics and improving the corresponding diagnostic accuracy, caused by the complex working environment of arc plasma generation systems, an online diagnostic method of arc plasma jet characteristics based on wavelet transform and linear regression theory was proposed. The wavelet denoising to improve the accuracy of plasma jet characteristics and the linear regression theory to predict the plasma arc voltage characteristics have been discussed. Hence, a comprehensive online diagnostic system of plasma jet characteristics has been designed by integrating the proposed method with LabVIEW virtual instrument technology. To verify the effectiveness of the online diagnostic method and system, the corresponding experiments have been conducted by using a homemade plasma generation system with specified working parameters. Sequence, the experimental results have been analyzed and discussed. The experimental results show that: (1) the proposed online diagnostic method based on wavelet transform and linear regression theory could effectively improve the accuracy and predict the jet characteristics of plasma jets; (2) the online jet characteristics of thermal plasma jet could be monitored by using the corresponding diagnostic system.


Introduction
The arc plasma jet, generated by ionizing the working gas (e.g., nitrogen, argon, air, mixed gases) in a plasma torch, has excellent characteristics, e.g., high thermal conversion efficiency, high temperature, high heat flux density, low environmental requirement, and low operating cost, etc. [1]. Due to its favorable jet characteristics, arc plasma jet has been used widely, such as hazardous waste treatment [2][3][4][5], plasma spraying [6], material processing [7], and powder preparation [8][9][10], etc. The variety of applications could have various requirements of the plasma jet characteristics, especially for some high-precision applications, e.g., material preparation and surface treatment.
The effects of different factors, e.g., plasma torch structure, power supply characteristics and operating parameters, on the stability and jet length of argon/nitrogen plasma jet have been investigated by Wenxia Pan et al. by using a homemade data acquisition system and laminar plasma generation system. The results showed that these factors significantly affected the plasma jet characteristics and that adjusting them could result in a long plasma jet with stable characteristics [11]. Similarly, Xiuquan Cao et al. studied the effects of plasma torch structure on the jet characteristics by using two typical laminar flow plasma torches. The results showed that different plasma torch structures could result in significantly different jet characteristics [12]. Senhui Liu investigated a new DC non-transferring arc plasma torch and found that it can generate a silent, stable, and ultra-long laminar plasma jet in the atmosphere. The jet length usually increases with increasing the output power and gas flow rate. Observations of temporal evolution of the plasma jet appearance and the voltage demonstrated that the jet is highly stable in the atmospheric environment [13].
Singh et al. explored the jet stability of a plasma torch by analyzing the light intensity and arc voltage fluctuations of the plasma jet. From the study, the autonomous frequencies of plasma jet were determined by the interaction of the arc, the shear layer, the hot-flowing gas and the surrounding cold gas by using the wavelet transform method [14]. Hlina et al. studied the fluctuation characteristics, kinetic properties, and velocity distribution of plasma jets under different operating conditions by using CCD imaging techniques [15][16][17]. Chumak O et al. investigated the effect of arc current and anode position on the fluctuation of the thermal plasma jet based on a statistical analysis of plasma jet images taken by a fast shutter camera. They showed that an arc current of 450 A is the best condition for generating a stable plasma jet and that placing the anode closer to the plasma jet results in a more stable generated jet [18]. Tiwari et al. used fast photography, emission spectroscopy, and arc dynamics to study the stability and internal structures of DC non-transferring arc plasma jets under different operating conditions. They found that the arc may extend into the jet region and significantly affect the plasma jet stability. The differences of thermodynamic and transport properties of nitrogen/argon/air can lead to significant differences of plasma jets [19][20][21]. Murashov I et al. used numerical simulations to study the effects of power supply parameters and anode geometry on the arc voltage and jet fluctuations. The optimal diffusion angle of the plasma torch to minimize the effect of power supply parameters on the plasma jet characteristics was obtained [22]. Lietz et al. investigated flow instability phenomena in the plasma jets by combining experiments and numerical simulations. Several flow instabilities in plasma jets, including vortex structure, shear layer instability, and jet edge instability were observed in the experiments. With the numerical simulations, the instability phenomena originate from the plasma electromagnetic field [23].
From above studies, current diagnostic methods of jet characteristic focus on the "online data collection & offline analysis" and numerical simulation. These methods could be used to investigate the effects of various factors on the jet characteristics for improving the plasma jet characteristics. However, the complex operating environment of the plasma generation system (high voltage, high current, strong magnetic field) could derive a large number of interference signals in the relevant basic data, reducing the accuracy of the plasma jet characteristics diagnosis results. Furthermore, the further improvement of plasma jet characteristics is limited by the inaccuracy of the diagnosis. Therefore, an online plasma jet characteristics diagnostic method based on wavelet denoising and linear regression theory was proposed to improve the accuracy of diagnosis results and implement online diagnosis of plasma jet characteristics. Firstly, the diagnostic method of plasma jet characteristics based on wavelet denoising and linear regression theory will be analyzed in this paper based on the experimental data. Next, a comprehensive online diagnostic system of plasma jet characteristics will be designed by integrating the proposed method with LabVIEW virtual instrument technology. Finally, the effectiveness of the online diagnostic method and system will be verified by conducting corresponding experiments using a homemade plasma generation system with specified working parameters.

Analysis of Wavelet Denoising Theory
For eliminating the noise in various frequency bands, the wavelet denoising is utilized by considering the distinct intensity distributions of noise and signal wavelet coefficients. By removing the noise from the diagnostic data, the signal resolution is enhanced. With the enhanced signal resolution, the diagnostic accuracy of jet characteristics and detection sensitivity of small fluctuations could be improved. For a better understanding of the wavelet denoising used to analyze the plasma jet characteristics, a detailed explanation will be given by taking the analysis of arc voltage characteristics as an example.

Noise Reduction Method Selection
The arc voltage signal can be influenced by the arc parameters, hydrodynamic fluctuations, the geometry of the chamber or in the current source When collecting the arc voltage signal from a plasma jet. However, the plasma jet is generated by using the direct current (DC) energy to ionize the working gas. Thus, we are more concerned with the DC component of arc voltage or on the component linearly dependent on the arc current and flow rate components and consider this to be a useful signal. As shown in Fig. 1a, multiple signal components with different frequencies were found in the original signal of arc voltage, indicating that there were some noises with various frequency scales in the original signal. Therefore, for analyzing the arc voltage characteristics of the plasma jet, the noises in the original arc voltage signal should be removed. As shown in Fig. 2, compared with the other wavelet denoising methods, e.g., continuous wavelet transform(CWT), discrete wavelet transform(DWT), and wavelet packet decomposition(WPD), etc., the multi-resolution analysis(MRA) method exhibits better denoising performance in removing noise at each frequency scale. Thus, the multi-resolution analysis method is considered to denoising the plasma arc voltage data. Multi-resolution denoising is the filtering of wavelet coefficients at different scales in order to remove noise and retain signal characteristics. As shown in Fig. 1b, the noise or interference signal of the arc voltage signal is effectively suppressed by this method.
As shown in Fig. 3, the acquired original signal is represented by S, while the low-frequency signals are represented by A 1 , A 2 , …, A n−1 , and A n , and the high-frequency components of the signal are represented by D 1 , D 2 , …, D n−1 , and D n . Thus, S n = A n + D n + D n−1 + ⋯ ⋯ + D 1 ,

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where n represents the decomposition layer number of the original signal, which will be specified in the next section. In the multi-resolution analysis, the low-frequency signal is decomposed layer by layer, while the high-frequency signal is not decomposed.

Wavelet Basis Function Selection
Wavelet basis functions are the functions involving the wavelet analysis to satisfy the wavelet conditions. The distribution of high and low-frequency wavelet coefficients after wavelet decomposition will be affected by these functions. Considering the orthogonality, regularity, tight support, vanishing moments, symmetry, etc., different wavelet basis functions could result in different effects. Hence, for effectively eliminating noise from the signals, it is important to choose a suitable wavelet basis functions. In addition, it is vital to determine an optimal wavelet order since higher order lead to smoother wavelet. The ideal wavelet basis function and optimal order can be determined by analyzing the denoised signal's signal-to-noise ratio and root-mean-square error of the denoised signal. Higher signal-to-noise ratio and lower rootmean-square error show superior denoising performance, which could improve the accuracy of original signal. Assuming that the original signal is a(n) and the denoised signal is b(n), the root mean square error (RMSE) and Signal-to-noise ratio (SNR) can be obtained by Eqs. (1) and (2), respectively:  where P s and P n are the power of the signal and the noise respectively.
Various wavelet basis functions, including Daubechies (dbN), Biorthogonal (biorNr. Nd), Coiflet (coifN), and SymletsA (symN) wavelet family, are commonly employed in signal processing. To compare the signal-to-noise ratio and root-mean-square error under different wavelet basis functions and orders, a wavelet denoising system was developed using LabVIEW virtual instrument's advanced signal processing toolkit. The system, as shown in Fig. 4, is not limited to the multiresolution decomposition denoising method used in this experiment. Instead, the "Wavelet Denoising. vi," "Wavelet Packet Analysis. vi," and "Multiresolution Analysis. vi" modules from the "Wavelet Analysis" section of the LabVIEW Advanced Signal Processing Toolkit are incorporated into the system. The integration of these modules into the system facilitates the adoption of wavelet analysis techniques. The corresponding denoising method can be selected by users to complete the signal processing according to their requirements. For instance, by clicking the "Multiresolution Analysis" button, the main interface of the wavelet denoising system appears, as shown in Fig. 4, with the original signal and the denoised low-frequency and high-frequency signals. The signal-to-noise ratio and root mean square error are calculated by the "MATLAB script" node shown in Fig. 5, which are used as evaluation criteria to determine the optimal wavelet basis function, optimal wavelet order, and the number of decomposition layers.
With the wavelet denoising system, the corresponding signal-to-noise ratio and rootmean-square error of the experimental data collected at an arc current of 120 A and a gas flow rate of 6 slm were analyzed using different wavelet basis functions and orders (as shown in Table 1). The corresponding results are shown in Figs. 6 and 7 respectively. It should be noted that the wavelet toolkit in LabVIEW program has some differences in the corresponding order of different wavelet basis functions. For instance, if the wavelet order is 1, the wavelet function corresponding to the Db wavelet family is db02, while that to the (2) SNR = 10 log 10 P s P n Fig. 4 Schematic diagram of multiresolution analysis in wavelet denoising system Bior wavelet family is Bior1-3. Hence, for providing a reference of the different wavelet basis functions and their respective orders, Table 1 was used to explain. From the Figs. 6 and 7, the coif wavelet family shows the highest signal-to-noise ratio and the lowest root-mean-square error. However, the additional factors beyond signal-to-noise ratio and root-mean-square error, such as higher-order vanishing moments, good regularity, symmetry, and perfect reconfiguration ability, should be considered when an appropriate wavelet basis function is selected. Therefore, both the coif and sym wavelet families are preferable for denoising plasma arc voltage signals. In addition, the

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oscillation may occur when the coif wavelets are used to deal with high-frequency signals, whereas the symlets wavelet family (improved by the Daubechies function) exhibits a better continuity, branch length, filter length, and symmetry. Thus, the symlets wavelet function is the best choice for denoising the arc voltage signal. Moreover, the sym4 wavelet in the sym wavelet family exhibits the highest signal-to-noise ratio and lowest root-mean-square error. Thus, the best number order is determined as 4.

Decomposition Layer Selection
In the process of wavelet denoising, the amount of filtered noise increases with the number of decomposition layers., At the same time, the detail levels in wavelet reconstruction loss increase, resulting in some loss of the actual signal. Conversely, if the decomposition level is too low, it may not be possible to separate the noise from the original signal. Therefore, choosing an optimal number of decomposition layers can have a significant influence on the denoising effect. Generally, the optimal number of decomposition layers depends on factors such as the signal-to-noise ratio and root mean square error of the original signal. From the reference [24]: Calculated: Generally, the value of r is always greater than 1. However, when r approaches 1, specifically when r is less than 1.1, it is considered that the noise has been significantly reduced. And, the optimal number of decomposition layers can be either k or k + 1. From the Table 2, the value of R is more reasonable when the number of decomposition layers is 4. Therefore, the number of decomposition layers can be either k = 4 or k = 5. In this paper, the optimal number of decomposition layers is chosen as 5.

Linear Regression Theory Analysis
The arc voltage of the plasma jet is a crucial indicator of the jet stability and strength of the plasma arc. It also influences the power, thermal efficiency, and mean enthalpy characteristics of the plasma jet. Thus, the arc voltage characteristics of the plasma jet will be analyzed in this paper. In addition, the accuracy of online diagnostic and controllability of the plasma jet characteristics could be improved by comprehending the trends and patterns of plasma jet characteristics and utilizing data prediction.
The relationship between continuous variables can be predicted or explained by using the linear regression analysis. It employs a least squares function known as a linear regression equation to model the relationship between one or more independent and dependent variables. The number of independent variables that affect the dependent variable should be considered when constructing the regression model function. For example, the arc voltage U of the plasma jet is primarily influenced by two independent variables: the arc current I and the gas flow rate G. As such, the basic multiple linear regression model developed is represented by Eq. (5): The intercept of the model is referred to as a, and the regression coefficients are known as b and c.
The arc voltage of the plasma jet is influenced by the experimental arc current and gas flow rate. Thus, when the basic model is established, it is necessary to consider the coupling relationships between these two parameters. As such, the basic model can be reformulated as shown in Eq. (6): For comparing the above two predictive models, the experimental data with an arc current of 120 A and a gas flow rate of 6 slm was used. These data are crucial for evaluating the accuracy of the models. The specific prediction method will be discussed in the Results and Analysis section. As shown in Fig. 8, predictive Value 1 was generated using the regression model displayed in Eq. (5) without considering the coupling relationship. In contrast, the coupling relationship is taken into account in the predictive Value 2, using the regression model shown in Eq. (6). From the Fig. 8, the predictive Value 2 has a more accurate prediction with a smaller error than the predictive Value 1. Therefore, the second

Design of Online Diagnostic System Based on LabVIEW
Based on the above theoretical analysis, an online diagnostic system based on LabVIEW virtual instrumentation technology is proposed. The main features of the proposed system include: • Excellent data processing and visualization capabilities.
• The accuracy and reliability of the collected original data can be improved online by using the above mentioned wavelet denoising technology. • The jet characteristics of the plasma jet could be predicted by using the regression model obtained through linear regression analysis. • With good scalability and flexibility, the proposed system can meet the various diagnostic requirements of plasma jet characteristics and different filtering methods.
The detailed design of the online diagnostic system, encompassing both the front panel interface and the back panel program, will be introduced following the completion of the hardware circuit design.
In this online diagnostic system, for obtaining the original signals, the hardware shown in Fig. 9 was used. A voltage drop module and a Photoelectric Isolation Plate were used to monitor the arc voltage with a voltage drop ratio K of 26 in the experiments. To the voltage drop module was used to drop the original arc voltage to a certain level (usually lower than Fig. 8 The predicted values and errors of the arc voltage with different prediction model (arc current = 120 A, gas flow rate = 6 slm) 10 V). The Photoelectric Isolation Plate with a ratio of 1:1 was used to isolate the diagnostic system and the working circuit for avoiding the interaction between the diagnostic system and the working circuit. Two temperature transmitter sensors were used to monitor the inlet and outlet cooling water temperatures of the plasma torch. And, a turbine water flow meter was used to monitor the flow rate of cooling water supplied to the plasma torch. With the inlet and outlet cooling water temperatures and mass flow rate, the thermal efficiency (η) and enthalpy (h 0 ) of the plasma torch could be calculated by using the Eq. (7) and (8): Here, U is the arc voltage, and I is the arc current, c is the specific heat capacity of water, m is the mass flow rate of cooling water, Δ t is the temperature difference between the inlet and outlet of cooling water, and G is the mass flow rate of the working gas. These mentioned original data were acquired automatically by a data acquisition card (NI-USB6210, The hardware circuits for monitoring the jet characteristics of a plasma torch have been discussed and designed. Table 3 shows the sensors and their main characteristics that were used in the online diagnostic system. The relationship between each sensor's output signal and the jet's characteristics in LabVIEW was determined with the assistance of these sensors. As shown in Fig. 10, the mapping relationship between the acquisitional signals and jet characteristics should be considered in the back panel of the LabVIEW program. In addition, the front panel design should prioritize interface simplicity, aesthetics, and good human-machine interaction.
According to their functions, several key modules, e.g., data denoising module, jet main characteristics display module, other characteristics display module, alarm module, raw data display module, and basic setting module, etc., in the LabVIEW program, as shown in Fig. 11, were designed and programmed with the mapping relationships shown in Fig. 10 and the method discussed in "Analysis of Wavelet Denoising Theory" and "Linear Regression Theory Analysis" sections. In addition, the "multiresolution  analysis" part of the wavelet denoising system was used as a sub-VI and added to the online diagnostic system. From the Fig. 11, The " jet main characteristics display module, "which displays the real-time main characteristics of the jet, such as arc voltage, power, thermal efficiency, and specific enthalpy were shown on the left of the designed interface. An " other characteristics display module " is featured on the central interface, which shows the plasma jet's other characteristics, such as current, inlet temperature, outlet temperature, water flow rate, and gas flow rate. The relevant plasma jet characteristics can be accessed by clicking on any of the five buttons below the other characteristics display module. The jet characteristics is shown as a green line on the waveform, and the average value of the characteristics is calculated in real-time and displayed as a white horizontal line at the top of the waveform. It should be noted that the displayed characteristics have been filtered by the data denoising module before being presented.
An "alarm module" is featured on the left bottom interface. If the collected raw data exceeds or falls below the set data size range, the indicator light is changed from green to red and flashes continuously. The "raw data display module, " which displays the real-time collected raw data of jet characteristics acquired by the acquisition card, is located at the top right of the interface. A "basic setting module" is featured on the bottom right interface, which enables users to customize the file name for saving the raw data on their local computer based on their specific requirements. The raw data sampling frequency and the number of samples can be set, and the size of data that exceeds or falls below the set range can be limited in the alarm module by users. Both the raw data before and after denoising are saved locally on the computer for future data analysis.
A fundamental model correlating the raw data with the plasma torch's arc voltage was created using MATLAB's linear regression analysis. This approach allows the arc voltage characteristics of the plasma jet to be predicted, which can serve as a crucial theoretical and research foundation for enhancing and regulating the plasma jet's characteristics, as well as advancing the development of plasma technology.

Experimental Setup
As shown in Fig. 12, the plasma generation system consisted of a gas injection subsystem, a cooling subsystem, a specified plasma power source, and a home-made plasma torch. In the gas injection subsystem, a reducing valve installed on the gas cylinder reduces the pressure of working gas to a certain level and feeds the working gas to a mass flow controller. Under the control of the mass flow controller, working gas is accurately supplied to plasma torch with a specified gas flow rate. The cooling subsystem uses a 24 kW industrial chiller to supply the cooling water to the plasma torch with a specified temperature of 15 °C in the experiments. The specified plasma power source transfers the three-phase alternating current to dc power by using an insulated gate bipolar translator module. In addition, an automatic ignition module was integrated with the power source to ignite the pilot arc with high-frequency and high-voltage power. Then, the anode arc attachment can be transferred to the anode from the pilot electrode by controlling the arc current.
As shown in Fig. 13, the plasma torch used in the experiments mainly consisted of a button cathode (1), a copper pilot electrode (2), three copper inter-electrodes (3), a copper 1 3 anode (4), some insulation rings, and sealing rings. The working gas was injected into the arc chamber between the button cathode and the copper pilot electrode. The cooling water was supplied to cool the button cathode firstly, then cool the pilot electrode and inter-electrodes in parallel, and cool the copper anode at last.

Experimental Conditions
Before the experiments, preliminary experiments had been conducted for determining the optimal experimental conditions by using the limiting method. With the preliminary   Table 4 by considering the jet stability and the erosion of electrodes. In addition, considering the cost and characteristics of working gas, the pure nitrogen (N 2 ) was chosen as the working gas.

Results and Discussion
The experiments were conducted by using the specified plasma generation system with specified experimental conditions. One image of the plasma jet working with an arc current of 90 A and a gas flow rate of 9 slm is shown in Fig. 14. With the experiments, the online diagnostic system of plasma jet characteristics was verified that it can monitor the online jet characteristics effectively. With the online diagnostic system, the arc voltage characteristics by using Wavelet Transform and Linear Regression Theory will be discussed in detail.

Analysis of Arc Voltage Characteristics Based on Wavelet Transform
The original signals and the denoising signals by using the wavelet transform discussed in "Analysis of Wavelet Denoising Theory" section are shown in Fig. 15 when the plasma generation system works with specified experimental conditions. It can be seen that the original signals show large fluctuations, while the noise level of the original signals was drastically reduced by the wavelet denoising algorithm. Thus, the wavelet denoising algorithm can effectively remove the noise from the original signals, which could improve the accuracy of the arc voltage characteristics. In addition, some outlier values were found in the wavelet-denoised signals, which may be caused by the extreme values or outliers in the original signals. During the denoising process, these extreme values and outliers may not be removed completely. Though there are few outlier values after the denoising process, the accuracy of the jet characteristics can be improved drastically.

Linear Regression-Based Analysis of Arc Voltage Characteristics
For improving the analysis efficiency, the average arc voltages with the acquisition frequency were calculated. More specifically, the arc voltage data obtained in one second was used to calculate the average arc voltage. According to the gas flow rate of the working gas, the experimental arc voltages were divided into three groups. Two of the groups were used to obtain the intercept and regression coefficients of the mode shown in Eq. (6) and predict  Arc current I=60A Gas flow rate G=6slm Arc current I=60A Gas flow rate G=9slm Arc current I=90A Gas flow rate G=6slm With the Eq. (9), the arc voltage with specified arc current and gas flow rate could be predicted. The experimental arc voltages and predicted arc voltages with specified working conditions were shown in Fig. 16. It can be seen that the predicted arc voltages are in accord with the experimental arc voltages with an average error of 2%. Thus, the Eq. (9) could predict the arc voltage characteristics effectively. Arc current I=90A Gas flow rate G=9slm Arc current I=120A Gas flow rate G=6slm Arc current I=120A Gas flow rate G=9slm The accuracy of arc plasma jet characteristics could be effectively improved by using the online diagnostic method based on wavelet transform and linear regression theory. However, further studies to investigate other characteristics, e.g., distributions of jet temperature and velocity, should be carried out. Moreover, the other influence factors should be considered in further study. With the online diagnostic system, the control method of the jet characteristics should be proposed further.