Contact‐free dielectric response measurement based on limit fitting by neural network

Correspondence State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, People’s Republic of China. Email: yljcqu@cqu.edu.cn Abstract Frequency domain spectroscopy (FDS) is widely used in analysis of insulation condition of dielectric material. The traditional contact method is susceptible to the impact of the contact state between the test sample and the electrode, which is also difficult to ensure the accuracy and repeatability of the test results. Using the method of sputtering gold on the surface of the sample is an effective way to avoid this problem though, this processing method is time-consuming and high-cost. This paper presents a method based on limit fitting by neural network to realize contact-free dielectric response measurement, which can eliminate the distortion effect of the air gap introduced by the uneven surface of dielectric material. Cross linked polyethylene (XLPE) is chosen as the test object to verify this method: in the domain of frequency = 1–1000 Hz, comparing with the test results of sample treatment by gold sputtering, the average error of frequency domain spectroscopy data in traditional method is 10%, while the new method is only 1%. Meanwhile, this paper analyses the cause of the errors in each method.


INTRODUCTION
Dielectric response technique is a non-destructive insulation testing method, which is widely used in analysis of dielectric characterization and diagnosis of equipment insulation state, including time-domain and frequency-domain dielectric response testing. The aim of frequency-domain dielectric response testing is to obtain frequency domain spectroscopy (FDS) by applying different frequency of sinusoidal voltage as excitation and calculate its relationship with response current. FDS not only intuitively reflects the polarization, conductivity, and loss characterization of dielectric materials, but also presents the relevant information of dielectric material's insulation state, such as moisture content and aging duration. By extracting FDS characteristic parameters, further diagnosis of the insulation health state of equipment can also be accomplished. [1][2][3][4][5]. However, the repeatability and accuracy of FDS parameters will be greatly affected by the arrangement of the electrode and dielectric sample. The traditional contact measurement method is to apply a certain contact pressure on the sample by electrode, which cannot completely eliminate the influence of small air gap between the sample and the electrode and the test repeatability and accuracy is often further reduced due to other problems like sample deformation and current distortion [5]. In this regard, scholars around the world have proposed various electrode arrangements and sample treatment methods to solve this contact problem, such as 'liquid' electrodes, semiconducting rubber electrode, comb electrodes etc. but will change the material properties, hard to fabricate and recycle, and yield increased loss factors at high frequencies, separately [6][7][8]. Sputtering nanosized gold on both sides of the sample is an effective method to solve the contact problem [9][10][11], which can make the sample surface equipotential to eliminate the influence of air gap and avoid the distortion of response current. Nevertheless, this method requires a lot of time and effort for sample treatment and is subject to the sample's sensitivity to temperature and high economic expense [5].
Contact-free measurement [6,12] is an ideal method to avoid the sample deformation caused by electrode extrusion. Chavez in [13] proves theoretically that the contact-free testing method can greatly improve the test accuracy through simulation calculation; X. Xu et al. proposed a contact-free measurement of solid dielectric response by air reference method

FIGURE 2
The three-dimensional schematic diagram of the dielectric with its unequal thickness (to clarify the meaning of h(x,y), the unevenness of the dielectric surface is slightly exaggerated) [5,14]. Although [5] confirms the advantage of test precision and repeatability. But the high dispersion of dielectric response on air gap, the sensitivity to the environmental factors and the composition of air like humidity, as well as the high demand of signal-to-noise ratio to testing equipment, will all limit the wide use of this method. This paper presents a various air gap contact-free dielectric response testing method, based on limit fitting by neural network, which has the potential to be widely used in various different circumstances. XLPE, a regular material used in high voltage condition [15,16] is chosen as the sample to verify the effectiveness of the method.

CONTACT-FREE MEASUREMENT AND AIR GAP IMPACT COMPENSATION
The contact-free measurement is to reserve a constant air gap d, like in Figure 1, to avoid a series of problems that will affect the accuracy and repeatability of the test, such as sample deformation, contact resistance, and current distortion caused by contact pressure [5]. The core of measurement is to compensate the influence of the reserved air gap d.
What the contact-free measurement measures is the joint response of the dielectric and the air gap, which can be considered as the response of the series connection of two capacitors. However, due to the uneven surface of the solid dielectric, the simple capacitor series expression will cause large errors. As shown in Figure 2, assuming (x,y) as a point in coordinates which is parallel to the electrode surface (electrode surface is assumed to be completely smooth), then the dielectric thickness of a certain point can be expressed as h(x,y). The dielectric capacitance dC 0 and air capacitance dC b corresponding to the unit area dxdy can be expressed as Equations (1) and (2), [5] separately: where˜b and ε 0 represent the complex permittivity of dielectric and air, respectively. D is the distance between the upper and lower electrode. The capacitance between unit surface of electrodes can be equivalent to the series connection of dC 0 and dC b , and the expression of the overall complex capacitance C m obtained by contact-free measurement is as follows: Substituting (1) and (2) into (3) can further obtain (4) If not consider the unevenness of the sample surface, in other words, the thickness of all parts of the surface is equal (h(x,y) = h 0 ), then the Equation (4) can be simplified as Equation (5), which is the series relationship of complex capacitors, where A represents the surface area of the dielectric.
However, the unevenness of the sample surface has a drastic impact on the test accuracy and repeatability, and to precisely obtain the distribution of h(x,y) requires a high-precision surface thickness scanner, which is obviously time-consuming, laborious and uneconomical. This paper will focus on how to invert the value of C b in high accuracy based on the contact-free test results of C m without the above problems.

Limit fitting
For any dielectric material, as shown in Figure 1, as the continuous reduction of distance D, the measured dielectric response and the calculated dielectric parameter will continuously approach the real response and value of the dielectric (the influence of air gap C 0 is continuously decreasing). It is conceivable that while D approaches the limit of the thickness of the dielectric itself (D→h(x,y)), the measurement result is certainly the true parameter of the dielectric, as shown in Figure 3 below. In the actual test, D can be reduced by moving the upper and lower electrodes to approximate h(x,y). As it can be observed from Figure 3, as D approximates h(x,y), tanσ m approximates tanσ b . In order to avoid a series of problems such as sample deformation and contact resistance, in physical or in actual contact-free measurement, changing D can reach max h(x,y) at most, that is the highest thickness of the dielectric surface (if below this values, electrodes will produce contact pressure).
As can be seen from Figure 3, there is still a big gap between the value measured on max h(x,y) with the reference value. The reason is that there is still a few of pit air gap between the electrode and the dielectric when D = max h(x,y). There is no doubt that if there are two imagined electrodes, the distance D between is exactly the same as the thickness of the dielectric h(x,y), the test result must be the true dielectric parameter of the dielectric. Although it can be hardly achieved physically (The self-made multifunctional electrode described in Section 3.1 has two highprecision polished copper electrodes, which means D can be considered as constant with respect to the (x,y) coordinates), but can be realized by the data fitting method, i.e. through the neural network and support vector machine described in Section 2.2, D is calculated in it to numerically approach h(x,y).
The following will further explain this limit fitting thought from the perspective of function and lay a mathematical foundation for the establishment of neural network to realize this thought. Initially, consider the expression of sputtered gold standard, in that it is considered that gold sputtering method can completely eliminate the ε 0 in Equation (4) introduced by air gap (the FDS result obtained by sputtering nano-sized gold on both sides of the sample is simplified as 'gold-sputtered standard' or symbol 'C b ' in this paper). Equation (6) can be obtained.
Expand˜b to ε1 + jε2, then the real part C b ′, imaginary part h(x,y) and 2 1 , respectively.
Similarly, from the Equation (4), the real part C m ′, imaginary part C m ′′ of C m can be obtained as: tanσ m is the ratio of Equations (7) and (8). It can be seen that if there is D→h(x,y), then C m →C b . Therefore, C m and C b expressed by Equations (4) and (6) have a functional relationship, which can be established by the limit fitting thought: Though D can hardly reach h(x,y) physically, as shown in Equations (7) and (8), C m ′, C m ′′ calculated by built-in algorithm in IDAX-300 or system identification method [2,3] from contact-free measurement, as well as variable D which can be directly and accurately controlled, do contain all the information needed to determine the true dielectric parameters of the dielectric material: ε 1 , ε 2 , h(x,y). Thus, the following describes the application of neural network as a specific method to realize limit fitting and by having the output of the neural network approach the true value of the dielectric parameter with sufficient accuracy.

Establishment of neural network model
A major advantage of the neural network model established in this paper is that it need not to get the exact h(x,y) for realizing D→h(x,y), and the idea of equivalent transformation functional expression can indirectly reach this point. Consider the right side of Equation (9) as an expression of function C b . Select C m ′, C m ′′, D obtained by contact-free measurement as the independent variables of the function (it has been proved in Section 2.1 that they contain all of the information needed for C b ). That is to say, the right side of Equation (9) can be deduced as an equivalent functional form of C b : According to functional analysis theory, [17][18][19] proved that under a wide range of conditions, the three-layer feedforward neural network model can approximate any function and its derivatives with arbitrary precision. Based on this, this paper expresses this function (Equation (10)) in the form of a ′′ are target outputs, tanσ b is obtained by the ratio of C b ′′ and C b ′. Actually, the structure of the neural network and its weights and threshold coefficients jointly express Equation (10) (also Equation (9)).
Therefore, the problem is transformed into establishing a suitable neural network model and a method to train its parameters to adapt to the application of decoupling the influence of air gap in contact-free measurement. First of all, assuming that FDS contain 11 frequency points from 1 to 1000 Hz, the general neural network model applies these 11 frequency points as different attributes of inputs, and the target outputs are also the true value of complex capacitance at these 11 frequency points. However, it seems that only one model is needed and the coupling relationship between various frequencies is considered as a large amount of data is introduced, it will lead to high-dimensional calculations and such a model will cause the neural network training algorithm fail to converge due to excessive information redundancy.
As shown in Figure 5, taking two frequency points as an example, if we focus on separately at 1000 and 110 Hz, the FDS has extremely strong monotonic uniqueness. If the information of two frequency points is used as input at the same time, it is obvious that there will be an overlap interval between the two curves. As points A, B in Figure 5, there is different D with the same tanσ b , which leads to the redundancy of information.
Thus, this paper establishes neural network for every frequency point as can be seen in Figure 4 above (assume that FDS contain 11 frequency points from 1 to 1000 Hz). Since the model is established, the training method of neural network

Definition of symbol Symbol
The output sum of hidden layer β j

Function sigmoid Sig
The threshold coefficient of the j th neuron in the output layer θ j The target output value of neural network The neural network parameters of the k th iteration, including Kernel space mapping function

Right vector w
Positive regularization parameter b parameters is further discussed. Use stochastic gradient descent method to initialize neural network parameters, which theoretically allows the training of neural network to reach arbitrary accuracy [20]. Table 1 shows the definition of the symbols used in the equation of neural network training.
During training, C m ′, C m ′′, D are input into the neural network through the input layer with calculation and then enter the hidden layer. Next, it is calculated with threshold coefficient and the operation of function sigmoid. The output C is obtained: C ideally is the true value of complex capacitance of the dielectric with air gap fully decoupled, but it must have a certain difference with the target output at the beginning of training, which is the gold-sputtered standard C b . E k and r, are thus defined to set the iteration step size h lm and convergence condition through LM algorithm.
According to LM algorithm, a larger error (E k , r) corresponds to a larger h lm , which can adapt to the dynamic changes of the algorithm process to speed up the convergence of the algorithm [21].
The convergence condition set by the algorithm is E k < 10 −4 , which means at this time, x (k) stops iterating and the model is built.
Another machine learning method, support vector machine (SVM) is also used. Compared with the aforementioned BP classic neural network to determine which model is more suitable for the implementation of the limit fitting method. Similar to the modelling establishment of BP neural network, we still use C m ′, C m ′′, D as inputs, and use E k to adjust the iteration step size. The difference is the SVM training method, which can be described by the following two equations: The basic principle is to calculate the input parameters C m ′, C m ′′, D through the kernel function and map them to a vector in the high-dimensional feature space. It is a way based on statistics to transform the function fitting problem into an optimization problem.

EXPERIMENTAL DESIGN FOR VALIDATION OF THE IMPROVEMENT OF THE METHOD
To verify the effectiveness of the measurement method described in this paper, XLPE is used as the test sample and the test results of gold-sputtered treatment of sample is used as standard (gold-sputtered standard). Comparing the contactfree measurement based on limit fitting by neural network and SVM as well as traditional contact measurement under different contact pressure, it is verified, to what extent, the limit fitting method improves the accuracy and repeatability. The test system designed is displaced in Figure 6.

Experimental setup
In order to realize the above method in Section 2, it is the key to accurately control the distance D between two electrodes. This research designs a self-made multifunctional electrode as shown in Figure 6. As the copper electrode is processed by the highprecision polishing, it can be considered that the unevenness The upper electrode is connected with a pressure sensor with an accuracy of 10 −2 kg to measure the contact pressure for verifying the influence of air gap and deformation on the test results. These two are connected by an L-shaped pressure arm from a sliding block, which is driven by a high-precision servo motor. Furthermore, D is measured by a high-precision displacement sensor whose accuracy is 10 −2 mm. To enable the neural network to decouple different thickness of air gap between the dielectric and the electrodes, it is necessary to change its value for measurement and input the test data into the neural network for training. For the verification and comparison experiment of traditional contact measurement, the downward movement of the upper electrode can be adjusted so that two electrodes can generate a constant contact pressure on the sample. The value of pressure and D can be precisely controlled by the electrode control box and displayed on its panel.
The electrode box shell is made of alloy, and an observation window made of conductive glass is designed to ensure the shielding of external electromagnetic interference, which is essential to ensure the measurement accuracy of the micro current especially in contact-free measurement. Meanwhile, in order to ensure the stability of air gap in contact-free measurement, O-ring seals are used at each opening to ensure its airtightness. If the test environment is not suitable for the measurement (for example, the humidity is too high), it can also be filled into dry inert gas such as nitrogen. IDAX-300, which is controlled by PC, is used to apply voltage excitation and collect response current to calculate C m ′, C m ′′ as well as C b ′, C b ′′.

Experimental procedure
25 XLPE samples with thicknesses of 1 and 2 mm were prepared for the experiment. In order to ensure the accuracy of the response current measurement to obtain accurate test data, the thickness of the XLPE sample should not exceed 10 mm. Randomly choose 20 of them as training set A; 5 of them as  Hz). Approximately 1-1000 Hz is a relatively common frequency range. This paper uses this frequency range to carry out experiments to verify the feasibility of the limit fitting method. The test method below 1 Hz or above 1000 Hz is completely consistent with this frequency range, and only differs in the value of the data obtained from the test, so the method in this article can be applied to other frequency points. FDS data was obtained by various air gap contact-free measurements. Recorded each group of (C′ m , C m ′′,D) as the input of the neural network and SVM model training. For the selection of D, it should be noted that the thickness of the air gap cannot be too large, otherwise two problems may occur: First, when the air gap is too large, it will weaken the information ratio of (C b ′,C b ′′), which will increase the training cost of neural network; second, it will cause the response current too small to be measured precisely and impose high demand of testing accuracy and filter function of the instrument. The tester used in this system is IDAX300, and all the curves in Figure 7 are measured by it. Under the same premise, the signal-to-noise (SNR) ratio will be greatly reduced, resulting in greater measurement errors. In this experiment, the variable air gap test interval was controlled within 1.00 mm, and the above-mentioned problems could be avoided according to amounts of experiments in the early stage. Therefore, if the thickness of the sample itself is 2 mm, the interval D should be [2,3.00] (unit: mm; selected test points in this interval and make their distribution uniform.) Because this experiment was carried out in Chongqing, the relative humidity of the air was high, which is around 60%, and the temperature was around 15 • C according to the temperature measured in our self-made electrode. After completing the contact-free measurement data collection of various air gaps, the sample in the training set A were sputtered with nano-sized gold and tested as training standard. Input those data into neural network and SVM for training according to the method described in Section 2.2 and iterated repeatedly until the convergence condition of algorithm was reached.
Then took the samples in the validation set B, randomly selected D (but should also meets the requirements of interval described above) for contact-free measurement. Input the measured data (C m ′, C m ′′,D) into the trained neural network and SVM to obtain the output (C n ′, C n ′′), (C s ′, C s ′′), separately. Afterwards, obtained FDS data (C p ′, C p ′′) of these samples in set B from traditional contact measurement. Adjusted different contact pressure by electrode control box to measure and calculate (C p ′, C p ′′). In order to ensure that the sample underwent elastic deformation but not plastic deformation, the contact pressure did not exceed 5.00 kg. Finally, the sample was treated with ion sputtering coating apparatus to obtain (C b ′, C b ′′) in different contact pressure. As C b is selected as the standard to calculate the error in this experiment, change the contact pressure is to make sure its reasonableness as a standard. Comparing (C n ′, C n ′′), (C s ′, C s ′′), (C p ′, C p ′′) with (C b ′, C b ′′) as well as tanσ from those three measurement method, analysed the error cause from the measurement. Table 2 shows the corresponding test results. Figure 7 shows the test results of a specific sample in set B from measurement of max h(x,y), contact-free measurement, traditional contact measurement, result of limit fitting by neural network, result of fitting by SVM, and gold-sputtered standard. Under different contact pressures, the average error of tanσ p and (C p ′, C p ′′) compared with gold-sputtered standard is 10%. While the average error of limit fitting by neural network (C′, C′′) and the error of tanσ is within 1% as can be shown in Figure 11 below (only slightly greater than 1% at individual frequency points), in other words, the accuracy level is greatly improved. In addition, the average error of fitting by SVM is about 3%, which is higher than the fitting method by neural network.

Error analysis of traditional contact measurement and gold-sputtered standard
It can be seen from Figure 8 that as the increase of the contact pressure, the error of the traditional contact measurement is smaller, but at the same time, the shape of FDS is deformed.
Gold-sputtered standard curve in Figure 8 is measured under different contact pressure from 0-5 kg. Within the allowable error range of IDAX-300, they completely overlap into one curve. It verifies that sputtering gold is an effective treatment to solve the contact problem caused by the air gap and prove its rationality as a standard in this experiment.
Combined with Figure 7, max h(x,y) curve, which is shown in Figure 9(a), the distance between two electrodes D reaches max h(x,y) with the accuracy of 10 −2 mm and the air gap still exists, which is the main cause of the error. It can be seen from Figure 9(b) that as the contact pressure of the electrode to the sample increases, the pit air gap decreases to a certain extent, which can eliminate a certain error caused by the pit air gap, so that the traditional contact measurement curve in Figure 7 is between the max h(x,y) curve and the gold-sputtered standard. However, due to the unevenness of the sample surface, it is impossible to make a pit-free air gap between the sample of a certain hardness and electrode through extrusion even if not considering whether the deformation of the sample can be recovered [5]. And this pit air gap as well as the deformation of the sample cause the error of traditional contact measurement, which is about 10% on average.
Nevertheless, Figure 8 shows that for XLPE samples, when the contact pressure is 4.77 kg, the average of test results at each frequency points is only about 1% (slightly greater than 1%), that is, the degree of influence from the pit air gap is greater than the one from sample deformation. Therefore, for occasions where the demand of accuracy level is not high, traditional contact measurement can be applied for its convenience, and the contact pressure is recommended to be as high as possible, unless it will cause the plastic deformation, to reduce the impact of the pit air gap as shown in Figure 9. But it must be admitted that the three curve shapes are changed compared with goldsputtered standard; in other words, as the pressure increases, the overall error decreases though, there will even be an increase in error at a certain frequency and there are two reasons for this error increase. First, additional contact resistance will be generated and increased as the increase of contact pressure [5]. In fact, this is also one of the reasons with higher contact pressure, the traditional contact measurement curve approaches the goldsputtered standard because of the increase of imaginary part of complex capacitance. The second reason is the deformation of the sample, which is shown in Figure 9.
In practice, traditional contact measurement uses the fixed method such as screw and electrode weight to randomly apply contact pressure to the measured dielectric. It is unknown whether it produces plastic deformation on the sample or is effective enough to eliminate the influence of air gap. The selfmade multifunctional electrode described in Section 3.1 provides a solution for its precise measurement and control of the contact pressure.

Result and analysis of limit fitting by neural network and SVM
Set A are the samples input to the neural network and SVM for parameter training. The convergence condition set in algorithm is E k < 10 −4 , and the training effect of 2 mm sample in set A is shown in Figure 10. The limit fitting output of neural network  Figure 10 almost coincides with the gold-sputtered standard curve. It can also be seen that the training result of SVM is not as good as the neural network. The effect of (C′, C′′) in set B for validation is shown in Figure 11 and the effect of tanσ can be seen in Figure 7.
Through analysis, the errors of C′, C′′, tanσ from neural network in set B are all within 1% (only slightly more than 1% at individual frequency points), but the error of SVM with standard is about 3%. The reason is that the SVM itself is mainly a model designed for classification. Although the function fitting can also be applied in this method, by transformation of a classification question, but the effect will be slightly inferior to that of the neural network.
The factors that cause errors in neural network include certain accidental errors in neural network training. The stochastic gradient descent method described in Section 2.2 can theoretically make the accuracy of the model meet any requirement, but it is subjected to the limit such as time and cost in actual operation. Training is stopped while the model parameters meet the demand of engineering application. As in this experiment, convergence condition is designed as E k < 10 −4 for data in set A. In addition, the input data itself to train the neural network has certain errors, including the limitation and errors of displacement sensor and distance adjustment instrument. The signal process and data analysis of IDAX-300 also limits to its own accuracy level.
Although the above-mentioned error causes still exist, compared with traditional contact measurement, contact-free measurement fundamentally solves the problem of sample surface contact. Meanwhile, because the amount of data for training the neural network is more than an order of magnitude larger than a single traditional contact measurement, the accidental error caused by the operation of the tester can be eliminated through big data analysis methods. Furthermore, after the establishment of neural network model, for one group of (C m ′, C m ′′, D), the time from data input to result output is realized within 1s, which means the decoupling operation of dielectric and air gap does not increase extra time-consuming relative to the total test time of several hours.
It is noteworthy that, based on the derivation and theoretical demonstration in Section 2.1, the limit fitting method as a test thought is applicable in different circumstances for various dielectric materials. The different types of dielectric material, different thicknesses, sizes, water content, aging degrees, even different air composition only cause different numerical values of C m ′, C m ′′ and their corresponding C b ′, C b ′′. No matter what change in the variables mentioned above, all the functional relationship described in Section 2.1 are valid and will not be changed. Limit fitting as a kind of test idea is not necessarily realized through the way of training and verifying neural network. Other more superior machine learning methods or mathematical models based on limit fitting can also be adopted and is expected to achieve the same or even better decoupling effect. The above-mentioned related researches are being carried out by the author and his research team.

CONCLUSION
This paper studies a contact-free test method for the dielectric properties of insulating materials. Starting from limit fitting idea, the neural network for each frequency point is established and is trained using the measurement results of various air gaps to achieve the solution of decoupling the parameters of dielectric and air gap. The comparison of results between this method and traditional contact measurement method shows that the test results of traditional test method are highly dispersed and are greatly affected by the contact pressure with 10% of average error; while the error of the method presented in this paper is only 1%, and the dispersion is tiny. This paper also compares the effect of the two fitting methods of neural network and SVM, and found that neural network is a more optimized solution.
According to the error analysis, the self-made multifunctional electrode can realize both contact-free measurements based on limit fitting and traditional one with sufficient accuracy.