Determination of the Correlation Coefficient Between Breakdown Voltages and Temperature for Polymers and Their Combinations with Transformer Oil
Abstract Generated abstract
This paper applies rank correlation analysis to experimental data on impulse breakdown voltages of polymer dielectrics and polymer, transformer oil insulation systems over a temperature range from minus 40 to plus 90 degrees Celsius. Using correlation coefficients for voltage application times from 2 x 10^-7 to 3 x 10^-6 seconds, it evaluates how strongly breakdown voltage varies with temperature for several polymers and for combinations in which the solid dielectric is placed at the pointed electrode. The results indicate that low-density polyethylene, polypropylene, and polytetrafluoroethylene show little temperature dependence, while polycaprolactam, polymethyl methacrylate, and especially polyisobutylene show decreasing breakdown voltage with increasing temperature. For combined insulation, the temperature dependence is found to be governed mainly by the dielectric located in the high-field region.
Full Text
DETERMINATION OF THE CORRELATION COEFFICIENT BETWEEN BREAKDOWN VOLTAGES AND TEMPERATURE FOR POLYMERS AND THEIR COMBINATIONS WITH TRANSFORMER OIL
Yu. I. Lekht
(Presented by the scientific seminar of the Research Institute of High Voltages)
Impulse electrical breakdown of polymer dielectrics and their combinations with transformer oil is determined to a considerable degree by the electrophysical properties of the polymers. However, it is not possible to reveal an approximate quantitative dependence of changes in impulse breakdown voltages as a function of temperature and of certain electrophysical properties because of the scatter of the experimental results. In this connection it is expedient to apply the method of correlation analysis to reveal dependences between breakdown voltages and the electrophysical characteristics of polymers. We carried out a correlation analysis of the impulse breakdown voltages of polymers and their combinations with transformer oil as a function of temperature.
In the present work, the quantitative expression of the closeness of the relationship between interrelated attributes was determined with the aid of the rank correlation coefficient [1]. To determine this coefficient, all values of the factorial attribute (temperature, dielectric permittivity, etc.) are written in increasing order, and the corresponding values of the resultant attribute (breakdown voltages) are written in decreasing order. The rank is determined for both attributes, i.e., the number of each attribute in the ranked series.
The value of the rank correlation coefficient is determined by the formula [1]
\[ r = 1 - \frac{6\sum d^2}{n(n^2 - 1)}, \tag{1} \]
where \(d\) is the difference between two ranks in adjacent series;
\(n\) is the number of measurements.
Table 1 gives the rank correlation coefficients between the impulse breakdown voltages of polymer dielectrics and temperature for voltage action times from \(2 \cdot 10^{-7}\) to \(3 \cdot 10^{-6}\) sec (\(U_{\text{br}}\) in kV, temperature in degrees Kelvin).
The minus sign before the correlation coefficients means that the relationship between breakdown voltages and temperature is inverse, i.e., as the temperature increases, \(U_{\text{br}}\) decreases.
As can be seen from Table 1, the rank coefficients between the breakdown voltages of low-density polyethylene, polypropylene, and poly-
tetrafluoroethylene and temperature are practically close to zero in the range from \(-40\) to \(+90^\circ\mathrm{C}\) for the investigated voltage application times. Consequently, the impulse breakdown voltages of the above-mentioned materials are practically independent of temperature. Insignificant deviations of the rank correlation coefficients from zero are probably associated with scatter in the breakdown voltages.
Table 1
| Material | Correlation coefficient | Correlation coefficient | Correlation coefficient |
|---|---|---|---|
| impulse-front duration, sec | impulse-front duration, sec | impulse-front duration, sec | |
| \(2\cdot10^{-7}\) | \(5\cdot10^{-7}\) | \(3\cdot10^{-6}\) | |
| Low-density polyethylene | 0.09 | 0.05 | 0.04 |
| Polypropylene | 0.12 | 0.10 | 0.06 |
| Polytetrafluoroethylene | 0.20 | 0.17 | 0.10 |
| Polycaprolactam | \(-0.35\) | \(-0.40\) | \(-0.57\) |
| Polymethyl methacrylate | \(-0.63\) | \(-0.74\) | \(-0.80\) |
| Polyisobutylene | \(-0.87\) | \(-0.9\) | \(-0.95\) |
As can be seen from Table 1, the breakdown voltages of polycaprolactam, polymethyl methacrylate, and polyisobutylene depend on temperature. As the latter increases, the breakdown voltages of these materials decrease. The strongest connection between breakdown voltages and temperature is found for polyisobutylene (correlation coefficient above 0.8), and the weakest for polycaprolactam (the correlation coefficient is 0.57 and lower). Moreover, as the voltage application time increases, the dependence of the impulse breakdown voltages on temperature becomes stronger. This is explained by the large influence of the change in the density of this material in the range from \(-40\) to \(+90^\circ\mathrm{C}\).
For the combined insulation “polymer—transformer oil,” with the solid dielectric located at the pointed electrode, the rank correlation coefficients between impulse breakdown voltages and temperature for voltage application times from \(2\cdot10^{-7}\) to \(3\cdot10^{-6}\) sec are given in Table 2.
As can be seen from Table 2, the correlation coefficients for such systems are approximately the same as the corresponding correlation coefficients for the solid polymer. The latter indicates that, in the breakdown of combined insulation, the main role is played by the dielectric located in the region of high field strengths.
A certain increase in the rank correlation coefficient has been noted for combined insulation in comparison with the correlation coefficients for the polymer dielectric. Probably, the increase in the correlation coefficients is connected with the fact that the temperature dependence of \(U_{\mathrm{br}}\) of transformer oil is stronger. And the oil, as a component part of the combined insulation, introduces changes into the total correlation coefficient.
The results of Table 2 make it possible to conclude that the impulse breakdown voltages of combined insulation consisting of polyethylene, polypropylene, and polytetrafluoroethylene at the pointed electrode and transformer oil at the plane electrode are practically independent of temperature. The breakdown voltages of the combination “polycaprolactam-
“transformer oil” and “polymethyl methacrylate–transformer oil” decreases with increasing temperature. Moreover, as the duration of voltage application increases, the dependence of \(U_{\mathrm{br}}\) on temperature becomes stronger. The relationship between breakdown voltages and temperature for the “polymethyl methacrylate–transformer oil” insulation is stronger than the relationship between \(U_{\mathrm{br}}\) and temperature for the “polycaprolactam–transformer oil” insulation.
Table 2
| Material | Correlation coefficient, rise-front duration \(2 \cdot 10^{-7}\) | Correlation coefficient, rise-front duration \(5 \cdot 10^{-7}\) | Correlation coefficient, pulse, sec \(3 \cdot 10^{-6}\) |
|---|---|---|---|
| Low-density polyethylene–transformer oil | \(-0.19\) | \(-0.16\) | \(-0.09\) |
| Polypropylene–transformer oil | \(-0.20\) | \(-0.18\) | \(-0.10\) |
| Polytetrafluoroethylene–transformer oil | \(-0.20\) | \(-0.17\) | \(-0.14\) |
| Polycaprolactam–transformer oil | \(-0.38\) | \(-0.40\) | \(-0.59\) |
| Polymethyl methacrylate–transformer oil | \(-0.50\) | \(-0.76\) | \(-0.80\) |
The correlation coefficients found between the impulse breakdown voltages and temperature of polymer dielectrics and their combinations with transformer oil make it possible to draw the following conclusions:
-
The impulse breakdown voltages of low-density polyethylene, polytetrafluoroethylene, and polypropylene are practically independent of temperature for voltage application times from \(2 \cdot 10^{-7}\) to \(3 \cdot 10^{-6}\) sec in the temperature range from \(-40\) to \(+90^\circ\mathrm{C}\). The impulse voltages of polycaprolactam decrease slightly with increasing temperature. A strong dependence of \(U_{\mathrm{br}}\) on temperature is observed for polymethyl methacrylate and polyisobutylene. The breakdown voltages of the latter decrease as the temperature rises.
-
The dependence of the impulse breakdown voltages of combined insulation on temperature is determined mainly by the temperature dependence of the dielectric located in the region of high field strengths.
REFERENCES
- I. G. Venetskii, G. S. Kil’dishev. Fundamentals of Mathematical Statistics. Gosstatizdat, Moscow, 1963.