On the Distribution of Breakdown Voltage in the Insulation of Enameled Wires
Abstract Generated abstract
The paper examines statistical modeling of breakdown voltage in enameled wire insulation for reliability calculations of electrical machine windings. It argues that a single extreme-value distribution may be inadequate when insulation failures arise from both microdefects and macrodefects, producing a bimodal distribution. Using breakdown tests on 1572 PETV wire specimens in wire-shot electrodes at industrial-frequency alternating voltage, the study fits the observed frequencies with superpositions of distribution laws. Agreement tests indicate that the distribution is described most satisfactorily by a superposition of two Weibull laws, or by a normal law combined with a Weibull law, supporting the treatment of distinct defect populations in insulation quality assessment.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR
named after S. M. Kirov
Volume 222 1975
ON THE DISTRIBUTION OF BREAKDOWN VOLTAGE IN THE INSULATION OF ENAMELED WIRES
Yu. P. POKHODKOV
(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)
Statistical assessment of insulation quality is finding increasingly broad application in connection with the solution of problems concerning the investigation and calculation of reliability indices for windings of electrical machines [1, 2, 3]. In this, the basis of most failure models for an insulation element is the expression
\[ P(u)=\int_{-\infty}^{u} f(U_{\mathrm{pr}})\,P(u)\,du, \tag{1} \]
where \(P(u)\) is the probability of breakdown of an insulation element;
\(f(U_{\mathrm{pr}})\) is the probability density of the breakdown voltage of the insulation element;
\(P(u)\) is the distribution function of the probabilities of switching overvoltages.
The accuracy of estimating the probability of breakdown of an insulation element depends to a large extent on the correctness of choosing the distribution law for approximating the series of breakdown-voltage values obtained experimentally.
When insulation specimens are tested for breakdown, the probability of occurrence of values \(U_{\mathrm{pr}}\) comparable with the values of switching overvoltages is very small, and for a reliable estimate of it a large number of specimens must be tested, which is not always possible and is not economical. In this connection, a well-known procedure is usually applied: the series of values \(U_{\mathrm{pr}}\) obtained on a comparatively small number of specimens is approximated by one of the known distribution laws of random variables, and then, according to this law, the probability of occurrence of those values of \(U_{\mathrm{pr}}\) that were not observed in the experiment is estimated.
In this case, as a rule, the agreement between the observed and approximating series is assessed without taking into account the degree of divergence of the series in the region of small values of \(U_{\mathrm{pr}}\), which are used for calculating the probability of breakdown of an insulation element. The use of such a procedure is justified only in those cases where there is confidence that the theoretical and experimental distributions are formed on the same basis. Thus, the distribution of breakdown-voltage values for insulation specimens of finite area should in principle be described by one of the laws of the distribution of extreme values [4]. This is due to the fact that breakdown of an insulation specimen of finite area occurs at the weakest point, and the value of \(U_{\mathrm{pr}}\) obtained in this case is the smallest of
all values of \(U_{\mathrm{br}}\) of the given specimen. However, it is not in all cases correct to use one of the extreme-value distribution laws (Weibull, double-exponential, limiting, etc.) to approximate the series of \(U_{\mathrm{br}}\). In particular, this applies to the insulation of enameled wires used in random windings of asynchronous motors. Breakdown of the insulation of winding enameled wires occurs at points of various weakenings and defects. The number of types of defects and weakenings in the insulation may be rather large, but from the point of view of the formation of the distribution of \(U_{\mathrm{br}}\) it seems advisable to divide all defects and weakenings into two groups: microdefects and macrodefects.
Microdefects may include various structural disturbances, inhomogeneities, and inclusions whose size is negligibly small in comparison with the insulation thickness.
Macrodefects may include gross mechanical damage (punctures, cuts, cracks, blisters, skips, etc.), conductive inclusions whose size is comparable with the insulation thickness.
The presence of defects of both groups in one sample of tested specimens may lead to a bimodal distribution, since the mean value of the breakdown voltage of insulation having gross mechanical damage differs substantially from the mean value of \(U_{\mathrm{br}}\) of insulation not having such damage [5].
Such distributions of \(U_{\mathrm{br}}\) may be observed not only in the breakdown of wire specimens that have undergone various technological or service effects, but also in the breakdown of wire specimens selected from coils (in the as-delivered condition).
Table 1 gives the values of the frequencies of the distribution of the breakdown voltage of PETV wire insulation with a diameter of \(1.08\) mm in the as-delivered condition.
Tests of the wire insulation for breakdown were carried out in “wire—shot” electrodes with an industrial-frequency alternating voltage. The voltage rise rate was \(1\ \mathrm{kV/sec}\). The length of the tested part of the specimen was 95 mm. The sample size in the tests was 1572 specimens.
The presence of a mode in the left part of the distribution indicates microdefects present in the wire insulation, which are point through-damage of the insulation [5]. The right mode of the distribution is apparently formed from the values of \(U_{\mathrm{br}}\) obtained during breakdown of specimens at microdefects.
Table 1 also gives values of the fitted frequencies, estimated by superposition of two laws. In the general case the fitted frequencies \((\tilde n_i)\) were estimated from the expression
\[ \tilde n_i = [q f_1(U_{\mathrm{br}}) + m f_2(U_{\mathrm{br}})] \Delta U \Sigma n_i, \tag{2} \]
where \(f_1(U_{\mathrm{br}})\) and \(f_2(U_{\mathrm{br}})\) are the probability distribution densities, respectively, of the first and second laws in the superposition;
\(q\) and \(m\) are, respectively, the fraction of each law in the superposition, \(q + m = 1\);
\(\Delta U\) is the discharge interval, \(\Delta U = 0.5\ \mathrm{kV}\);
\(\Sigma n_i\) is the sample size in the tests, \(\Sigma n_i = 1572\) specimens.
Approximation of the right-hand part of the distribution by the Weibull law with parameters \(\alpha_2 = 5.17,\ C_2 = 0.000371\) makes it possible to determine the values \(m\) and \(q\)
\[ m = \frac{\Sigma \tilde n_{i,\mathrm{right}}}{\Sigma n_i}, \tag{3} \]
Table 1
| Discharge value $U_i$ (kV) | Observed frequency $n_i$ | Estimate of equalizing frequencies by superpositions of distribution laws: two Weibull laws $\tilde n_{i1}$ | Estimate of equalizing frequencies by superpositions of distribution laws: truncated normal law and Weibull law $\tilde n_{i2}$ | Estimate of equalizing frequencies by superpositions of distribution laws: normal law and Weibull law $\tilde n_{i3}$ |
|---|---|---|---|---|
| 0.25 | 23 | 22.9 | 38.4 | 35.4 |
| 0.75 | 92 | 80 | 81.2 | 75.2 |
| 1.25 | 91 | 105.4 | 105 | 97.3 |
| 1.75 | 96 | 88.4 | 92.9 | 86.6 |
| 2.25 | 74 | 76.6 | 78.6 | 75.6 |
| 2.75 | 102 | 89.3 | 88 | 87.1 |
| 3.25 | 153 | 142.9 | 141.9 | 141.7 |
| 3.75 | 188 | 212 | 212 | 212 |
| 4.25 | 242 | 259 | 259 | 259 |
| 4.75 | 238 | 250 | 250 | 250 |
| 5.25 | 177 | 177 | 177 | 177 |
| 5.75 | 75 | 76.6 | 76.6 | 76.6 |
| 6.25 | 21 | 20.4 | 20.4 | 20.4 |
| Assessment of the agreement of distributions by the $\chi^2$ criterion | Assessment of the agreement of distributions by the $\chi^2$ criterion | Assessment of the agreement of distributions by the $\chi^2$ criterion | Assessment of the agreement of distributions by the $\chi^2$ criterion | Assessment of the agreement of distributions by the $\chi^2$ criterion |
| $\dfrac{|\chi^2-\nu|}{\sqrt{2\nu}}\leq 3$ | $0.53<3$ | $2.39<3$ | $1.29<3$ |
where $\sum \tilde n_{i\text{right}}$ is the sum of the equalizing frequencies estimated according to the Weibull law with parameters $\alpha_2$ and $C_2$.
For this batch of tested specimens $m=0.805$, $q=0.195$.
The parameters of the law approximating the left-hand part of the overall distribution are determined from the frequency distribution in the left-hand part of the distribution
\[ n_{i\text{left}}=n_i-\tilde n_{i\text{right}} . \]
In the distribution presented, its left-hand part is described sufficiently well by any of three laws—Weibull, truncated normal, and normal. Accordingly, the overall distribution of breakdown voltage of the insulation of PETV enameled wire of 1.08 mm can be approximated by the following superpositions of laws:
1) by a superposition of two Weibull laws
\[ f(U_{\text{br}})=0.195\,\alpha_1 C_1 U^{\alpha_1-1}\cdot e^{-C_1U^{\alpha_1}} +0.805\,\alpha_2 C_2 U^{\alpha_2-1}\cdot e^{-C_2U^{\alpha_2}}, \tag{4} \]
where $\alpha_1=2.3$; $C_1=0.36$; $\alpha_2=5.17$; $C_2=0.00037$; 2) by a superposition of a truncated normal law and a Weibull law
\[ f(U_{\text{br}})=0.195\cdot C\cdot e^{-\frac{(U-\overline U)^2}{2\sigma^2}} +0.805\cdot \alpha_2 C_2 U^{\alpha_2-1}\cdot e^{-C_2U^{\alpha_2}}, \tag{5} \]
where $C=0.608$; $\overline U=1.25$; $\sigma=0.715$;
3) by a superposition of a normal law and the Weibull distribution law
\[ f(U_{\mathrm{br}})=0.195\,\frac{1}{\sqrt{2\pi}\cdot\sigma}\cdot e^{-\frac{(U-\bar U)^2}{2\sigma^2}} +0.805\,\alpha_2 C_2 U^{-\alpha_2-1}\cdot e^{-C_2 U^{\alpha_2}}. \tag{6} \]
An assessment of the agreement between the experimental and approximating distributions, carried out by the \(\chi^2\) criterion using Romanovsky’s rule [6], showed (Table 1) that the resulting distribution is described most satisfactorily by the superposition of two Weibull laws, as well as by the superposition of the normal law and the Weibull law.
REFERENCES
-
O. D. Gol’dberg. Quality and Reliability of Asynchronous Motors. Moscow, “Energiya,” 1968.
-
Kh. S. Fishman. Statistical model and assessment of the reliability of the insulation of electrical machines. In: Power Engineering and Electrification of KPI. Karaganda, 1972.
-
N. A. Kozyrev. Insulation of Electrical Machines and Methods for Its Testing. Moscow, “Energiya,” 1963.
-
E. J. Gumbel. Statistics of Extremes. Moscow, “Mir,” 1965.
-
Yu. P. Pokholkov, V. I. Derevyanko. On a method for determining the defectiveness of winding wires used in low-voltage electrical equipment. Proceedings of TPI, vol. 284, Tomsk, TSU Publishing House, 1974.
-
A. K. Mitropol’skii. The Technique of Statistical Computations. Moscow, “Nauka,” 1971.