Theoretical Distribution of Breakdown Voltage in Turn Insulation of Electrical Machine Windings
Abstract Generated abstract
This paper develops a probabilistic model for the breakdown voltage distribution of turn-to-turn insulation in electrical machine windings, motivated by the difficulty of estimating rare weak points from small test samples by formal curve fitting. It relates enamel layer formation to normal thickness distributions, applies extreme-value distributions to breakdown in shot-bed and twist tests, and represents damaged winding wire as a mixture of undamaged and damaged elementary sections. Using experimental breakdown data, the analysis estimates a damage probability of about 0.010 to 0.015 and models damaged-section breakdown voltages by a normal distribution with mean 2.25 kV and standard deviation 1.00 kV. The resulting mixture formulation is used to derive an expression for a lower bound on winding reliability under dense conductor packing.
Full Text
THEORETICAL DISTRIBUTION OF THE BREAKDOWN VOLTAGE OF TURN-TO-TURN INSULATION IN ELECTRICAL MACHINE WINDINGS
E. K. STREL’BITSKII
(The work was presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
In analyzing changes in the quality of wire at various stages of the technological process of manufacturing a winding, the problem arises of determining defectiveness, i.e., the fraction of weak points, from the shape of the breakdown-voltage distribution curve. Since even with a comparatively low winding reliability on the order of 0.9 the fraction of defective pieces of wire of standard length is \(10^{-4} \div 10^{-5}\), and a reasonable number of specimens in testing should not exceed several hundred, the formal use, as fitting curves, of theoretical distributions of the Pearson type and their extrapolation into the region of small breakdown-voltage values may lead to large errors.
In such cases the possibilities of a formal approach are exhausted; therefore it is necessary to consider the mechanism underlying the formation of random values of the breakdown voltage.
The choice of a suitable distribution may prove to be a difficult problem, since the analytical form of the distribution depends on the failure mechanism of the particular device. As a result of extensive experimental work it has been shown that there exists a small number of probability distributions whose applicability is almost universal. This conclusion is based above all on the principle of finding the simplest functional forms of distribution. The basis for such an approach is the common circumstance that simple assumptions make it possible to obtain broadly applicable results [1].
Indeed, the formation of the enamel-coating film is produced by a simple operation—passing the wire through a die; and the infliction of damage results from impacts of a hammer on the end parts and from pressing the wire into a slot by means of a slot-insertion plate and an opposing wedge.
Let us consider mathematical algorithms that model these operations.
The technological process of applying enamel insulation consists in repeatedly passing the wire through dies of increasing diameter. Since the amplitude of the wire’s oscillations before the die is considerably greater than the gap between the inner surface of the die and the wire, the thickness of one-sided insulation in one pass varies in the interval \([0; 2\delta]\), where \(\delta\) is the mean layer thickness. The information given provides grounds for assuming the distribution of the one-sided thickness of one layer to be uniform in the interval \([0; 2\delta]\). With increasing
for the number of layers, the distribution of the sum of the breakdown voltages of the layers is asymptotically normal, with parameters
\[ m_{\Delta}=r\bar{\delta}\quad \text{and}\quad \sigma_{\Delta}^{2}=\frac{r}{3}\cdot \bar{\delta}^{2}, \]
where \(r\) is the number of layers.
When wire is tested in a shot bed and in twists, breakdown occurs at the point having the smallest thickness. Therefore, in these cases the distribution of the breakdown voltage of an undamaged wire will be the distribution of extreme values of a normal distribution [2].
For experimental verification, distributions of the breakdown voltages of PEVL wire, \(\varnothing\ 0.87\), were obtained in the following cases:
a. Two insulated wires placed crosswise. The distribution is normal with parameters: \(m_u=16\ \text{kV}\), \(\sigma_u=2.9\ \text{kV}\).
b. An insulated wire and a bare wire placed crosswise. The distribution is normal with parameters: \(m_u=8\ \text{kV}\), \(\sigma_u=2\ \text{kV}\).
c. An insulated wire \(125\ \text{mm}\) long in a shot bed with a diameter of \(2.5\ \text{mm}\).
The distribution corresponds to the distribution of extreme values of the distribution \(\delta\) for a sample size \(N_{\text{dr}}=120\).
d. A twist \(125\ \text{mm}\) long made of two insulated wires. The distribution corresponds to the distribution of extreme values of the distribution \(a\) for a sample size \(N_{\text{skr}}=130\).
It is characteristic that in the last case
\[ \frac{l_{\text{pr}}}{d_{\text{pr}}}\approx N_{\text{skr}}, \]
i.e., it may be assumed that the breakdown voltages of neighboring sections in the twist are not correlated with one another.
When a winding is laid, some elementary sections are damaged—the thickness is reduced. The fraction of damaged sections and the distribution law of the depth of damage are extremely difficult to establish by direct experiment. Therefore, we proceeded by making an a priori assumption about the distribution law, followed by a goodness-of-fit check, again keeping in mind above all simple distributions. Thus, exponential and normal laws were tested, as well as a deterministic value. The most suitable proved to be the variant with a normally distributed depth of damage.
It is natural to seek the density of the distribution of the thickness of elementary sections of damaged wire in the form
\[ \varphi(\Delta)=p\varphi_{\text{p}}(\Delta)+(1-p)\varphi_{\text{n}}(\Delta), \tag{1} \]
where \(p\) is the probability of damage to an elementary section; \(\varphi_{\text{p}}(\Delta)\), \(\varphi_{\text{n}}(\Delta)\) are the distribution densities of the thickness of damaged and undamaged elementary sections, respectively.
The parameters of the distribution \(\varphi_{\text{n}}(\Delta)\) are known to us. The parameter \(p\) can be determined in the following way. The density of the distribution of breakdown voltages of a segment of undamaged wire in a shot bed at the point lying in the right-hand part of the distribution curve is:
\[ f_{\text{n}}(\tilde{U}_{\text{np}})= N\varphi_{\text{n}}(\tilde{U}_{\text{np}}) \left[ 1-\int_{0}^{\tilde{U}_{\text{np}}}\varphi_{\text{n}}(U_{\text{np}})\,du \right]^{N-1}, \tag{2} \]
where \(U_{\text{np}}\) is the mode.
For a damaged wire, the density at the same point
\[ f_{\mathrm{p}}(\widetilde{U}_{\mathrm{np}})= N\,[p\varphi_{\mathrm{p}}(u)+(1-p)\varphi_{\mathrm{n}}(u)] \times \left[ 1-(1-p)\int_{0}^{\widetilde{u}_{\mathrm{np}}}\varphi_{\mathrm{n}}(u)\,du -p\int_{0}^{\widetilde{u}_{\mathrm{np}}}\varphi_{\mathrm{p}}(u)\,du \right]^{N-1}. \tag{3} \]
As will be shown below, \(p\) is small, and
\[ \int_{0}^{\widetilde{u}_{\mathrm{np}}}\varphi_{\mathrm{p}}(u)\,du \to 1. \]
Dividing (3) by (2), and taking the above into account, we have
\[ \frac{f_{\mathrm{p}}(\widetilde{u}_{\mathrm{np}})} {i_{\mathrm{n}}(\widetilde{u}_{\mathrm{np}})} \simeq \left[ \frac{ 1-\displaystyle\int_{0}^{\widetilde{u}_{\mathrm{np}}}\varphi_{\mathrm{n}}(u)\,du-p }{ 1-\displaystyle\int_{0}^{\widetilde{u}_{\mathrm{np}}}\varphi_{\mathrm{n}}(u)\,du } \right]^{N-1}. \]
Hence
\[ p= \left[ 1-\int_{0}^{\widetilde{u}_{\mathrm{np}}}\varphi(u)\,du \right] \left[ 1- \left( \frac{f_{\mathrm{p}}(\widetilde{u}_{\mathrm{np}})} {i_{\mathrm{n}}(\widetilde{u}_{\mathrm{np}})} \right)^{\frac{1}{N-1}} \right]. \tag{4} \]
The calculations carried out showed that, first, \(p=0.010\div0.015\), and second, the distribution \(\varphi_{\mathrm{p}}(u_{\mathrm{p}})\) is shifted strongly to the left in comparison with the distribution \(\varphi_{\mathrm{n}}(u_{\mathrm{n}})\). Therefore, in the region of small values of \(u\),
\[ \varphi_{\mathrm{n}}(u)\simeq 0;\qquad \int_{0}^{u}\varphi_{\mathrm{n}}(u)\,du\simeq 0, \]
which gives grounds for replacing formula (3) by its approximate expression
\[ f_{\mathrm{p}}(u_{\mathrm{p}})\simeq Np\,\varphi_{\mathrm{p}}(u) \left[ 1-p\int_{0}^{u}\varphi_{\mathrm{p}}(u)\,du \right]^{N-1}. \tag{5} \]
Although expression (5) has the form of a density of the distribution of extreme values, because of the smallness of \(p\) in the interval \([0;\widetilde{u}_{\mathrm{np}}]\), where \(\widetilde{u}_{\mathrm{np}}\) is the median of the distribution \(\varphi_{\mathrm{p}}(u_{\mathrm{p}})\),
\[ \left[ 1-p\int_{0}^{u}\varphi_{\mathrm{p}}(u)\,du \right]^{N-1} \le (1-0.005)^{N-1} \]
and \(f_{\mathrm{p}}(u_{\mathrm{p}})\), to within the factor \(Np\), repeats \(\varphi_{\mathrm{p}}(u)\).
The latter circumstance is very important. It gives us grounds to regard the part of the distribution \(f_{\mathrm{p}}(u_{\mathrm{p}})\) lying to the left of the mode \(u_{\mathrm{np}}\) as the undistorted distribution \(\varphi_{\mathrm{p}}(u)\) and, thus, to find the parameters of the distribution \(\varphi_{\mathrm{p}}(u)\).
Processing the experimental distribution of breakdown voltages of wire removed from machines and tested in bulk made it possible to establish that \(\varphi_{\mathrm{p}}(u)\) is the density of a normal distribution with parameters \(m_{\mathrm{p}}=2.25\ \mathrm{kV}\), \(\sigma_{\mathrm{p}}=1.00\ \mathrm{kV}\).
The decrease of \(\sigma_{\mathrm{p}}\) in comparison with \(\sigma_{\mathrm{n}}\) can be explained by the fact that the dimensions of the working part of the winding tool are considerably larger than the dimensions of an elementary section. Therefore, when damage occurs, the depth of damage is leveled out.
The distribution density of the breakdown voltages of elementary sections in a twist will be a mixture of three distribution densities.
- The convolution of the distributions of the breakdown voltages of two undamaged sections, with weight \((1-p)^2\),
\[ \psi_1(u)=\frac{1}{\sqrt{2\pi}\,\Sigma_1}\exp\left[-\frac{(u-2m_n)^2}{2\Sigma_1^2}\right], \tag{6} \]
where \(\Sigma_1^2=2\sigma_n^2\).
- The convolution of the distributions of the breakdown voltages of a damaged and an undamaged section, with weight \(2p(1-p)\),
\[ \psi_2(u)=\frac{1}{\sqrt{2\pi}\,\Sigma_2}\exp\left[-\frac{(u-m_n-m_p)^2}{2\Sigma_2^2}\right], \tag{7} \]
where \(\Sigma_2^2=\sigma_n^2+\sigma_p^2\).
- The convolution of the distributions of the breakdown voltages of two damaged sections, with weight \(p^2\),
\[ \psi_3(u)=\frac{1}{\sqrt{2\pi}\,\Sigma_3}\exp\left[-\frac{(u-2m_p)^2}{2\Sigma_3^2}\right], \tag{8} \]
where \(\Sigma_3^2=2\sigma_p^2\).
The lower bound of the winding reliability, calculated under the assumption of dense packing of the conductors:
\[ R_n(u_{\mathrm{pr}})=\left[1-(1-p)^2F\left(\frac{u_{\mathrm{pr}}-2m_n}{\Sigma_1}\right)- \tag{9} \]
\[ {}-2p(1-p)F\left(\frac{u_{\mathrm{pr}}-m_n-m_p}{\Sigma_2}\right) -p^2F\left(\frac{u_{\mathrm{pr}}-2m_p}{\Sigma_3}\right)\right]^{(N-1)n}, \]
where \(F(x)\) is the Laplace function;
\(u_{\mathrm{pr}}\) is the maximum voltage applied to a pair of conductors in the machine;
\(n\) is the number of pairs of conductors of length \(125\ \mathrm{mm}\) in the machine.
References
- D. Lloyd, M. Lipow. Reliability. Soviet Radio, 1964.
- E. Gumbel. Statistics of Extremes. Mir, 1965.