Testing of Insulating Materials Required for Predicting Reliability During the Run-In Period
Abstract Generated abstract
This paper examines statistical methods for testing electrical machine insulation in order to estimate reliability during the burn-in period. It relates turn-insulation reliability to the defectiveness of wire, breakdown probability in wire twists, and detection factors, then compares approaches based on tests of assembled machines, sequential analysis, direct determination of wire defectiveness, and regression analysis of breakdowns at elevated voltage. The authors argue that the choice of method should depend on expected defectiveness, required accuracy, and feasible test volume, with sequential analysis reducing test scope for production reliability screening and regression analysis providing a practical way to estimate very small defectiveness values.
Full Text
TESTS OF INSULATING MATERIALS REQUIRED FOR PREDICTING RELIABILITY DURING THE BURN-IN PERIOD
O. P. MURAVLEV, Yu. P. POKHOLKOV, E. K. STREL'BITSKII
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The reliability of turn insulation during the burn-in period is determined by the expression
\[ R = (1 - \varepsilon m q_{\mathrm{pr}}^{2})^{N} = (1 - \varepsilon q_{\mathrm{skr}})^{N} \tag{1} \]
where \(q_{\mathrm{pr}}\) is the fraction of defective wires of length 1;
\(m\) is the coefficient of detection of defects in a twist of wires of length 1;
\(\varepsilon\) is the coefficient of detection of a potential weak point in the machine;
\(N\) is the number of paired groupings of wires in the machine;
\(q_{\mathrm{skr}}\) is the probability of breakdown in a twist.
\[ q_{\mathrm{skr}} = m q_{\mathrm{pr}}^{2}. \tag{2} \]
Since, as the defectiveness of the wire decreases, \(R\) asymptotically approaches unity, the accuracy of determining reliability must be referred to the quantity
\[ 1 - R = Q, \tag{3} \]
i.e., to the failure rate.
Depending on the specific conditions, the magnitude of reliability can be estimated by determining \(Q\), \(q_{\mathrm{skr}}\), or \(q_{\mathrm{pr}}\). Below are set forth test methods for estimating these quantities and the areas of their application, which ensure a minimum cost and labor input for the tests.
It is advisable to use reliability estimation from the results of tests of assembled machines in the following cases:
- To determine the reliability level of large-series and mass production.
- To evaluate the effectiveness of measures when such changes are made in the insulation design and in the technology of winding-and-insulating operations that there is little risk of mass rejects.
The advantage of this method is that, first, it is combined with acceptance tests, and second, it has high reliability owing to direct measurement of the fraction of defective machines.
Let us define \(Q_{\mathrm{isp}}\) as
\[ Q_{\mathrm{isp}} = \int_{0}^{u_{\mathrm{isp}}} \lambda e^{-\lambda u}\,du, \tag{4} \]
where \(U_{\text{test}}\) is the test voltage;
\(\hat{\lambda}\) is the parameter of the exponential distribution of the breakdown voltages of machines. \(\hat{\lambda}\) is determined from equation (4).
The failure rate of asynchronous motors during the burn-in period is
\[ Q=\int_{0}^{u_0}\lambda e^{-\lambda u}\,du-Q_{\text{test}}, \tag{5} \]
where \(u_0\) is the voltage corresponding to the minimum failure-density value, which corresponds to the end of the burn-in period.
If, in the investigated lot of \(n\) machines at the test station, \(r\) failures have been recorded, then the confidence limits for \(Q_{\text{test}}\) are calculated by the formulas [1]:
for \(r=0\)
\[ Q_{\text{test.l}}=0 \qquad Q_{\text{test.u}}=\frac{R_0}{n}, \tag{6} \]
for \(r\ne 0\)
\[ Q_{\text{test.l}}=\frac{r}{nR_1} \qquad Q_{\text{test.u}}=\frac{r}{nR_2}, \tag{7} \]
where \(R_0, R_1, R_2\) are determined from Table 18 in [1].
From the obtained values \(Q_{\text{test.l}}\) and \(Q_{\text{test.u}}\), on the basis of (5), \(Q_{\text{l}}\) and \(Q_{\text{u}}\) and the confidence limits for the reliability of the turn insulation are calculated.
When it is necessary to determine reliability quickly, especially if reliability is high, it is advisable to use sequential analysis.
A characteristic feature of this method is that the number of observations \(n\) is not specified in advance. Sequential analysis gives an exact rule for making, at any moment of testing, one of three decisions: 1) to consider that the insulation has an acceptable defectiveness; 2) to reject it; 3) to continue testing.
Sequential tests are carried out as follows. Let \(q\) be the true value of the parameter; two hypotheses are tested: the null hypothesis \(H_0\), that \(q=q_0\), and the alternative hypothesis \(H_1\), that \(q=q_1\).
The quantity \(q_1\) is the specified maximum acceptable value of the parameter, and \(q_0<q_1\) is an arbitrarily chosen quantity. As the criterion for making a decision, the quantity \(p(r)\) is taken, which in our case is determined by the formula
\[ p(r)=\left(\frac{q_1}{q_0}\right)^r e^{-n(q_1-q_0)}, \tag{8} \]
which is compared with the constants \(A\) and \(B\):
\[ A=\frac{1-\beta}{\alpha'}, \qquad B=\frac{\beta}{1-\alpha'}, \tag{9} \]
where \(\beta\) is the risk of accepting a lot with \(q>q_1\);
\(\alpha'\) is the risk of rejecting a lot with \(q<q_0\).
The decision rule is as follows:
1) accept if \(p(r)\le B\);
2) reject if \(p(r)\ge A\);
3) continue testing if \(B<p(r)<A\).
Usually one chooses \(\alpha'=\beta=0.10\ldots 0.01\), the value \(q_1\) from the condition of the permissible failure rate, and \(q_0=(0.1\ldots 0.5)q_1\).
The expected number of tests is an important characteristic of the test plan. A. Wald showed that the plan considered above leads to a reduction in the volume of testing by no less than 47 percent in comparison with the usual criterion [2].
Determining reliability by testing machines ceases to be effective in the case of a substantial change in design or technology, which is characteristic of experimental production. In this case, as a rule, it is impossible to manufacture a sufficiently large number of machines, and the risk is high of incurring large losses from possible rejects. Moreover, we only ascertain the actual magnitude of unreliability, but are deprived of the possibility of establishing the causes of its occurrence.
In these cases it is advisable to estimate the defectiveness of the wire \(q_{\mathrm{pr}}\) or \(q_{\mathrm{skr}}\), with subsequent determination of reliability from (1).
The defectiveness of the wire \(q_{\mathrm{pr}}\) is determined by breaking down the wire insulation into crumbs. We shall set out a method for determining the required test volumes for this case.
Taking into account the binomial distribution of defective insulation, on the basis of the Moivre–Laplace limit theorem one may write an expression for the probability that the mean value \(q\) lies in the interval \([q(1-\varepsilon), q(1+\varepsilon)]\):
\[ P\{q-\varepsilon q \leq \overline{q} \leq q+\varepsilon q\} \simeq \Phi\left(\frac{\varepsilon q\sqrt{n}}{\sqrt{q(1-q)}}\right)=\alpha, \tag{10} \]
whence
\[ n=\frac{z_{\alpha}^{2}(1-q)}{\varepsilon^{2}q}. \tag{11} \]
Here \(z_{\alpha}\) is the argument of the Laplace function corresponding to the probability \(\alpha\); it is determined from Table V [3].
The sample size calculated by formula (11) depends on the assumed value of the defectiveness \(q\), the required accuracy of determination \(\varepsilon\), and the confidence probability \(\alpha\). The latter is chosen on the basis of specific requirements and is usually \(\alpha=0.90\ldots0.95\), \(\varepsilon=0.1\ldots0.4\). The number of necessary tests \(n\) turns out to be relatively large, but this is not an obstacle when the experiment is of low labor intensity. Thus, for \(\alpha=0.9\), \(\varepsilon=0.2\), and \(q_{\mathrm{pr}}=0.20\ldots0.05\), the sample size is \(n=250\ldots1250\).
To calculate the coefficient of coincidence of defects in a winding \(m\), it is necessary to determine the value \(q_{\mathrm{skr}}\), which is considerably smaller than \(Q_{\mathrm{isp}}\) and \(q_{\mathrm{pr}}\); therefore the methods considered above require a very large number of tests. The electrical strength and defectiveness of insulation are continuous quantities, and therefore it is advisable to use the additional information obtained in testing at increased voltage \(u_{\mathrm{isp}}>u_{\mathrm{rab}}\).
The maximum voltage \(u_k\) is selected so that the resulting dependence of the number of breakdowns \(n_i\) on \(u\) can be approximated by a straight line with a sufficient degree of accuracy.
To estimate \(q_{\mathrm{skr}}\) by the least-squares method, the parameters \(a\) and \(b\) of the regression line are calculated, whose equation is
\[ \hat{n}_i=a+bu. \tag{12} \]
The confidence limits for \(a\) and \(b\) are determined with the aid of the standard deviations \(S_a\) and \(S_b\):
\[ a-tS_a \leq a \leq a+tS_b \quad\text{and}\quad b-tS_b \leq b \leq b+tS_b, \]
where \(t\) is the argument of Student’s distribution; it is determined from Table XVI [3] as a function of the confidence probability \(\alpha\) and the degrees of freedom \(Y\).
\(S_a\) and \(S_b\) are determined from the formulas
\[ S_a=\frac{S}{\sqrt{n'-2}}, \tag{13} \]
\[ S_b=\frac{\sqrt{n'}\,S}{\sqrt{(n'-2)\sum (u_i-\overline{u})^2}}, \tag{14} \]
where
\[ S=\sqrt{\frac{1}{n'}\sum_{1}^{n'}(n_i-\widehat{n}_i)^2}, \tag{15} \]
\(n'\) is the number of points from which the regression line is determined.
Figure 1 shows the calculated regression line—1, and the boundary lines—2 for \(a_{\max}\) and \(b_{\max}\), and 3 for \(a_{\min}\) and \(b_{\min}\), for twists of APEV-2 wire of diameter \(0.83\) mm.
Fig. 1.
The defectiveness value is determined from the formula
\[ \overline{q}=\left(\frac{u_{rab}-O_{c_1}}{u_k-O_{c_1}}\right)^2\frac{\sum n_i}{n}. \tag{16} \]
The lower limit of defectiveness is
\[ q_n=\left(\frac{u_{rab}-O_{c_3}}{u_k-O_{c_3}}\right)^2\frac{\sum n_i}{n}. \tag{17} \]
and the upper limit is
\[ q_b=\left(\frac{u_{rab}-O_{c_2}}{u_k-O_{c_2}}\right)^2\frac{\sum n_i}{n}, \tag{18} \]
where \(\sum n_i\) is the number of breakdowns of the insulation at voltage \(u<u_k\);
\(n\) is the size of the sample batch.
The coefficient of coincidence of defects in a twist is
\[ m=\frac{q_{skr}}{q_{pr}^{\,2}}. \tag{19} \]
and its variance is
\[ D_m=D_{skr}+4D_{pr}. \tag{20} \]
The scope of testing when regression analysis is used is reduced by tens of times.
Conclusions
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The method for determining the defectiveness of insulating materials should be chosen on the basis of the specific conditions: the expected magnitude of the defectiveness, the required accuracy, and the possible scope and duration of the tests.
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The application of sequential analysis substantially reduces the scope of testing when establishing the lower reliability level of the electrical machines being manufactured.
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The classical method of determining the sample size is advisable when relatively high accuracy is required and the tests are not labor-intensive.
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Regression analysis makes it possible to use additional information on insulation breakdown by a voltage higher than the operating voltage. This method can be recommended for determining small values of defectiveness.
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The methods considered are applicable for testing any insulation of various electrical machines.
References
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Ya. B. Shor. Statistical Methods of Analysis and Quality and Reliability Control. Sovetskoe Radio Publishing House, Moscow, 1962.
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A. Wald. Sequential Analysis. Fizmatgiz, Moscow, 1960.
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A. K. Mitropolsky. Technique of Statistical Computations. Fizmatgiz, Moscow, 1961.