Method for Assessing Insulation Defectiveness in Low-Voltage Electric Machines
Abstract Generated abstract
The paper proposes a method for estimating insulation defectiveness in low-voltage electrical machines, motivated by the view that winding breakdown during service or testing occurs only where defects are present. Instead of testing insulation removed from unimpregnated stators, the method determines distances between nearest defects by breakdown tests on conductor pairs, including in finished or impregnated windings, and distinguishes flashover between defects from insulation puncture by comparing primary and secondary breakdown voltages. Using a Poisson model for defect occurrence and an exponential distribution of interdefect distances, the authors derive probability formulas and an iterative calculation scheme for separating flashover and puncture events and estimating mean defect spacing. The approach is extended to frame insulation and to tests on adjacent turns in finished machines.
Full Text
A Method for Estimating the Defectiveness of the Insulation of Low-Voltage Electrical Machines
A. P. Matyalis, E. K. Strelbitsky
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
The accumulated experimental material gives grounds to assert that, in service and during tests, breakdown of the insulation of the windings of low-voltage electrical machines can occur only in the presence of a defect. By a defect here is meant an elementary section of insulation whose breakdown voltage corresponds to the flashover voltage over a length equal to the insulation thickness. We shall call defectiveness the number of defects per unit length or area of insulation.
A defect may be present in the as-delivered state (foreign inclusions, exposed spots), arise during the manufacture of the winding (cuts, punctures), and form as a result of aging (cracks, delamination).
The reliability of a winding, for a known design and specified operating conditions, is completely determined by defectiveness. Therefore, obtaining the initial data for calculating the probability of failure-free operation and the gamma-percentile life reduces to determining the number of defects.
Previously, the level of defectiveness was determined as a characteristic of defectiveness from the results of tests of insulation removed from the slots of an unimpregnated stator. Turn insulation is tested in wire-to-shot or wire-to-plane electrodes, and ground insulation in needle-to-plane electrodes.
A disadvantage of this method is that it is inapplicable for estimating the defectiveness of the insulation of impregnated windings.
In the present work, for estimating defectiveness it is proposed to determine the distances between nearest defects by breaking down pairs of conductors in the machine or outside the machine. The correspondence between the two indicated test methods is due to the fact that the Poisson distribution of the number of defects on a section of wire leads to an exponential distribution of the distances between defects. However, the use in calculations of distances and flashover voltages makes it possible to dispense with the conventional concept of the length (area) of an elementary section and brings the failure model closer to the actual mechanism of breakdown.
The maximum overvoltages between turns of a winding do not exceed 1 kV; however, in tests at such a voltage, only a small fraction of the distances between defects will be broken down. To obtain reliable statistics, it is necessary either to increase the number of samples or
Table 1
| \(U_i\) (kV) | \multicolumn{17}{c}{\(\Delta U_j = U_j - U_{21}\) (kV)} | \(n_i\) |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | 0—0.4 | 0.4—0.8 | 0.8—1.2 | 1.2—1.6 | 1.6—2.0 | 2.0—2.4 | 2.4—2.8 | 2.8—3.2 | 3.2—3.6 | 3.6—4.0 | 4.0—4.4 | 4.4—4.8 | 4.8—5.2 | 5.2—5.6 | 5.6—6.0 | 6.0—6.4 | 6.4—6.8 | |
| below 2.0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — |
| 2.0—2.4 | 1 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 1 |
| 2.4—2.8 | 1 | — | 1 | — | — | — | — | 1 | — | — | — | — | — | — | — | — | — | 2 |
| 2.8—3.2 | 1 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 2 |
| 3.2—3.6 | 3 | — | — | — | 1 | — | — | — | — | — | — | — | — | — | — | — | — | 4 |
| 3.6—4.0 | 4 | — | — | — | — | — | 1 | — | — | — | — | — | — | — | — | — | — | 7 |
| 4.0—4.4 | 5 | 1 | — | — | 2 | 1 | — | 2 | — | 1 | — | — | — | — | — | — | — | 13 |
| 4.4—4.8 | 10 | — | 1 | — | 2 | — | 1 | — | 1 | 1 | — | 4 | — | — | — | — | — | 20 |
| 4.8—5.2 | 7 | 1 | — | — | 2 | — | 3 | 1 | — | 4 | 2 | 3 | 2 | — | — | — | — | 26 |
| 5.2—5.6 | 7 | 3 | 2 | — | — | 2 | 2 | 6 | 6 | 2 | 4 | 5 | 3 | 8 | 6 | — | — | 51 |
| 5.6—6.0 | 2 | 2 | 2 | — | 2 | 1 | 1 | 3 | 4 | 6 | 3 | 8 | 8 | 5 | 6 | — | — | 54 |
| 6.0—6.4 | 2 | — | 1 | — | 1 | 2 | 5 | 5 | 5 | 3 | 9 | 8 | 16 | 7 | 9 | — | — | 80 |
| 6.4—6.8 | — | 1 | 1 | — | — | 1 | 1 | 2 | 3 | 3 | 4 | 1 | 2 | 4 | 5 | 2 | 4 | 34 |
| 6.8—7.2 | — | — | — | — | — | — | — | — | — | — | — | — | — | 1 | 1 | 1 | 2 | 5 |
| 7.2—7.6 | — | — | — | — | — | — | 1 | — | — | — | — | — | — | — | — | — | — | 1 |
| \(\Sigma n_j\) | 43 | 4 | 3 | 8 | 10 | 9 | 15 | 20 | 19 | 20 | 22 | 29 | 31 | 25 | 21 | 10 | 6 | |
increase the test voltage. As the voltage is increased, insulation that has no damage may be punctured. To recognize such cases, after the first puncture the voltage is applied again. In the case of a flashover between defects, the secondary breakdown voltage differs only slightly (by up to \(0.4\div0.8\) kV) from the primary one. In the case of insulation puncture, the secondary voltage decreases considerably.
Table 1 gives the results of cassette tests of pairs of touching PETV wires with a diameter of \(0.85\) mm, one of which had one artificial defect. In the table one can clearly see the regions of points with
\[ \Delta U = U_1 - U_2 \le 0.6\div0.8\ \text{kV}, \]
corresponding to flashovers, and with
\[ \Delta U > 0.8\ \text{kV}, \]
corresponding to insulation punctures.
The probability of flashover from a cut to a defect at voltage \(U_k\) is
\[ P_{\mathrm{r}}(U_k)=1-e^{-\lambda 2x_k}, \tag{1} \]
where
\[ x_k \text{ is the distance corresponding to } U_k; \]
\[ \lambda=\frac{1}{x} \text{ is the defectiveness;} \]
\[ x \text{ is the mean distance between defects.} \]
The probability of insulation puncture by voltage \(U_k\), if there are no defects bridged by this voltage, is
\[ Q(U_k)=\int_{0}^{U_k} f_{2x}(U)\,dU, \tag{2} \]
where
\[ f_{2x}(U) \text{ is the distribution density of the breakdown voltage of an insulation section of length } 2x_k. \]
The probability that, when the voltage is raised from 0 to \(U_{k-1}\), no flashover will occur, and that in the interval \(U_{k-1}\)—\(U_k\) an insulation puncture will occur, is
\[ q_k' = e^{-2\lambda x_{k-1}}\int_{U_{k-1}}^{U_k} f_{2x}(U)\,dU . \tag{3} \]
The probability that, when the voltage is raised from 0 to \(U_{k-1}\), no puncture will occur, and that in the interval \(U_{k-1}\)—\(U_k\) a flashover will occur is
\[ p_k^1=\left(1-\int_{0}^{U_{k-1}} f_{2x}(U)\,dU\right) \left(e^{-2\lambda x_{k-1}}-e^{-2\lambda x_k}\right) \tag{4} \]
The last two probabilities are known from experiment. The probabilities of interest to us can be obtained from expressions (3) and (4):
\[ q_k=\int_{U_{k-1}}^{U_k} f_{2x}(U)\,dU = \frac{q_k'}{e^{-2\lambda x_{k-1}}}, \tag{5} \]
\[ p_k=e^{-2\lambda x_{k-1}}-e^{-2\lambda x_k} =\frac{p_k'}{1-\displaystyle\int_0^{U_{k-1}} f_{2x}(U)\,dU} \tag{6} \]
Since the quantities \(e^{-2\lambda x_{k-1}}\) and \(\displaystyle\int_0^{U_{k-1}} f_{2x}(U)\,dU\) are unknown, an iterative calculation by recurrence formulas is proposed, with the following taken for the first interval:
\[ q_1=\int_0^{U_1} f_{2x}(U)\,dU \simeq q_1' \tag{7} \]
\[ p_1=1-e^{-2\lambda x_1}\simeq p_1' \tag{8} \]
The complete calculation scheme is given in Table 2. From the value found for \(\overline P_T(U_k)\), the mean distance between defects is determined from expression (1).
Table 2
| \(p_1'\) | \(p_2'\) | \(p_3'\) | \(\ldots\) | \(p_k'\) | |
|---|---|---|---|---|---|
| \(q_1'\) | \(q_2'\) | \(q_3'\) | \(\ldots\) | \(q_k'\) | |
| \(p_1\simeq p_1'\) | \(p_2=\dfrac{p_2'}{1-q_1}\) | \(p_3=\dfrac{p_3'}{1-q_1-q_2}\) | \(\ldots\) | \(p_k=\dfrac{p_k'}{1-q_1-\ldots-q_{k-1}}\) | \(\displaystyle\sum_{i=1}^{k} p_i=P_T(U_k)\) |
| \(q_1\simeq q_1'\) | \(q_2=\dfrac{q_2'}{1-p_1}\) | \(q_3=\dfrac{q_3'}{1-p_1-p_2}\) | \(\ldots\) | \(q_k=\dfrac{q_k'}{1-p_1-\ldots-p_{k-1}}\) | \(\displaystyle\sum_{i=1}^{k} q_i=Q(U_k)\) |
The defectiveness of the frame insulation is determined analogously. In this case the defectiveness \(\lambda\) expresses the number of defects per unit area and is determined from the expression
\[ P_T(U_k)=1-e^{-\lambda S_k}. \tag{9} \]
Here \(S_k\) is the area of insulation within which the search for a defect at voltage \(U_k\) is carried out. This area is determined by the value \(x_k\) and depends on the type and dimensions of the test electrodes.
In conclusion, we note that when the method described is used to determine the defectiveness of turn insulation directly on a finished machine by testing two adjacent turns, the search for a defect is carried out among all defects existing on one of the specimens. As a result, formula (1) takes the form
\[ P_T(U_k)=1-e^{-2\lambda^2 l x_k}, \tag{10} \]
where \(l\) is the length of the tested part of the turn.