Assessment of Defectiveness of Electrical Machine Insulation by the Flashover Voltage of Samples
Abstract Generated abstract
This paper proposes a method for assessing defects in the insulation of low-voltage electrical machine windings, including impregnated windings for which earlier extracted-sample methods are not applicable. It models defects as insulation sections whose breakdown voltage corresponds to flashover over the insulation thickness, assumes a Poisson distribution of defects and an exponential distribution of distances between them, and estimates defectiveness from breakdown tests of conductor pairs. The authors distinguish flashover between defects from insulation puncture by comparing primary and repeated breakdown voltages, then derive probability relations and an iterative calculation scheme for separating flashover and puncture statistics. The method yields an estimate of the mean distance between defects, and thus the defectiveness of turn and frame insulation.
Full Text
Assessment of Insulation Defectiveness in Electrical Machines by the Flashover Voltage of Samples
A. P. MATYALIS, E. K. STRELBITSKY
(Submitted by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
The accumulated experimental material gives grounds to assert that, in operation and during testing, breakdown of the insulation of 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 thickness of the insulation. Defectiveness will be understood as the number of defects per unit length or area of insulation.
A defect may be present in the as-supplied state (foreign inclusions, accumulations), may arise during the manufacture of the winding (cuts, punctures), and may form as a result of aging (cracks, delamination).
The reliability of a winding, for a known design and specified operating conditions, is determined entirely by its defectiveness. Therefore, obtaining initial data for calculating the probability of failure-free operation and the gamma-percentile service 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 extracted from the slots of an unimpregnated stator. Turn insulation is tested in wire—shot or wire—plane electrodes, and ground insulation in needle—plane electrodes.
A drawback of this method is that it is not applicable for assessing the defectiveness of the insulation of impregnated windings.
In the present work, to assess defectiveness it is proposed to determine the distances between the nearest defects by breakdown of pairs of conductors in the machine or outside the machine. The correspondence between the two indicated testing 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 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 breakdown mechanism.
The maximum overvoltages between the turns of a winding do not exceed 1 kV; however, when testing at such a voltage, only a small fraction of the distances between defects will break down. To obtain reliable statistics it is necessary either to increase the number of samples or to increase the test voltage. When increasing the volta-
insulation that has no damage may be punctured. To recognize such cases after the first puncture, the voltage is applied a second time. In the case of flashover between defects, the secondary puncture voltage differs only slightly (by up to \(0.4 \div 0.8\) 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 0.85 mm in diameter, one of which had one artificial defect. The table clearly shows regions of points with
\[ \Delta U = U_1 - U_2 < 0.6\ \text{kV}, \]
corresponding to flashovers, and with
\[ \Delta U > 0.8\ \text{kV}, \]
corresponding to insulation punctures.
The probability of flashover from a notch to a defect at voltage \(U_k\) is equal to
\[ P_T\{U_k\}=P\{x \leq 2x_k\}=1-e^{-\lambda 2x_k}, \tag{1} \]
where \(x_k\) is the distance corresponding to \(U_k\);
\[ \lambda=\frac{1}{\bar{x}} \]
is the defectiveness;
\(\bar{x}\) is the mean distance between defects.
The probability of puncture at this voltage, under the condition that the voltage is fully applied to the insulation, is equal to
\[ Q\{U_k\}=\int_{0}^{U_k} f_{2x}(U)\,dU, \tag{2} \]
where \(f_{2x}(U)\) is the density of the distribution of the puncture voltage of an insulation section of length \(2x_k\).
The probability that, when the voltage is raised from 0 to \(U_i\), no flashover will occur, and that in the interval \((U_i, U_{i+1})\) insulation puncture will occur, is equal to
\[ q_{i+1}=P\{U_i<U_{\mathrm{pr}}<U_{i+1}\} = e^{-2\lambda x_i}\int_{U_i}^{U_{i+1}} f_{2x}(U)\,dU . \tag{3} \]
The probability that, when the voltage is raised from 0 to \(U_i\), no puncture will occur, and that in the interval \((U_i, U_{i+1})\) flashover will occur,
\[ p_{i+1}=P\{U_i<U_{\mathrm{per}}<U_{i+1}\} =\left(e^{-2\lambda x_i}-e^{-2\lambda x_{i+1}}\right) \left(1-\int_{0}^{U_i} f_{2x}(U)\,dU\right). \tag{4} \]
The last two probabilities are known from experiment. The probabilities of interest to us can be obtained from expressions (3) and (4):
\[ q_{i+1}=P\{U_i<U_{\mathrm{pr}}<U_{i+1}\}e^{2\lambda x_i}, \tag{5} \]
\[ p_{i+1}= \frac{P\{U_i<U_{\mathrm{per}}<U_{i+1}\}} {1-\displaystyle\int_{0}^{U_i} f_{2x}(U)\,dU}. \tag{6} \]
Since the quantities \(e^{2\lambda x_i}\) and \(\displaystyle\int_{0}^{U_i} f_{2x}(U)\,dU\) are unknown, an iterative calculation by recurrence formulas is proposed; for the first interval it is assumed
Table 1
| \(U_i\) (kV) | \(\Delta U_j = U_i - U_i'\) (kV): 0.0–0.4 | 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 | \(\sum n_i\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| up to 2.5 | - | - | 1 | - | - | - | - | 1 | - | 2 | 2 | 1 | 1 | - | - | - | - | 8 |
| 2.0–2.4 | - | - | - | - | - | - | 1 | - | 3 | 2 | - | 1 | - | - | - | - | - | 4 |
| 2.4–2.8 | - | - | - | - | 1 | - | 3 | 2 | 1 | 5 | 1 | - | - | - | - | - | - | 8 |
| 2.8–3.2 | - | - | - | 1 | - | 2 | 5 | - | 5 | 2 | 1 | - | - | - | - | - | - | 8 |
| 3.2–3.6 | - | - | 1 | - | 1 | 6 | 3 | 3 | 5 | 2 | 1 | - | - | - | - | - | - | 10 |
| 3.6–4.0 | - | - | - | - | 1 | 2 | 4 | 5 | 3 | 6 | 3 | 3 | - | - | - | - | - | 9 |
| 4.0–4.4 | - | - | - | - | 2 | 6 | 9 | 6 | 4 | 2 | 8 | 1 | - | - | - | - | - | 15 |
| 4.4–4.8 | - | 1 | 2 | 1 | - | 6 | 3 | 2 | 5 | 1 | 5 | 3 | - | - | - | - | - | 20 |
| 4.8–5.2 | 1 | - | 4 | - | 1 | 2 | 8 | 6 | 3 | 8 | 3 | 3 | 2 | - | - | - | - | 19 |
| 5.2–5.6 | 3 | 1 | 2 | 3 | 3 | 5 | 4 | 6 | 4 | 2 | 3 | 8 | 3 | - | - | - | - | 20 |
| 5.6–6.0 | 2 | 1 | 6 | 3 | 2 | 2 | 8 | 3 | 5 | 6 | 5 | 8 | 8 | - | - | - | - | 22 |
| 6.0–6.4 | 2 | - | 3 | 9 | 1 | 5 | 8 | 5 | 3 | 3 | 8 | 8 | 16 | 4 | - | - | - | 29 |
| 6.4–6.8 | - | - | 3 | 4 | 1 | 2 | 1 | 2 | 5 | 4 | 1 | 1 | 2 | 7 | 5 | - | - | 31 |
| 6.8–7.2 | - | - | - | - | - | 1 | 1 | - | 3 | 3 | - | - | 1 | 1 | 2 | 2 | - | 25 |
| 7.2–7.6 | - | - | - | - | - | - | 1 | - | - | - | - | - | - | - | - | - | - | 21 |
| \(\sum n_j\) | 43 | 4 | 8 | 8 | 10 | 9 | 15 | 20 | 19 | 20 | 22 | 29 | 31 | 25 | 21 | 10 | 6 |
Table 2
| $p'_1$ | $p'_2$ | $p'_3$ | $\ldots$ | $p'_k$ | $\displaystyle \sum_{i=1}^{k} p_i = P_T(U_k)$ |
| $q'_1$ | $q'_2$ | $q'_3$ | $\ldots$ | $q'_k$ | $\displaystyle \sum_{i=1}^{k} q_i = Q(U_k)$ |
| $p_1 \approx p'_1$ | $\displaystyle p_2 = \frac{p'_2}{1-q_1}$ | $\displaystyle p_3 = \frac{p'_3}{1-q_1-q_2}$ | $\ldots$ | $\displaystyle p_k = \frac{p'_k}{1-q_1-\ldots-q_{k-1}}$ | $\displaystyle \sum_{i=1}^{k} p_i = P_T(U_k)$ |
| $q_1 \approx q'_1$ | $\displaystyle q_2 = \frac{q'_2}{1-p_1}$ | $\displaystyle q_3 = \frac{q'_3}{1-p_1-p_2}$ | $\ldots$ | $\displaystyle q_k = \frac{q'_k}{1-p_1-\ldots-p_{k-1}}$ | $\displaystyle \sum_{i=1}^{k} q_i = Q(U_k)$ |
\[ q_1 = \int_0^{U_1} f_{2x}(U')\, dU' \simeq P\{U_0 < U_{\mathrm{br}} < U_1\}, \tag{7} \]
\[ p_1 = 1 - e^{-2l x_1} \simeq P\{U_0 < U_{\mathrm{br}} < U_1\}. \tag{8} \]
The complete calculation scheme is given in Table 2. From the found value of \(P_{\mathrm{T}}(U_k)\), expression (1) is used to determine the mean distance between defects.
The defectiveness of the frame insulation is determined analogously.