RELIABILITY MODEL OF FRAME INSULATION

A. P. MATYALIS, E. K. STRELBITSKII

(Presented by the scientific seminar of the Department of Electrical Machines and by the Electrical Machines Department of the Scientific Research Institute of AEM)

A distinctive feature of frame insulation, which has a substantial influence on the form of its mathematical interpretation, is its multilayer character. The presence of several heterogeneous layers with random parameters leads to uncertainty both with respect to the distribution of voltages and with respect to the sequence of breakdown of the layers. Additional difficulties are created by the complexity of the form of the distribution of breakdown voltages of elementary sections of individual layers. The distributions of breakdown voltages of elementary sections, as a rule, are mixtures of the voltage distributions of damaged and undamaged sections with greatly differing mean values. Therefore, in the general case an analytical representation is difficult, and a numerical probabilistic model is preferable.

In [1] we described an algorithm, developed by us, for applying the Monte Carlo method to calculate the probability of defective frame insulation.

The advantage of the statistical testing method consists in a sufficiently high calculation speed on a digital computer and in its independence from the complexity of representing the distributions of the breakdown voltages of the layers entering into the composition. An obvious shortcoming is the impossibility of carrying out a variance analysis aimed at ranking the individual causes of breakdown according to the degree of their influence on the accident rate of the frame insulation, with subsequent development of recommendations for eliminating weak points. In addition, it is very difficult to introduce into the calculation scheme the arising overvoltage as a random variable.

Therefore, for a number of practically important cases an analytical method has been developed, set forth below. For definiteness, let us consider the method for the concrete example of the frame insulation of motors of series A2, A3, and A4. The frame insulation of these motors consists of a layer of film electrotechnical cardboard (cardboard–lavsan), a layer of lavsan, and the enamel coating of the wire.

The voltage falling on the \(i\)-th layer is equal to

\[ V_i = V \frac{d_i}{\varepsilon_i \sum_j \frac{d_j}{\varepsilon_j}} = V \cdot \kappa_i, \tag{1} \]

where \(V\) is the arising phase overvoltage,

\[ \kappa_i = \frac{d_i}{\varepsilon_i \sum_j \frac{d_j}{\varepsilon_j}} . \]

The mean value of the overvoltage falling on the \(i\)-th layer is

\[ \bar V_i=\bar V \bar\kappa_i, \tag{2} \]

where

\[ \bar\kappa_i=\frac{\dfrac{\bar d_i}{\bar\varepsilon_i}}{\displaystyle \sum_j \frac{d_j}{\varepsilon_j}}. \]

The variance of the voltage of the \(i\)-th layer, accurate to second-order terms, is

\[ D(V_i)=\bar\kappa_i^{\,2}D(V)+D_{\mathrm{add}}(V_i), \tag{3} \]

where \(D(V)\) is the variance of the phase overvoltage;
\(D_{\mathrm{add}}(V)\) is the additional variance of the voltage of the \(i\)-th layer, caused by fluctuations of \(d\) and \(\varepsilon\).

The additional variance is equal to the scalar product of the vector of squares of the influence coefficients of the thicknesses and dielectric constants on the voltage of the \(i\)-th layer and the vector of the corresponding variances. In view of the closeness of the mean values for different materials, one may operate with averaged values of \(\varepsilon\) and \(D(\varepsilon)\) without substantial loss of accuracy. The formula for calculating the additional variance is

\[ D_{\mathrm{add}}(V_i)= \frac{\bar V_i^{\,2}}{\left(\sum_j \bar d_j\right)^4} \left\{ D(d_i)\left(\sum_{j\ne i}\bar d_j\right)^2+ \right. \]

\[ \left. +(\bar d)^2\sum_{j\ne i}D(d_j) +\frac{D(\varepsilon)(\bar d_i)^2}{\varepsilon^2} \left(\sum_{j\ne i}\bar d_j\right)^2 +\frac{(\bar d_i)^2D(\varepsilon)}{\varepsilon^2} \sum_{j\ne i}(\bar d_j)^2 \right\}. \tag{4} \]

Calculations carried out by formula (4) showed the possibility of neglecting the quantity \(D_{\mathrm{add}}(V_i)\) in comparison with \(D(V_i)\).

When the applied voltage \(V_i\) exceeds the breakdown voltage of the \(i\)-th layer \(U_i\), this layer is punctured, and the applied voltage \(V_i\) is distributed among the remaining layers according to formula (1).

The probability of breakdown of the entire chain under a single overvoltage impulse (for simplicity, take a three-layer composition) [1] is

\[ P_1(ABC)=P(AB)P_{AB}(C)+P(AC)P_{AC}(B)+P(BC)P_{BC}(A)- \]

\[ -P(A)P_A(B)P_A(C)-P(B)P_B(A)P_B(C)- \]

\[ -P(C)P_C(A)P_C(B)+P(A)P(B)P(C), \tag{5} \]

where

\[ P(AB)=P(A)P_A(B)+P(B)P_B(A)-P(A)P(B); \]

\[ P(AC)=P(A)P_A(C)+P(C)P_C(A)-P(A)P(C); \]

\[ P(BC)=P(B)P_B(C)+P(C)P_C(B)-P(B)P(C). \]

The probability of breakdown over \(\nu\) overvoltage impulses is

\[ P_\nu(ABC)=1-\left[1-P_1(ABC)\right]^\nu. \tag{6} \]

Probabilities of the type \(P(A)\), entering into expression (5), can be calculated from the formulas

\[ P(A)=P\{V_i\ge U_i\}\cdot P\{V_i>U_{i\min}\}; \]

\[ P\{V_i\ge U_i\}=F\left(\frac{\bar V_i-\bar U_i}{\sigma_{\Sigma i}}\right); \tag{7} \]

\[ P\{V_i>U_{i\min}\}=F\left(\frac{\bar V_i-U_{i\min}}{\sigma_{Vi}}\right); \]

\[ \sigma_{\Sigma i}=\sqrt{D(V_i)+D(U_i)}. \]

\(U_{i \min}\)—the minimum breakdown voltage of the \(i\)-th layer;
\(U_{i \min}=327\) V for an air layer.

Calculation by formulas (5) assumes homogeneity of the sections of the individual layers and of the entire frame-insulation composition. In reality, some sections of the layers are damaged and are characterized by sharply reduced breakdown voltages. A study of histograms of the breakdown voltages of layers of sheet insulation and conductor insulation gives grounds for representing each of the distributions as a mixture of distributions of the form

\[ f_i(U_i)=p_i f_{1i}(U)+(1-p_i)f_{2i}(U), \tag{8} \]

where \(p_i\) is the fraction of sections of the \(i\)-th layer having damage, i.e., the probability of damage to a section;

\(f_{1i}(U)\) is the distribution density of the breakdown voltages of damaged sections, i.e., the distribution density of the breakdown voltages of air or impregnating varnish at the corresponding thicknesses;

\(f_{2i}(U)\) is the distribution density of the breakdown voltages of undamaged sections.

Formula (5) determines the probability of breakdown by a unit impulse of one elementary section, that is, a section of such area within which the breakdown voltage at any point may be regarded as unchanged. Such a formulation is somewhat vague because of the possibility of surface flashover. Therefore the area of an elementary section can be determined only with some probability that breakdown will occur within the area of the elementary section to which the voltage is applied. It is not difficult to see that the size of the section will depend both on the number of layers and on the parameters of the layers, which, with a large variety of section types, leads to an inordinate complication of the calculations.

To avoid unnecessary complications associated with an exact calculation of negligibly small probabilities of breakdown of individual types of sections, let us carry out a screening analysis. At this stage we may arbitrarily specify the size of the elementary section, observing only the condition

\[ nP_S\{V>U\}\gg P_{nS}\{V>U\}, \]

where \(P_S\{V>U\}\) is the probability of breakdown of an elementary section of area \(S\);

\(P_{nS}\{V>U\}\) is the probability of breakdown of an elementary section of area \(nS\).

If there are two types of sections of each layer in a composition consisting of \(n\) layers, the number of types of sections of the composition is \(2n\).

Damage to insulation layers generally cannot be regarded as independent, since when the winding is placed in the slots the inner layer of lavsan may be damaged only from the side of the inner cavity of the slot (by the insertion tool) or from the side of the stator stack (by burrs). In both cases this is associated with obligatory damage to the adjacent layer: in the first case, to the upper layer of lavsan; in the second, to the electrical cardboard. Tests we carried out on frame insulation at various stages of the technological process of manufacturing the winding at the “Sibelektromotor” plant showed that damage to the inner layer of lavsan is determined mainly by combined damage to two layers of lavsan at the stage of placing the winding in the machine slots. The fractions of damage types for each layer, relative to the total number of damages to the layer, are given in Table 1.

Table 1

Table 2 gives the possible types of sections of four-layer frame insulation of AO2-series asynchronous motors, indicating the probability of their occurrence and of breakdown by a unit pulse applicable to the production technology of the Sibelektromotor plant. The following notation is adopted here: \(\bar A\) is a damaged section, \(A\) is an undamaged section.

Table 2

Analysis of Table 2 gives grounds for drawing the following conclusion. Sections containing at least one undamaged lavsan layer have vanish-

vanishingly small probability of breakdown. Only sections containing damaged lavsan layers are subject to further analysis.

When electroinsulating materials are tested in “needle—plane” electrodes, breakdown rarely occurs at the point of contact of the needle. The breakdown site is at a distance \(x\) from the needle point; its mean value \(\bar{x}\) and standard deviation \(\sigma_x\) are determined by the electric strength and the degree of homogeneity of the material. Regardless of the presence of damage to the layer, within an area of radius \(\bar{x}\) there is also undamaged insulation and, consequently, if the breakdowns of any insulation sections under the applied voltage \(V\) are independent, breakdown is possible both at the damage site and in the undamaged insulation. This circumstance leads to uncertainty in the distribution of voltage along the breakdown path.

Let us assume that each breakdown path has its own voltage distribution, which does not depend on the state of adjacent sections of the insulation layers, nor on the fact that some damages of individual layers belong to different breakdown paths.

Table 3

Let us determine the probability of breakdown of the frame insulation if there is one coincident damage of both lavsan layers. The possible breakdown paths of layer \(A\) (wire enamel) and \(D\) (electrocardboard) are presented in Table 3.

The probability of breakdown of the frame insulation through damaged sites depends on the distance of the damages in layers \(A\) and \(D\) from the damage point of the lavsan layers; therefore it is necessary to consider the state of the wire enamel and the electrocardboard in the neighborhoods of the lavsan damage. To do this, let us divide insulation layers \(A\) and \(D\) into rings of width \(h\) with center at the puncture point of the lavsan layers and mean radius

\[ r_{Ai}=r_{Di}=\frac{h}{2}+(i-1)h, \tag{9} \]

where \(i\) is the ring number.

The value of \(h\) is determined by the required accuracy of the calculations.

The probability of damage to an elementary section of insulation with area \(\pi h^2\) is determined from the condition:

\[ 1-Q=(1-q)^{\frac{S}{\pi h^2}}; \tag{10} \]

where

\(q\) is the probability of damage to an elementary section;

\(Q\) is the probability of the presence of at least one damage over the area \(S\).

The value \(Q\) is determined from the results of tests of the frame insulation.

The conditional probability of breakdown of the frame insulation in the presence of through damage to the lavsan layers is equal to the probability of breakdown along at least one of the paths indicated in Table 3. Let us consider each of these paths.

1. The probability of the appearance of sections of undamaged enamel and electrocardboard in the neighborhood of the damage to the lavsan layers is equal to 1, therefore

Consequently, the probability of breakdown along the path of the undamaged layers \(A\) and \(D\) is determined as the probability of breakdown of a three-layer composition

\[ P_1 = 1 - \left[1 - P_1\left(A\overline{B}\,\overline{C}D\right)\right]^{\nu}. \tag{11} \]

1. In determining the probability of breakdown along a damage path, it is necessary to take into account the distance of the defective elementary section from the point of damage to the lavsan layers, as well as the possibility that damage is hidden by the impregnating varnish. If the value \(h\) is small, \(0.1 \ldots 0.5\) mm, then it may be assumed that the occurrence of more than one damage on an elementary section is impossible. Then the events consisting in the occurrence, on an elementary section of a layer, of a varnish-filled damage and of an unfilled damage on this section are incompatible. Therefore the probability of breakdown along the damage path of an elementary section of enamel, located at a distance \(r_{Ai}\) from the place of damage to the lavsan layers and of the undamaged electrocardboard, is equal to the sum

\[ q_A q_{\mathrm{l}} p'_{2i} + q_A(1-q_{\mathrm{l}})p_{2i}, \]

where \(q_A\) is the probability of damage to an elementary section of layer \(A\);
\(q_{\mathrm{l}}\) is the probability that the damage is hidden by the impregnating varnish;
\(p'_{2i}\) and \(p_{2i}\) are the conditional probabilities of breakdown along the above-mentioned path in the presence, on an elementary section of layer \(A\), of a varnish-filled damage and of an unfilled damage, respectively.

The number of elementary sections of layer \(A\), located at a distance \(r_{Ai}\) from the damaged lavsan layers, is determined by the ratio of the area of the \(i\)-th ring to the area of an elementary section

\[ n_i=\frac{2\pi r_{Ai}h}{\pi h^2}=2i-1. \tag{12} \]

Using the condition of independence of breakdown of the frame insulation along any of the possible paths and applying the formula for the probability of the product of independent events, we obtain the probability of breakdown of the frame insulation along at least one of the paths of the damaged section of the enamel layer and the undamaged electrocardboard

\[ P_2 = 1 - \prod_{i=1}^{i} \left\{1-q_A\left[q_{\mathrm{l}}p'_{2i}+(1-q_{\mathrm{l}})p_{2i}\right]\right\}^{n_i}. \tag{13} \]

1. The probability of breakdown of the frame insulation along the path of the damaged section of electrocardboard and the undamaged enamel layer is determined analogously to case 2.

\[ P_3 = 1 - \prod_{i=1}^{i} \left\{1-q_D\left[q_{\mathrm{l}}p'_{3i}+(1-q_{\mathrm{l}})p_{3i}\right]\right\}^{n_i}. \tag{14} \]

Here \(p'_{3i}\) and \(p_{3i}\) are the conditional probabilities of breakdown along the path of damage to layer \(D\) and the undamaged layer \(A\), in the presence, on an elementary section of layer \(D\), of damage filled with varnish and not filled with varnish, respectively.

It is assumed here that simultaneous damage to all layers of the slot cell in a machine that has passed the control tests does not occur.

1. Suppose that every elementary section of layers \(A\) and \(D\) is damaged. Then breakdown paths will exist through the damages

from each elementary section of one layer to all elementary sections of the second layer. Let us single out those paths for which

\[ r_{iD}+r_{iA}=j\cdot h=\mathrm{const},\quad j=1,2,3\ldots \]

Breakdown along any of these paths is equally probable, and their number is determined by the expression

\[ n_j=\sum_{i=1}^{j}(2i-1)[2j-(2i-1)]. \tag{15} \]

In reality, the elementary sections are damaged only with a certain probability, and part of the damage is filled with impregnating varnish. The event space determining the existence of breakdown paths through damaged places for any elementary sections of layers \(A\) and \(D\) is presented in Table 4.

Table 4

The probability of breakdown of the frame insulation along at least one of the paths through damaged layers \(A\) and \(D\), in the presence of coincident damages of the lavsan layers, is calculated by the formula

\[ P_4=1-\prod_{j=1}^{j}\{1-q_Aq_D[q_{\text{л}}^2p'_{4j}+(1-q_{\text{л}})q_{\text{л}}p'_{4Aj}+ \]

\[ +(1-q_{\text{л}})q_{\text{л}}p'_{4Dj}+(1-q_{\text{л}})^2p_{4j}]\}^{n_j}. \tag{16} \]

The probability of breakdown of the frame insulation at the location of coincident damages of the lavsan layers, in the presence of such damage, is equal to

\[ P\{\Pi_i\overline{B}\,\overline{C}\}=1-(1-P_1)(1-P_2)(1-P_3)(1-P_4). \tag{17} \]

Finally, let us determine the probability that damage to the Lavsan layers, displaced by some distance, will occur in the machine. To do this, divide the area of layer \(B\) into rings of width \(h_B\) with their center at the point of damage of layer \(C\). Then the probability that in the machine there is damage to layer \(B\), displaced by a distance \(r_{Bi}\) from the damage to layer \(C\), is equal to

\[ Q_{Bri}=1-(1-q_B)^{2i-1}; \tag{18} \]

\[ r_{Bi}=\frac{h_B}{2}+(i-1)h_B. \]

The probability of breakdown of the frame insulation along a path of displaced defects in the Lavsan layers, in the presence of damage to layer \(C\), is determined in the same way as for coincident damages, with the only difference that in each case the breakdown path is lengthened by \(r_{Bi}\), and the right-hand side of formulas (11, 13, 14, 16) is multiplied by \(Q_{Bri}\).

The unconditional probability of breakdown of the frame insulation of the machine is determined as the probability of breakdown along at least one of the possible paths,

\[ P_k=1-\left[1-P\{\Pi/\overline{B}\,\overline{C}\}\right]^{n_{\overline{B}\,\overline{C}}} \prod_{i=1}^{l}\left[1-P\{\Pi/\overline{B}_{ri}\,\overline{C}\}\right]^{n_{\overline{C}}}, \tag{19} \]

where \(n_{\overline{B}\,\overline{C}}\) is the mathematical expectation of the number of coincident damages of the Lavsan layers;

\(n_{\overline{C}}\) is the mathematical expectation of the number of damages of layer \(C\).

Calculations of the probability of breakdown of the frame insulation carried out by the above method showed that it is determined mainly by the probability of breakdown through coincident Lavsan defects. Table 5 gives the results of calculating the probability of breakdown of the frame insulation of motors of the A02 series for \(v=10000\) and \(q_l=0.7\), indicating the error when breakdown paths through impregnated defects and along displaced damages in the Lavsan layers are excluded from the calculation.

Table 5

Comparing the results obtained, we see that in practical calculations it is possible to exclude from consideration damages filled with varnish during the impregnation process, and also (for small defectiveness) breakdown paths through displaced damages of the Lavsan layers. In this case the volume of calculation is reduced severalfold with an insignificant loss of accuracy.

Thus, the initial data for calculating the reliability of the frame insulation of asynchronous motors are the defectiveness of its individual layers. In this case we must obtain information on the probability of coincidence of damages of the Lavsan layers.

Several test schemes can be proposed that meet this requirement.

1. The inner layer of lavsan, extracted from the slots after the winding has been laid, is tested. In the histogram of breakdown voltages, two regions are clearly distinguished:

a) a region characterizing the frequency of through damage (the mean breakdown voltage of a through defect is equal to the breakdown voltage of an air gap of the corresponding thickness);

b) a region characterizing the frequency of undamaged sections of lavsan.

The probability of joint damage to two layers of lavsan is determined as the probability of damage to the inner layer of lavsan, after subtracting the fraction of damage to this layer at the stages of the technological process prior to laying the winding in the machine slots. The disadvantage of this scheme is the large volume of tests associated with dismantling machines. The number of machines that must be dismantled in order to obtain a sufficient number of frame-insulation specimens is many times greater than the number of machines dismantled to estimate the damageability of turn insulation.

1. To calculate the reliability of frame insulation, quality-control data on the magnitude of rejects due to frame insulation may be used. This method is based on the idea that failures during the running-in period are a continuation of the detection of hidden defects begun at the test station.

Of course, such a method must be based on a preliminary study of the frequency of damage to individual layers, but only quality-control data are used directly for estimating reliability.

Table 6

When using this method it is necessary to bear in mind that there are types of defects in frame insulation that are completely detected and rejected at the test station. These defects are not associated with damage to the layers. Usually these are shifts of individual layers and of the box as a whole. The frequency of such rejects at the Sibelektromotor plant is shown in Table 6. Naturally, when individual layers are tested, such defects are not detected.

The mean number of coincident defects in lavsan is determined from the expression

\[ P_{k\,\mathrm{test}} = 1- \left[ 1- P_{\mathrm{test}}\left(\Pi_i/\overline{B}\,\overline{C}\right) \right]^{n_{\overline{B}\,\overline{C}}}, \tag{20} \]

where \(n_{\overline{B}\,\overline{C}}\) is the mean number of coincident defects in the lavsan layers;

\(P_{k\,\mathrm{test}}\) is the frequency of frame-insulation rejects due to damage to the insulation of the slot box;

\(P_{\mathrm{test}}\{\Pi/\overline{B}\,\overline{C}\}\) is the probability of breakdown of the frame insulation at the location of damage to two layers of lavsan, in the presence of one such damage, determined by the method described above for

\[ V = U_{\mathrm{test}}, \qquad \sigma_{\Sigma}=\sigma_U, \qquad \nu=1, \qquad q_l=0. \]

When calculating the reliability of the insulation from quality-control data, it should be taken into account that in a machine that has passed control tests there are no defects detected by the test voltage. Taking this into account

circumstances and, when the probabilities of breakdown of damages filled with varnish and the probabilities of breakdown along displaced defects in the lavsan layers are excluded from consideration, the calculation formulas take the form

\[ P_1=\{1-[1-P_1(\bar A \bar B \bar C D)]^y\}(1-P_{1\mathrm{test}}); \]

\[ P_2=1-\prod_{i=1}^{i}\{[1-q_A(1-q_L)p_{2i}]^{n_i}(1-p_{2i\mathrm{test}})\}; \]

\[ P_3=1-\prod_{i=1}^{i}\{[1-q_D(1-q_L)p_{3i}]^{n_i}(1-p_{3i\mathrm{test}})\}; \tag{21} \]

\[ P_4=1-\prod_{j=1}^{j}\{[1-q_Aq_D(1-q_L)^2p_{4j}]^{n_j}(1-p_{4j\mathrm{test}})\}; \]

\[ P_k=1-[1-P\{\Pi_i/\bar B\bar C\}]^{n_{\bar B\bar C}[1-p_{\mathrm{test}}\{\Pi_i/\bar B\bar C\}]}. \]

Here \(P_{1\mathrm{test}}, p_{2i\mathrm{test}}, p_{3i\mathrm{test}}, p_{4j\mathrm{test}}, P_{\mathrm{test}}\{\Pi_i/\bar B\bar C\}\) are the conditional probabilities of breakdown of the frame insulation by the test voltage along the corresponding paths when such breakdown paths are present.

References

1. E. K. Strelbitskii, A. S. Gitman, Yu. P. Pokholkov. Study of the frame insulation of asynchronous motors with random windings by the Monte Carlo method. Izvestiya TPI, vol. 160, 1966.